:: MATRIX13 semantic presentation
begin
theorem Th1: :: MATRIX13:1
for D being non empty set
for n, m being Nat
for A being Matrix of n,m,D holds
( ( n = 0 implies m = 0 ) iff ( len A = n & width A = m ) )
proof
let D be non empty set ; ::_thesis: for n, m being Nat
for A being Matrix of n,m,D holds
( ( n = 0 implies m = 0 ) iff ( len A = n & width A = m ) )
let n, m be Nat; ::_thesis: for A being Matrix of n,m,D holds
( ( n = 0 implies m = 0 ) iff ( len A = n & width A = m ) )
let A be Matrix of n,m,D; ::_thesis: ( ( n = 0 implies m = 0 ) iff ( len A = n & width A = m ) )
thus ( ( n = 0 implies m = 0 ) implies ( len A = n & width A = m ) ) ::_thesis: ( len A = n & width A = m & n = 0 implies m = 0 )
proof
assume A1: ( n = 0 implies m = 0 ) ; ::_thesis: ( len A = n & width A = m )
percases ( n = 0 or n > 0 ) ;
supposeA2: n = 0 ; ::_thesis: ( len A = n & width A = m )
then len A = 0 by MATRIX_1:def_2;
hence ( len A = n & width A = m ) by A1, A2, MATRIX_1:def_3; ::_thesis: verum
end;
supposeA3: n > 0 ; ::_thesis: ( len A = n & width A = m )
len A = n by MATRIX_1:def_2;
hence ( len A = n & width A = m ) by A3, MATRIX_1:20; ::_thesis: verum
end;
end;
end;
thus ( len A = n & width A = m & n = 0 implies m = 0 ) by MATRIX_1:def_3; ::_thesis: verum
end;
theorem Th2: :: MATRIX13:2
for n being Nat
for K being Field
for M being Matrix of n,K holds
( M is Lower_Triangular_Matrix of n,K iff M @ is Upper_Triangular_Matrix of n,K )
proof
let n be Nat; ::_thesis: for K being Field
for M being Matrix of n,K holds
( M is Lower_Triangular_Matrix of n,K iff M @ is Upper_Triangular_Matrix of n,K )
let K be Field; ::_thesis: for M being Matrix of n,K holds
( M is Lower_Triangular_Matrix of n,K iff M @ is Upper_Triangular_Matrix of n,K )
let M be Matrix of n,K; ::_thesis: ( M is Lower_Triangular_Matrix of n,K iff M @ is Upper_Triangular_Matrix of n,K )
thus ( M is Lower_Triangular_Matrix of n,K implies M @ is Upper_Triangular_Matrix of n,K ) ::_thesis: ( M @ is Upper_Triangular_Matrix of n,K implies M is Lower_Triangular_Matrix of n,K )
proof
assume A1: M is Lower_Triangular_Matrix of n,K ; ::_thesis: M @ is Upper_Triangular_Matrix of n,K
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_(M_@)_&_i_>_j_holds_
(M_@)_*_(i,j)_=_0._K
let i, j be Nat; ::_thesis: ( [i,j] in Indices (M @) & i > j implies (M @) * (i,j) = 0. K )
assume that
A2: [i,j] in Indices (M @) and
A3: i > j ; ::_thesis: (M @) * (i,j) = 0. K
A4: [j,i] in Indices M by A2, MATRIX_1:def_6;
then M * (j,i) = 0. K by A1, A3, MATRIX_2:def_4;
hence (M @) * (i,j) = 0. K by A4, MATRIX_1:def_6; ::_thesis: verum
end;
hence M @ is Upper_Triangular_Matrix of n,K by MATRIX_2:def_3; ::_thesis: verum
end;
assume A5: M @ is Upper_Triangular_Matrix of n,K ; ::_thesis: M is Lower_Triangular_Matrix of n,K
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_M_&_i_<_j_holds_
M_*_(i,j)_=_0._K
let i, j be Nat; ::_thesis: ( [i,j] in Indices M & i < j implies M * (i,j) = 0. K )
assume that
A6: [i,j] in Indices M and
A7: i < j ; ::_thesis: M * (i,j) = 0. K
[j,i] in Indices (M @) by A6, MATRIX_1:def_6;
then (M @) * (j,i) = 0. K by A5, A7, MATRIX_2:def_3;
hence M * (i,j) = 0. K by A6, MATRIX_1:def_6; ::_thesis: verum
end;
hence M is Lower_Triangular_Matrix of n,K by MATRIX_2:def_4; ::_thesis: verum
end;
theorem Th3: :: MATRIX13:3
for n being Nat
for K being Field
for M being Matrix of n,K holds diagonal_of_Matrix M = diagonal_of_Matrix (M @)
proof
let n be Nat; ::_thesis: for K being Field
for M being Matrix of n,K holds diagonal_of_Matrix M = diagonal_of_Matrix (M @)
let K be Field; ::_thesis: for M being Matrix of n,K holds diagonal_of_Matrix M = diagonal_of_Matrix (M @)
let M be Matrix of n,K; ::_thesis: diagonal_of_Matrix M = diagonal_of_Matrix (M @)
set DM = diagonal_of_Matrix M;
set DM9 = diagonal_of_Matrix (M @);
A1: len (diagonal_of_Matrix M) = n by MATRIX_3:def_10;
A2: now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_len_(diagonal_of_Matrix_M)_holds_
(diagonal_of_Matrix_M)_._i_=_(diagonal_of_Matrix_(M_@))_._i
let i be Nat; ::_thesis: ( 1 <= i & i <= len (diagonal_of_Matrix M) implies (diagonal_of_Matrix M) . i = (diagonal_of_Matrix (M @)) . i )
assume that
A3: 1 <= i and
A4: i <= len (diagonal_of_Matrix M) ; ::_thesis: (diagonal_of_Matrix M) . i = (diagonal_of_Matrix (M @)) . i
i in NAT by ORDINAL1:def_12;
then A5: i in Seg n by A1, A3, A4;
then A6: (diagonal_of_Matrix (M @)) . i = (M @) * (i,i) by MATRIX_3:def_10;
Indices M = [:(Seg n),(Seg n):] by MATRIX_1:24;
then A7: [i,i] in Indices M by A5, ZFMISC_1:87;
(diagonal_of_Matrix M) . i = M * (i,i) by A5, MATRIX_3:def_10;
hence (diagonal_of_Matrix M) . i = (diagonal_of_Matrix (M @)) . i by A7, A6, MATRIX_1:def_6; ::_thesis: verum
end;
len (diagonal_of_Matrix (M @)) = n by MATRIX_3:def_10;
hence diagonal_of_Matrix M = diagonal_of_Matrix (M @) by A1, A2, FINSEQ_1:14; ::_thesis: verum
end;
theorem Th4: :: MATRIX13:4
for n being Nat
for perm being Element of Permutations n st perm <> idseq n holds
( ex i being Nat st
( i in Seg n & perm . i > i ) & ex j being Nat st
( j in Seg n & perm . j < j ) )
proof
let n be Nat; ::_thesis: for perm being Element of Permutations n st perm <> idseq n holds
( ex i being Nat st
( i in Seg n & perm . i > i ) & ex j being Nat st
( j in Seg n & perm . j < j ) )
let p be Element of Permutations n; ::_thesis: ( p <> idseq n implies ( ex i being Nat st
( i in Seg n & p . i > i ) & ex j being Nat st
( j in Seg n & p . j < j ) ) )
assume A1: p <> idseq n ; ::_thesis: ( ex i being Nat st
( i in Seg n & p . i > i ) & ex j being Nat st
( j in Seg n & p . j < j ) )
reconsider p9 = p as Permutation of (Seg n) by MATRIX_2:def_9;
dom p9 = Seg n by FUNCT_2:52;
then consider x being set such that
A2: x in Seg n and
A3: p . x <> x by A1, FUNCT_1:17;
consider i being Element of NAT such that
A4: i = x and
A5: 1 <= i and
A6: i <= n by A2;
now__::_thesis:_ex_j_being_Nat_st_
(_j_in_Seg_n_&_p_._j_>_j_)
percases ( p . i > i or p . i < i ) by A3, A4, XXREAL_0:1;
suppose p . i > i ; ::_thesis: ex j being Nat st
( j in Seg n & p . j > j )
hence ex j being Nat st
( j in Seg n & p . j > j ) by A2, A4; ::_thesis: verum
end;
supposeA7: p . i < i ; ::_thesis: ex j being Nat st
( j in Seg n & p . j > j )
then reconsider i1 = i - 1 as Nat by NAT_1:20;
thus ex j being Nat st
( j in Seg n & p . j > j ) ::_thesis: verum
proof
reconsider pS = p9 .: (Seg i) as finite set ;
A8: dom p9 = Seg n by FUNCT_2:52;
Seg i c= Seg n by A6, FINSEQ_1:5;
then Seg i,pS are_equipotent by A8, CARD_1:33;
then A9: card (Seg i) = card pS by CARD_1:5;
assume A10: for j being Nat st j in Seg n holds
p . j <= j ; ::_thesis: contradiction
p .: (Seg i) c= Seg i1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in p .: (Seg i) or x in Seg i1 )
assume x in p .: (Seg i) ; ::_thesis: x in Seg i1
then consider y being set such that
A11: y in dom p9 and
A12: y in Seg i and
A13: p . y = x by FUNCT_1:def_6;
consider j being Element of NAT such that
A14: j = y and
1 <= j and
A15: j <= i by A12;
percases ( j = i or j < i ) by A15, XXREAL_0:1;
suppose j = i ; ::_thesis: x in Seg i1
then p . j < i1 + 1 by A7;
then A16: p . j <= i1 by NAT_1:13;
A17: rng p9 = Seg n by FUNCT_2:def_3;
p . j in rng p by A11, A14, FUNCT_1:def_3;
then p . j >= 1 by A17, FINSEQ_1:1;
hence x in Seg i1 by A13, A14, A16, FINSEQ_1:1; ::_thesis: verum
end;
supposeA18: j < i ; ::_thesis: x in Seg i1
p . j <= j by A10, A11, A14;
then p . j < i1 + 1 by A18, XXREAL_0:2;
then A19: p . j <= i1 by NAT_1:13;
A20: rng p9 = Seg n by FUNCT_2:def_3;
p . j in rng p by A11, A14, FUNCT_1:def_3;
then p . j >= 1 by A20, FINSEQ_1:1;
hence x in Seg i1 by A13, A14, A19, FINSEQ_1:1; ::_thesis: verum
end;
end;
end;
then A21: card pS <= card (Seg i1) by NAT_1:43;
card (Seg i) = i by FINSEQ_1:57;
then i1 + 1 <= i1 by A9, A21, FINSEQ_1:57;
hence contradiction by NAT_1:13; ::_thesis: verum
end;
end;
end;
end;
hence ex j being Nat st
( j in Seg n & p . j > j ) ; ::_thesis: ex j being Nat st
( j in Seg n & p . j < j )
A22: n in NAT by ORDINAL1:def_12;
percases ( p . i < i or p . i > i ) by A3, A4, XXREAL_0:1;
suppose p . i < i ; ::_thesis: ex j being Nat st
( j in Seg n & p . j < j )
hence ex j being Nat st
( j in Seg n & p . j < j ) by A2, A4; ::_thesis: verum
end;
supposeA23: p . i > i ; ::_thesis: ex j being Nat st
( j in Seg n & p . j < j )
thus ex j being Nat st
( j in Seg n & p . j < j ) ::_thesis: verum
proof
set NI = nat_interval (i,n);
reconsider pN = p9 .: (nat_interval (i,n)) as finite set ;
A24: i in nat_interval (i,n) by A22, A6, SGRAPH1:2;
assume A25: for j being Nat st j in Seg n holds
p . j >= j ; ::_thesis: contradiction
A26: pN c= nat_interval (i,n)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in pN or x in nat_interval (i,n) )
A27: rng p9 = Seg n by FUNCT_2:def_3;
assume x in pN ; ::_thesis: x in nat_interval (i,n)
then consider j being set such that
A28: j in dom p9 and
A29: j in nat_interval (i,n) and
A30: p . j = x by FUNCT_1:def_6;
reconsider j = j as Nat by A29;
reconsider pj = p . j as Element of NAT by ORDINAL1:def_12;
A31: j <= pj by A25, A28;
i <= j by A22, A29, SGRAPH1:2;
then A32: i <= pj by A31, XXREAL_0:2;
pj in rng p9 by A28, FUNCT_1:def_3;
then pj <= n by A27, FINSEQ_1:1;
hence x in nat_interval (i,n) by A22, A30, A32, SGRAPH1:1; ::_thesis: verum
end;
dom p9 = Seg n by FUNCT_2:52;
then nat_interval (i,n),pN are_equipotent by A22, A5, CARD_1:33, SGRAPH1:4;
then card (nat_interval (i,n)) = card pN by CARD_1:5;
then nat_interval (i,n) = pN by A26, CARD_FIN:1;
then consider j being set such that
A33: j in dom p9 and
A34: j in nat_interval (i,n) and
A35: p . j = i by A24, FUNCT_1:def_6;
reconsider j = j as Nat by A34;
A36: i <= j by A22, A34, SGRAPH1:2;
j <= i by A25, A33, A35;
hence contradiction by A23, A35, A36, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
end;
theorem Th5: :: MATRIX13:5
for n being Nat
for K being Field
for M being Matrix of n,K
for perm being Element of Permutations n st perm <> idseq n & ( M is Lower_Triangular_Matrix of n,K or M is Upper_Triangular_Matrix of n,K ) holds
(Path_product M) . perm = 0. K
proof
let n be Nat; ::_thesis: for K being Field
for M being Matrix of n,K
for perm being Element of Permutations n st perm <> idseq n & ( M is Lower_Triangular_Matrix of n,K or M is Upper_Triangular_Matrix of n,K ) holds
(Path_product M) . perm = 0. K
let K be Field; ::_thesis: for M being Matrix of n,K
for perm being Element of Permutations n st perm <> idseq n & ( M is Lower_Triangular_Matrix of n,K or M is Upper_Triangular_Matrix of n,K ) holds
(Path_product M) . perm = 0. K
let M be Matrix of n,K; ::_thesis: for perm being Element of Permutations n st perm <> idseq n & ( M is Lower_Triangular_Matrix of n,K or M is Upper_Triangular_Matrix of n,K ) holds
(Path_product M) . perm = 0. K
let p be Element of Permutations n; ::_thesis: ( p <> idseq n & ( M is Lower_Triangular_Matrix of n,K or M is Upper_Triangular_Matrix of n,K ) implies (Path_product M) . p = 0. K )
assume that
A1: p <> idseq n and
A2: ( M is Lower_Triangular_Matrix of n,K or M is Upper_Triangular_Matrix of n,K ) ; ::_thesis: (Path_product M) . p = 0. K
reconsider p9 = p as Permutation of (Seg n) by MATRIX_2:def_9;
set PP = Path_product M;
set PATH = Path_matrix (p,M);
now__::_thesis:_ex_i_being_Nat_st_
(_i_in_dom_(Path_matrix_(p,M))_&_(Path_matrix_(p,M))_._i_=_0._K_)
percases ( M is Lower_Triangular_Matrix of n,K or M is Upper_Triangular_Matrix of n,K ) by A2;
supposeA3: M is Lower_Triangular_Matrix of n,K ; ::_thesis: ex i being Nat st
( i in dom (Path_matrix (p,M)) & (Path_matrix (p,M)) . i = 0. K )
A4: rng p9 = Seg n by FUNCT_2:def_3;
A5: Indices M = [:(Seg n),(Seg n):] by MATRIX_1:24;
consider i being Nat such that
A6: i in Seg n and
A7: p . i > i by A1, Th4;
reconsider Pi = p . i as Nat ;
dom p9 = Seg n by FUNCT_2:52;
then p9 . i in Seg n by A6, A4, FUNCT_1:def_3;
then [i,Pi] in [:(Seg n),(Seg n):] by A6, ZFMISC_1:87;
then A8: M * (i,Pi) = 0. K by A3, A7, A5, MATRIX_2:def_4;
len (Path_matrix (p,M)) = n by MATRIX_3:def_7;
then A9: dom (Path_matrix (p,M)) = Seg n by FINSEQ_1:def_3;
then (Path_matrix (p,M)) . i = M * (i,Pi) by A6, MATRIX_3:def_7;
hence ex i being Nat st
( i in dom (Path_matrix (p,M)) & (Path_matrix (p,M)) . i = 0. K ) by A6, A9, A8; ::_thesis: verum
end;
supposeA10: M is Upper_Triangular_Matrix of n,K ; ::_thesis: ex i being Nat st
( i in dom (Path_matrix (p,M)) & (Path_matrix (p,M)) . i = 0. K )
A11: rng p9 = Seg n by FUNCT_2:def_3;
A12: Indices M = [:(Seg n),(Seg n):] by MATRIX_1:24;
consider i being Nat such that
A13: i in Seg n and
A14: p . i < i by A1, Th4;
reconsider Pi = p . i as Nat ;
dom p9 = Seg n by FUNCT_2:52;
then p9 . i in Seg n by A13, A11, FUNCT_1:def_3;
then [i,Pi] in [:(Seg n),(Seg n):] by A13, ZFMISC_1:87;
then A15: M * (i,Pi) = 0. K by A10, A14, A12, MATRIX_2:def_3;
len (Path_matrix (p,M)) = n by MATRIX_3:def_7;
then A16: dom (Path_matrix (p,M)) = Seg n by FINSEQ_1:def_3;
then (Path_matrix (p,M)) . i = M * (i,Pi) by A13, MATRIX_3:def_7;
hence ex i being Nat st
( i in dom (Path_matrix (p,M)) & (Path_matrix (p,M)) . i = 0. K ) by A13, A16, A15; ::_thesis: verum
end;
end;
end;
then Product (Path_matrix (p,M)) = 0. K by FVSUM_1:82;
then A17: (Path_product M) . p = - ((0. K),p) by MATRIX_3:def_8;
( - ((0. K),p) = 0. K or - ((0. K),p) = - (0. K) ) by MATRIX_2:def_13;
hence (Path_product M) . p = 0. K by A17, RLVECT_1:12; ::_thesis: verum
end;
theorem Th6: :: MATRIX13:6
for n being Nat
for K being Field
for M being Matrix of n,K
for I being Element of Permutations n st I = idseq n holds
diagonal_of_Matrix M = Path_matrix (I,M)
proof
let n be Nat; ::_thesis: for K being Field
for M being Matrix of n,K
for I being Element of Permutations n st I = idseq n holds
diagonal_of_Matrix M = Path_matrix (I,M)
let K be Field; ::_thesis: for M being Matrix of n,K
for I being Element of Permutations n st I = idseq n holds
diagonal_of_Matrix M = Path_matrix (I,M)
let M be Matrix of n,K; ::_thesis: for I being Element of Permutations n st I = idseq n holds
diagonal_of_Matrix M = Path_matrix (I,M)
let I be Element of Permutations n; ::_thesis: ( I = idseq n implies diagonal_of_Matrix M = Path_matrix (I,M) )
assume A1: I = idseq n ; ::_thesis: diagonal_of_Matrix M = Path_matrix (I,M)
set P = Path_matrix (I,M);
set D = diagonal_of_Matrix M;
A2: len (Path_matrix (I,M)) = n by MATRIX_3:def_7;
A3: now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_n_holds_
(Path_matrix_(I,M))_._i_=_(diagonal_of_Matrix_M)_._i
let i be Nat; ::_thesis: ( 1 <= i & i <= n implies (Path_matrix (I,M)) . i = (diagonal_of_Matrix M) . i )
assume that
A4: 1 <= i and
A5: i <= n ; ::_thesis: (Path_matrix (I,M)) . i = (diagonal_of_Matrix M) . i
i in NAT by ORDINAL1:def_12;
then A6: i in Seg n by A4, A5;
then A7: I . i = i by A1, FINSEQ_2:49;
i in dom (Path_matrix (I,M)) by A2, A6, FINSEQ_1:def_3;
then (Path_matrix (I,M)) . i = M * (i,i) by A7, MATRIX_3:def_7;
hence (Path_matrix (I,M)) . i = (diagonal_of_Matrix M) . i by A6, MATRIX_3:def_10; ::_thesis: verum
end;
len (diagonal_of_Matrix M) = n by MATRIX_3:def_10;
hence diagonal_of_Matrix M = Path_matrix (I,M) by A2, A3, FINSEQ_1:14; ::_thesis: verum
end;
theorem Th7: :: MATRIX13:7
for n being Nat
for K being Field
for M being Upper_Triangular_Matrix of n,K holds Det M = the multF of K $$ (diagonal_of_Matrix M)
proof
let n be Nat; ::_thesis: for K being Field
for M being Upper_Triangular_Matrix of n,K holds Det M = the multF of K $$ (diagonal_of_Matrix M)
let K be Field; ::_thesis: for M being Upper_Triangular_Matrix of n,K holds Det M = the multF of K $$ (diagonal_of_Matrix M)
let M be Upper_Triangular_Matrix of n,K; ::_thesis: Det M = the multF of K $$ (diagonal_of_Matrix M)
set aa = the addF of K;
set mm = the multF of K;
set P = Permutations n;
set F = FinOmega (Permutations n);
set PP = Path_product M;
idseq n is Element of (Group_of_Perm n) by MATRIX_2:20;
then reconsider I = idseq n as Element of Permutations n by MATRIX_2:def_10;
len (Permutations n) = n by MATRIX_2:18;
then A1: I is even by MATRIX_2:25;
Permutations n is finite by MATRIX_2:26;
then reconsider II = {I}, PI = (Permutations n) \ {I} as Element of Fin (Permutations n) by FINSUB_1:def_5;
A2: FinOmega (Permutations n) = Permutations n by MATRIX_2:26, MATRIX_2:def_14;
now__::_thesis:_the_addF_of_K_$$_((FinOmega_(Permutations_n)),(Path_product_M))_=_the_addF_of_K_$$_(II,(Path_product_M))
percases ( PI = {} or PI <> {} ) ;
suppose PI = {} ; ::_thesis: the addF of K $$ ((FinOmega (Permutations n)),(Path_product M)) = the addF of K $$ (II,(Path_product M))
then Permutations n c= II by XBOOLE_1:37;
hence the addF of K $$ ((FinOmega (Permutations n)),(Path_product M)) = the addF of K $$ (II,(Path_product M)) by A2, XBOOLE_0:def_10; ::_thesis: verum
end;
supposeA3: PI <> {} ; ::_thesis: the addF of K $$ ((FinOmega (Permutations n)),(Path_product M)) = the addF of K $$ (II,(Path_product M))
A4: (Path_product M) .: PI c= {(0. K)}
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in (Path_product M) .: PI or y in {(0. K)} )
assume y in (Path_product M) .: PI ; ::_thesis: y in {(0. K)}
then consider x being set such that
A5: x in dom (Path_product M) and
A6: x in PI and
A7: y = (Path_product M) . x by FUNCT_1:def_6;
reconsider f = x as Element of Permutations n by A5;
not f in {I} by A6, XBOOLE_0:def_5;
then f <> I by TARSKI:def_1;
then (Path_product M) . f = 0. K by Th5;
hence y in {(0. K)} by A7, TARSKI:def_1; ::_thesis: verum
end;
A8: 0. K = the_unity_wrt the addF of K by FVSUM_1:7;
dom (Path_product M) = Permutations n by FUNCT_2:def_1;
then (Path_product M) .: PI = {(0. K)} by A3, A4, ZFMISC_1:33;
then A9: the addF of K $$ (PI,(Path_product M)) = 0. K by A8, FVSUM_1:8, SETWOP_2:8;
A10: PI \/ II = II \/ (Permutations n) by XBOOLE_1:39;
A11: II \/ (Permutations n) = Permutations n by XBOOLE_1:12;
PI misses II by XBOOLE_1:79;
hence the addF of K $$ ((FinOmega (Permutations n)),(Path_product M)) = ( the addF of K $$ (II,(Path_product M))) + (0. K) by A2, A3, A9, A10, A11, SETWOP_2:4
.= the addF of K $$ (II,(Path_product M)) by RLVECT_1:def_4 ;
::_thesis: verum
end;
end;
end;
hence Det M = (Path_product M) . I by SETWISEO:17
.= - (( the multF of K "**" (Path_matrix (I,M))),I) by MATRIX_3:def_8
.= the multF of K "**" (Path_matrix (I,M)) by A1, MATRIX_2:def_13
.= the multF of K "**" (diagonal_of_Matrix M) by Th6 ;
::_thesis: verum
end;
theorem Th8: :: MATRIX13:8
for n being Nat
for K being Field
for M being Lower_Triangular_Matrix of n,K holds Det M = the multF of K $$ (diagonal_of_Matrix M)
proof
let n be Nat; ::_thesis: for K being Field
for M being Lower_Triangular_Matrix of n,K holds Det M = the multF of K $$ (diagonal_of_Matrix M)
let K be Field; ::_thesis: for M being Lower_Triangular_Matrix of n,K holds Det M = the multF of K $$ (diagonal_of_Matrix M)
let M be Lower_Triangular_Matrix of n,K; ::_thesis: Det M = the multF of K $$ (diagonal_of_Matrix M)
A1: Det M = Det (M @) by MATRIXR2:43;
M @ is Upper_Triangular_Matrix of n,K by Th2;
hence Det M = the multF of K $$ (diagonal_of_Matrix (M @)) by A1, Th7
.= the multF of K $$ (diagonal_of_Matrix M) by Th3 ;
::_thesis: verum
end;
theorem Th9: :: MATRIX13:9
for X being finite set
for n being Nat holds card { Y where Y is Subset of X : card Y = n } = (card X) choose n
proof
let X be finite set ; ::_thesis: for n being Nat holds card { Y where Y is Subset of X : card Y = n } = (card X) choose n
let n be Nat; ::_thesis: card { Y where Y is Subset of X : card Y = n } = (card X) choose n
reconsider N = n as Element of NAT by ORDINAL1:def_12;
set YY = { Y where Y is Subset of X : card Y = n } ;
set CH = Choose (X,N,1,0);
deffunc H1( set ) -> set = (X --> 0) +* ($1 --> 1);
consider f being Function such that
A1: ( dom f = { Y where Y is Subset of X : card Y = n } & ( for x being set st x in { Y where Y is Subset of X : card Y = n } holds
f . x = H1(x) ) ) from FUNCT_1:sch_3();
A2: Choose (X,N,1,0) c= rng f
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Choose (X,N,1,0) or y in rng f )
A3: dom (X --> 0) = X by FUNCOP_1:13;
assume y in Choose (X,N,1,0) ; ::_thesis: y in rng f
then consider g being Function of X,{0,1} such that
A4: g = y and
A5: card (g " {1}) = n by CARD_FIN:def_1;
X = dom g by FUNCT_2:def_1;
then reconsider Y = g " {1} as Subset of X by RELAT_1:132;
A6: Y in { Y where Y is Subset of X : card Y = n } by A5;
A7: now__::_thesis:_for_x_being_set_st_x_in_dom_g_holds_
g_._x_=_H1(Y)_._x
let x be set ; ::_thesis: ( x in dom g implies g . x = H1(Y) . x )
assume A8: x in dom g ; ::_thesis: g . x = H1(Y) . x
now__::_thesis:_g_._x_=_H1(Y)_._x
percases ( x in Y or not x in Y ) ;
supposeA9: x in Y ; ::_thesis: g . x = H1(Y) . x
then g . x in {1} by FUNCT_1:def_7;
then A10: g . x = 1 by TARSKI:def_1;
A11: (Y --> 1) . x = 1 by A9, FUNCOP_1:7;
x in dom (Y --> 1) by A9, FUNCOP_1:13;
hence g . x = H1(Y) . x by A11, A10, FUNCT_4:13; ::_thesis: verum
end;
supposeA12: not x in Y ; ::_thesis: g . x = H1(Y) . x
then not g . x in {1} by A8, FUNCT_1:def_7;
then A13: g . x <> 1 by TARSKI:def_1;
A14: rng g c= {0,1} by RELAT_1:def_19;
A15: (X --> 0) . x = 0 by A8, FUNCOP_1:7;
A16: dom (Y --> 1) = Y by FUNCOP_1:13;
g . x in rng g by A8, FUNCT_1:def_3;
then g . x = 0 by A14, A13, TARSKI:def_2;
hence g . x = H1(Y) . x by A12, A15, A16, FUNCT_4:11; ::_thesis: verum
end;
end;
end;
hence g . x = H1(Y) . x ; ::_thesis: verum
end;
dom (Y --> 1) = Y by FUNCOP_1:13;
then A17: dom H1(Y) = X \/ Y by A3, FUNCT_4:def_1
.= X by XBOOLE_1:12 ;
dom g = X by FUNCT_2:def_1;
then H1(Y) = g by A17, A7, FUNCT_1:2;
then f . Y = g by A1, A6;
hence y in rng f by A1, A4, A6, FUNCT_1:def_3; ::_thesis: verum
end;
for x1, x2 being set st x1 in dom f & x2 in dom f & f . x1 = f . x2 holds
x1 = x2
proof
let x1, x2 be set ; ::_thesis: ( x1 in dom f & x2 in dom f & f . x1 = f . x2 implies x1 = x2 )
assume that
A18: x1 in dom f and
A19: x2 in dom f and
A20: f . x1 = f . x2 ; ::_thesis: x1 = x2
consider Y2 being Subset of X such that
A21: x2 = Y2 and
A22: card Y2 = n by A1, A19;
consider Y1 being Subset of X such that
A23: x1 = Y1 and
A24: card Y1 = n by A1, A18;
Y1 c= Y2
proof
A25: dom (Y1 --> 1) = Y1 by FUNCOP_1:13;
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in Y1 or y in Y2 )
assume A26: y in Y1 ; ::_thesis: y in Y2
(Y1 --> 1) . y = 1 by A26, FUNCOP_1:7;
then A27: H1(Y1) . y = 1 by A26, A25, FUNCT_4:13;
A28: H1(Y1) = f . x1 by A1, A18, A23;
A29: dom (Y2 --> 1) = Y2 by FUNCOP_1:13;
assume A30: not y in Y2 ; ::_thesis: contradiction
(X --> 0) . y = 0 by A26, FUNCOP_1:7;
then H1(Y2) . y = 0 by A30, A29, FUNCT_4:11;
hence contradiction by A1, A19, A20, A21, A27, A28; ::_thesis: verum
end;
hence x1 = x2 by A23, A24, A21, A22, CARD_FIN:1; ::_thesis: verum
end;
then A31: f is one-to-one by FUNCT_1:def_4;
rng f c= Choose (X,N,1,0)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng f or y in Choose (X,N,1,0) )
assume y in rng f ; ::_thesis: y in Choose (X,N,1,0)
then consider x being set such that
A32: x in dom f and
A33: f . x = y by FUNCT_1:def_3;
consider Y being Subset of X such that
A34: x = Y and
A35: card Y = n by A1, A32;
Y \/ X = X by XBOOLE_1:12;
then H1(Y) in Choose (X,N,1,0) by A35, CARD_FIN:17;
hence y in Choose (X,N,1,0) by A1, A32, A33, A34; ::_thesis: verum
end;
then rng f = Choose (X,N,1,0) by A2, XBOOLE_0:def_10;
then { Y where Y is Subset of X : card Y = n } , Choose (X,N,1,0) are_equipotent by A1, A31, WELLORD2:def_4;
then card { Y where Y is Subset of X : card Y = n } = card (Choose (X,N,1,0)) by CARD_1:5;
hence card { Y where Y is Subset of X : card Y = n } = (card X) choose n by CARD_FIN:16; ::_thesis: verum
end;
theorem Th10: :: MATRIX13:10
for n being Nat holds card (2Set (Seg n)) = n choose 2
proof
let n be Nat; ::_thesis: card (2Set (Seg n)) = n choose 2
{ Y where Y is Subset of (Seg n) : card Y = 2 } = 2Set (Seg n) by SGRAPH1:def_2;
hence card (2Set (Seg n)) = (card (Seg n)) choose 2 by Th9
.= n choose 2 by FINSEQ_1:57 ;
::_thesis: verum
end;
theorem :: MATRIX13:11
for n being Nat
for R being Element of Permutations n st R = Rev (idseq n) holds
( R is even iff (n choose 2) mod 2 = 0 )
proof
let n be Nat; ::_thesis: for R being Element of Permutations n st R = Rev (idseq n) holds
( R is even iff (n choose 2) mod 2 = 0 )
let r be Element of Permutations n; ::_thesis: ( r = Rev (idseq n) implies ( r is even iff (n choose 2) mod 2 = 0 ) )
assume A1: r = Rev (idseq n) ; ::_thesis: ( r is even iff (n choose 2) mod 2 = 0 )
percases ( n < 2 or n >= 2 ) ;
supposeA2: n < 2 ; ::_thesis: ( r is even iff (n choose 2) mod 2 = 0 )
then ( n = 0 or n = 1 ) by NAT_1:23;
then (n * (n - 1)) / 2 = 0 ;
then n choose 2 = 0 by STIRL2_1:51;
hence ( r is even iff (n choose 2) mod 2 = 0 ) by A2, LAPLACE:11, NAT_D:26; ::_thesis: verum
end;
supposeA3: n >= 2 ; ::_thesis: ( r is even iff (n choose 2) mod 2 = 0 )
set CH = n choose 2;
reconsider n2 = n - 2 as Nat by A3, NAT_1:21;
reconsider R = r as Element of Permutations (n2 + 2) ;
set K = the Fanoian Field;
set S = 2Set (Seg (n2 + 2));
A4: FinOmega (2Set (Seg (n2 + 2))) = 2Set (Seg (n2 + 2)) by MATRIX_2:def_14;
idseq (n2 + 2) is Element of (Group_of_Perm (n2 + 2)) by MATRIX_2:20;
then reconsider I = idseq (n2 + 2) as Element of Permutations (n2 + 2) by MATRIX_2:def_10;
set D = { s where s is Element of 2Set (Seg (n2 + 2)) : ( s in 2Set (Seg (n2 + 2)) & (Part_sgn (I, the Fanoian Field)) . s <> (Part_sgn (R, the Fanoian Field)) . s ) } ;
A5: { s where s is Element of 2Set (Seg (n2 + 2)) : ( s in 2Set (Seg (n2 + 2)) & (Part_sgn (I, the Fanoian Field)) . s <> (Part_sgn (R, the Fanoian Field)) . s ) } c= 2Set (Seg (n2 + 2))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { s where s is Element of 2Set (Seg (n2 + 2)) : ( s in 2Set (Seg (n2 + 2)) & (Part_sgn (I, the Fanoian Field)) . s <> (Part_sgn (R, the Fanoian Field)) . s ) } or x in 2Set (Seg (n2 + 2)) )
assume x in { s where s is Element of 2Set (Seg (n2 + 2)) : ( s in 2Set (Seg (n2 + 2)) & (Part_sgn (I, the Fanoian Field)) . s <> (Part_sgn (R, the Fanoian Field)) . s ) } ; ::_thesis: x in 2Set (Seg (n2 + 2))
then ex s being Element of 2Set (Seg (n2 + 2)) st
( x = s & s in 2Set (Seg (n2 + 2)) & (Part_sgn (I, the Fanoian Field)) . s <> (Part_sgn (R, the Fanoian Field)) . s ) ;
hence x in 2Set (Seg (n2 + 2)) ; ::_thesis: verum
end;
then reconsider D = { s where s is Element of 2Set (Seg (n2 + 2)) : ( s in 2Set (Seg (n2 + 2)) & (Part_sgn (I, the Fanoian Field)) . s <> (Part_sgn (R, the Fanoian Field)) . s ) } as finite set ;
2Set (Seg (n2 + 2)) c= D
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in 2Set (Seg (n2 + 2)) or x in D )
assume x in 2Set (Seg (n2 + 2)) ; ::_thesis: x in D
then reconsider s = x as Element of 2Set (Seg (n2 + 2)) ;
consider i, j being Nat such that
A6: i in Seg (n2 + 2) and
A7: j in Seg (n2 + 2) and
A8: i < j and
A9: s = {i,j} by MATRIX11:1;
A10: I . j = j by A7, FUNCT_1:17;
reconsider i9 = i, j9 = j, n29 = n2 as Element of NAT by ORDINAL1:def_12;
A11: j9 <= n2 + 2 by A7, FINSEQ_1:1;
i9 <= n29 + 2 by A6, FINSEQ_1:1;
then reconsider ni = ((n29 + 2) - i9) + 1, nj = ((n29 + 2) - j9) + 1 as Element of NAT by A11, FINSEQ_5:1;
ni in Seg (n2 + 2) by A6, FINSEQ_5:2;
then A12: I . ni = ni by FUNCT_1:17;
A13: len (idseq (n2 + 2)) = n29 + 2 by CARD_1:def_7;
then i in dom I by A6, FINSEQ_1:def_3;
then A14: R . i9 = I . ni by A1, A13, FINSEQ_5:58;
j in dom I by A7, A13, FINSEQ_1:def_3;
then A15: R . j9 = I . nj by A1, A13, FINSEQ_5:58;
nj in Seg (n2 + 2) by A7, FINSEQ_5:2;
then A16: I . nj = nj by FUNCT_1:17;
I . i = i by A6, FUNCT_1:17;
then A17: (Part_sgn (I, the Fanoian Field)) . s = 1_ the Fanoian Field by A6, A7, A8, A9, A10, MATRIX11:def_1;
((n29 + 2) + 1) - i9 > ((n29 + 2) + 1) - j9 by A8, XREAL_1:15;
then (Part_sgn (R, the Fanoian Field)) . s = - (1_ the Fanoian Field) by A6, A7, A8, A9, A14, A15, A12, A16, MATRIX11:def_1;
then (Part_sgn (I, the Fanoian Field)) . s <> (Part_sgn (R, the Fanoian Field)) . s by A17, MATRIX11:22;
hence x in D ; ::_thesis: verum
end;
then A18: 2Set (Seg (n2 + 2)) = D by A5, XBOOLE_0:def_10;
A19: card (2Set (Seg (n2 + 2))) = n choose 2 by Th10;
percases ( (n choose 2) mod 2 = 0 or (n choose 2) mod 2 = 1 ) by NAT_D:12;
supposeA20: (n choose 2) mod 2 = 0 ; ::_thesis: ( r is even iff (n choose 2) mod 2 = 0 )
A21: sgn (I, the Fanoian Field) = 1_ the Fanoian Field by MATRIX11:12;
sgn (I, the Fanoian Field) = sgn (R, the Fanoian Field) by A18, A4, A19, A20, MATRIX11:7;
hence ( r is even iff (n choose 2) mod 2 = 0 ) by A20, A21, MATRIX11:23; ::_thesis: verum
end;
supposeA22: (n choose 2) mod 2 = 1 ; ::_thesis: ( r is even iff (n choose 2) mod 2 = 0 )
A23: sgn (I, the Fanoian Field) = 1_ the Fanoian Field by MATRIX11:12;
sgn (R, the Fanoian Field) = - (sgn (I, the Fanoian Field)) by A18, A4, A19, A22, MATRIX11:7;
hence ( r is even iff (n choose 2) mod 2 = 0 ) by A22, A23, MATRIX11:23; ::_thesis: verum
end;
end;
end;
end;
end;
theorem Th12: :: MATRIX13:12
for n being Nat
for K being Field
for M being Matrix of n,K
for R being Permutation of (Seg n) st R = Rev (idseq n) & ( for i, j being Nat st i in Seg n & j in Seg n & i + j <= n holds
M * (i,j) = 0. K ) holds
M * R is Upper_Triangular_Matrix of n,K
proof
let n be Nat; ::_thesis: for K being Field
for M being Matrix of n,K
for R being Permutation of (Seg n) st R = Rev (idseq n) & ( for i, j being Nat st i in Seg n & j in Seg n & i + j <= n holds
M * (i,j) = 0. K ) holds
M * R is Upper_Triangular_Matrix of n,K
let K be Field; ::_thesis: for M being Matrix of n,K
for R being Permutation of (Seg n) st R = Rev (idseq n) & ( for i, j being Nat st i in Seg n & j in Seg n & i + j <= n holds
M * (i,j) = 0. K ) holds
M * R is Upper_Triangular_Matrix of n,K
let M be Matrix of n,K; ::_thesis: for R being Permutation of (Seg n) st R = Rev (idseq n) & ( for i, j being Nat st i in Seg n & j in Seg n & i + j <= n holds
M * (i,j) = 0. K ) holds
M * R is Upper_Triangular_Matrix of n,K
let R be Permutation of (Seg n); ::_thesis: ( R = Rev (idseq n) & ( for i, j being Nat st i in Seg n & j in Seg n & i + j <= n holds
M * (i,j) = 0. K ) implies M * R is Upper_Triangular_Matrix of n,K )
assume that
A1: R = Rev (idseq n) and
A2: for i, j being Nat st i in Seg n & j in Seg n & i + j <= n holds
M * (i,j) = 0. K ; ::_thesis: M * R is Upper_Triangular_Matrix of n,K
set I = idseq n;
set MR = M * R;
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_(M_*_R)_&_i_>_j_holds_
(M_*_R)_*_(i,j)_=_0._K
let i, j be Nat; ::_thesis: ( [i,j] in Indices (M * R) & i > j implies (M * R) * (i,j) = 0. K )
assume that
A3: [i,j] in Indices (M * R) and
A4: i > j ; ::_thesis: (M * R) * (i,j) = 0. K
reconsider i9 = i as Element of NAT by ORDINAL1:def_12;
A5: Indices (M * R) = [:(Seg n),(Seg n):] by MATRIX_1:24;
then A6: i in Seg n by A3, ZFMISC_1:87;
then i <= n by FINSEQ_1:1;
then reconsider ni = (n - i9) + 1 as Element of NAT by FINSEQ_5:1;
A7: ni in Seg n by A6, FINSEQ_5:2;
then A8: (idseq n) . ni = ni by FUNCT_1:17;
(n + 1) - i < (n + 1) - j by A4, XREAL_1:15;
then ni + j < ((n + 1) - j) + j by XREAL_1:8;
then A9: ni + j <= n by NAT_1:13;
j in Seg n by A3, A5, ZFMISC_1:87;
then A10: M * (ni,j) = 0. K by A2, A7, A9;
A11: len (idseq n) = n by CARD_1:def_7;
A12: Indices M = [:(Seg n),(Seg n):] by MATRIX_1:24;
dom (idseq n) = Seg (len (idseq n)) by FINSEQ_1:def_3;
then R . i = (idseq n) . ni by A1, A6, A11, FINSEQ_5:58;
hence (M * R) * (i,j) = 0. K by A3, A5, A12, A8, A10, MATRIX11:def_4; ::_thesis: verum
end;
hence M * R is Upper_Triangular_Matrix of n,K by MATRIX_2:def_3; ::_thesis: verum
end;
theorem Th13: :: MATRIX13:13
for n being Nat
for K being Field
for M being Matrix of n,K
for R being Permutation of (Seg n) st R = Rev (idseq n) & ( for i, j being Nat st i in Seg n & j in Seg n & i + j > n + 1 holds
M * (i,j) = 0. K ) holds
M * R is Lower_Triangular_Matrix of n,K
proof
let n be Nat; ::_thesis: for K being Field
for M being Matrix of n,K
for R being Permutation of (Seg n) st R = Rev (idseq n) & ( for i, j being Nat st i in Seg n & j in Seg n & i + j > n + 1 holds
M * (i,j) = 0. K ) holds
M * R is Lower_Triangular_Matrix of n,K
let K be Field; ::_thesis: for M being Matrix of n,K
for R being Permutation of (Seg n) st R = Rev (idseq n) & ( for i, j being Nat st i in Seg n & j in Seg n & i + j > n + 1 holds
M * (i,j) = 0. K ) holds
M * R is Lower_Triangular_Matrix of n,K
let M be Matrix of n,K; ::_thesis: for R being Permutation of (Seg n) st R = Rev (idseq n) & ( for i, j being Nat st i in Seg n & j in Seg n & i + j > n + 1 holds
M * (i,j) = 0. K ) holds
M * R is Lower_Triangular_Matrix of n,K
let R be Permutation of (Seg n); ::_thesis: ( R = Rev (idseq n) & ( for i, j being Nat st i in Seg n & j in Seg n & i + j > n + 1 holds
M * (i,j) = 0. K ) implies M * R is Lower_Triangular_Matrix of n,K )
assume that
A1: R = Rev (idseq n) and
A2: for i, j being Nat st i in Seg n & j in Seg n & i + j > n + 1 holds
M * (i,j) = 0. K ; ::_thesis: M * R is Lower_Triangular_Matrix of n,K
set I = idseq n;
set MR = M * R;
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_(M_*_R)_&_i_<_j_holds_
(M_*_R)_*_(i,j)_=_0._K
let i, j be Nat; ::_thesis: ( [i,j] in Indices (M * R) & i < j implies (M * R) * (i,j) = 0. K )
assume that
A3: [i,j] in Indices (M * R) and
A4: i < j ; ::_thesis: (M * R) * (i,j) = 0. K
reconsider i9 = i as Element of NAT by ORDINAL1:def_12;
A5: Indices (M * R) = [:(Seg n),(Seg n):] by MATRIX_1:24;
then A6: i in Seg n by A3, ZFMISC_1:87;
then i <= n by FINSEQ_1:1;
then reconsider ni = (n - i9) + 1 as Element of NAT by FINSEQ_5:1;
(n + 1) - i > (n + 1) - j by A4, XREAL_1:15;
then A7: ni + j > ((n + 1) - j) + j by XREAL_1:8;
A8: len (idseq n) = n by CARD_1:def_7;
A9: Indices M = [:(Seg n),(Seg n):] by MATRIX_1:24;
A10: ni in Seg n by A6, FINSEQ_5:2;
then A11: (idseq n) . ni = ni by FUNCT_1:17;
j in Seg n by A3, A5, ZFMISC_1:87;
then A12: M * (ni,j) = 0. K by A2, A7, A10;
dom (idseq n) = Seg (len (idseq n)) by FINSEQ_1:def_3;
then R . i = (idseq n) . ni by A1, A6, A8, FINSEQ_5:58;
hence (M * R) * (i,j) = 0. K by A3, A5, A9, A11, A12, MATRIX11:def_4; ::_thesis: verum
end;
hence M * R is Lower_Triangular_Matrix of n,K by MATRIX_2:def_4; ::_thesis: verum
end;
theorem :: MATRIX13:14
for n being Nat
for K being Field
for M being Matrix of n,K
for R being Element of Permutations n st R = Rev (idseq n) & ( for i, j being Nat st i in Seg n & j in Seg n & i + j <= n holds
M * (i,j) = 0. K or for i, j being Nat st i in Seg n & j in Seg n & i + j > n + 1 holds
M * (i,j) = 0. K ) holds
Det M = - (( the multF of K "**" (Path_matrix (R,M))),R)
proof
let n be Nat; ::_thesis: for K being Field
for M being Matrix of n,K
for R being Element of Permutations n st R = Rev (idseq n) & ( for i, j being Nat st i in Seg n & j in Seg n & i + j <= n holds
M * (i,j) = 0. K or for i, j being Nat st i in Seg n & j in Seg n & i + j > n + 1 holds
M * (i,j) = 0. K ) holds
Det M = - (( the multF of K "**" (Path_matrix (R,M))),R)
let K be Field; ::_thesis: for M being Matrix of n,K
for R being Element of Permutations n st R = Rev (idseq n) & ( for i, j being Nat st i in Seg n & j in Seg n & i + j <= n holds
M * (i,j) = 0. K or for i, j being Nat st i in Seg n & j in Seg n & i + j > n + 1 holds
M * (i,j) = 0. K ) holds
Det M = - (( the multF of K "**" (Path_matrix (R,M))),R)
let M be Matrix of n,K; ::_thesis: for R being Element of Permutations n st R = Rev (idseq n) & ( for i, j being Nat st i in Seg n & j in Seg n & i + j <= n holds
M * (i,j) = 0. K or for i, j being Nat st i in Seg n & j in Seg n & i + j > n + 1 holds
M * (i,j) = 0. K ) holds
Det M = - (( the multF of K "**" (Path_matrix (R,M))),R)
let R be Element of Permutations n; ::_thesis: ( R = Rev (idseq n) & ( for i, j being Nat st i in Seg n & j in Seg n & i + j <= n holds
M * (i,j) = 0. K or for i, j being Nat st i in Seg n & j in Seg n & i + j > n + 1 holds
M * (i,j) = 0. K ) implies Det M = - (( the multF of K "**" (Path_matrix (R,M))),R) )
assume that
A1: R = Rev (idseq n) and
A2: ( for i, j being Nat st i in Seg n & j in Seg n & i + j <= n holds
M * (i,j) = 0. K or for i, j being Nat st i in Seg n & j in Seg n & i + j > n + 1 holds
M * (i,j) = 0. K ) ; ::_thesis: Det M = - (( the multF of K "**" (Path_matrix (R,M))),R)
set mm = the multF of K;
idseq n is Element of (Group_of_Perm n) by MATRIX_2:20;
then reconsider I = idseq n as Element of Permutations n by MATRIX_2:def_10;
set X = Seg n;
reconsider r = Rev (idseq n) as Permutation of (Seg n) by A1, MATRIX_2:def_9;
set Mr = M * r;
set PR = Path_matrix (R,M);
set PI = Path_matrix (I,(M * r));
A3: len (Path_matrix (R,M)) = n by MATRIX_3:def_7;
A4: len (Path_matrix (I,(M * r))) = n by MATRIX_3:def_7;
A5: now__::_thesis:_the_multF_of_K_"**"_(Path_matrix_(I,(M_*_r)))_=_the_multF_of_K_"**"_(Path_matrix_(R,M))
percases ( n < 1 or n >= 1 ) ;
supposeA6: n < 1 ; ::_thesis: the multF of K "**" (Path_matrix (I,(M * r))) = the multF of K "**" (Path_matrix (R,M))
then A7: Path_matrix (R,M) = {} by A3, NAT_1:14;
Path_matrix (I,(M * r)) = {} by A4, A6, NAT_1:14;
hence the multF of K "**" (Path_matrix (I,(M * r))) = the multF of K "**" (Path_matrix (R,M)) by A7; ::_thesis: verum
end;
supposeA8: n >= 1 ; ::_thesis: the multF of K "**" (Path_matrix (I,(M * r))) = the multF of K "**" (Path_matrix (R,M))
rng I = Seg n by RELAT_1:45;
then A9: rng (Rev (idseq n)) = Seg n by FINSEQ_5:57;
reconsider PRR = (Path_matrix (R,M)) * R as FinSequence of K by A3, A8, MATRIX_7:34;
A10: dom r = Seg n by FUNCT_2:52;
dom (Path_matrix (R,M)) = Seg n by A3, FINSEQ_1:def_3;
then A11: dom PRR = dom R by A1, A9, RELAT_1:27;
A12: now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_k_<=_len_(Path_matrix_(I,(M_*_r)))_holds_
(Path_matrix_(I,(M_*_r)))_._k_=_PRR_._k
A13: dom (Path_matrix (R,M)) = Seg n by A3, FINSEQ_1:def_3;
A14: dom (Path_matrix (I,(M * r))) = Seg n by A4, FINSEQ_1:def_3;
let k be Nat; ::_thesis: ( 1 <= k & k <= len (Path_matrix (I,(M * r))) implies (Path_matrix (I,(M * r))) . k = PRR . k )
assume that
A15: 1 <= k and
A16: k <= len (Path_matrix (I,(M * r))) ; ::_thesis: (Path_matrix (I,(M * r))) . k = PRR . k
k in NAT by ORDINAL1:def_12;
then A17: k in Seg n by A4, A15, A16;
then A18: (n - k) + 1 in Seg n by FINSEQ_5:2;
I . k = k by A17, FUNCT_1:17;
then A19: (Path_matrix (I,(M * r))) . k = (M * r) * (k,k) by A17, A14, MATRIX_3:def_7;
A20: len (idseq n) = n by CARD_1:def_7;
then r . k = I . ((n - k) + 1) by A10, A17, FINSEQ_5:def_3;
then A21: r . k = (n - k) + 1 by A18, FUNCT_1:17;
A22: Indices M = [:(Seg n),(Seg n):] by MATRIX_1:24;
then [k,k] in Indices M by A17, ZFMISC_1:87;
then consider m being Nat such that
A23: r . k = m and
A24: [m,k] in Indices M and
A25: (M * r) * (k,k) = M * (m,k) by MATRIX11:37;
A26: m in Seg n by A22, A24, ZFMISC_1:87;
then A27: (n - m) + 1 in Seg n by FINSEQ_5:2;
r . m = I . ((n - m) + 1) by A10, A20, A26, FINSEQ_5:def_3;
then A28: R . m = k by A1, A23, A27, A21, FUNCT_1:17;
m in Seg n by A22, A24, ZFMISC_1:87;
then (Path_matrix (R,M)) . m = M * (m,k) by A13, A28, MATRIX_3:def_7;
hence (Path_matrix (I,(M * r))) . k = PRR . k by A1, A11, A10, A17, A23, A25, A19, FUNCT_1:12; ::_thesis: verum
end;
n is Element of NAT by ORDINAL1:def_12;
then len PRR = n by A1, A11, A10, FINSEQ_1:def_3;
hence the multF of K "**" (Path_matrix (I,(M * r))) = the multF of K "**" (Path_matrix (R,M)) by A4, A3, A8, A12, FINSEQ_1:14, MATRIX_7:33; ::_thesis: verum
end;
end;
end;
( M * r is Upper_Triangular_Matrix of n,K or M * r is Lower_Triangular_Matrix of n,K ) by A2, Th12, Th13;
then A29: the multF of K $$ (diagonal_of_Matrix (M * r)) = Det (M * r) by Th7, Th8
.= - ((Det M),R) by A1, MATRIX11:46 ;
then A30: - ((Det M),R) = the multF of K $$ (Path_matrix (R,M)) by A5, Th6;
percases ( R is even or R is odd ) ;
suppose R is even ; ::_thesis: Det M = - (( the multF of K "**" (Path_matrix (R,M))),R)
then - ((Det M),R) = Det M by MATRIX_2:def_13;
hence Det M = - (( the multF of K "**" (Path_matrix (R,M))),R) by A5, A29, Th6; ::_thesis: verum
end;
supposeA31: R is odd ; ::_thesis: Det M = - (( the multF of K "**" (Path_matrix (R,M))),R)
then A32: - (( the multF of K "**" (Path_matrix (R,M))),R) = - ( the multF of K "**" (Path_matrix (R,M))) by MATRIX_2:def_13;
- ((Det M),R) = - (Det M) by A31, MATRIX_2:def_13;
then (- (( the multF of K "**" (Path_matrix (R,M))),R)) + (- (Det M)) = 0. K by A30, A32, VECTSP_1:19;
hence Det M = - (( the multF of K "**" (Path_matrix (R,M))),R) by VECTSP_1:19; ::_thesis: verum
end;
end;
end;
theorem Th15: :: MATRIX13:15
for n being Nat
for K being Field
for M being Matrix of n,K
for M1, M2 being Upper_Triangular_Matrix of n,K st M = M1 * M2 holds
( M is Upper_Triangular_Matrix of n,K & diagonal_of_Matrix M = mlt ((diagonal_of_Matrix M1),(diagonal_of_Matrix M2)) )
proof
let n be Nat; ::_thesis: for K being Field
for M being Matrix of n,K
for M1, M2 being Upper_Triangular_Matrix of n,K st M = M1 * M2 holds
( M is Upper_Triangular_Matrix of n,K & diagonal_of_Matrix M = mlt ((diagonal_of_Matrix M1),(diagonal_of_Matrix M2)) )
let K be Field; ::_thesis: for M being Matrix of n,K
for M1, M2 being Upper_Triangular_Matrix of n,K st M = M1 * M2 holds
( M is Upper_Triangular_Matrix of n,K & diagonal_of_Matrix M = mlt ((diagonal_of_Matrix M1),(diagonal_of_Matrix M2)) )
let M be Matrix of n,K; ::_thesis: for M1, M2 being Upper_Triangular_Matrix of n,K st M = M1 * M2 holds
( M is Upper_Triangular_Matrix of n,K & diagonal_of_Matrix M = mlt ((diagonal_of_Matrix M1),(diagonal_of_Matrix M2)) )
reconsider N = n as Element of NAT by ORDINAL1:def_12;
let M1, M2 be Upper_Triangular_Matrix of n,K; ::_thesis: ( M = M1 * M2 implies ( M is Upper_Triangular_Matrix of n,K & diagonal_of_Matrix M = mlt ((diagonal_of_Matrix M1),(diagonal_of_Matrix M2)) ) )
assume A1: M = M1 * M2 ; ::_thesis: ( M is Upper_Triangular_Matrix of n,K & diagonal_of_Matrix M = mlt ((diagonal_of_Matrix M1),(diagonal_of_Matrix M2)) )
set SS = [:(Seg n),(Seg n):];
set KK = the carrier of K;
A2: len M2 = n by MATRIX_1:24;
A3: width M1 = n by MATRIX_1:24;
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_M_&_i_>_j_holds_
0._K_=_M_*_(i,j)
set n0 = n |-> (0. K);
let i, j be Nat; ::_thesis: ( [i,j] in Indices M & i > j implies 0. K = M * (i,j) )
assume that
A4: [i,j] in Indices M and
A5: i > j ; ::_thesis: 0. K = M * (i,j)
set C = Col (M2,j);
set L = Line (M1,i);
reconsider L9 = Line (M1,i), C9 = Col (M2,j) as Element of N -tuples_on the carrier of K by MATRIX_1:24;
set m = mlt (L9,C9);
A6: len (mlt (L9,C9)) = n by CARD_1:def_7;
A7: now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_k_<=_n_holds_
(mlt_(L9,C9))_._k_=_(n_|->_(0._K))_._k
let k be Nat; ::_thesis: ( 1 <= k & k <= n implies (mlt (L9,C9)) . b1 = (n |-> (0. K)) . b1 )
assume that
A8: 1 <= k and
A9: k <= n ; ::_thesis: (mlt (L9,C9)) . b1 = (n |-> (0. K)) . b1
A10: k in NAT by ORDINAL1:def_12;
then A11: k in Seg n by A8, A9;
then A12: (n |-> (0. K)) . k = 0. K by FINSEQ_2:57;
A13: Indices M2 = [:(Seg n),(Seg n):] by MATRIX_1:24;
A14: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_1:24;
A15: (Line (M1,i)) . k = M1 * (i,k) by A3, A11, MATRIX_1:def_7;
A16: Indices M = [:(Seg n),(Seg n):] by MATRIX_1:24;
then i in Seg n by A4, ZFMISC_1:87;
then A17: [i,k] in Indices M1 by A11, A14, ZFMISC_1:87;
j in Seg n by A4, A16, ZFMISC_1:87;
then A18: [k,j] in Indices M2 by A11, A13, ZFMISC_1:87;
A19: dom (mlt (L9,C9)) = Seg n by A6, FINSEQ_1:def_3;
A20: k in Seg n by A8, A9, A10;
dom M2 = Seg n by A2, FINSEQ_1:def_3;
then A21: (Col (M2,j)) . k = M2 * (k,j) by A11, MATRIX_1:def_8;
percases ( k <= j or k > j ) ;
suppose k <= j ; ::_thesis: (mlt (L9,C9)) . b1 = (n |-> (0. K)) . b1
then k < i by A5, XXREAL_0:2;
then A22: M1 * (i,k) = 0. K by A17, MATRIX_2:def_3;
(mlt (L9,C9)) . k = (M1 * (i,k)) * (M2 * (k,j)) by A15, A21, A19, A20, FVSUM_1:60;
hence (mlt (L9,C9)) . k = (n |-> (0. K)) . k by A12, A22, VECTSP_1:12; ::_thesis: verum
end;
suppose k > j ; ::_thesis: (mlt (L9,C9)) . b1 = (n |-> (0. K)) . b1
then A23: M2 * (k,j) = 0. K by A18, MATRIX_2:def_3;
(mlt (L9,C9)) . k = (M1 * (i,k)) * (M2 * (k,j)) by A15, A21, A19, A20, FVSUM_1:60;
hence (mlt (L9,C9)) . k = (n |-> (0. K)) . k by A12, A23, VECTSP_1:12; ::_thesis: verum
end;
end;
end;
len (n |-> (0. K)) = n by CARD_1:def_7;
then mlt (L9,C9) = n |-> (0. K) by A6, A7, FINSEQ_1:14;
hence 0. K = (Line (M1,i)) "*" (Col (M2,j)) by MATRIX_3:11
.= M * (i,j) by A1, A3, A2, A4, MATRIX_3:def_4 ;
::_thesis: verum
end;
hence M is Upper_Triangular_Matrix of n,K by MATRIX_2:def_3; ::_thesis: diagonal_of_Matrix M = mlt ((diagonal_of_Matrix M1),(diagonal_of_Matrix M2))
set D2 = diagonal_of_Matrix M2;
set D1 = diagonal_of_Matrix M1;
set DM = diagonal_of_Matrix M;
A24: len (diagonal_of_Matrix M2) = n by MATRIX_3:def_10;
len (diagonal_of_Matrix M1) = n by MATRIX_3:def_10;
then reconsider D19 = diagonal_of_Matrix M1, D29 = diagonal_of_Matrix M2 as Element of N -tuples_on the carrier of K by A24, FINSEQ_2:92;
set m = mlt (D19,D29);
A25: len (mlt (D19,D29)) = n by CARD_1:def_7;
A26: now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_n_holds_
(mlt_(D19,D29))_._i_=_(diagonal_of_Matrix_M)_._i
set aa = the addF of K;
let i be Nat; ::_thesis: ( 1 <= i & i <= n implies (mlt (D19,D29)) . i = (diagonal_of_Matrix M) . i )
assume that
A27: 1 <= i and
A28: i <= n ; ::_thesis: (mlt (D19,D29)) . i = (diagonal_of_Matrix M) . i
i in NAT by ORDINAL1:def_12;
then A29: i in Seg n by A27, A28;
then A30: (diagonal_of_Matrix M) . i = M * (i,i) by MATRIX_3:def_10;
set C = Col (M2,i);
set L = Line (M1,i);
reconsider L9 = Line (M1,i), C9 = Col (M2,i) as Element of N -tuples_on the carrier of K by MATRIX_1:24;
set mLC = mlt (L9,C9);
A31: the addF of K is having_a_unity by FVSUM_1:8;
Indices M = [:(Seg n),(Seg n):] by MATRIX_1:24;
then [i,i] in Indices M by A29, ZFMISC_1:87;
then A32: (diagonal_of_Matrix M) . i = (Line (M1,i)) "*" (Col (M2,i)) by A1, A3, A2, A30, MATRIX_3:def_4;
A33: (diagonal_of_Matrix M2) . i = M2 * (i,i) by A29, MATRIX_3:def_10;
A34: (diagonal_of_Matrix M1) . i = M1 * (i,i) by A29, MATRIX_3:def_10;
len (mlt (L9,C9)) = n by CARD_1:def_7;
then consider f being Function of NAT, the carrier of K such that
A35: f . 1 = (mlt (L9,C9)) . 1 and
A36: for k being Element of NAT st 0 <> k & k < n holds
f . (k + 1) = the addF of K . ((f . k),((mlt (L9,C9)) . (k + 1))) and
A37: (diagonal_of_Matrix M) . i = f . n by A27, A28, A32, A31, FINSOP_1:def_1;
defpred S1[ Nat] means ( 1 <= $1 & $1 <= n implies ( ( $1 < i implies f . $1 = 0. K ) & ( $1 >= i implies f . $1 = (mlt (D19,D29)) . i ) ) );
i in dom (mlt (D19,D29)) by A25, A29, FINSEQ_1:def_3;
then A38: (mlt (D19,D29)) . i = (M1 * (i,i)) * (M2 * (i,i)) by A34, A33, FVSUM_1:60;
A39: for j being Nat st j in Seg n holds
( ( j <> i implies (mlt (L9,C9)) . j = 0. K ) & ( j = i implies (mlt (L9,C9)) . j = (mlt (D19,D29)) . i ) )
proof
i in NAT by ORDINAL1:def_12;
then A40: i in Seg n by A27, A28;
let j be Nat; ::_thesis: ( j in Seg n implies ( ( j <> i implies (mlt (L9,C9)) . j = 0. K ) & ( j = i implies (mlt (L9,C9)) . j = (mlt (D19,D29)) . i ) ) )
assume A41: j in Seg n ; ::_thesis: ( ( j <> i implies (mlt (L9,C9)) . j = 0. K ) & ( j = i implies (mlt (L9,C9)) . j = (mlt (D19,D29)) . i ) )
A42: (Line (M1,i)) . j = M1 * (i,j) by A3, A41, MATRIX_1:def_7;
Indices M1 = [:(Seg n),(Seg n):] by MATRIX_1:24;
then A43: [i,j] in Indices M1 by A41, A40, ZFMISC_1:87;
dom M2 = Seg n by A2, FINSEQ_1:def_3;
then A44: (Col (M2,i)) . j = M2 * (j,i) by A41, MATRIX_1:def_8;
Indices M2 = [:(Seg n),(Seg n):] by MATRIX_1:24;
then A45: [j,i] in Indices M2 by A41, A40, ZFMISC_1:87;
percases ( i <> j or i = j ) ;
supposeA46: i <> j ; ::_thesis: ( ( j <> i implies (mlt (L9,C9)) . j = 0. K ) & ( j = i implies (mlt (L9,C9)) . j = (mlt (D19,D29)) . i ) )
then ( i < j or j < i ) by XXREAL_0:1;
then A47: ( M1 * (i,j) = 0. K or M2 * (j,i) = 0. K ) by A43, A45, MATRIX_2:def_3;
(mlt (L9,C9)) . j = (M1 * (i,j)) * (M2 * (j,i)) by A41, A42, A44, FVSUM_1:61;
hence ( ( j <> i implies (mlt (L9,C9)) . j = 0. K ) & ( j = i implies (mlt (L9,C9)) . j = (mlt (D19,D29)) . i ) ) by A46, A47, VECTSP_1:12; ::_thesis: verum
end;
suppose i = j ; ::_thesis: ( ( j <> i implies (mlt (L9,C9)) . j = 0. K ) & ( j = i implies (mlt (L9,C9)) . j = (mlt (D19,D29)) . i ) )
hence ( ( j <> i implies (mlt (L9,C9)) . j = 0. K ) & ( j = i implies (mlt (L9,C9)) . j = (mlt (D19,D29)) . i ) ) by A38, A41, A42, A44, FVSUM_1:61; ::_thesis: verum
end;
end;
end;
A48: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A49: S1[k] ; ::_thesis: S1[k + 1]
set k1 = k + 1;
assume that
A50: 1 <= k + 1 and
A51: k + 1 <= n ; ::_thesis: ( ( k + 1 < i implies f . (k + 1) = 0. K ) & ( k + 1 >= i implies f . (k + 1) = (mlt (D19,D29)) . i ) )
A52: k + 1 in Seg n by A50, A51;
percases ( k = 0 or k > 0 ) ;
suppose k = 0 ; ::_thesis: ( ( k + 1 < i implies f . (k + 1) = 0. K ) & ( k + 1 >= i implies f . (k + 1) = (mlt (D19,D29)) . i ) )
then ( ( f . (k + 1) = 0. K & k + 1 < i ) or ( f . (k + 1) = (mlt (L9,C9)) . (k + 1) & k + 1 = i ) ) by A27, A39, A35, A52, XXREAL_0:1;
hence ( ( k + 1 < i implies f . (k + 1) = 0. K ) & ( k + 1 >= i implies f . (k + 1) = (mlt (D19,D29)) . i ) ) by A39, A52; ::_thesis: verum
end;
supposeA53: k > 0 ; ::_thesis: ( ( k + 1 < i implies f . (k + 1) = 0. K ) & ( k + 1 >= i implies f . (k + 1) = (mlt (D19,D29)) . i ) )
A54: k in NAT by ORDINAL1:def_12;
k < n by A51, NAT_1:13;
then A55: f . (k + 1) = the addF of K . ((f . k),((mlt (L9,C9)) . (k + 1))) by A36, A53, A54;
percases ( k + 1 < i or k + 1 = i or k + 1 > i ) by XXREAL_0:1;
supposeA56: k + 1 < i ; ::_thesis: ( ( k + 1 < i implies f . (k + 1) = 0. K ) & ( k + 1 >= i implies f . (k + 1) = (mlt (D19,D29)) . i ) )
then f . (k + 1) = (0. K) + (0. K) by A39, A49, A51, A52, A53, A55, NAT_1:13, NAT_1:14;
hence ( ( k + 1 < i implies f . (k + 1) = 0. K ) & ( k + 1 >= i implies f . (k + 1) = (mlt (D19,D29)) . i ) ) by A56, RLVECT_1:def_4; ::_thesis: verum
end;
supposeA57: k + 1 = i ; ::_thesis: ( ( k + 1 < i implies f . (k + 1) = 0. K ) & ( k + 1 >= i implies f . (k + 1) = (mlt (D19,D29)) . i ) )
then f . (k + 1) = (0. K) + ((M1 * (i,i)) * (M2 * (i,i))) by A38, A39, A49, A51, A52, A53, A55, NAT_1:13, NAT_1:14;
hence ( ( k + 1 < i implies f . (k + 1) = 0. K ) & ( k + 1 >= i implies f . (k + 1) = (mlt (D19,D29)) . i ) ) by A38, A57, RLVECT_1:def_4; ::_thesis: verum
end;
supposeA58: k + 1 > i ; ::_thesis: ( ( k + 1 < i implies f . (k + 1) = 0. K ) & ( k + 1 >= i implies f . (k + 1) = (mlt (D19,D29)) . i ) )
then f . (k + 1) = ((M1 * (i,i)) * (M2 * (i,i))) + (0. K) by A38, A39, A49, A51, A52, A53, A55, NAT_1:13, NAT_1:14;
hence ( ( k + 1 < i implies f . (k + 1) = 0. K ) & ( k + 1 >= i implies f . (k + 1) = (mlt (D19,D29)) . i ) ) by A38, A58, RLVECT_1:def_4; ::_thesis: verum
end;
end;
end;
end;
end;
A59: 1 <= n by A27, A28, NAT_1:14;
A60: S1[ 0 ] ;
for k being Nat holds S1[k] from NAT_1:sch_2(A60, A48);
hence (mlt (D19,D29)) . i = (diagonal_of_Matrix M) . i by A28, A37, A59; ::_thesis: verum
end;
len (diagonal_of_Matrix M) = n by MATRIX_3:def_10;
hence diagonal_of_Matrix M = mlt ((diagonal_of_Matrix M1),(diagonal_of_Matrix M2)) by A25, A26, FINSEQ_1:14; ::_thesis: verum
end;
theorem :: MATRIX13:16
for n being Nat
for K being Field
for M being Matrix of n,K
for M1, M2 being Lower_Triangular_Matrix of n,K st M = M1 * M2 holds
( M is Lower_Triangular_Matrix of n,K & diagonal_of_Matrix M = mlt ((diagonal_of_Matrix M1),(diagonal_of_Matrix M2)) )
proof
let n be Nat; ::_thesis: for K being Field
for M being Matrix of n,K
for M1, M2 being Lower_Triangular_Matrix of n,K st M = M1 * M2 holds
( M is Lower_Triangular_Matrix of n,K & diagonal_of_Matrix M = mlt ((diagonal_of_Matrix M1),(diagonal_of_Matrix M2)) )
let K be Field; ::_thesis: for M being Matrix of n,K
for M1, M2 being Lower_Triangular_Matrix of n,K st M = M1 * M2 holds
( M is Lower_Triangular_Matrix of n,K & diagonal_of_Matrix M = mlt ((diagonal_of_Matrix M1),(diagonal_of_Matrix M2)) )
let M be Matrix of n,K; ::_thesis: for M1, M2 being Lower_Triangular_Matrix of n,K st M = M1 * M2 holds
( M is Lower_Triangular_Matrix of n,K & diagonal_of_Matrix M = mlt ((diagonal_of_Matrix M1),(diagonal_of_Matrix M2)) )
reconsider N = n as Element of NAT by ORDINAL1:def_12;
let M1, M2 be Lower_Triangular_Matrix of n,K; ::_thesis: ( M = M1 * M2 implies ( M is Lower_Triangular_Matrix of n,K & diagonal_of_Matrix M = mlt ((diagonal_of_Matrix M1),(diagonal_of_Matrix M2)) ) )
assume A1: M = M1 * M2 ; ::_thesis: ( M is Lower_Triangular_Matrix of n,K & diagonal_of_Matrix M = mlt ((diagonal_of_Matrix M1),(diagonal_of_Matrix M2)) )
A2: width M2 = n by MATRIX_1:24;
A3: len M2 = n by MATRIX_1:24;
A4: width M1 = n by MATRIX_1:24;
A5: now__::_thesis:_M_@_=_(M2_@)_*_(M1_@)
percases ( n = 0 or n > 0 ) ;
supposeA6: n = 0 ; ::_thesis: M @ = (M2 @) * (M1 @)
then len ((M2 @) * (M1 @)) = 0 by MATRIX_1:24;
then A7: (M2 @) * (M1 @) = {} ;
len (M @) = 0 by A6, MATRIX_1:24;
hence M @ = (M2 @) * (M1 @) by A7; ::_thesis: verum
end;
suppose n > 0 ; ::_thesis: M @ = (M2 @) * (M1 @)
hence M @ = (M2 @) * (M1 @) by A1, A4, A2, A3, MATRIX_3:22; ::_thesis: verum
end;
end;
end;
set D29 = diagonal_of_Matrix (M2 @);
set D2 = diagonal_of_Matrix M2;
set D19 = diagonal_of_Matrix (M1 @);
set D1 = diagonal_of_Matrix M1;
A8: len (diagonal_of_Matrix M2) = n by MATRIX_3:def_10;
len (diagonal_of_Matrix M1) = n by MATRIX_3:def_10;
then reconsider d1 = diagonal_of_Matrix M1, d2 = diagonal_of_Matrix M2 as Element of N -tuples_on the carrier of K by A8, FINSEQ_2:92;
A9: M2 @ is Upper_Triangular_Matrix of n,K by Th2;
A10: M1 @ is Upper_Triangular_Matrix of n,K by Th2;
then diagonal_of_Matrix (M @) = mlt ((diagonal_of_Matrix (M2 @)),(diagonal_of_Matrix (M1 @))) by A5, A9, Th15;
then A11: diagonal_of_Matrix M = mlt ((diagonal_of_Matrix (M2 @)),(diagonal_of_Matrix (M1 @))) by Th3
.= mlt ((diagonal_of_Matrix M2),(diagonal_of_Matrix (M1 @))) by Th3
.= mlt (d2,d1) by Th3
.= mlt ((diagonal_of_Matrix M1),(diagonal_of_Matrix M2)) by FVSUM_1:63 ;
M @ is Upper_Triangular_Matrix of n,K by A5, A10, A9, Th15;
hence ( M is Lower_Triangular_Matrix of n,K & diagonal_of_Matrix M = mlt ((diagonal_of_Matrix M1),(diagonal_of_Matrix M2)) ) by A11, Th2; ::_thesis: verum
end;
begin
definition
let D be non empty set ;
let M be Matrix of D;
let n, m be Nat;
let nt be Element of n -tuples_on NAT;
let mt be Element of m -tuples_on NAT;
func Segm (M,nt,mt) -> Matrix of n,m,D means :Def1: :: MATRIX13:def 1
for i, j being Nat st [i,j] in Indices it holds
it * (i,j) = M * ((nt . i),(mt . j));
existence
ex b1 being Matrix of n,m,D st
for i, j being Nat st [i,j] in Indices b1 holds
b1 * (i,j) = M * ((nt . i),(mt . j))
proof
reconsider m9 = m, n9 = n as Element of NAT by ORDINAL1:def_12;
deffunc H1( set , set ) -> Element of D = M * ((nt . $1),(mt . $2));
ex S being Matrix of n9,m9,D st
for i, j being Nat st [i,j] in Indices S holds
S * (i,j) = H1(i,j) from MATRIX_1:sch_1();
then consider S being Matrix of n9,m9,D such that
A1: for i, j being Nat st [i,j] in Indices S holds
S * (i,j) = H1(i,j) ;
reconsider S = S as Matrix of n,m,D ;
take S ; ::_thesis: for i, j being Nat st [i,j] in Indices S holds
S * (i,j) = M * ((nt . i),(mt . j))
thus for i, j being Nat st [i,j] in Indices S holds
S * (i,j) = M * ((nt . i),(mt . j)) by A1; ::_thesis: verum
end;
uniqueness
for b1, b2 being Matrix of n,m,D st ( for i, j being Nat st [i,j] in Indices b1 holds
b1 * (i,j) = M * ((nt . i),(mt . j)) ) & ( for i, j being Nat st [i,j] in Indices b2 holds
b2 * (i,j) = M * ((nt . i),(mt . j)) ) holds
b1 = b2
proof
let S1, S2 be Matrix of n,m,D; ::_thesis: ( ( for i, j being Nat st [i,j] in Indices S1 holds
S1 * (i,j) = M * ((nt . i),(mt . j)) ) & ( for i, j being Nat st [i,j] in Indices S2 holds
S2 * (i,j) = M * ((nt . i),(mt . j)) ) implies S1 = S2 )
assume that
A2: for i, j being Nat st [i,j] in Indices S1 holds
S1 * (i,j) = M * ((nt . i),(mt . j)) and
A3: for i, j being Nat st [i,j] in Indices S2 holds
S2 * (i,j) = M * ((nt . i),(mt . j)) ; ::_thesis: S1 = S2
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_S1_holds_
S1_*_(i,j)_=_S2_*_(i,j)
let i, j be Nat; ::_thesis: ( [i,j] in Indices S1 implies S1 * (i,j) = S2 * (i,j) )
assume A4: [i,j] in Indices S1 ; ::_thesis: S1 * (i,j) = S2 * (i,j)
A5: [i,j] in Indices S2 by A4, MATRIX_1:26;
reconsider i9 = i, j9 = j as Element of NAT by ORDINAL1:def_12;
S1 * (i,j) = M * ((nt . i9),(mt . j9)) by A2, A4;
hence S1 * (i,j) = S2 * (i,j) by A3, A5; ::_thesis: verum
end;
hence S1 = S2 by MATRIX_1:27; ::_thesis: verum
end;
end;
:: deftheorem Def1 defines Segm MATRIX13:def_1_:_
for D being non empty set
for M being Matrix of D
for n, m being Nat
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT
for b7 being Matrix of n,m,D holds
( b7 = Segm (M,nt,mt) iff for i, j being Nat st [i,j] in Indices b7 holds
b7 * (i,j) = M * ((nt . i),(mt . j)) );
theorem Th17: :: MATRIX13:17
for D being non empty set
for n, m, i, j being Nat
for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st [:(rng nt),(rng mt):] c= Indices A holds
( [i,j] in Indices (Segm (A,nt,mt)) iff [(nt . i),(mt . j)] in Indices A )
proof
let D be non empty set ; ::_thesis: for n, m, i, j being Nat
for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st [:(rng nt),(rng mt):] c= Indices A holds
( [i,j] in Indices (Segm (A,nt,mt)) iff [(nt . i),(mt . j)] in Indices A )
let n, m, i, j be Nat; ::_thesis: for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st [:(rng nt),(rng mt):] c= Indices A holds
( [i,j] in Indices (Segm (A,nt,mt)) iff [(nt . i),(mt . j)] in Indices A )
let A be Matrix of D; ::_thesis: for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st [:(rng nt),(rng mt):] c= Indices A holds
( [i,j] in Indices (Segm (A,nt,mt)) iff [(nt . i),(mt . j)] in Indices A )
let nt be Element of n -tuples_on NAT; ::_thesis: for mt being Element of m -tuples_on NAT st [:(rng nt),(rng mt):] c= Indices A holds
( [i,j] in Indices (Segm (A,nt,mt)) iff [(nt . i),(mt . j)] in Indices A )
let mt be Element of m -tuples_on NAT; ::_thesis: ( [:(rng nt),(rng mt):] c= Indices A implies ( [i,j] in Indices (Segm (A,nt,mt)) iff [(nt . i),(mt . j)] in Indices A ) )
set S = Segm (A,nt,mt);
A1: dom mt = Seg m by FINSEQ_2:124;
assume A2: [:(rng nt),(rng mt):] c= Indices A ; ::_thesis: ( [i,j] in Indices (Segm (A,nt,mt)) iff [(nt . i),(mt . j)] in Indices A )
thus ( [i,j] in Indices (Segm (A,nt,mt)) implies [(nt . i),(mt . j)] in Indices A ) ::_thesis: ( [(nt . i),(mt . j)] in Indices A implies [i,j] in Indices (Segm (A,nt,mt)) )
proof
A3: dom mt = Seg m by FINSEQ_2:124;
assume A4: [i,j] in Indices (Segm (A,nt,mt)) ; ::_thesis: [(nt . i),(mt . j)] in Indices A
then A5: j in Seg (width (Segm (A,nt,mt))) by ZFMISC_1:87;
[i,j] in [:(Seg n),(Seg (width (Segm (A,nt,mt)))):] by A4, MATRIX_1:25;
then A6: i in Seg n by ZFMISC_1:87;
then A7: n <> 0 ;
dom nt = Seg n by FINSEQ_2:124;
then A8: nt . i in rng nt by A6, FUNCT_1:def_3;
width (Segm (A,nt,mt)) = m by Th1, A7;
then mt . j in rng mt by A5, A3, FUNCT_1:def_3;
then [(nt . i),(mt . j)] in [:(rng nt),(rng mt):] by A8, ZFMISC_1:87;
hence [(nt . i),(mt . j)] in Indices A by A2; ::_thesis: verum
end;
assume A9: [(nt . i),(mt . j)] in Indices A ; ::_thesis: [i,j] in Indices (Segm (A,nt,mt))
A10: j in dom mt
proof
assume not j in dom mt ; ::_thesis: contradiction
then mt . j = {} by FUNCT_1:def_2;
then 0 in Seg (width A) by A9, ZFMISC_1:87;
hence contradiction by FINSEQ_1:1; ::_thesis: verum
end;
A11: i in dom nt
proof
assume not i in dom nt ; ::_thesis: contradiction
then A12: nt . i = {} by FUNCT_1:def_2;
dom A = Seg (len A) by FINSEQ_1:def_3;
then 0 in Seg (len A) by A9, A12, ZFMISC_1:87;
hence contradiction by FINSEQ_1:1; ::_thesis: verum
end;
then n <> 0 ;
then A13: width (Segm (A,nt,mt)) = m by Th1;
dom nt = Seg n by FINSEQ_2:124;
then [i,j] in [:(Seg n),(Seg (width (Segm (A,nt,mt)))):] by A11, A10, A13, A1, ZFMISC_1:87;
hence [i,j] in Indices (Segm (A,nt,mt)) by MATRIX_1:25; ::_thesis: verum
end;
theorem Th18: :: MATRIX13:18
for D being non empty set
for n, m being Nat
for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT holds
not ( [:(rng nt),(rng mt):] c= Indices A & ( n = 0 implies m = 0 ) & ( m = 0 implies n = 0 ) & not (Segm (A,nt,mt)) @ = Segm ((A @),mt,nt) )
proof
let D be non empty set ; ::_thesis: for n, m being Nat
for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT holds
not ( [:(rng nt),(rng mt):] c= Indices A & ( n = 0 implies m = 0 ) & ( m = 0 implies n = 0 ) & not (Segm (A,nt,mt)) @ = Segm ((A @),mt,nt) )
let n, m be Nat; ::_thesis: for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT holds
not ( [:(rng nt),(rng mt):] c= Indices A & ( n = 0 implies m = 0 ) & ( m = 0 implies n = 0 ) & not (Segm (A,nt,mt)) @ = Segm ((A @),mt,nt) )
let A be Matrix of D; ::_thesis: for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT holds
not ( [:(rng nt),(rng mt):] c= Indices A & ( n = 0 implies m = 0 ) & ( m = 0 implies n = 0 ) & not (Segm (A,nt,mt)) @ = Segm ((A @),mt,nt) )
let nt be Element of n -tuples_on NAT; ::_thesis: for mt being Element of m -tuples_on NAT holds
not ( [:(rng nt),(rng mt):] c= Indices A & ( n = 0 implies m = 0 ) & ( m = 0 implies n = 0 ) & not (Segm (A,nt,mt)) @ = Segm ((A @),mt,nt) )
let mt be Element of m -tuples_on NAT; ::_thesis: not ( [:(rng nt),(rng mt):] c= Indices A & ( n = 0 implies m = 0 ) & ( m = 0 implies n = 0 ) & not (Segm (A,nt,mt)) @ = Segm ((A @),mt,nt) )
assume that
A1: [:(rng nt),(rng mt):] c= Indices A and
A2: ( n = 0 iff m = 0 ) ; ::_thesis: (Segm (A,nt,mt)) @ = Segm ((A @),mt,nt)
set A9 = A @ ;
set SA = Segm (A,nt,mt);
set SA9 = Segm ((A @),mt,nt);
percases ( n = 0 or n > 0 ) ;
supposeA3: n = 0 ; ::_thesis: (Segm (A,nt,mt)) @ = Segm ((A @),mt,nt)
then width (Segm (A,nt,mt)) = 0 by A2, Th1;
then len ((Segm (A,nt,mt)) @) = 0 by MATRIX_1:def_6;
then A4: (Segm (A,nt,mt)) @ = {} ;
len (Segm ((A @),mt,nt)) = 0 by A2, A3, Th1;
hence (Segm (A,nt,mt)) @ = Segm ((A @),mt,nt) by A4; ::_thesis: verum
end;
supposeA5: n > 0 ; ::_thesis: (Segm (A,nt,mt)) @ = Segm ((A @),mt,nt)
then A6: width (Segm (A,nt,mt)) = m by Th1;
A7: width (Segm ((A @),mt,nt)) = n by A2, Th1;
A8: now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_((Segm_(A,nt,mt))_@)_holds_
((Segm_(A,nt,mt))_@)_*_(i,j)_=_(Segm_((A_@),mt,nt))_*_(i,j)
A9: Indices (Segm ((A @),mt,nt)) = [:(Seg m),(Seg n):] by A7, MATRIX_1:25;
let i, j be Nat; ::_thesis: ( [i,j] in Indices ((Segm (A,nt,mt)) @) implies ((Segm (A,nt,mt)) @) * (i,j) = (Segm ((A @),mt,nt)) * (i,j) )
assume A10: [i,j] in Indices ((Segm (A,nt,mt)) @) ; ::_thesis: ((Segm (A,nt,mt)) @) * (i,j) = (Segm ((A @),mt,nt)) * (i,j)
reconsider i9 = i, j9 = j as Element of NAT by ORDINAL1:def_12;
A11: [j9,i9] in Indices (Segm (A,nt,mt)) by A10, MATRIX_1:def_6;
then A12: ((Segm (A,nt,mt)) @) * (i9,j9) = (Segm (A,nt,mt)) * (j9,i9) by MATRIX_1:def_6;
Indices (Segm (A,nt,mt)) = [:(Seg n),(Seg m):] by A6, MATRIX_1:25;
then A13: j9 in Seg n by A11, ZFMISC_1:87;
i9 in Seg m by A6, A11, ZFMISC_1:87;
then A14: [i9,j9] in Indices (Segm ((A @),mt,nt)) by A13, A9, ZFMISC_1:87;
A15: (Segm (A,nt,mt)) * (j9,i9) = A * ((nt . j),(mt . i)) by A11, Def1;
[(nt . j),(mt . i)] in Indices A by A1, A11, Th17;
then ((Segm (A,nt,mt)) @) * (i9,j9) = (A @) * ((mt . i),(nt . j)) by A15, A12, MATRIX_1:def_6;
hence ((Segm (A,nt,mt)) @) * (i,j) = (Segm ((A @),mt,nt)) * (i,j) by A14, Def1; ::_thesis: verum
end;
len (Segm ((A @),mt,nt)) = m by A2, Th1;
then A16: len ((Segm (A,nt,mt)) @) = len (Segm ((A @),mt,nt)) by A2, A5, A6, MATRIX_2:10;
len (Segm (A,nt,mt)) = n by A5, Th1;
hence (Segm (A,nt,mt)) @ = Segm ((A @),mt,nt) by A2, A5, A6, A7, A16, A8, MATRIX_1:21, MATRIX_2:10; ::_thesis: verum
end;
end;
end;
theorem Th19: :: MATRIX13:19
for D being non empty set
for m, n being Nat
for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st [:(rng nt),(rng mt):] c= Indices A & ( m = 0 implies n = 0 ) holds
Segm (A,nt,mt) = (Segm ((A @),mt,nt)) @
proof
let D be non empty set ; ::_thesis: for m, n being Nat
for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st [:(rng nt),(rng mt):] c= Indices A & ( m = 0 implies n = 0 ) holds
Segm (A,nt,mt) = (Segm ((A @),mt,nt)) @
let m, n be Nat; ::_thesis: for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st [:(rng nt),(rng mt):] c= Indices A & ( m = 0 implies n = 0 ) holds
Segm (A,nt,mt) = (Segm ((A @),mt,nt)) @
let A be Matrix of D; ::_thesis: for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st [:(rng nt),(rng mt):] c= Indices A & ( m = 0 implies n = 0 ) holds
Segm (A,nt,mt) = (Segm ((A @),mt,nt)) @
let nt be Element of n -tuples_on NAT; ::_thesis: for mt being Element of m -tuples_on NAT st [:(rng nt),(rng mt):] c= Indices A & ( m = 0 implies n = 0 ) holds
Segm (A,nt,mt) = (Segm ((A @),mt,nt)) @
let mt be Element of m -tuples_on NAT; ::_thesis: ( [:(rng nt),(rng mt):] c= Indices A & ( m = 0 implies n = 0 ) implies Segm (A,nt,mt) = (Segm ((A @),mt,nt)) @ )
assume that
A1: [:(rng nt),(rng mt):] c= Indices A and
A2: ( m = 0 implies n = 0 ) ; ::_thesis: Segm (A,nt,mt) = (Segm ((A @),mt,nt)) @
set S9 = Segm ((A @),mt,nt);
set S = Segm (A,nt,mt);
percases ( n = 0 or n > 0 ) ;
supposeA3: n = 0 ; ::_thesis: Segm (A,nt,mt) = (Segm ((A @),mt,nt)) @
( len (Segm ((A @),mt,nt)) = 0 or ( len (Segm ((A @),mt,nt)) > 0 & len (Segm ((A @),mt,nt)) = m ) ) by MATRIX_1:def_2;
then width (Segm ((A @),mt,nt)) = 0 by A3, MATRIX_1:23, MATRIX_1:def_3;
then A4: len ((Segm ((A @),mt,nt)) @) = 0 by MATRIX_1:def_6;
len (Segm (A,nt,mt)) = 0 by A3, MATRIX_1:def_2;
then Segm (A,nt,mt) = {} ;
hence Segm (A,nt,mt) = (Segm ((A @),mt,nt)) @ by A4; ::_thesis: verum
end;
supposeA5: n > 0 ; ::_thesis: Segm (A,nt,mt) = (Segm ((A @),mt,nt)) @
then A6: width (Segm (A,nt,mt)) = m by Th1;
len (Segm (A,nt,mt)) = n by A5, Th1;
then ((Segm (A,nt,mt)) @) @ = Segm (A,nt,mt) by A2, A5, A6, MATRIX_2:13;
hence Segm (A,nt,mt) = (Segm ((A @),mt,nt)) @ by A1, A2, A5, Th18; ::_thesis: verum
end;
end;
end;
theorem Th20: :: MATRIX13:20
for D being non empty set
for A being Matrix of 1,D holds A = <*<*(A * (1,1))*>*>
proof
let D be non empty set ; ::_thesis: for A being Matrix of 1,D holds A = <*<*(A * (1,1))*>*>
let A be Matrix of 1,D; ::_thesis: A = <*<*(A * (1,1))*>*>
reconsider AA = <*<*(A * (1,1))*>*> as Matrix of 1,D by MATRIX_1:15;
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_A_holds_
AA_*_(i,j)_=_A_*_(i,j)
A1: Indices A = [:(Seg 1),(Seg 1):] by MATRIX_1:24;
let i, j be Nat; ::_thesis: ( [i,j] in Indices A implies AA * (i,j) = A * (i,j) )
assume A2: [i,j] in Indices A ; ::_thesis: AA * (i,j) = A * (i,j)
j in {1} by A2, A1, FINSEQ_1:2, ZFMISC_1:87;
then A3: j = 1 by TARSKI:def_1;
i in {1} by A2, A1, FINSEQ_1:2, ZFMISC_1:87;
then i = 1 by TARSKI:def_1;
hence AA * (i,j) = A * (i,j) by A3, MATRIX_2:5; ::_thesis: verum
end;
hence A = <*<*(A * (1,1))*>*> by MATRIX_1:27; ::_thesis: verum
end;
theorem Th21: :: MATRIX13:21
for D being non empty set
for n, m being Nat
for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st n = 1 & m = 1 holds
Segm (A,nt,mt) = <*<*(A * ((nt . 1),(mt . 1)))*>*>
proof
let D be non empty set ; ::_thesis: for n, m being Nat
for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st n = 1 & m = 1 holds
Segm (A,nt,mt) = <*<*(A * ((nt . 1),(mt . 1)))*>*>
let n, m be Nat; ::_thesis: for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st n = 1 & m = 1 holds
Segm (A,nt,mt) = <*<*(A * ((nt . 1),(mt . 1)))*>*>
let A be Matrix of D; ::_thesis: for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st n = 1 & m = 1 holds
Segm (A,nt,mt) = <*<*(A * ((nt . 1),(mt . 1)))*>*>
let nt be Element of n -tuples_on NAT; ::_thesis: for mt being Element of m -tuples_on NAT st n = 1 & m = 1 holds
Segm (A,nt,mt) = <*<*(A * ((nt . 1),(mt . 1)))*>*>
let mt be Element of m -tuples_on NAT; ::_thesis: ( n = 1 & m = 1 implies Segm (A,nt,mt) = <*<*(A * ((nt . 1),(mt . 1)))*>*> )
A1: 1 in Seg 1 ;
assume that
A2: n = 1 and
A3: m = 1 ; ::_thesis: Segm (A,nt,mt) = <*<*(A * ((nt . 1),(mt . 1)))*>*>
Indices (Segm (A,nt,mt)) = [:(Seg 1),(Seg 1):] by A2, A3, MATRIX_1:24;
then [1,1] in Indices (Segm (A,nt,mt)) by A1, ZFMISC_1:87;
then (Segm (A,nt,mt)) * (1,1) = A * ((nt . 1),(mt . 1)) by Def1;
hence Segm (A,nt,mt) = <*<*(A * ((nt . 1),(mt . 1)))*>*> by A2, A3, Th20; ::_thesis: verum
end;
theorem Th22: :: MATRIX13:22
for D being non empty set
for A being Matrix of 2,D holds A = ((A * (1,1)),(A * (1,2))) ][ ((A * (2,1)),(A * (2,2)))
proof
let D be non empty set ; ::_thesis: for A being Matrix of 2,D holds A = ((A * (1,1)),(A * (1,2))) ][ ((A * (2,1)),(A * (2,2)))
let A be Matrix of 2,D; ::_thesis: A = ((A * (1,1)),(A * (1,2))) ][ ((A * (2,1)),(A * (2,2)))
reconsider AA = ((A * (1,1)),(A * (1,2))) ][ ((A * (2,1)),(A * (2,2))) as Matrix of 2,D ;
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_A_holds_
AA_*_(i,j)_=_A_*_(i,j)
A1: Indices A = [:(Seg 2),(Seg 2):] by MATRIX_1:24;
let i, j be Nat; ::_thesis: ( [i,j] in Indices A implies AA * (i,j) = A * (i,j) )
assume A2: [i,j] in Indices A ; ::_thesis: AA * (i,j) = A * (i,j)
j in {1,2} by A2, A1, FINSEQ_1:2, ZFMISC_1:87;
then A3: ( j = 1 or j = 2 ) by TARSKI:def_2;
i in {1,2} by A2, A1, FINSEQ_1:2, ZFMISC_1:87;
then ( i = 1 or i = 2 ) by TARSKI:def_2;
hence AA * (i,j) = A * (i,j) by A3, MATRIX_2:6; ::_thesis: verum
end;
hence A = ((A * (1,1)),(A * (1,2))) ][ ((A * (2,1)),(A * (2,2))) by MATRIX_1:27; ::_thesis: verum
end;
theorem Th23: :: MATRIX13:23
for D being non empty set
for n, m being Nat
for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st n = 2 & m = 2 holds
Segm (A,nt,mt) = ((A * ((nt . 1),(mt . 1))),(A * ((nt . 1),(mt . 2)))) ][ ((A * ((nt . 2),(mt . 1))),(A * ((nt . 2),(mt . 2))))
proof
let D be non empty set ; ::_thesis: for n, m being Nat
for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st n = 2 & m = 2 holds
Segm (A,nt,mt) = ((A * ((nt . 1),(mt . 1))),(A * ((nt . 1),(mt . 2)))) ][ ((A * ((nt . 2),(mt . 1))),(A * ((nt . 2),(mt . 2))))
let n, m be Nat; ::_thesis: for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st n = 2 & m = 2 holds
Segm (A,nt,mt) = ((A * ((nt . 1),(mt . 1))),(A * ((nt . 1),(mt . 2)))) ][ ((A * ((nt . 2),(mt . 1))),(A * ((nt . 2),(mt . 2))))
let A be Matrix of D; ::_thesis: for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st n = 2 & m = 2 holds
Segm (A,nt,mt) = ((A * ((nt . 1),(mt . 1))),(A * ((nt . 1),(mt . 2)))) ][ ((A * ((nt . 2),(mt . 1))),(A * ((nt . 2),(mt . 2))))
let nt be Element of n -tuples_on NAT; ::_thesis: for mt being Element of m -tuples_on NAT st n = 2 & m = 2 holds
Segm (A,nt,mt) = ((A * ((nt . 1),(mt . 1))),(A * ((nt . 1),(mt . 2)))) ][ ((A * ((nt . 2),(mt . 1))),(A * ((nt . 2),(mt . 2))))
let mt be Element of m -tuples_on NAT; ::_thesis: ( n = 2 & m = 2 implies Segm (A,nt,mt) = ((A * ((nt . 1),(mt . 1))),(A * ((nt . 1),(mt . 2)))) ][ ((A * ((nt . 2),(mt . 1))),(A * ((nt . 2),(mt . 2)))) )
set S = Segm (A,nt,mt);
set I = Indices (Segm (A,nt,mt));
assume that
A1: n = 2 and
A2: m = 2 ; ::_thesis: Segm (A,nt,mt) = ((A * ((nt . 1),(mt . 1))),(A * ((nt . 1),(mt . 2)))) ][ ((A * ((nt . 2),(mt . 1))),(A * ((nt . 2),(mt . 2))))
A3: Indices (Segm (A,nt,mt)) = [:(Seg 2),(Seg 2):] by A1, A2, MATRIX_1:24;
A4: 2 in Seg 2 ;
then [2,2] in Indices (Segm (A,nt,mt)) by A3, ZFMISC_1:87;
then A5: (Segm (A,nt,mt)) * (2,2) = A * ((nt . 2),(mt . 2)) by Def1;
A6: 1 in Seg 2 ;
then [1,1] in Indices (Segm (A,nt,mt)) by A3, ZFMISC_1:87;
then A7: (Segm (A,nt,mt)) * (1,1) = A * ((nt . 1),(mt . 1)) by Def1;
[2,1] in Indices (Segm (A,nt,mt)) by A6, A4, A3, ZFMISC_1:87;
then A8: (Segm (A,nt,mt)) * (2,1) = A * ((nt . 2),(mt . 1)) by Def1;
[1,2] in Indices (Segm (A,nt,mt)) by A6, A4, A3, ZFMISC_1:87;
then (Segm (A,nt,mt)) * (1,2) = A * ((nt . 1),(mt . 2)) by Def1;
hence Segm (A,nt,mt) = ((A * ((nt . 1),(mt . 1))),(A * ((nt . 1),(mt . 2)))) ][ ((A * ((nt . 2),(mt . 1))),(A * ((nt . 2),(mt . 2)))) by A1, A2, A7, A8, A5, Th22; ::_thesis: verum
end;
theorem Th24: :: MATRIX13:24
for D being non empty set
for m, i, n being Nat
for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st i in Seg n & rng mt c= Seg (width A) holds
Line ((Segm (A,nt,mt)),i) = (Line (A,(nt . i))) * mt
proof
let D be non empty set ; ::_thesis: for m, i, n being Nat
for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st i in Seg n & rng mt c= Seg (width A) holds
Line ((Segm (A,nt,mt)),i) = (Line (A,(nt . i))) * mt
let m, i, n be Nat; ::_thesis: for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st i in Seg n & rng mt c= Seg (width A) holds
Line ((Segm (A,nt,mt)),i) = (Line (A,(nt . i))) * mt
let A be Matrix of D; ::_thesis: for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st i in Seg n & rng mt c= Seg (width A) holds
Line ((Segm (A,nt,mt)),i) = (Line (A,(nt . i))) * mt
let nt be Element of n -tuples_on NAT; ::_thesis: for mt being Element of m -tuples_on NAT st i in Seg n & rng mt c= Seg (width A) holds
Line ((Segm (A,nt,mt)),i) = (Line (A,(nt . i))) * mt
let mt be Element of m -tuples_on NAT; ::_thesis: ( i in Seg n & rng mt c= Seg (width A) implies Line ((Segm (A,nt,mt)),i) = (Line (A,(nt . i))) * mt )
set S = Segm (A,nt,mt);
set Li = Line ((Segm (A,nt,mt)),i);
set LA = Line (A,(nt . i));
assume that
A1: i in Seg n and
A2: rng mt c= Seg (width A) ; ::_thesis: Line ((Segm (A,nt,mt)),i) = (Line (A,(nt . i))) * mt
n <> 0 by A1;
then A3: width (Segm (A,nt,mt)) = m by Th1;
then len (Line ((Segm (A,nt,mt)),i)) = m by MATRIX_1:def_7;
then A4: dom (Line ((Segm (A,nt,mt)),i)) = Seg m by FINSEQ_1:def_3;
A5: dom mt = Seg m by FINSEQ_2:124;
len (Line (A,(nt . i))) = width A by MATRIX_1:def_7;
then dom (Line (A,(nt . i))) = Seg (width A) by FINSEQ_1:def_3;
then A6: dom ((Line (A,(nt . i))) * mt) = dom mt by A2, RELAT_1:27;
now__::_thesis:_for_x_being_set_st_x_in_dom_(Line_((Segm_(A,nt,mt)),i))_holds_
(Line_((Segm_(A,nt,mt)),i))_._x_=_((Line_(A,(nt_._i)))_*_mt)_._x
let x be set ; ::_thesis: ( x in dom (Line ((Segm (A,nt,mt)),i)) implies (Line ((Segm (A,nt,mt)),i)) . x = ((Line (A,(nt . i))) * mt) . x )
assume A7: x in dom (Line ((Segm (A,nt,mt)),i)) ; ::_thesis: (Line ((Segm (A,nt,mt)),i)) . x = ((Line (A,(nt . i))) * mt) . x
consider k being Element of NAT such that
A8: k = x and
1 <= k and
k <= m by A4, A7;
A9: (Line ((Segm (A,nt,mt)),i)) . k = (Segm (A,nt,mt)) * (i,k) by A3, A4, A7, A8, MATRIX_1:def_7;
[i,k] in [:(Seg n),(Seg (width (Segm (A,nt,mt)))):] by A1, A3, A4, A7, A8, ZFMISC_1:87;
then A10: [i,k] in Indices (Segm (A,nt,mt)) by MATRIX_1:25;
mt . k in rng mt by A5, A4, A7, A8, FUNCT_1:def_3;
then A11: (Line (A,(nt . i))) . (mt . k) = A * ((nt . i),(mt . k)) by A2, MATRIX_1:def_7;
((Line (A,(nt . i))) * mt) . k = (Line (A,(nt . i))) . (mt . k) by A6, A5, A4, A7, A8, FUNCT_1:12;
hence (Line ((Segm (A,nt,mt)),i)) . x = ((Line (A,(nt . i))) * mt) . x by A8, A11, A10, A9, Def1; ::_thesis: verum
end;
hence Line ((Segm (A,nt,mt)),i) = (Line (A,(nt . i))) * mt by A6, A5, A4, FUNCT_1:2; ::_thesis: verum
end;
theorem Th25: :: MATRIX13:25
for D being non empty set
for m, i, n, j being Nat
for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st i in Seg n & j in Seg n & nt . i = nt . j holds
Line ((Segm (A,nt,mt)),i) = Line ((Segm (A,nt,mt)),j)
proof
let D be non empty set ; ::_thesis: for m, i, n, j being Nat
for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st i in Seg n & j in Seg n & nt . i = nt . j holds
Line ((Segm (A,nt,mt)),i) = Line ((Segm (A,nt,mt)),j)
let m, i, n, j be Nat; ::_thesis: for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st i in Seg n & j in Seg n & nt . i = nt . j holds
Line ((Segm (A,nt,mt)),i) = Line ((Segm (A,nt,mt)),j)
let A be Matrix of D; ::_thesis: for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st i in Seg n & j in Seg n & nt . i = nt . j holds
Line ((Segm (A,nt,mt)),i) = Line ((Segm (A,nt,mt)),j)
let nt be Element of n -tuples_on NAT; ::_thesis: for mt being Element of m -tuples_on NAT st i in Seg n & j in Seg n & nt . i = nt . j holds
Line ((Segm (A,nt,mt)),i) = Line ((Segm (A,nt,mt)),j)
let mt be Element of m -tuples_on NAT; ::_thesis: ( i in Seg n & j in Seg n & nt . i = nt . j implies Line ((Segm (A,nt,mt)),i) = Line ((Segm (A,nt,mt)),j) )
set S = Segm (A,nt,mt);
set Li = Line ((Segm (A,nt,mt)),i);
set Lj = Line ((Segm (A,nt,mt)),j);
assume that
A1: i in Seg n and
A2: j in Seg n and
A3: nt . i = nt . j ; ::_thesis: Line ((Segm (A,nt,mt)),i) = Line ((Segm (A,nt,mt)),j)
A4: now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_k_<=_width_(Segm_(A,nt,mt))_holds_
(Line_((Segm_(A,nt,mt)),i))_._k_=_(Line_((Segm_(A,nt,mt)),j))_._k
let k be Nat; ::_thesis: ( 1 <= k & k <= width (Segm (A,nt,mt)) implies (Line ((Segm (A,nt,mt)),i)) . k = (Line ((Segm (A,nt,mt)),j)) . k )
assume that
A5: 1 <= k and
A6: k <= width (Segm (A,nt,mt)) ; ::_thesis: (Line ((Segm (A,nt,mt)),i)) . k = (Line ((Segm (A,nt,mt)),j)) . k
k in NAT by ORDINAL1:def_12;
then A7: k in Seg (width (Segm (A,nt,mt))) by A5, A6;
then [i,k] in [:(Seg n),(Seg (width (Segm (A,nt,mt)))):] by A1, ZFMISC_1:87;
then [i,k] in Indices (Segm (A,nt,mt)) by MATRIX_1:25;
then A8: (Segm (A,nt,mt)) * (i,k) = A * ((nt . i),(mt . k)) by Def1;
[j,k] in [:(Seg n),(Seg (width (Segm (A,nt,mt)))):] by A2, A7, ZFMISC_1:87;
then [j,k] in Indices (Segm (A,nt,mt)) by MATRIX_1:25;
then A9: (Segm (A,nt,mt)) * (j,k) = A * ((nt . j),(mt . k)) by Def1;
(Segm (A,nt,mt)) * (i,k) = (Line ((Segm (A,nt,mt)),i)) . k by A7, MATRIX_1:def_7;
hence (Line ((Segm (A,nt,mt)),i)) . k = (Line ((Segm (A,nt,mt)),j)) . k by A3, A7, A8, A9, MATRIX_1:def_7; ::_thesis: verum
end;
A10: len (Line ((Segm (A,nt,mt)),j)) = width (Segm (A,nt,mt)) by MATRIX_1:def_7;
len (Line ((Segm (A,nt,mt)),i)) = width (Segm (A,nt,mt)) by MATRIX_1:def_7;
hence Line ((Segm (A,nt,mt)),i) = Line ((Segm (A,nt,mt)),j) by A10, A4, FINSEQ_1:14; ::_thesis: verum
end;
theorem Th26: :: MATRIX13:26
for i, n, j being Nat
for K being Field
for nt, nt1 being Element of n -tuples_on NAT
for M being Matrix of K st i in Seg n & j in Seg n & nt . i = nt . j & i <> j holds
Det (Segm (M,nt,nt1)) = 0. K
proof
let i, n, j be Nat; ::_thesis: for K being Field
for nt, nt1 being Element of n -tuples_on NAT
for M being Matrix of K st i in Seg n & j in Seg n & nt . i = nt . j & i <> j holds
Det (Segm (M,nt,nt1)) = 0. K
let K be Field; ::_thesis: for nt, nt1 being Element of n -tuples_on NAT
for M being Matrix of K st i in Seg n & j in Seg n & nt . i = nt . j & i <> j holds
Det (Segm (M,nt,nt1)) = 0. K
let nt, nt1 be Element of n -tuples_on NAT; ::_thesis: for M being Matrix of K st i in Seg n & j in Seg n & nt . i = nt . j & i <> j holds
Det (Segm (M,nt,nt1)) = 0. K
let M be Matrix of K; ::_thesis: ( i in Seg n & j in Seg n & nt . i = nt . j & i <> j implies Det (Segm (M,nt,nt1)) = 0. K )
assume that
A1: i in Seg n and
A2: j in Seg n and
A3: nt . i = nt . j and
A4: i <> j ; ::_thesis: Det (Segm (M,nt,nt1)) = 0. K
A5: ( i < j or j < i ) by A4, XXREAL_0:1;
Line ((Segm (M,nt,nt1)),i) = Line ((Segm (M,nt,nt1)),j) by A1, A2, A3, Th25;
hence Det (Segm (M,nt,nt1)) = 0. K by A1, A2, A5, MATRIX11:50; ::_thesis: verum
end;
theorem Th27: :: MATRIX13:27
for n being Nat
for K being Field
for nt, nt1 being Element of n -tuples_on NAT
for M being Matrix of K st not nt is one-to-one holds
Det (Segm (M,nt,nt1)) = 0. K
proof
let n be Nat; ::_thesis: for K being Field
for nt, nt1 being Element of n -tuples_on NAT
for M being Matrix of K st not nt is one-to-one holds
Det (Segm (M,nt,nt1)) = 0. K
let K be Field; ::_thesis: for nt, nt1 being Element of n -tuples_on NAT
for M being Matrix of K st not nt is one-to-one holds
Det (Segm (M,nt,nt1)) = 0. K
let nt, nt1 be Element of n -tuples_on NAT; ::_thesis: for M being Matrix of K st not nt is one-to-one holds
Det (Segm (M,nt,nt1)) = 0. K
let M be Matrix of K; ::_thesis: ( not nt is one-to-one implies Det (Segm (M,nt,nt1)) = 0. K )
assume not nt is one-to-one ; ::_thesis: Det (Segm (M,nt,nt1)) = 0. K
then consider x, y being set such that
A1: x in dom nt and
A2: y in dom nt and
A3: nt . x = nt . y and
A4: x <> y by FUNCT_1:def_4;
A5: dom nt = Seg n by FINSEQ_2:124;
then consider i being Element of NAT such that
A6: x = i and
A7: 1 <= i and
A8: i <= n by A1;
consider j being Element of NAT such that
A9: y = j and
A10: 1 <= j and
A11: j <= n by A2, A5;
A12: j in Seg n by A10, A11;
i in Seg n by A7, A8;
hence Det (Segm (M,nt,nt1)) = 0. K by A3, A4, A6, A9, A12, Th26; ::_thesis: verum
end;
theorem Th28: :: MATRIX13:28
for D being non empty set
for n, j, m being Nat
for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st j in Seg m & rng nt c= Seg (len A) holds
Col ((Segm (A,nt,mt)),j) = (Col (A,(mt . j))) * nt
proof
let D be non empty set ; ::_thesis: for n, j, m being Nat
for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st j in Seg m & rng nt c= Seg (len A) holds
Col ((Segm (A,nt,mt)),j) = (Col (A,(mt . j))) * nt
let n, j, m be Nat; ::_thesis: for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st j in Seg m & rng nt c= Seg (len A) holds
Col ((Segm (A,nt,mt)),j) = (Col (A,(mt . j))) * nt
let A be Matrix of D; ::_thesis: for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st j in Seg m & rng nt c= Seg (len A) holds
Col ((Segm (A,nt,mt)),j) = (Col (A,(mt . j))) * nt
let nt be Element of n -tuples_on NAT; ::_thesis: for mt being Element of m -tuples_on NAT st j in Seg m & rng nt c= Seg (len A) holds
Col ((Segm (A,nt,mt)),j) = (Col (A,(mt . j))) * nt
let mt be Element of m -tuples_on NAT; ::_thesis: ( j in Seg m & rng nt c= Seg (len A) implies Col ((Segm (A,nt,mt)),j) = (Col (A,(mt . j))) * nt )
set S = Segm (A,nt,mt);
set Cj = Col ((Segm (A,nt,mt)),j);
set CA = Col (A,(mt . j));
assume that
A1: j in Seg m and
A2: rng nt c= Seg (len A) ; ::_thesis: Col ((Segm (A,nt,mt)),j) = (Col (A,(mt . j))) * nt
len (Col (A,(mt . j))) = len A by MATRIX_1:def_8;
then dom (Col (A,(mt . j))) = Seg (len A) by FINSEQ_1:def_3;
then A3: dom ((Col (A,(mt . j))) * nt) = dom nt by A2, RELAT_1:27;
A4: dom nt = Seg n by FINSEQ_2:124;
A5: len (Segm (A,nt,mt)) = n by MATRIX_1:def_2;
then A6: dom (Segm (A,nt,mt)) = Seg n by FINSEQ_1:def_3;
len (Col ((Segm (A,nt,mt)),j)) = n by A5, MATRIX_1:def_8;
then A7: dom (Col ((Segm (A,nt,mt)),j)) = Seg n by FINSEQ_1:def_3;
A8: dom A = Seg (len A) by FINSEQ_1:def_3;
now__::_thesis:_for_x_being_set_st_x_in_dom_(Col_((Segm_(A,nt,mt)),j))_holds_
(Col_((Segm_(A,nt,mt)),j))_._x_=_((Col_(A,(mt_._j)))_*_nt)_._x
let x be set ; ::_thesis: ( x in dom (Col ((Segm (A,nt,mt)),j)) implies (Col ((Segm (A,nt,mt)),j)) . x = ((Col (A,(mt . j))) * nt) . x )
assume A9: x in dom (Col ((Segm (A,nt,mt)),j)) ; ::_thesis: (Col ((Segm (A,nt,mt)),j)) . x = ((Col (A,(mt . j))) * nt) . x
consider k being Element of NAT such that
A10: k = x and
A11: 1 <= k and
A12: k <= n by A7, A9;
A13: (Col ((Segm (A,nt,mt)),j)) . k = (Segm (A,nt,mt)) * (k,j) by A6, A7, A9, A10, MATRIX_1:def_8;
nt . k in rng nt by A4, A7, A9, A10, FUNCT_1:def_3;
then A14: (Col (A,(mt . j))) . (nt . k) = A * ((nt . k),(mt . j)) by A2, A8, MATRIX_1:def_8;
[k,j] in [:(Seg n),(Seg m):] by A1, A7, A9, A10, ZFMISC_1:87;
then A15: [k,j] in Indices (Segm (A,nt,mt)) by A6, A11, A12, Th1;
((Col (A,(mt . j))) * nt) . k = (Col (A,(mt . j))) . (nt . k) by A3, A4, A7, A9, A10, FUNCT_1:12;
hence (Col ((Segm (A,nt,mt)),j)) . x = ((Col (A,(mt . j))) * nt) . x by A10, A14, A15, A13, Def1; ::_thesis: verum
end;
hence Col ((Segm (A,nt,mt)),j) = (Col (A,(mt . j))) * nt by A3, A4, A7, FUNCT_1:2; ::_thesis: verum
end;
theorem Th29: :: MATRIX13:29
for D being non empty set
for n, i, m, j being Nat
for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st i in Seg m & j in Seg m & mt . i = mt . j holds
Col ((Segm (A,nt,mt)),i) = Col ((Segm (A,nt,mt)),j)
proof
let D be non empty set ; ::_thesis: for n, i, m, j being Nat
for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st i in Seg m & j in Seg m & mt . i = mt . j holds
Col ((Segm (A,nt,mt)),i) = Col ((Segm (A,nt,mt)),j)
let n, i, m, j be Nat; ::_thesis: for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st i in Seg m & j in Seg m & mt . i = mt . j holds
Col ((Segm (A,nt,mt)),i) = Col ((Segm (A,nt,mt)),j)
let A be Matrix of D; ::_thesis: for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st i in Seg m & j in Seg m & mt . i = mt . j holds
Col ((Segm (A,nt,mt)),i) = Col ((Segm (A,nt,mt)),j)
let nt be Element of n -tuples_on NAT; ::_thesis: for mt being Element of m -tuples_on NAT st i in Seg m & j in Seg m & mt . i = mt . j holds
Col ((Segm (A,nt,mt)),i) = Col ((Segm (A,nt,mt)),j)
let mt be Element of m -tuples_on NAT; ::_thesis: ( i in Seg m & j in Seg m & mt . i = mt . j implies Col ((Segm (A,nt,mt)),i) = Col ((Segm (A,nt,mt)),j) )
set S = Segm (A,nt,mt);
set Ci = Col ((Segm (A,nt,mt)),i);
set Cj = Col ((Segm (A,nt,mt)),j);
assume that
A1: i in Seg m and
A2: j in Seg m and
A3: mt . i = mt . j ; ::_thesis: Col ((Segm (A,nt,mt)),i) = Col ((Segm (A,nt,mt)),j)
A4: now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_k_<=_len_(Segm_(A,nt,mt))_holds_
(Col_((Segm_(A,nt,mt)),i))_._k_=_(Col_((Segm_(A,nt,mt)),j))_._k
let k be Nat; ::_thesis: ( 1 <= k & k <= len (Segm (A,nt,mt)) implies (Col ((Segm (A,nt,mt)),i)) . k = (Col ((Segm (A,nt,mt)),j)) . k )
assume that
A5: 1 <= k and
A6: k <= len (Segm (A,nt,mt)) ; ::_thesis: (Col ((Segm (A,nt,mt)),i)) . k = (Col ((Segm (A,nt,mt)),j)) . k
k in NAT by ORDINAL1:def_12;
then A7: k in Seg (len (Segm (A,nt,mt))) by A5, A6;
then A8: k in Seg n by MATRIX_1:def_2;
then n <> 0 ;
then A9: width (Segm (A,nt,mt)) = m by Th1;
[k,j] in [:(Seg n),(Seg m):] by A2, A8, ZFMISC_1:87;
then [k,j] in Indices (Segm (A,nt,mt)) by A9, MATRIX_1:25;
then A10: (Segm (A,nt,mt)) * (k,j) = A * ((nt . k),(mt . j)) by Def1;
[k,i] in [:(Seg n),(Seg m):] by A1, A8, ZFMISC_1:87;
then [k,i] in Indices (Segm (A,nt,mt)) by A9, MATRIX_1:25;
then A11: (Segm (A,nt,mt)) * (k,i) = A * ((nt . k),(mt . i)) by Def1;
A12: k in dom (Segm (A,nt,mt)) by A7, FINSEQ_1:def_3;
then (Segm (A,nt,mt)) * (k,i) = (Col ((Segm (A,nt,mt)),i)) . k by MATRIX_1:def_8;
hence (Col ((Segm (A,nt,mt)),i)) . k = (Col ((Segm (A,nt,mt)),j)) . k by A3, A12, A11, A10, MATRIX_1:def_8; ::_thesis: verum
end;
A13: len (Col ((Segm (A,nt,mt)),j)) = len (Segm (A,nt,mt)) by MATRIX_1:def_8;
len (Col ((Segm (A,nt,mt)),i)) = len (Segm (A,nt,mt)) by MATRIX_1:def_8;
hence Col ((Segm (A,nt,mt)),i) = Col ((Segm (A,nt,mt)),j) by A13, A4, FINSEQ_1:14; ::_thesis: verum
end;
theorem Th30: :: MATRIX13:30
for i, m, j being Nat
for K being Field
for mt, mt1 being Element of m -tuples_on NAT
for M being Matrix of K st i in Seg m & j in Seg m & mt . i = mt . j & i <> j holds
Det (Segm (M,mt1,mt)) = 0. K
proof
let i, m, j be Nat; ::_thesis: for K being Field
for mt, mt1 being Element of m -tuples_on NAT
for M being Matrix of K st i in Seg m & j in Seg m & mt . i = mt . j & i <> j holds
Det (Segm (M,mt1,mt)) = 0. K
let K be Field; ::_thesis: for mt, mt1 being Element of m -tuples_on NAT
for M being Matrix of K st i in Seg m & j in Seg m & mt . i = mt . j & i <> j holds
Det (Segm (M,mt1,mt)) = 0. K
let mt, mt1 be Element of m -tuples_on NAT; ::_thesis: for M being Matrix of K st i in Seg m & j in Seg m & mt . i = mt . j & i <> j holds
Det (Segm (M,mt1,mt)) = 0. K
let M be Matrix of K; ::_thesis: ( i in Seg m & j in Seg m & mt . i = mt . j & i <> j implies Det (Segm (M,mt1,mt)) = 0. K )
assume that
A1: i in Seg m and
A2: j in Seg m and
A3: mt . i = mt . j and
A4: i <> j ; ::_thesis: Det (Segm (M,mt1,mt)) = 0. K
A5: ( i < j or j < i ) by A4, XXREAL_0:1;
set S = Segm (M,mt1,mt);
A6: width (Segm (M,mt1,mt)) = m by MATRIX_1:24;
then A7: Col ((Segm (M,mt1,mt)),j) = Line (((Segm (M,mt1,mt)) @),j) by A2, MATRIX_2:15;
Col ((Segm (M,mt1,mt)),i) = Line (((Segm (M,mt1,mt)) @),i) by A1, A6, MATRIX_2:15;
hence 0. K = Det ((Segm (M,mt1,mt)) @) by A1, A2, A3, A7, A5, Th29, MATRIX11:50
.= Det (Segm (M,mt1,mt)) by MATRIXR2:43 ;
::_thesis: verum
end;
theorem Th31: :: MATRIX13:31
for m being Nat
for K being Field
for mt, mt1 being Element of m -tuples_on NAT
for M being Matrix of K st not mt is one-to-one holds
Det (Segm (M,mt1,mt)) = 0. K
proof
let m be Nat; ::_thesis: for K being Field
for mt, mt1 being Element of m -tuples_on NAT
for M being Matrix of K st not mt is one-to-one holds
Det (Segm (M,mt1,mt)) = 0. K
let K be Field; ::_thesis: for mt, mt1 being Element of m -tuples_on NAT
for M being Matrix of K st not mt is one-to-one holds
Det (Segm (M,mt1,mt)) = 0. K
let mt, mt1 be Element of m -tuples_on NAT; ::_thesis: for M being Matrix of K st not mt is one-to-one holds
Det (Segm (M,mt1,mt)) = 0. K
let M be Matrix of K; ::_thesis: ( not mt is one-to-one implies Det (Segm (M,mt1,mt)) = 0. K )
assume not mt is one-to-one ; ::_thesis: Det (Segm (M,mt1,mt)) = 0. K
then consider x, y being set such that
A1: x in dom mt and
A2: y in dom mt and
A3: mt . x = mt . y and
A4: x <> y by FUNCT_1:def_4;
A5: dom mt = Seg m by FINSEQ_2:124;
then consider i being Element of NAT such that
A6: x = i and
A7: 1 <= i and
A8: i <= m by A1;
consider j being Element of NAT such that
A9: y = j and
A10: 1 <= j and
A11: j <= m by A2, A5;
A12: j in Seg m by A10, A11;
i in Seg m by A7, A8;
hence Det (Segm (M,mt1,mt)) = 0. K by A3, A4, A6, A9, A12, Th30; ::_thesis: verum
end;
theorem Th32: :: MATRIX13:32
for n being Nat
for nt, nt1 being Element of n -tuples_on NAT st nt is one-to-one & nt1 is one-to-one & rng nt = rng nt1 holds
ex perm being Permutation of (Seg n) st nt1 = nt * perm
proof
let n be Nat; ::_thesis: for nt, nt1 being Element of n -tuples_on NAT st nt is one-to-one & nt1 is one-to-one & rng nt = rng nt1 holds
ex perm being Permutation of (Seg n) st nt1 = nt * perm
let nt, nt1 be Element of n -tuples_on NAT; ::_thesis: ( nt is one-to-one & nt1 is one-to-one & rng nt = rng nt1 implies ex perm being Permutation of (Seg n) st nt1 = nt * perm )
assume that
A1: nt is one-to-one and
A2: nt1 is one-to-one and
A3: rng nt = rng nt1 ; ::_thesis: ex perm being Permutation of (Seg n) st nt1 = nt * perm
reconsider nt9 = nt " as Function ;
A4: dom nt9 = rng nt1 by A1, A3, FUNCT_1:33;
A5: dom nt = Seg n by FINSEQ_2:124;
rng nt9 = dom nt by A1, FUNCT_1:33;
then A6: rng (nt9 * nt1) = Seg n by A5, A4, RELAT_1:28;
dom nt1 = Seg n by FINSEQ_2:124;
then dom (nt9 * nt1) = Seg n by A4, RELAT_1:27;
then reconsider nn = nt9 * nt1 as Function of (Seg n),(Seg n) by A6, FUNCT_2:1;
( nn is one-to-one & nn is onto ) by A1, A2, A6, FUNCT_2:def_3;
then reconsider nn = nn as Permutation of (Seg n) ;
take nn ; ::_thesis: nt1 = nt * nn
thus nt * nn = (nt * nt9) * nt1 by RELAT_1:36
.= (id (rng nt1)) * nt1 by A1, A3, FUNCT_1:39
.= nt1 by RELAT_1:54 ; ::_thesis: verum
end;
theorem Th33: :: MATRIX13:33
for D being non empty set
for m, n being Nat
for A being Matrix of D
for nt1, nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT
for f being Function of (Seg n),(Seg n) st nt1 = nt * f holds
Segm (A,nt1,mt) = (Segm (A,nt,mt)) * f
proof
let D be non empty set ; ::_thesis: for m, n being Nat
for A being Matrix of D
for nt1, nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT
for f being Function of (Seg n),(Seg n) st nt1 = nt * f holds
Segm (A,nt1,mt) = (Segm (A,nt,mt)) * f
let m, n be Nat; ::_thesis: for A being Matrix of D
for nt1, nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT
for f being Function of (Seg n),(Seg n) st nt1 = nt * f holds
Segm (A,nt1,mt) = (Segm (A,nt,mt)) * f
let A be Matrix of D; ::_thesis: for nt1, nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT
for f being Function of (Seg n),(Seg n) st nt1 = nt * f holds
Segm (A,nt1,mt) = (Segm (A,nt,mt)) * f
let nt1, nt be Element of n -tuples_on NAT; ::_thesis: for mt being Element of m -tuples_on NAT
for f being Function of (Seg n),(Seg n) st nt1 = nt * f holds
Segm (A,nt1,mt) = (Segm (A,nt,mt)) * f
let mt be Element of m -tuples_on NAT; ::_thesis: for f being Function of (Seg n),(Seg n) st nt1 = nt * f holds
Segm (A,nt1,mt) = (Segm (A,nt,mt)) * f
let f be Function of (Seg n),(Seg n); ::_thesis: ( nt1 = nt * f implies Segm (A,nt1,mt) = (Segm (A,nt,mt)) * f )
assume A1: nt1 = nt * f ; ::_thesis: Segm (A,nt1,mt) = (Segm (A,nt,mt)) * f
set S = Segm (A,nt,mt);
set S1 = Segm (A,nt1,mt);
set Sf = (Segm (A,nt,mt)) * f;
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_(Segm_(A,nt1,mt))_holds_
(Segm_(A,nt1,mt))_*_(i,j)_=_((Segm_(A,nt,mt))_*_f)_*_(i,j)
let i, j be Nat; ::_thesis: ( [i,j] in Indices (Segm (A,nt1,mt)) implies (Segm (A,nt1,mt)) * (i,j) = ((Segm (A,nt,mt)) * f) * (i,j) )
assume A2: [i,j] in Indices (Segm (A,nt1,mt)) ; ::_thesis: (Segm (A,nt1,mt)) * (i,j) = ((Segm (A,nt,mt)) * f) * (i,j)
Indices (Segm (A,nt1,mt)) = [:(Seg n),(Seg (width (Segm (A,nt1,mt)))):] by MATRIX_1:25;
then A3: i in Seg n by A2, ZFMISC_1:87;
Indices (Segm (A,nt1,mt)) = Indices (Segm (A,nt,mt)) by MATRIX_1:26;
then consider k being Nat such that
A4: f . i = k and
A5: [k,j] in Indices (Segm (A,nt,mt)) and
A6: ((Segm (A,nt,mt)) * f) * (i,j) = (Segm (A,nt,mt)) * (k,j) by A2, MATRIX11:37;
reconsider i9 = i, j9 = j, k9 = k as Element of NAT by ORDINAL1:def_12;
Seg n = dom nt1 by FINSEQ_2:124;
then nt1 . i9 = nt . (f . i) by A1, A3, FUNCT_1:12;
hence (Segm (A,nt1,mt)) * (i,j) = A * ((nt . k9),(mt . j9)) by A2, A4, Def1
.= ((Segm (A,nt,mt)) * f) * (i,j) by A5, A6, Def1 ;
::_thesis: verum
end;
hence Segm (A,nt1,mt) = (Segm (A,nt,mt)) * f by MATRIX_1:27; ::_thesis: verum
end;
theorem Th34: :: MATRIX13:34
for D being non empty set
for n, m being Nat
for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt1, mt being Element of m -tuples_on NAT
for f being Function of (Seg m),(Seg m) st mt1 = mt * f holds
(Segm (A,nt,mt1)) @ = ((Segm (A,nt,mt)) @) * f
proof
let D be non empty set ; ::_thesis: for n, m being Nat
for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt1, mt being Element of m -tuples_on NAT
for f being Function of (Seg m),(Seg m) st mt1 = mt * f holds
(Segm (A,nt,mt1)) @ = ((Segm (A,nt,mt)) @) * f
let n, m be Nat; ::_thesis: for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt1, mt being Element of m -tuples_on NAT
for f being Function of (Seg m),(Seg m) st mt1 = mt * f holds
(Segm (A,nt,mt1)) @ = ((Segm (A,nt,mt)) @) * f
let A be Matrix of D; ::_thesis: for nt being Element of n -tuples_on NAT
for mt1, mt being Element of m -tuples_on NAT
for f being Function of (Seg m),(Seg m) st mt1 = mt * f holds
(Segm (A,nt,mt1)) @ = ((Segm (A,nt,mt)) @) * f
let nt be Element of n -tuples_on NAT; ::_thesis: for mt1, mt being Element of m -tuples_on NAT
for f being Function of (Seg m),(Seg m) st mt1 = mt * f holds
(Segm (A,nt,mt1)) @ = ((Segm (A,nt,mt)) @) * f
let mt1, mt be Element of m -tuples_on NAT; ::_thesis: for f being Function of (Seg m),(Seg m) st mt1 = mt * f holds
(Segm (A,nt,mt1)) @ = ((Segm (A,nt,mt)) @) * f
let f be Function of (Seg m),(Seg m); ::_thesis: ( mt1 = mt * f implies (Segm (A,nt,mt1)) @ = ((Segm (A,nt,mt)) @) * f )
assume A1: mt1 = mt * f ; ::_thesis: (Segm (A,nt,mt1)) @ = ((Segm (A,nt,mt)) @) * f
set S = Segm (A,nt,mt);
set S1 = Segm (A,nt,mt1);
percases ( m = 0 or n = 0 or ( m > 0 & n > 0 ) ) ;
supposeA2: ( m = 0 or n = 0 ) ; ::_thesis: (Segm (A,nt,mt1)) @ = ((Segm (A,nt,mt)) @) * f
len (Segm (A,nt,mt)) = n by MATRIX_1:def_2;
then width (Segm (A,nt,mt)) = 0 by A2, Th1, MATRIX_1:def_3;
then len ((Segm (A,nt,mt)) @) = 0 by MATRIX_1:def_6;
then A3: (Segm (A,nt,mt)) @ = {} ;
len (Segm (A,nt,mt1)) = n by MATRIX_1:def_2;
then width (Segm (A,nt,mt1)) = 0 by A2, Th1, MATRIX_1:def_3;
then len ((Segm (A,nt,mt1)) @) = 0 by MATRIX_1:def_6;
then (Segm (A,nt,mt1)) @ = {} ;
hence (Segm (A,nt,mt1)) @ = ((Segm (A,nt,mt)) @) * f by A3; ::_thesis: verum
end;
supposeA4: ( m > 0 & n > 0 ) ; ::_thesis: (Segm (A,nt,mt1)) @ = ((Segm (A,nt,mt)) @) * f
then A5: width (Segm (A,nt,mt1)) = m by Th1;
then A6: len ((Segm (A,nt,mt1)) @) = m by A4, MATRIX_2:10;
len (Segm (A,nt,mt1)) = n by A4, Th1;
then A7: width ((Segm (A,nt,mt1)) @) = n by A4, A5, MATRIX_2:10;
A8: width (Segm (A,nt,mt)) = m by A4, Th1;
len (Segm (A,nt,mt)) = n by A4, Th1;
then A9: width ((Segm (A,nt,mt)) @) = n by A4, A8, MATRIX_2:10;
len ((Segm (A,nt,mt)) @) = m by A4, A8, MATRIX_2:10;
then reconsider S9 = (Segm (A,nt,mt)) @ , S19 = (Segm (A,nt,mt1)) @ as Matrix of m,n,D by A4, A9, A7, A6, MATRIX_1:20;
set Sf = S9 * f;
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_S19_holds_
S19_*_(i,j)_=_(S9_*_f)_*_(i,j)
let i, j be Nat; ::_thesis: ( [i,j] in Indices S19 implies S19 * (i,j) = (S9 * f) * (i,j) )
assume A10: [i,j] in Indices S19 ; ::_thesis: S19 * (i,j) = (S9 * f) * (i,j)
A11: [j,i] in Indices (Segm (A,nt,mt1)) by A10, MATRIX_1:def_6;
then A12: S19 * (i,j) = (Segm (A,nt,mt1)) * (j,i) by MATRIX_1:def_6;
Indices S19 = [:(Seg m),(Seg n):] by A7, MATRIX_1:25;
then A13: i in Seg m by A10, ZFMISC_1:87;
Indices S19 = Indices S9 by MATRIX_1:26;
then consider k being Nat such that
A14: f . i = k and
A15: [k,j] in Indices S9 and
A16: (S9 * f) * (i,j) = S9 * (k,j) by A10, MATRIX11:37;
reconsider i9 = i, j9 = j, k9 = k as Element of NAT by ORDINAL1:def_12;
Seg m = dom mt1 by FINSEQ_2:124;
then mt1 . i9 = mt . (f . i) by A1, A13, FUNCT_1:12;
then A17: (Segm (A,nt,mt1)) * (j9,i9) = A * ((nt . j9),(mt . k9)) by A14, A11, Def1;
A18: [j,k] in Indices (Segm (A,nt,mt)) by A15, MATRIX_1:def_6;
then S9 * (k,j) = (Segm (A,nt,mt)) * (j,k) by MATRIX_1:def_6;
hence S19 * (i,j) = (S9 * f) * (i,j) by A16, A18, A12, A17, Def1; ::_thesis: verum
end;
hence (Segm (A,nt,mt1)) @ = ((Segm (A,nt,mt)) @) * f by MATRIX_1:27; ::_thesis: verum
end;
end;
end;
theorem Th35: :: MATRIX13:35
for n being Nat
for K being Field
for nt1, nt2, nt being Element of n -tuples_on NAT
for M being Matrix of K
for perm being Element of Permutations n st nt1 = nt2 * perm holds
( Det (Segm (M,nt1,nt)) = - ((Det (Segm (M,nt2,nt))),perm) & Det (Segm (M,nt,nt1)) = - ((Det (Segm (M,nt,nt2))),perm) )
proof
let n be Nat; ::_thesis: for K being Field
for nt1, nt2, nt being Element of n -tuples_on NAT
for M being Matrix of K
for perm being Element of Permutations n st nt1 = nt2 * perm holds
( Det (Segm (M,nt1,nt)) = - ((Det (Segm (M,nt2,nt))),perm) & Det (Segm (M,nt,nt1)) = - ((Det (Segm (M,nt,nt2))),perm) )
let K be Field; ::_thesis: for nt1, nt2, nt being Element of n -tuples_on NAT
for M being Matrix of K
for perm being Element of Permutations n st nt1 = nt2 * perm holds
( Det (Segm (M,nt1,nt)) = - ((Det (Segm (M,nt2,nt))),perm) & Det (Segm (M,nt,nt1)) = - ((Det (Segm (M,nt,nt2))),perm) )
let nt1, nt2, nt be Element of n -tuples_on NAT; ::_thesis: for M being Matrix of K
for perm being Element of Permutations n st nt1 = nt2 * perm holds
( Det (Segm (M,nt1,nt)) = - ((Det (Segm (M,nt2,nt))),perm) & Det (Segm (M,nt,nt1)) = - ((Det (Segm (M,nt,nt2))),perm) )
let M be Matrix of K; ::_thesis: for perm being Element of Permutations n st nt1 = nt2 * perm holds
( Det (Segm (M,nt1,nt)) = - ((Det (Segm (M,nt2,nt))),perm) & Det (Segm (M,nt,nt1)) = - ((Det (Segm (M,nt,nt2))),perm) )
let perm be Element of Permutations n; ::_thesis: ( nt1 = nt2 * perm implies ( Det (Segm (M,nt1,nt)) = - ((Det (Segm (M,nt2,nt))),perm) & Det (Segm (M,nt,nt1)) = - ((Det (Segm (M,nt,nt2))),perm) ) )
assume A1: nt1 = nt2 * perm ; ::_thesis: ( Det (Segm (M,nt1,nt)) = - ((Det (Segm (M,nt2,nt))),perm) & Det (Segm (M,nt,nt1)) = - ((Det (Segm (M,nt,nt2))),perm) )
reconsider Perm = perm as Permutation of (Seg n) by MATRIX_2:def_9;
Segm (M,nt1,nt) = (Segm (M,nt2,nt)) * Perm by A1, Th33;
hence Det (Segm (M,nt1,nt)) = - ((Det (Segm (M,nt2,nt))),perm) by MATRIX11:46; ::_thesis: Det (Segm (M,nt,nt1)) = - ((Det (Segm (M,nt,nt2))),perm)
thus Det (Segm (M,nt,nt1)) = Det ((Segm (M,nt,nt1)) @) by MATRIXR2:43
.= Det (((Segm (M,nt,nt2)) @) * Perm) by A1, Th34
.= - ((Det ((Segm (M,nt,nt2)) @)),perm) by MATRIX11:46
.= - ((Det (Segm (M,nt,nt2))),perm) by MATRIXR2:43 ; ::_thesis: verum
end;
Lm1: for n being Nat
for nt, nt1 being Element of n -tuples_on NAT st rng nt = rng nt1 & nt is one-to-one holds
nt1 is one-to-one
proof
let n be Nat; ::_thesis: for nt, nt1 being Element of n -tuples_on NAT st rng nt = rng nt1 & nt is one-to-one holds
nt1 is one-to-one
let nt, nt1 be Element of n -tuples_on NAT; ::_thesis: ( rng nt = rng nt1 & nt is one-to-one implies nt1 is one-to-one )
assume that
A1: rng nt = rng nt1 and
A2: nt is one-to-one ; ::_thesis: nt1 is one-to-one
A3: len nt1 = n by CARD_1:def_7;
len nt = n by CARD_1:def_7;
hence nt1 is one-to-one by A1, A2, A3, FINSEQ_4:61; ::_thesis: verum
end;
theorem Th36: :: MATRIX13:36
for n being Nat
for K being Field
for M being Matrix of K
for nt, nt1, nt9, nt19 being Element of n -tuples_on NAT st rng nt = rng nt9 & rng nt1 = rng nt19 & not Det (Segm (M,nt,nt1)) = Det (Segm (M,nt9,nt19)) holds
Det (Segm (M,nt,nt1)) = - (Det (Segm (M,nt9,nt19)))
proof
let n be Nat; ::_thesis: for K being Field
for M being Matrix of K
for nt, nt1, nt9, nt19 being Element of n -tuples_on NAT st rng nt = rng nt9 & rng nt1 = rng nt19 & not Det (Segm (M,nt,nt1)) = Det (Segm (M,nt9,nt19)) holds
Det (Segm (M,nt,nt1)) = - (Det (Segm (M,nt9,nt19)))
let K be Field; ::_thesis: for M being Matrix of K
for nt, nt1, nt9, nt19 being Element of n -tuples_on NAT st rng nt = rng nt9 & rng nt1 = rng nt19 & not Det (Segm (M,nt,nt1)) = Det (Segm (M,nt9,nt19)) holds
Det (Segm (M,nt,nt1)) = - (Det (Segm (M,nt9,nt19)))
let M be Matrix of K; ::_thesis: for nt, nt1, nt9, nt19 being Element of n -tuples_on NAT st rng nt = rng nt9 & rng nt1 = rng nt19 & not Det (Segm (M,nt,nt1)) = Det (Segm (M,nt9,nt19)) holds
Det (Segm (M,nt,nt1)) = - (Det (Segm (M,nt9,nt19)))
let nt, nt1, nt9, nt19 be Element of n -tuples_on NAT; ::_thesis: ( rng nt = rng nt9 & rng nt1 = rng nt19 & not Det (Segm (M,nt,nt1)) = Det (Segm (M,nt9,nt19)) implies Det (Segm (M,nt,nt1)) = - (Det (Segm (M,nt9,nt19))) )
assume that
A1: rng nt = rng nt9 and
A2: rng nt1 = rng nt19 ; ::_thesis: ( Det (Segm (M,nt,nt1)) = Det (Segm (M,nt9,nt19)) or Det (Segm (M,nt,nt1)) = - (Det (Segm (M,nt9,nt19))) )
set S19 = Segm (M,nt,nt19);
set S9 = Segm (M,nt9,nt19);
set S = Segm (M,nt,nt1);
percases ( not nt is one-to-one or not nt1 is one-to-one or ( nt is one-to-one & nt1 is one-to-one ) ) ;
supposeA3: ( not nt is one-to-one or not nt1 is one-to-one ) ; ::_thesis: ( Det (Segm (M,nt,nt1)) = Det (Segm (M,nt9,nt19)) or Det (Segm (M,nt,nt1)) = - (Det (Segm (M,nt9,nt19))) )
then A4: Det (Segm (M,nt,nt1)) = 0. K by Th27, Th31;
( not nt9 is one-to-one or not nt19 is one-to-one ) by A1, A2, A3, Lm1;
hence ( Det (Segm (M,nt,nt1)) = Det (Segm (M,nt9,nt19)) or Det (Segm (M,nt,nt1)) = - (Det (Segm (M,nt9,nt19))) ) by A4, Th27, Th31; ::_thesis: verum
end;
supposeA5: ( nt is one-to-one & nt1 is one-to-one ) ; ::_thesis: ( Det (Segm (M,nt,nt1)) = Det (Segm (M,nt9,nt19)) or Det (Segm (M,nt,nt1)) = - (Det (Segm (M,nt9,nt19))) )
then nt19 is one-to-one by A2, Lm1;
then consider perm1 being Permutation of (Seg n) such that
A6: nt1 = nt19 * perm1 by A2, A5, Th32;
nt9 is one-to-one by A1, A5, Lm1;
then consider perm being Permutation of (Seg n) such that
A7: nt = nt9 * perm by A1, A5, Th32;
reconsider perm = perm, perm1 = perm1 as Element of Permutations n by MATRIX_2:def_9;
percases ( perm1 is even or perm1 is odd ) ;
supposeA8: perm1 is even ; ::_thesis: ( Det (Segm (M,nt,nt1)) = Det (Segm (M,nt9,nt19)) or Det (Segm (M,nt,nt1)) = - (Det (Segm (M,nt9,nt19))) )
Det (Segm (M,nt,nt1)) = - ((Det (Segm (M,nt,nt19))),perm1) by A6, Th35;
then A9: Det (Segm (M,nt,nt1)) = Det (Segm (M,nt,nt19)) by A8, MATRIX_2:def_13;
Det (Segm (M,nt,nt19)) = - ((Det (Segm (M,nt9,nt19))),perm) by A7, Th35;
hence ( Det (Segm (M,nt,nt1)) = Det (Segm (M,nt9,nt19)) or Det (Segm (M,nt,nt1)) = - (Det (Segm (M,nt9,nt19))) ) by A9, MATRIX_2:def_13; ::_thesis: verum
end;
supposeA10: perm1 is odd ; ::_thesis: ( Det (Segm (M,nt,nt1)) = Det (Segm (M,nt9,nt19)) or Det (Segm (M,nt,nt1)) = - (Det (Segm (M,nt9,nt19))) )
Det (Segm (M,nt,nt19)) = - ((Det (Segm (M,nt9,nt19))),perm) by A7, Th35;
then A11: ( Det (Segm (M,nt,nt19)) = Det (Segm (M,nt9,nt19)) or Det (Segm (M,nt,nt19)) = - (Det (Segm (M,nt9,nt19))) ) by MATRIX_2:def_13;
Det (Segm (M,nt,nt1)) = - ((Det (Segm (M,nt,nt19))),perm1) by A6, Th35;
then Det (Segm (M,nt,nt1)) = - (Det (Segm (M,nt,nt19))) by A10, MATRIX_2:def_13;
then ( Det (Segm (M,nt,nt1)) = - (Det (Segm (M,nt9,nt19))) or Det (Segm (M,nt,nt1)) = (0. K) + (- (- (Det (Segm (M,nt9,nt19))))) ) by A11, RLVECT_1:def_4;
then ( Det (Segm (M,nt,nt1)) = - (Det (Segm (M,nt9,nt19))) or Det (Segm (M,nt,nt1)) = (0. K) - (- (Det (Segm (M,nt9,nt19)))) ) ;
then ( Det (Segm (M,nt,nt1)) = - (Det (Segm (M,nt9,nt19))) or (Det (Segm (M,nt,nt1))) + (- (Det (Segm (M,nt9,nt19)))) = 0. K ) by VECTSP_2:2;
hence ( Det (Segm (M,nt,nt1)) = Det (Segm (M,nt9,nt19)) or Det (Segm (M,nt,nt1)) = - (Det (Segm (M,nt9,nt19))) ) by VECTSP_1:19; ::_thesis: verum
end;
end;
end;
end;
end;
theorem Th37: :: MATRIX13:37
for D being non empty set
for n9, m9, n, m being Nat
for A9 being Matrix of n9,m9,D
for F, Fmt being FinSequence of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st len F = width A9 & Fmt = F * mt & [:(rng nt),(rng mt):] c= Indices A9 holds
for i, j being Nat st nt " {j} = {i} holds
RLine ((Segm (A9,nt,mt)),i,Fmt) = Segm ((RLine (A9,j,F)),nt,mt)
proof
let D be non empty set ; ::_thesis: for n9, m9, n, m being Nat
for A9 being Matrix of n9,m9,D
for F, Fmt being FinSequence of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st len F = width A9 & Fmt = F * mt & [:(rng nt),(rng mt):] c= Indices A9 holds
for i, j being Nat st nt " {j} = {i} holds
RLine ((Segm (A9,nt,mt)),i,Fmt) = Segm ((RLine (A9,j,F)),nt,mt)
let n9, m9, n, m be Nat; ::_thesis: for A9 being Matrix of n9,m9,D
for F, Fmt being FinSequence of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st len F = width A9 & Fmt = F * mt & [:(rng nt),(rng mt):] c= Indices A9 holds
for i, j being Nat st nt " {j} = {i} holds
RLine ((Segm (A9,nt,mt)),i,Fmt) = Segm ((RLine (A9,j,F)),nt,mt)
let A9 be Matrix of n9,m9,D; ::_thesis: for F, Fmt being FinSequence of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st len F = width A9 & Fmt = F * mt & [:(rng nt),(rng mt):] c= Indices A9 holds
for i, j being Nat st nt " {j} = {i} holds
RLine ((Segm (A9,nt,mt)),i,Fmt) = Segm ((RLine (A9,j,F)),nt,mt)
let F, Fmt be FinSequence of D; ::_thesis: for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st len F = width A9 & Fmt = F * mt & [:(rng nt),(rng mt):] c= Indices A9 holds
for i, j being Nat st nt " {j} = {i} holds
RLine ((Segm (A9,nt,mt)),i,Fmt) = Segm ((RLine (A9,j,F)),nt,mt)
let nt be Element of n -tuples_on NAT; ::_thesis: for mt being Element of m -tuples_on NAT st len F = width A9 & Fmt = F * mt & [:(rng nt),(rng mt):] c= Indices A9 holds
for i, j being Nat st nt " {j} = {i} holds
RLine ((Segm (A9,nt,mt)),i,Fmt) = Segm ((RLine (A9,j,F)),nt,mt)
let mt be Element of m -tuples_on NAT; ::_thesis: ( len F = width A9 & Fmt = F * mt & [:(rng nt),(rng mt):] c= Indices A9 implies for i, j being Nat st nt " {j} = {i} holds
RLine ((Segm (A9,nt,mt)),i,Fmt) = Segm ((RLine (A9,j,F)),nt,mt) )
assume that
A1: len F = width A9 and
A2: Fmt = F * mt and
A3: [:(rng nt),(rng mt):] c= Indices A9 ; ::_thesis: for i, j being Nat st nt " {j} = {i} holds
RLine ((Segm (A9,nt,mt)),i,Fmt) = Segm ((RLine (A9,j,F)),nt,mt)
let i, j be Nat; ::_thesis: ( nt " {j} = {i} implies RLine ((Segm (A9,nt,mt)),i,Fmt) = Segm ((RLine (A9,j,F)),nt,mt) )
assume A4: nt " {j} = {i} ; ::_thesis: RLine ((Segm (A9,nt,mt)),i,Fmt) = Segm ((RLine (A9,j,F)),nt,mt)
A5: i in {i} by TARSKI:def_1;
then A6: i in dom nt by A4, FUNCT_1:def_7;
then nt . i in rng nt by FUNCT_1:def_3;
then ( rng mt = {} or [:(rng nt),(rng mt):] <> {} ) ;
then A7: rng mt c= Seg (width A9) by A3, XBOOLE_1:2, ZFMISC_1:114;
nt . i in {j} by A4, A5, FUNCT_1:def_7;
then A8: nt . i = j by TARSKI:def_1;
set R = RLine (A9,j,F);
set SR = Segm ((RLine (A9,j,F)),nt,mt);
set S = Segm (A9,nt,mt);
A9: dom mt = Seg m by FINSEQ_2:124;
set RS = RLine ((Segm (A9,nt,mt)),i,Fmt);
A10: Indices (Segm ((RLine (A9,j,F)),nt,mt)) = Indices (Segm (A9,nt,mt)) by MATRIX_1:26;
dom F = Seg (width A9) by A1, FINSEQ_1:def_3;
then A11: dom Fmt = dom mt by A2, A7, RELAT_1:27;
A12: n <> 0 by A6;
then width (Segm (A9,nt,mt)) = m by MATRIX_1:23;
then A13: len Fmt = width (Segm (A9,nt,mt)) by A11, A9, FINSEQ_1:def_3;
A14: Indices A9 = Indices (RLine (A9,j,F)) by MATRIX_1:26;
now__::_thesis:_for_k,_l_being_Nat_st_[k,l]_in_Indices_(Segm_((RLine_(A9,j,F)),nt,mt))_holds_
(RLine_((Segm_(A9,nt,mt)),i,Fmt))_*_(k,l)_=_(Segm_((RLine_(A9,j,F)),nt,mt))_*_(k,l)
A15: dom nt = Seg n by FINSEQ_2:124;
let k, l be Nat; ::_thesis: ( [k,l] in Indices (Segm ((RLine (A9,j,F)),nt,mt)) implies (RLine ((Segm (A9,nt,mt)),i,Fmt)) * (b1,b2) = (Segm ((RLine (A9,j,F)),nt,mt)) * (b1,b2) )
assume A16: [k,l] in Indices (Segm ((RLine (A9,j,F)),nt,mt)) ; ::_thesis: (RLine ((Segm (A9,nt,mt)),i,Fmt)) * (b1,b2) = (Segm ((RLine (A9,j,F)),nt,mt)) * (b1,b2)
A17: Indices (Segm (A9,nt,mt)) = [:(Seg n),(Seg m):] by A12, MATRIX_1:23;
then A18: k in Seg n by A10, A16, ZFMISC_1:87;
reconsider k9 = k, l9 = l as Element of NAT by ORDINAL1:def_12;
A19: (Segm ((RLine (A9,j,F)),nt,mt)) * (k,l) = (RLine (A9,j,F)) * ((nt . k9),(mt . l9)) by A16, Def1;
A20: [(nt . k9),(mt . l9)] in Indices A9 by A3, A14, A16, Th17;
A21: l in dom mt by A9, A10, A16, A17, ZFMISC_1:87;
percases ( k = i or k <> i ) ;
supposeA22: k = i ; ::_thesis: (RLine ((Segm (A9,nt,mt)),i,Fmt)) * (b1,b2) = (Segm ((RLine (A9,j,F)),nt,mt)) * (b1,b2)
then A23: (RLine ((Segm (A9,nt,mt)),i,Fmt)) * (k,l) = Fmt . l by A13, A10, A16, MATRIX11:def_3;
(Segm ((RLine (A9,j,F)),nt,mt)) * (k,l) = F . (mt . l) by A1, A8, A19, A20, A22, MATRIX11:def_3;
hence (RLine ((Segm (A9,nt,mt)),i,Fmt)) * (k,l) = (Segm ((RLine (A9,j,F)),nt,mt)) * (k,l) by A2, A21, A23, FUNCT_1:13; ::_thesis: verum
end;
suppose k <> i ; ::_thesis: (RLine ((Segm (A9,nt,mt)),i,Fmt)) * (b1,b2) = (Segm ((RLine (A9,j,F)),nt,mt)) * (b1,b2)
then not k in nt " {j} by A4, TARSKI:def_1;
then not nt . k in {j} by A18, A15, FUNCT_1:def_7;
then A24: nt . k <> j by TARSKI:def_1;
then A25: (Segm ((RLine (A9,j,F)),nt,mt)) * (k,l) = A9 * ((nt . k9),(mt . l9)) by A1, A19, A20, MATRIX11:def_3;
(RLine ((Segm (A9,nt,mt)),i,Fmt)) * (k,l) = (Segm (A9,nt,mt)) * (k,l) by A8, A13, A10, A16, A24, MATRIX11:def_3;
hence (RLine ((Segm (A9,nt,mt)),i,Fmt)) * (k,l) = (Segm ((RLine (A9,j,F)),nt,mt)) * (k,l) by A10, A16, A25, Def1; ::_thesis: verum
end;
end;
end;
hence RLine ((Segm (A9,nt,mt)),i,Fmt) = Segm ((RLine (A9,j,F)),nt,mt) by MATRIX_1:27; ::_thesis: verum
end;
theorem Th38: :: MATRIX13:38
for D being non empty set
for n9, m9, m, n being Nat
for A9 being Matrix of n9,m9,D
for mt being Element of m -tuples_on NAT
for F being FinSequence of D
for i being Nat
for nt being Element of n -tuples_on NAT st not i in rng nt & [:(rng nt),(rng mt):] c= Indices A9 holds
Segm (A9,nt,mt) = Segm ((RLine (A9,i,F)),nt,mt)
proof
let D be non empty set ; ::_thesis: for n9, m9, m, n being Nat
for A9 being Matrix of n9,m9,D
for mt being Element of m -tuples_on NAT
for F being FinSequence of D
for i being Nat
for nt being Element of n -tuples_on NAT st not i in rng nt & [:(rng nt),(rng mt):] c= Indices A9 holds
Segm (A9,nt,mt) = Segm ((RLine (A9,i,F)),nt,mt)
let n9, m9, m, n be Nat; ::_thesis: for A9 being Matrix of n9,m9,D
for mt being Element of m -tuples_on NAT
for F being FinSequence of D
for i being Nat
for nt being Element of n -tuples_on NAT st not i in rng nt & [:(rng nt),(rng mt):] c= Indices A9 holds
Segm (A9,nt,mt) = Segm ((RLine (A9,i,F)),nt,mt)
let A9 be Matrix of n9,m9,D; ::_thesis: for mt being Element of m -tuples_on NAT
for F being FinSequence of D
for i being Nat
for nt being Element of n -tuples_on NAT st not i in rng nt & [:(rng nt),(rng mt):] c= Indices A9 holds
Segm (A9,nt,mt) = Segm ((RLine (A9,i,F)),nt,mt)
let mt be Element of m -tuples_on NAT; ::_thesis: for F being FinSequence of D
for i being Nat
for nt being Element of n -tuples_on NAT st not i in rng nt & [:(rng nt),(rng mt):] c= Indices A9 holds
Segm (A9,nt,mt) = Segm ((RLine (A9,i,F)),nt,mt)
let F be FinSequence of D; ::_thesis: for i being Nat
for nt being Element of n -tuples_on NAT st not i in rng nt & [:(rng nt),(rng mt):] c= Indices A9 holds
Segm (A9,nt,mt) = Segm ((RLine (A9,i,F)),nt,mt)
let i be Nat; ::_thesis: for nt being Element of n -tuples_on NAT st not i in rng nt & [:(rng nt),(rng mt):] c= Indices A9 holds
Segm (A9,nt,mt) = Segm ((RLine (A9,i,F)),nt,mt)
let nt be Element of n -tuples_on NAT; ::_thesis: ( not i in rng nt & [:(rng nt),(rng mt):] c= Indices A9 implies Segm (A9,nt,mt) = Segm ((RLine (A9,i,F)),nt,mt) )
assume that
A1: not i in rng nt and
A2: [:(rng nt),(rng mt):] c= Indices A9 ; ::_thesis: Segm (A9,nt,mt) = Segm ((RLine (A9,i,F)),nt,mt)
set S = Segm (A9,nt,mt);
set R = RLine (A9,i,F);
set SR = Segm ((RLine (A9,i,F)),nt,mt);
percases ( len F <> width A9 or len F = width A9 ) ;
suppose len F <> width A9 ; ::_thesis: Segm (A9,nt,mt) = Segm ((RLine (A9,i,F)),nt,mt)
hence Segm (A9,nt,mt) = Segm ((RLine (A9,i,F)),nt,mt) by MATRIX11:def_3; ::_thesis: verum
end;
supposeA3: len F = width A9 ; ::_thesis: Segm (A9,nt,mt) = Segm ((RLine (A9,i,F)),nt,mt)
A4: Indices (Segm ((RLine (A9,i,F)),nt,mt)) = Indices (Segm (A9,nt,mt)) by MATRIX_1:26;
now__::_thesis:_for_k,_m_being_Nat_st_[k,m]_in_Indices_(Segm_((RLine_(A9,i,F)),nt,mt))_holds_
(Segm_((RLine_(A9,i,F)),nt,mt))_*_(k,m)_=_(Segm_(A9,nt,mt))_*_(k,m)
A5: dom nt = Seg n by FINSEQ_2:124;
let k, m be Nat; ::_thesis: ( [k,m] in Indices (Segm ((RLine (A9,i,F)),nt,mt)) implies (Segm ((RLine (A9,i,F)),nt,mt)) * (k,m) = (Segm (A9,nt,mt)) * (k,m) )
assume A6: [k,m] in Indices (Segm ((RLine (A9,i,F)),nt,mt)) ; ::_thesis: (Segm ((RLine (A9,i,F)),nt,mt)) * (k,m) = (Segm (A9,nt,mt)) * (k,m)
Indices (Segm ((RLine (A9,i,F)),nt,mt)) = [:(Seg n),(Seg (width (Segm ((RLine (A9,i,F)),nt,mt)))):] by MATRIX_1:25;
then k in Seg n by A6, ZFMISC_1:87;
then A7: i <> nt . k by A1, A5, FUNCT_1:def_3;
reconsider K = k, M = m as Element of NAT by ORDINAL1:def_12;
[(nt . K),(mt . M)] in Indices A9 by A2, A4, A6, Th17;
then A8: A9 * ((nt . K),(mt . M)) = (RLine (A9,i,F)) * ((nt . K),(mt . M)) by A3, A7, MATRIX11:def_3;
(Segm (A9,nt,mt)) * (K,M) = A9 * ((nt . K),(mt . M)) by A4, A6, Def1;
hence (Segm ((RLine (A9,i,F)),nt,mt)) * (k,m) = (Segm (A9,nt,mt)) * (k,m) by A6, A8, Def1; ::_thesis: verum
end;
hence Segm (A9,nt,mt) = Segm ((RLine (A9,i,F)),nt,mt) by MATRIX_1:27; ::_thesis: verum
end;
end;
end;
theorem Th39: :: MATRIX13:39
for D being non empty set
for n9, m9, m, i, n, j being Nat
for A9 being Matrix of n9,m9,D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st i in rng nt & [:(rng nt),(rng mt):] c= Indices A9 holds
ex nt1 being Element of n -tuples_on NAT st
( rng nt1 = ((rng nt) \ {i}) \/ {j} & Segm ((RLine (A9,i,(Line (A9,j)))),nt,mt) = Segm (A9,nt1,mt) )
proof
let D be non empty set ; ::_thesis: for n9, m9, m, i, n, j being Nat
for A9 being Matrix of n9,m9,D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st i in rng nt & [:(rng nt),(rng mt):] c= Indices A9 holds
ex nt1 being Element of n -tuples_on NAT st
( rng nt1 = ((rng nt) \ {i}) \/ {j} & Segm ((RLine (A9,i,(Line (A9,j)))),nt,mt) = Segm (A9,nt1,mt) )
let n9, m9, m, i, n, j be Nat; ::_thesis: for A9 being Matrix of n9,m9,D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st i in rng nt & [:(rng nt),(rng mt):] c= Indices A9 holds
ex nt1 being Element of n -tuples_on NAT st
( rng nt1 = ((rng nt) \ {i}) \/ {j} & Segm ((RLine (A9,i,(Line (A9,j)))),nt,mt) = Segm (A9,nt1,mt) )
let A9 be Matrix of n9,m9,D; ::_thesis: for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st i in rng nt & [:(rng nt),(rng mt):] c= Indices A9 holds
ex nt1 being Element of n -tuples_on NAT st
( rng nt1 = ((rng nt) \ {i}) \/ {j} & Segm ((RLine (A9,i,(Line (A9,j)))),nt,mt) = Segm (A9,nt1,mt) )
let nt be Element of n -tuples_on NAT; ::_thesis: for mt being Element of m -tuples_on NAT st i in rng nt & [:(rng nt),(rng mt):] c= Indices A9 holds
ex nt1 being Element of n -tuples_on NAT st
( rng nt1 = ((rng nt) \ {i}) \/ {j} & Segm ((RLine (A9,i,(Line (A9,j)))),nt,mt) = Segm (A9,nt1,mt) )
let mt be Element of m -tuples_on NAT; ::_thesis: ( i in rng nt & [:(rng nt),(rng mt):] c= Indices A9 implies ex nt1 being Element of n -tuples_on NAT st
( rng nt1 = ((rng nt) \ {i}) \/ {j} & Segm ((RLine (A9,i,(Line (A9,j)))),nt,mt) = Segm (A9,nt1,mt) ) )
assume that
A1: i in rng nt and
A2: [:(rng nt),(rng mt):] c= Indices A9 ; ::_thesis: ex nt1 being Element of n -tuples_on NAT st
( rng nt1 = ((rng nt) \ {i}) \/ {j} & Segm ((RLine (A9,i,(Line (A9,j)))),nt,mt) = Segm (A9,nt1,mt) )
defpred S1[ set , set ] means for k being Nat st k = $1 holds
( ( nt . k = i implies $2 = j ) & ( nt . k <> i implies $2 = nt . k ) );
A3: for k being Nat st k in Seg n holds
ex x being Element of NAT st S1[k,x]
proof
let k be Nat; ::_thesis: ( k in Seg n implies ex x being Element of NAT st S1[k,x] )
assume k in Seg n ; ::_thesis: ex x being Element of NAT st S1[k,x]
percases ( nt . k = i or nt . k <> i ) ;
supposeA4: nt . k = i ; ::_thesis: ex x being Element of NAT st S1[k,x]
reconsider J = j as Element of NAT by ORDINAL1:def_12;
take J ; ::_thesis: S1[k,J]
thus S1[k,J] by A4; ::_thesis: verum
end;
supposeA5: nt . k <> i ; ::_thesis: ex x being Element of NAT st S1[k,x]
reconsider ntk = nt . k as Element of NAT by ORDINAL1:def_12;
take ntk ; ::_thesis: S1[k,ntk]
thus S1[k,ntk] by A5; ::_thesis: verum
end;
end;
end;
consider p being FinSequence of NAT such that
A6: dom p = Seg n and
A7: for k being Nat st k in Seg n holds
S1[k,p . k] from FINSEQ_1:sch_5(A3);
n in NAT by ORDINAL1:def_12;
then len p = n by A6, FINSEQ_1:def_3;
then reconsider p9 = p as Element of n -tuples_on NAT by FINSEQ_2:92;
A8: rng p9 c= ((rng nt) \ {i}) \/ {j}
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng p9 or y in ((rng nt) \ {i}) \/ {j} )
A9: dom nt = Seg n by FINSEQ_2:124;
assume y in rng p9 ; ::_thesis: y in ((rng nt) \ {i}) \/ {j}
then consider x being set such that
A10: x in dom p and
A11: p . x = y by FUNCT_1:def_3;
consider k being Element of NAT such that
A12: x = k and
1 <= k and
k <= n by A6, A10;
( ( p . k = j & nt . k = i ) or ( p . k = nt . k & nt . k <> i & nt . k in rng nt ) ) by A6, A7, A10, A12, A9, FUNCT_1:def_3;
then ( p . k in {j} or ( p . k in rng nt & not p . k in {i} ) ) by TARSKI:def_1;
then ( p . k in {j} or p . k in (rng nt) \ {i} ) by XBOOLE_0:def_5;
hence y in ((rng nt) \ {i}) \/ {j} by A11, A12, XBOOLE_0:def_3; ::_thesis: verum
end;
take p9 ; ::_thesis: ( rng p9 = ((rng nt) \ {i}) \/ {j} & Segm ((RLine (A9,i,(Line (A9,j)))),nt,mt) = Segm (A9,p9,mt) )
((rng nt) \ {i}) \/ {j} c= rng p9
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in ((rng nt) \ {i}) \/ {j} or y in rng p9 )
assume A13: y in ((rng nt) \ {i}) \/ {j} ; ::_thesis: y in rng p9
percases ( y in (rng nt) \ {i} or y in {j} ) by A13, XBOOLE_0:def_3;
supposeA14: y in (rng nt) \ {i} ; ::_thesis: y in rng p9
then consider x being set such that
A15: x in dom nt and
A16: nt . x = y by FUNCT_1:def_3;
A17: dom nt = Seg n by FINSEQ_2:124;
then consider k being Element of NAT such that
A18: x = k and
1 <= k and
k <= n by A15;
not y in {i} by A14, XBOOLE_0:def_5;
then y <> i by TARSKI:def_1;
then p . k = y by A7, A15, A16, A17, A18;
hence y in rng p9 by A6, A15, A17, A18, FUNCT_1:def_3; ::_thesis: verum
end;
suppose y in {j} ; ::_thesis: y in rng p9
then A19: y = j by TARSKI:def_1;
consider x being set such that
A20: x in dom nt and
A21: nt . x = i by A1, FUNCT_1:def_3;
A22: dom nt = Seg n by FINSEQ_2:124;
then consider k being Element of NAT such that
A23: x = k and
1 <= k and
k <= n by A20;
p . k = j by A7, A20, A21, A22, A23;
hence y in rng p9 by A6, A20, A22, A23, A19, FUNCT_1:def_3; ::_thesis: verum
end;
end;
end;
hence rng p9 = ((rng nt) \ {i}) \/ {j} by A8, XBOOLE_0:def_10; ::_thesis: Segm ((RLine (A9,i,(Line (A9,j)))),nt,mt) = Segm (A9,p9,mt)
set S = Segm (A9,p9,mt);
set LA = Line (A9,j);
set R = RLine (A9,i,(Line (A9,j)));
set SR = Segm ((RLine (A9,i,(Line (A9,j)))),nt,mt);
A24: Indices A9 = Indices (RLine (A9,i,(Line (A9,j)))) by MATRIX_1:26;
A25: len (Line (A9,j)) = width A9 by MATRIX_1:def_7;
A26: Indices (Segm (A9,p9,mt)) = Indices (Segm ((RLine (A9,i,(Line (A9,j)))),nt,mt)) by MATRIX_1:26;
now__::_thesis:_for_k,_l_being_Nat_st_[k,l]_in_Indices_(Segm_((RLine_(A9,i,(Line_(A9,j)))),nt,mt))_holds_
(Segm_((RLine_(A9,i,(Line_(A9,j)))),nt,mt))_*_(k,l)_=_(Segm_(A9,p9,mt))_*_(k,l)
let k, l be Nat; ::_thesis: ( [k,l] in Indices (Segm ((RLine (A9,i,(Line (A9,j)))),nt,mt)) implies (Segm ((RLine (A9,i,(Line (A9,j)))),nt,mt)) * (k,l) = (Segm (A9,p9,mt)) * (k,l) )
assume A27: [k,l] in Indices (Segm ((RLine (A9,i,(Line (A9,j)))),nt,mt)) ; ::_thesis: (Segm ((RLine (A9,i,(Line (A9,j)))),nt,mt)) * (k,l) = (Segm (A9,p9,mt)) * (k,l)
reconsider K = k, L = l as Element of NAT by ORDINAL1:def_12;
A28: (Segm ((RLine (A9,i,(Line (A9,j)))),nt,mt)) * (K,L) = (RLine (A9,i,(Line (A9,j)))) * ((nt . K),(mt . L)) by A27, Def1;
Indices (Segm ((RLine (A9,i,(Line (A9,j)))),nt,mt)) = [:(Seg n),(Seg (width (Segm ((RLine (A9,i,(Line (A9,j)))),nt,mt)))):] by MATRIX_1:25;
then A29: K in Seg n by A27, ZFMISC_1:87;
A30: ( nt . K = i or nt . K <> i ) ;
A31: [(nt . K),(mt . L)] in Indices A9 by A2, A24, A27, Th17;
then mt . L in Seg (width A9) by ZFMISC_1:87;
then ( ( (Segm ((RLine (A9,i,(Line (A9,j)))),nt,mt)) * (K,L) = (Line (A9,j)) . (mt . L) & p9 . K = j & (Line (A9,j)) . (mt . L) = A9 * (j,(mt . L)) ) or ( (Segm ((RLine (A9,i,(Line (A9,j)))),nt,mt)) * (K,L) = A9 * ((nt . K),(mt . L)) & p9 . K = nt . K ) ) by A7, A25, A31, A28, A30, A29, MATRIX11:def_3, MATRIX_1:def_7;
hence (Segm ((RLine (A9,i,(Line (A9,j)))),nt,mt)) * (k,l) = (Segm (A9,p9,mt)) * (k,l) by A26, A27, Def1; ::_thesis: verum
end;
hence Segm ((RLine (A9,i,(Line (A9,j)))),nt,mt) = Segm (A9,p9,mt) by MATRIX_1:27; ::_thesis: verum
end;
theorem Th40: :: MATRIX13:40
for D being non empty set
for n9, m9, i being Nat
for A9 being Matrix of n9,m9,D
for F being FinSequence of D st not i in Seg (len A9) holds
RLine (A9,i,F) = A9
proof
let D be non empty set ; ::_thesis: for n9, m9, i being Nat
for A9 being Matrix of n9,m9,D
for F being FinSequence of D st not i in Seg (len A9) holds
RLine (A9,i,F) = A9
let n9, m9, i be Nat; ::_thesis: for A9 being Matrix of n9,m9,D
for F being FinSequence of D st not i in Seg (len A9) holds
RLine (A9,i,F) = A9
let A9 be Matrix of n9,m9,D; ::_thesis: for F being FinSequence of D st not i in Seg (len A9) holds
RLine (A9,i,F) = A9
let F be FinSequence of D; ::_thesis: ( not i in Seg (len A9) implies RLine (A9,i,F) = A9 )
assume A1: not i in Seg (len A9) ; ::_thesis: RLine (A9,i,F) = A9
set R = RLine (A9,i,F);
percases ( len F = width A9 or len F <> width A9 ) ;
supposeA2: len F = width A9 ; ::_thesis: RLine (A9,i,F) = A9
A3: now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_k_<=_len_A9_holds_
(RLine_(A9,i,F))_._k_=_A9_._k
let k be Nat; ::_thesis: ( 1 <= k & k <= len A9 implies (RLine (A9,i,F)) . k = A9 . k )
assume that
A4: 1 <= k and
A5: k <= len A9 ; ::_thesis: (RLine (A9,i,F)) . k = A9 . k
k in NAT by ORDINAL1:def_12;
then A6: k in Seg (len A9) by A4, A5;
A7: len A9 = n9 by MATRIX_1:def_2;
then A8: (RLine (A9,i,F)) . k = Line ((RLine (A9,i,F)),k) by A6, MATRIX_2:8;
Line ((RLine (A9,i,F)),k) = Line (A9,k) by A1, A6, A7, MATRIX11:28;
hence (RLine (A9,i,F)) . k = A9 . k by A6, A7, A8, MATRIX_2:8; ::_thesis: verum
end;
len A9 = len (RLine (A9,i,F)) by A2, MATRIX11:def_3;
hence RLine (A9,i,F) = A9 by A3, FINSEQ_1:14; ::_thesis: verum
end;
suppose len F <> width A9 ; ::_thesis: RLine (A9,i,F) = A9
hence RLine (A9,i,F) = A9 by MATRIX11:def_3; ::_thesis: verum
end;
end;
end;
definition
let n, m be Nat;
let K be Field;
let M be Matrix of n,m,K;
let a be Element of K;
:: original: *
redefine funca * M -> Matrix of n,m,K;
coherence
a * M is Matrix of n,m,K
proof
A1: len M = n by MATRIX_1:def_2;
A2: len (a * M) = len M by MATRIX_3:def_5;
percases ( n = 0 or n > 0 ) ;
supposeA3: n = 0 ; ::_thesis: a * M is Matrix of n,m,K
then a * M = {} by A2, MATRIX_1:def_2;
hence a * M is Matrix of n,m,K by A3, MATRIX_1:13; ::_thesis: verum
end;
supposeA4: n > 0 ; ::_thesis: a * M is Matrix of n,m,K
A5: width (a * M) = width M by MATRIX_3:def_5;
width M = m by A4, MATRIX_1:23;
hence a * M is Matrix of n,m,K by A1, A2, A5, MATRIX_2:7; ::_thesis: verum
end;
end;
end;
end;
theorem Th41: :: MATRIX13:41
for n, m being Nat
for K being Field
for a being Element of K
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT
for M being Matrix of K st [:(rng nt),(rng mt):] c= Indices M holds
a * (Segm (M,nt,mt)) = Segm ((a * M),nt,mt)
proof
let n, m be Nat; ::_thesis: for K being Field
for a being Element of K
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT
for M being Matrix of K st [:(rng nt),(rng mt):] c= Indices M holds
a * (Segm (M,nt,mt)) = Segm ((a * M),nt,mt)
let K be Field; ::_thesis: for a being Element of K
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT
for M being Matrix of K st [:(rng nt),(rng mt):] c= Indices M holds
a * (Segm (M,nt,mt)) = Segm ((a * M),nt,mt)
let a be Element of K; ::_thesis: for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT
for M being Matrix of K st [:(rng nt),(rng mt):] c= Indices M holds
a * (Segm (M,nt,mt)) = Segm ((a * M),nt,mt)
let nt be Element of n -tuples_on NAT; ::_thesis: for mt being Element of m -tuples_on NAT
for M being Matrix of K st [:(rng nt),(rng mt):] c= Indices M holds
a * (Segm (M,nt,mt)) = Segm ((a * M),nt,mt)
let mt be Element of m -tuples_on NAT; ::_thesis: for M being Matrix of K st [:(rng nt),(rng mt):] c= Indices M holds
a * (Segm (M,nt,mt)) = Segm ((a * M),nt,mt)
let M be Matrix of K; ::_thesis: ( [:(rng nt),(rng mt):] c= Indices M implies a * (Segm (M,nt,mt)) = Segm ((a * M),nt,mt) )
set Sa = Segm ((a * M),nt,mt);
set S = Segm (M,nt,mt);
set aS = a * (Segm (M,nt,mt));
A1: Indices (a * M) = Indices M by MATRIXR1:18;
A2: Indices (Segm ((a * M),nt,mt)) = Indices (Segm (M,nt,mt)) by MATRIX_1:26;
assume A3: [:(rng nt),(rng mt):] c= Indices M ; ::_thesis: a * (Segm (M,nt,mt)) = Segm ((a * M),nt,mt)
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_(Segm_((a_*_M),nt,mt))_holds_
(Segm_((a_*_M),nt,mt))_*_(i,j)_=_(a_*_(Segm_(M,nt,mt)))_*_(i,j)
let i, j be Nat; ::_thesis: ( [i,j] in Indices (Segm ((a * M),nt,mt)) implies (Segm ((a * M),nt,mt)) * (i,j) = (a * (Segm (M,nt,mt))) * (i,j) )
assume A4: [i,j] in Indices (Segm ((a * M),nt,mt)) ; ::_thesis: (Segm ((a * M),nt,mt)) * (i,j) = (a * (Segm (M,nt,mt))) * (i,j)
A5: (a * (Segm (M,nt,mt))) * (i,j) = a * ((Segm (M,nt,mt)) * (i,j)) by A2, A4, MATRIX_3:def_5;
reconsider i9 = i, j9 = j as Element of NAT by ORDINAL1:def_12;
A6: (Segm ((a * M),nt,mt)) * (i9,j9) = (a * M) * ((nt . i),(mt . j)) by A4, Def1;
A7: (Segm (M,nt,mt)) * (i9,j9) = M * ((nt . i),(mt . j)) by A2, A4, Def1;
[(nt . i9),(mt . j9)] in Indices M by A3, A1, A4, Th17;
hence (Segm ((a * M),nt,mt)) * (i,j) = (a * (Segm (M,nt,mt))) * (i,j) by A6, A5, A7, MATRIX_3:def_5; ::_thesis: verum
end;
hence a * (Segm (M,nt,mt)) = Segm ((a * M),nt,mt) by MATRIX_1:27; ::_thesis: verum
end;
theorem Th42: :: MATRIX13:42
for D being non empty set
for n, m being Nat
for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st nt = idseq (len A) & mt = idseq (width A) holds
Segm (A,nt,mt) = A
proof
let D be non empty set ; ::_thesis: for n, m being Nat
for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st nt = idseq (len A) & mt = idseq (width A) holds
Segm (A,nt,mt) = A
let n, m be Nat; ::_thesis: for A being Matrix of D
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st nt = idseq (len A) & mt = idseq (width A) holds
Segm (A,nt,mt) = A
let A be Matrix of D; ::_thesis: for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT st nt = idseq (len A) & mt = idseq (width A) holds
Segm (A,nt,mt) = A
let nt be Element of n -tuples_on NAT; ::_thesis: for mt being Element of m -tuples_on NAT st nt = idseq (len A) & mt = idseq (width A) holds
Segm (A,nt,mt) = A
let mt be Element of m -tuples_on NAT; ::_thesis: ( nt = idseq (len A) & mt = idseq (width A) implies Segm (A,nt,mt) = A )
set S = Segm (A,nt,mt);
assume that
A1: nt = idseq (len A) and
A2: mt = idseq (width A) ; ::_thesis: Segm (A,nt,mt) = A
A3: len nt = n by CARD_1:def_7;
A4: len (idseq (width A)) = width A by CARD_1:def_7;
A5: len (idseq (len A)) = len A by CARD_1:def_7;
A6: len mt = m by CARD_1:def_7;
percases ( n = 0 or n > 0 ) ;
supposeA7: n = 0 ; ::_thesis: Segm (A,nt,mt) = A
then A8: len (Segm (A,nt,mt)) = 0 by MATRIX_1:def_2;
A = {} by A1, A7;
hence Segm (A,nt,mt) = A by A8; ::_thesis: verum
end;
supposeA9: n > 0 ; ::_thesis: Segm (A,nt,mt) = A
then A10: width (Segm (A,nt,mt)) = m by Th1;
then A11: Indices (Segm (A,nt,mt)) = [:(Seg n),(Seg m):] by MATRIX_1:25;
A12: now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_(Segm_(A,nt,mt))_holds_
(Segm_(A,nt,mt))_*_(i,j)_=_A_*_(i,j)
let i, j be Nat; ::_thesis: ( [i,j] in Indices (Segm (A,nt,mt)) implies (Segm (A,nt,mt)) * (i,j) = A * (i,j) )
assume A13: [i,j] in Indices (Segm (A,nt,mt)) ; ::_thesis: (Segm (A,nt,mt)) * (i,j) = A * (i,j)
reconsider i9 = i, j9 = j as Element of NAT by ORDINAL1:def_12;
j in Seg m by A10, A13, ZFMISC_1:87;
then A14: mt . j9 = j by A2, A6, A4, FINSEQ_2:49;
i in Seg n by A11, A13, ZFMISC_1:87;
then nt . i9 = i by A1, A3, A5, FINSEQ_2:49;
hence (Segm (A,nt,mt)) * (i,j) = A * (i,j) by A13, A14, Def1; ::_thesis: verum
end;
len (Segm (A,nt,mt)) = n by A9, Th1;
hence Segm (A,nt,mt) = A by A1, A2, A3, A6, A5, A4, A10, A12, MATRIX_1:21; ::_thesis: verum
end;
end;
end;
registration
cluster empty finite without_zero for Element of bool NAT;
existence
ex b1 being Subset of NAT st
( b1 is empty & b1 is without_zero & b1 is finite )
proof
{} NAT is without_zero by MEASURE6:def_2;
hence ex b1 being Subset of NAT st
( b1 is empty & b1 is without_zero & b1 is finite ) ; ::_thesis: verum
end;
cluster non empty finite without_zero for Element of bool NAT;
existence
ex b1 being Subset of NAT st
( not b1 is empty & b1 is without_zero & b1 is finite )
proof
{1} is Subset of NAT ;
hence ex b1 being Subset of NAT st
( not b1 is empty & b1 is without_zero & b1 is finite ) ; ::_thesis: verum
end;
end;
registration
let n be Nat;
cluster Seg n -> without_zero ;
coherence
Seg n is without_zero
proof
not 0 in Seg n by FINSEQ_1:1;
hence Seg n is without_zero by MEASURE6:def_2; ::_thesis: verum
end;
end;
registration
let X be without_zero set ;
let Y be set ;
clusterX \ Y -> without_zero ;
coherence
X \ Y is without_zero
proof
assume 0 in X \ Y ; :: according to MEASURE6:def_2 ::_thesis: contradiction
hence contradiction ; ::_thesis: verum
end;
end;
definition
let i be Nat;
:: original: {
redefine func{i} -> Subset of NAT;
coherence
{i} is Subset of NAT
proof
i in NAT by ORDINAL1:def_12;
hence {i} is Subset of NAT by ZFMISC_1:31; ::_thesis: verum
end;
let j be Nat;
:: original: {
redefine func{i,j} -> Subset of NAT;
coherence
{i,j} is Subset of NAT
proof
A1: j in NAT by ORDINAL1:def_12;
i in NAT by ORDINAL1:def_12;
hence {i,j} is Subset of NAT by A1, ZFMISC_1:32; ::_thesis: verum
end;
end;
theorem Th43: :: MATRIX13:43
for N being finite without_zero Subset of NAT ex k being Nat st N c= Seg k
proof
let N be finite without_zero Subset of NAT; ::_thesis: ex k being Nat st N c= Seg k
consider k being Nat such that
A1: for n being Nat st n in N holds
n <= k by STIRL2_1:56;
take k ; ::_thesis: N c= Seg k
thus N c= Seg k ::_thesis: verum
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in N or x in Seg k )
assume A2: x in N ; ::_thesis: x in Seg k
then consider n being Element of NAT such that
A3: n = x ;
A4: n >= 1 by A2, A3, NAT_1:14;
n <= k by A1, A2, A3;
hence x in Seg k by A3, A4; ::_thesis: verum
end;
end;
definition
let N be finite without_zero Subset of NAT;
:: original: Sgm
redefine func Sgm N -> Element of (card N) -tuples_on NAT;
coherence
Sgm N is Element of (card N) -tuples_on NAT
proof
ex k being Nat st N c= Seg k by Th43;
then len (Sgm N) = card N by FINSEQ_3:39;
hence Sgm N is Element of (card N) -tuples_on NAT by FINSEQ_2:92; ::_thesis: verum
end;
end;
definition
let D be non empty set ;
let A be Matrix of D;
let P, Q be finite without_zero Subset of NAT;
func Segm (A,P,Q) -> Matrix of card P, card Q,D equals :: MATRIX13:def 2
Segm (A,(Sgm P),(Sgm Q));
coherence
Segm (A,(Sgm P),(Sgm Q)) is Matrix of card P, card Q,D ;
end;
:: deftheorem defines Segm MATRIX13:def_2_:_
for D being non empty set
for A being Matrix of D
for P, Q being finite without_zero Subset of NAT holds Segm (A,P,Q) = Segm (A,(Sgm P),(Sgm Q));
theorem Th44: :: MATRIX13:44
for D being non empty set
for i0, j0 being non zero Nat
for A being Matrix of D holds Segm (A,{i0},{j0}) = <*<*(A * (i0,j0))*>*>
proof
let D be non empty set ; ::_thesis: for i0, j0 being non zero Nat
for A being Matrix of D holds Segm (A,{i0},{j0}) = <*<*(A * (i0,j0))*>*>
let i0, j0 be non zero Nat; ::_thesis: for A being Matrix of D holds Segm (A,{i0},{j0}) = <*<*(A * (i0,j0))*>*>
let A be Matrix of D; ::_thesis: Segm (A,{i0},{j0}) = <*<*(A * (i0,j0))*>*>
A1: card {j0} = 1 by CARD_1:30;
A2: Sgm {i0} = <*i0*> by FINSEQ_3:44;
A3: <*j0*> . 1 = j0 by FINSEQ_1:40;
A4: <*i0*> . 1 = i0 by FINSEQ_1:40;
A5: Sgm {j0} = <*j0*> by FINSEQ_3:44;
card {i0} = 1 by CARD_1:30;
hence Segm (A,{i0},{j0}) = <*<*(A * (i0,j0))*>*> by A1, A2, A5, A4, A3, Th21; ::_thesis: verum
end;
theorem Th45: :: MATRIX13:45
for D being non empty set
for i0, j0, n0, m0 being non zero Nat
for A being Matrix of D st i0 < j0 & n0 < m0 holds
Segm (A,{i0,j0},{n0,m0}) = ((A * (i0,n0)),(A * (i0,m0))) ][ ((A * (j0,n0)),(A * (j0,m0)))
proof
let D be non empty set ; ::_thesis: for i0, j0, n0, m0 being non zero Nat
for A being Matrix of D st i0 < j0 & n0 < m0 holds
Segm (A,{i0,j0},{n0,m0}) = ((A * (i0,n0)),(A * (i0,m0))) ][ ((A * (j0,n0)),(A * (j0,m0)))
let i0, j0, n0, m0 be non zero Nat; ::_thesis: for A being Matrix of D st i0 < j0 & n0 < m0 holds
Segm (A,{i0,j0},{n0,m0}) = ((A * (i0,n0)),(A * (i0,m0))) ][ ((A * (j0,n0)),(A * (j0,m0)))
let A be Matrix of D; ::_thesis: ( i0 < j0 & n0 < m0 implies Segm (A,{i0,j0},{n0,m0}) = ((A * (i0,n0)),(A * (i0,m0))) ][ ((A * (j0,n0)),(A * (j0,m0))) )
assume that
A1: i0 < j0 and
A2: n0 < m0 ; ::_thesis: Segm (A,{i0,j0},{n0,m0}) = ((A * (i0,n0)),(A * (i0,m0))) ][ ((A * (j0,n0)),(A * (j0,m0)))
A3: card {n0,m0} = 2 by A2, CARD_2:57;
A4: Sgm {n0,m0} = <*n0,m0*> by A2, FINSEQ_3:45;
then A5: (Sgm {n0,m0}) . 1 = n0 by FINSEQ_1:44;
A6: Sgm {i0,j0} = <*i0,j0*> by A1, FINSEQ_3:45;
then A7: (Sgm {i0,j0}) . 1 = i0 by FINSEQ_1:44;
A8: (Sgm {i0,j0}) . 2 = j0 by A6, FINSEQ_1:44;
A9: (Sgm {n0,m0}) . 2 = m0 by A4, FINSEQ_1:44;
card {i0,j0} = 2 by A1, CARD_2:57;
hence Segm (A,{i0,j0},{n0,m0}) = ((A * (i0,n0)),(A * (i0,m0))) ][ ((A * (j0,n0)),(A * (j0,m0))) by A3, A5, A9, A7, A8, Th23; ::_thesis: verum
end;
theorem Th46: :: MATRIX13:46
for D being non empty set
for A being Matrix of D holds Segm (A,(Seg (len A)),(Seg (width A))) = A
proof
let D be non empty set ; ::_thesis: for A being Matrix of D holds Segm (A,(Seg (len A)),(Seg (width A))) = A
let A be Matrix of D; ::_thesis: Segm (A,(Seg (len A)),(Seg (width A))) = A
A1: Sgm (Seg (width A)) = idseq (width A) by FINSEQ_3:48;
Sgm (Seg (len A)) = idseq (len A) by FINSEQ_3:48;
hence Segm (A,(Seg (len A)),(Seg (width A))) = A by A1, Th42; ::_thesis: verum
end;
theorem Th47: :: MATRIX13:47
for D being non empty set
for i being Nat
for A being Matrix of D
for P, Q being finite without_zero Subset of NAT st i in Seg (card P) & Q c= Seg (width A) holds
Line ((Segm (A,P,Q)),i) = (Line (A,((Sgm P) . i))) * (Sgm Q)
proof
let D be non empty set ; ::_thesis: for i being Nat
for A being Matrix of D
for P, Q being finite without_zero Subset of NAT st i in Seg (card P) & Q c= Seg (width A) holds
Line ((Segm (A,P,Q)),i) = (Line (A,((Sgm P) . i))) * (Sgm Q)
let i be Nat; ::_thesis: for A being Matrix of D
for P, Q being finite without_zero Subset of NAT st i in Seg (card P) & Q c= Seg (width A) holds
Line ((Segm (A,P,Q)),i) = (Line (A,((Sgm P) . i))) * (Sgm Q)
let A be Matrix of D; ::_thesis: for P, Q being finite without_zero Subset of NAT st i in Seg (card P) & Q c= Seg (width A) holds
Line ((Segm (A,P,Q)),i) = (Line (A,((Sgm P) . i))) * (Sgm Q)
let P, Q be finite without_zero Subset of NAT; ::_thesis: ( i in Seg (card P) & Q c= Seg (width A) implies Line ((Segm (A,P,Q)),i) = (Line (A,((Sgm P) . i))) * (Sgm Q) )
assume that
A1: i in Seg (card P) and
A2: Q c= Seg (width A) ; ::_thesis: Line ((Segm (A,P,Q)),i) = (Line (A,((Sgm P) . i))) * (Sgm Q)
rng (Sgm Q) = Q by A2, FINSEQ_1:def_13;
hence Line ((Segm (A,P,Q)),i) = (Line (A,((Sgm P) . i))) * (Sgm Q) by A1, A2, Th24; ::_thesis: verum
end;
theorem Th48: :: MATRIX13:48
for D being non empty set
for i being Nat
for A being Matrix of D
for P being finite without_zero Subset of NAT st i in Seg (card P) holds
Line ((Segm (A,P,(Seg (width A)))),i) = Line (A,((Sgm P) . i))
proof
let D be non empty set ; ::_thesis: for i being Nat
for A being Matrix of D
for P being finite without_zero Subset of NAT st i in Seg (card P) holds
Line ((Segm (A,P,(Seg (width A)))),i) = Line (A,((Sgm P) . i))
let i be Nat; ::_thesis: for A being Matrix of D
for P being finite without_zero Subset of NAT st i in Seg (card P) holds
Line ((Segm (A,P,(Seg (width A)))),i) = Line (A,((Sgm P) . i))
let A be Matrix of D; ::_thesis: for P being finite without_zero Subset of NAT st i in Seg (card P) holds
Line ((Segm (A,P,(Seg (width A)))),i) = Line (A,((Sgm P) . i))
let P be finite without_zero Subset of NAT; ::_thesis: ( i in Seg (card P) implies Line ((Segm (A,P,(Seg (width A)))),i) = Line (A,((Sgm P) . i)) )
assume A1: i in Seg (card P) ; ::_thesis: Line ((Segm (A,P,(Seg (width A)))),i) = Line (A,((Sgm P) . i))
set S = Seg (width A);
set sP = Sgm P;
len (Line (A,((Sgm P) . i))) = width A by MATRIX_1:def_7;
then A2: dom (Line (A,((Sgm P) . i))) = Seg (width A) by FINSEQ_1:def_3;
Sgm (Seg (width A)) = idseq (width A) by FINSEQ_3:48;
then (Line (A,((Sgm P) . i))) * (Sgm (Seg (width A))) = Line (A,((Sgm P) . i)) by A2, RELAT_1:52;
hence Line ((Segm (A,P,(Seg (width A)))),i) = Line (A,((Sgm P) . i)) by A1, Th47; ::_thesis: verum
end;
theorem Th49: :: MATRIX13:49
for D being non empty set
for j being Nat
for A being Matrix of D
for Q, P being finite without_zero Subset of NAT st j in Seg (card Q) & P c= Seg (len A) holds
Col ((Segm (A,P,Q)),j) = (Col (A,((Sgm Q) . j))) * (Sgm P)
proof
let D be non empty set ; ::_thesis: for j being Nat
for A being Matrix of D
for Q, P being finite without_zero Subset of NAT st j in Seg (card Q) & P c= Seg (len A) holds
Col ((Segm (A,P,Q)),j) = (Col (A,((Sgm Q) . j))) * (Sgm P)
let j be Nat; ::_thesis: for A being Matrix of D
for Q, P being finite without_zero Subset of NAT st j in Seg (card Q) & P c= Seg (len A) holds
Col ((Segm (A,P,Q)),j) = (Col (A,((Sgm Q) . j))) * (Sgm P)
let A be Matrix of D; ::_thesis: for Q, P being finite without_zero Subset of NAT st j in Seg (card Q) & P c= Seg (len A) holds
Col ((Segm (A,P,Q)),j) = (Col (A,((Sgm Q) . j))) * (Sgm P)
let Q, P be finite without_zero Subset of NAT; ::_thesis: ( j in Seg (card Q) & P c= Seg (len A) implies Col ((Segm (A,P,Q)),j) = (Col (A,((Sgm Q) . j))) * (Sgm P) )
assume that
A1: j in Seg (card Q) and
A2: P c= Seg (len A) ; ::_thesis: Col ((Segm (A,P,Q)),j) = (Col (A,((Sgm Q) . j))) * (Sgm P)
rng (Sgm P) = P by A2, FINSEQ_1:def_13;
hence Col ((Segm (A,P,Q)),j) = (Col (A,((Sgm Q) . j))) * (Sgm P) by A1, A2, Th28; ::_thesis: verum
end;
theorem Th50: :: MATRIX13:50
for D being non empty set
for j being Nat
for A being Matrix of D
for Q being finite without_zero Subset of NAT st j in Seg (card Q) holds
Col ((Segm (A,(Seg (len A)),Q)),j) = Col (A,((Sgm Q) . j))
proof
let D be non empty set ; ::_thesis: for j being Nat
for A being Matrix of D
for Q being finite without_zero Subset of NAT st j in Seg (card Q) holds
Col ((Segm (A,(Seg (len A)),Q)),j) = Col (A,((Sgm Q) . j))
let j be Nat; ::_thesis: for A being Matrix of D
for Q being finite without_zero Subset of NAT st j in Seg (card Q) holds
Col ((Segm (A,(Seg (len A)),Q)),j) = Col (A,((Sgm Q) . j))
let A be Matrix of D; ::_thesis: for Q being finite without_zero Subset of NAT st j in Seg (card Q) holds
Col ((Segm (A,(Seg (len A)),Q)),j) = Col (A,((Sgm Q) . j))
let Q be finite without_zero Subset of NAT; ::_thesis: ( j in Seg (card Q) implies Col ((Segm (A,(Seg (len A)),Q)),j) = Col (A,((Sgm Q) . j)) )
assume A1: j in Seg (card Q) ; ::_thesis: Col ((Segm (A,(Seg (len A)),Q)),j) = Col (A,((Sgm Q) . j))
set S = Seg (len A);
set sQ = Sgm Q;
len (Col (A,((Sgm Q) . j))) = len A by MATRIX_1:def_8;
then A2: dom (Col (A,((Sgm Q) . j))) = Seg (len A) by FINSEQ_1:def_3;
Sgm (Seg (len A)) = idseq (len A) by FINSEQ_3:48;
then (Col (A,((Sgm Q) . j))) * (Sgm (Seg (len A))) = Col (A,((Sgm Q) . j)) by A2, RELAT_1:52;
hence Col ((Segm (A,(Seg (len A)),Q)),j) = Col (A,((Sgm Q) . j)) by A1, Th49; ::_thesis: verum
end;
theorem Th51: :: MATRIX13:51
for D being non empty set
for i being Nat
for A being Matrix of D holds Segm (A,((Seg (len A)) \ {i}),(Seg (width A))) = Del (A,i)
proof
let D be non empty set ; ::_thesis: for i being Nat
for A being Matrix of D holds Segm (A,((Seg (len A)) \ {i}),(Seg (width A))) = Del (A,i)
let i be Nat; ::_thesis: for A being Matrix of D holds Segm (A,((Seg (len A)) \ {i}),(Seg (width A))) = Del (A,i)
let A be Matrix of D; ::_thesis: Segm (A,((Seg (len A)) \ {i}),(Seg (width A))) = Del (A,i)
set SLA = Seg (len A);
set Si = (Seg (len A)) \ {i};
set S = Segm (A,((Seg (len A)) \ {i}),(Seg (width A)));
A1: dom A = Seg (len A) by FINSEQ_1:def_3;
percases ( not i in dom A or i in dom A ) ;
supposeA2: not i in dom A ; ::_thesis: Segm (A,((Seg (len A)) \ {i}),(Seg (width A))) = Del (A,i)
then A3: Del (A,i) = A by FINSEQ_3:104;
(Seg (len A)) \ {i} = Seg (len A) by A1, A2, ZFMISC_1:57;
hence Segm (A,((Seg (len A)) \ {i}),(Seg (width A))) = Del (A,i) by A3, Th46; ::_thesis: verum
end;
supposeA4: i in dom A ; ::_thesis: Segm (A,((Seg (len A)) \ {i}),(Seg (width A))) = Del (A,i)
then consider m being Nat such that
A5: len A = m + 1 and
A6: len (Del (A,i)) = m by FINSEQ_3:104;
reconsider m = m as Element of NAT by ORDINAL1:def_12;
card (Seg (len A)) = m + 1 by A5, FINSEQ_1:57;
then A7: card ((Seg (len A)) \ {i}) = m by A1, A4, STIRL2_1:55;
A8: now__::_thesis:_for_j_being_Nat_st_1_<=_j_&_j_<=_m_holds_
(Del_(A,i))_._j_=_(Segm_(A,((Seg_(len_A))_\_{i}),(Seg_(width_A))))_._j
reconsider A9 = A as Matrix of m + 1, width A,D by A5, MATRIX_1:20;
let j be Nat; ::_thesis: ( 1 <= j & j <= m implies (Del (A,i)) . j = (Segm (A,((Seg (len A)) \ {i}),(Seg (width A)))) . j )
assume that
A9: 1 <= j and
A10: j <= m ; ::_thesis: (Del (A,i)) . j = (Segm (A,((Seg (len A)) \ {i}),(Seg (width A)))) . j
j in NAT by ORDINAL1:def_12;
then A11: j in Seg m by A9, A10;
A12: dom A = Seg (len A) by FINSEQ_1:def_3;
A13: Del (A,i) = A * (Sgm ((Seg (len A)) \ {i})) by A1, FINSEQ_3:def_2;
A14: dom (Del (A,i)) = Seg m by A6, FINSEQ_1:def_3;
then A15: (Sgm ((Seg (len A)) \ {i})) . j in dom A by A11, A13, FUNCT_1:11;
(Del (A,i)) . j = A9 . ((Sgm ((Seg (len A)) \ {i})) . j) by A14, A11, A13, FUNCT_1:12;
hence (Del (A,i)) . j = Line (A9,((Sgm ((Seg (len A)) \ {i})) . j)) by A5, A15, A12, MATRIX_2:8
.= Line ((Segm (A,((Seg (len A)) \ {i}),(Seg (width A)))),j) by A7, A11, Th48
.= (Segm (A,((Seg (len A)) \ {i}),(Seg (width A)))) . j by A7, A11, MATRIX_2:8 ;
::_thesis: verum
end;
len (Segm (A,((Seg (len A)) \ {i}),(Seg (width A)))) = m by A7, MATRIX_1:def_2;
hence Segm (A,((Seg (len A)) \ {i}),(Seg (width A))) = Del (A,i) by A6, A8, FINSEQ_1:14; ::_thesis: verum
end;
end;
end;
theorem Th52: :: MATRIX13:52
for i being Nat
for K being Field
for M being Matrix of K holds Segm (M,(Seg (len M)),((Seg (width M)) \ {i})) = DelCol (M,i)
proof
let i be Nat; ::_thesis: for K being Field
for M being Matrix of K holds Segm (M,(Seg (len M)),((Seg (width M)) \ {i})) = DelCol (M,i)
let K be Field; ::_thesis: for M being Matrix of K holds Segm (M,(Seg (len M)),((Seg (width M)) \ {i})) = DelCol (M,i)
let M be Matrix of K; ::_thesis: Segm (M,(Seg (len M)),((Seg (width M)) \ {i})) = DelCol (M,i)
set SW = Seg (width M);
set Si = (Seg (width M)) \ {i};
set SL = Seg (len M);
set SEGM = Segm (M,(Seg (len M)),((Seg (width M)) \ {i}));
set D = DelCol (M,i);
card (Seg (len M)) = len M by FINSEQ_1:57;
then A1: len (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) = len M by MATRIX_1:def_2;
A2: now__::_thesis:_for_j_being_Nat_st_1_<=_j_&_j_<=_len_M_holds_
(Segm_(M,(Seg_(len_M)),((Seg_(width_M))_\_{i})))_._j_=_(DelCol_(M,i))_._j
let j be Nat; ::_thesis: ( 1 <= j & j <= len M implies (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) . j = (DelCol (M,i)) . j )
assume that
A3: 1 <= j and
A4: j <= len M ; ::_thesis: (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) . j = (DelCol (M,i)) . j
j in NAT by ORDINAL1:def_12;
then A5: j in Seg (len M) by A3, A4;
then A6: j in dom M by FINSEQ_1:def_3;
Sgm (Seg (len M)) = idseq (len M) by FINSEQ_3:48;
then A7: (Sgm (Seg (len M))) . j = j by A5, FINSEQ_2:49;
len (Line (M,j)) = width M by MATRIX_1:def_7;
then A8: dom (Line (M,j)) = Seg (width M) by FINSEQ_1:def_3;
A9: card (Seg (len M)) = len M by FINSEQ_1:57;
then A10: Line ((Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))),j) = (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) . j by A5, MATRIX_2:8;
Line ((Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))),j) = (Line (M,((Sgm (Seg (len M))) . j))) * (Sgm ((Seg (width M)) \ {i})) by A9, A5, Th47, XBOOLE_1:36;
then (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) . j = Del ((Line (M,j)),i) by A7, A10, A8, FINSEQ_3:def_2;
hence (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) . j = (DelCol (M,i)) . j by A6, MATRIX_2:def_5; ::_thesis: verum
end;
len (DelCol (M,i)) = len M by MATRIX_2:def_5;
hence Segm (M,(Seg (len M)),((Seg (width M)) \ {i})) = DelCol (M,i) by A1, A2, FINSEQ_1:14; ::_thesis: verum
end;
theorem Th53: :: MATRIX13:53
for X being set
for P being finite without_zero Subset of NAT holds (Sgm P) " X is finite without_zero Subset of NAT
proof
let X be set ; ::_thesis: for P being finite without_zero Subset of NAT holds (Sgm P) " X is finite without_zero Subset of NAT
let P be finite without_zero Subset of NAT; ::_thesis: (Sgm P) " X is finite without_zero Subset of NAT
A1: (Sgm P) " X c= dom (Sgm P) by RELAT_1:132;
ex k being Nat st P c= Seg k by Th43;
then dom (Sgm P) = Seg (card P) by FINSEQ_3:40;
then not 0 in (Sgm P) " X by A1;
hence (Sgm P) " X is finite without_zero Subset of NAT by A1, MEASURE6:def_2, XBOOLE_1:1; ::_thesis: verum
end;
theorem Th54: :: MATRIX13:54
for X being set
for P being finite without_zero Subset of NAT st X c= P holds
Sgm X = (Sgm P) * (Sgm ((Sgm P) " X))
proof
let X be set ; ::_thesis: for P being finite without_zero Subset of NAT st X c= P holds
Sgm X = (Sgm P) * (Sgm ((Sgm P) " X))
let P be finite without_zero Subset of NAT; ::_thesis: ( X c= P implies Sgm X = (Sgm P) * (Sgm ((Sgm P) " X)) )
assume A1: X c= P ; ::_thesis: Sgm X = (Sgm P) * (Sgm ((Sgm P) " X))
A2: (Sgm P) " X c= dom (Sgm P) by RELAT_1:132;
consider n being Nat such that
A3: P c= Seg n by Th43;
A4: rng (Sgm P) = P by A3, FINSEQ_1:def_13;
A5: n in NAT by ORDINAL1:def_12;
rng ((Sgm P) | ((Sgm P) " X)) = (Sgm P) .: ((Sgm P) " X) by RELAT_1:115
.= X by A1, A4, FUNCT_1:77 ;
hence Sgm X = (Sgm P) * (Sgm ((Sgm P) " X)) by A3, A5, A2, GRAPH_2:3; ::_thesis: verum
end;
Lm2: for X being set
for P being finite without_zero Subset of NAT st X c= P holds
card X = card ((Sgm P) " X)
proof
let X be set ; ::_thesis: for P being finite without_zero Subset of NAT st X c= P holds
card X = card ((Sgm P) " X)
let P be finite without_zero Subset of NAT; ::_thesis: ( X c= P implies card X = card ((Sgm P) " X) )
assume A1: X c= P ; ::_thesis: card X = card ((Sgm P) " X)
A2: ex n being Nat st P c= Seg n by Th43;
then rng (Sgm P) = P by FINSEQ_1:def_13;
then A3: (Sgm P) .: ((Sgm P) " X) = X by A1, FUNCT_1:77;
A4: (Sgm P) " X c= dom (Sgm P) by RELAT_1:132;
Sgm P is one-to-one by A2, FINSEQ_3:92;
then X,(Sgm P) " X are_equipotent by A3, A4, CARD_1:33;
hence card X = card ((Sgm P) " X) by CARD_1:5; ::_thesis: verum
end;
theorem Th55: :: MATRIX13:55
for X, Y being set
for D being non empty set
for A being Matrix of D
for P, Q being finite without_zero Subset of NAT holds [:((Sgm P) " X),((Sgm Q) " Y):] c= Indices (Segm (A,P,Q))
proof
let X, Y be set ; ::_thesis: for D being non empty set
for A being Matrix of D
for P, Q being finite without_zero Subset of NAT holds [:((Sgm P) " X),((Sgm Q) " Y):] c= Indices (Segm (A,P,Q))
let D be non empty set ; ::_thesis: for A being Matrix of D
for P, Q being finite without_zero Subset of NAT holds [:((Sgm P) " X),((Sgm Q) " Y):] c= Indices (Segm (A,P,Q))
let A be Matrix of D; ::_thesis: for P, Q being finite without_zero Subset of NAT holds [:((Sgm P) " X),((Sgm Q) " Y):] c= Indices (Segm (A,P,Q))
let P, Q be finite without_zero Subset of NAT; ::_thesis: [:((Sgm P) " X),((Sgm Q) " Y):] c= Indices (Segm (A,P,Q))
set SP = Sgm P;
set SQ = Sgm Q;
set I = Indices (Segm (A,P,Q));
A1: now__::_thesis:_[:(Seg_(card_P)),(Seg_(card_Q)):]_=_Indices_(Segm_(A,P,Q))
percases ( card P = 0 or card P > 0 ) ;
supposeA2: card P = 0 ; ::_thesis: [:(Seg (card P)),(Seg (card Q)):] = Indices (Segm (A,P,Q))
then Seg (card P) = {} ;
then [:(Seg (card P)),(Seg (card Q)):] = {} by ZFMISC_1:90;
hence [:(Seg (card P)),(Seg (card Q)):] = Indices (Segm (A,P,Q)) by A2, MATRIX_1:22; ::_thesis: verum
end;
suppose card P > 0 ; ::_thesis: [:(Seg (card P)),(Seg (card Q)):] = Indices (Segm (A,P,Q))
hence [:(Seg (card P)),(Seg (card Q)):] = Indices (Segm (A,P,Q)) by MATRIX_1:23; ::_thesis: verum
end;
end;
end;
ex m being Nat st Q c= Seg m by Th43;
then dom (Sgm Q) = Seg (card Q) by FINSEQ_3:40;
then A3: (Sgm Q) " Y c= Seg (card Q) by RELAT_1:132;
ex n being Nat st P c= Seg n by Th43;
then dom (Sgm P) = Seg (card P) by FINSEQ_3:40;
then (Sgm P) " X c= Seg (card P) by RELAT_1:132;
hence [:((Sgm P) " X),((Sgm Q) " Y):] c= Indices (Segm (A,P,Q)) by A3, A1, ZFMISC_1:96; ::_thesis: verum
end;
theorem Th56: :: MATRIX13:56
for D being non empty set
for A being Matrix of D
for P, P1, Q, Q1, P2, Q2 being finite without_zero Subset of NAT st P c= P1 & Q c= Q1 & P2 = (Sgm P1) " P & Q2 = (Sgm Q1) " Q holds
( [:(rng (Sgm P2)),(rng (Sgm Q2)):] c= Indices (Segm (A,P1,Q1)) & Segm ((Segm (A,P1,Q1)),P2,Q2) = Segm (A,P,Q) )
proof
let D be non empty set ; ::_thesis: for A being Matrix of D
for P, P1, Q, Q1, P2, Q2 being finite without_zero Subset of NAT st P c= P1 & Q c= Q1 & P2 = (Sgm P1) " P & Q2 = (Sgm Q1) " Q holds
( [:(rng (Sgm P2)),(rng (Sgm Q2)):] c= Indices (Segm (A,P1,Q1)) & Segm ((Segm (A,P1,Q1)),P2,Q2) = Segm (A,P,Q) )
let A be Matrix of D; ::_thesis: for P, P1, Q, Q1, P2, Q2 being finite without_zero Subset of NAT st P c= P1 & Q c= Q1 & P2 = (Sgm P1) " P & Q2 = (Sgm Q1) " Q holds
( [:(rng (Sgm P2)),(rng (Sgm Q2)):] c= Indices (Segm (A,P1,Q1)) & Segm ((Segm (A,P1,Q1)),P2,Q2) = Segm (A,P,Q) )
let P, P1, Q, Q1, P2, Q2 be finite without_zero Subset of NAT; ::_thesis: ( P c= P1 & Q c= Q1 & P2 = (Sgm P1) " P & Q2 = (Sgm Q1) " Q implies ( [:(rng (Sgm P2)),(rng (Sgm Q2)):] c= Indices (Segm (A,P1,Q1)) & Segm ((Segm (A,P1,Q1)),P2,Q2) = Segm (A,P,Q) ) )
assume that
A1: P c= P1 and
A2: Q c= Q1 and
A3: P2 = (Sgm P1) " P and
A4: Q2 = (Sgm Q1) " Q ; ::_thesis: ( [:(rng (Sgm P2)),(rng (Sgm Q2)):] c= Indices (Segm (A,P1,Q1)) & Segm ((Segm (A,P1,Q1)),P2,Q2) = Segm (A,P,Q) )
set SA = Segm (A,P1,Q1);
card P = card P2 by A1, A3, Lm2;
then reconsider SAA = Segm ((Segm (A,P1,Q1)),P2,Q2) as Matrix of card P, card Q,D by A2, A4, Lm2;
set Sq2 = Sgm Q2;
set Sp2 = Sgm P2;
set Sq1 = Sgm Q1;
set Sp1 = Sgm P1;
set S = Segm (A,P,Q);
A5: ex q2 being Nat st Q2 c= Seg q2 by Th43;
then A6: rng (Sgm Q2) = Q2 by FINSEQ_1:def_13;
A7: ex p2 being Nat st P2 c= Seg p2 by Th43;
then rng (Sgm P2) = P2 by FINSEQ_1:def_13;
hence A8: [:(rng (Sgm P2)),(rng (Sgm Q2)):] c= Indices (Segm (A,P1,Q1)) by A3, A4, A6, Th55; ::_thesis: Segm ((Segm (A,P1,Q1)),P2,Q2) = Segm (A,P,Q)
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_(Segm_(A,P,Q))_holds_
(Segm_(A,P,Q))_*_(i,j)_=_SAA_*_(i,j)
A9: (Sgm Q1) * (Sgm Q2) = Sgm Q by A2, A4, Th54;
let i, j be Nat; ::_thesis: ( [i,j] in Indices (Segm (A,P,Q)) implies (Segm (A,P,Q)) * (i,j) = SAA * (i,j) )
assume A10: [i,j] in Indices (Segm (A,P,Q)) ; ::_thesis: (Segm (A,P,Q)) * (i,j) = SAA * (i,j)
A11: [i,j] in Indices SAA by A10, MATRIX_1:26;
then A12: j in Seg (width SAA) by ZFMISC_1:87;
reconsider Sp2i = (Sgm P2) . i, Sq2j = (Sgm Q2) . j as Element of NAT by ORDINAL1:def_12;
A13: (Sgm P1) * (Sgm P2) = Sgm P by A1, A3, Th54;
Indices SAA = [:(Seg (card P2)),(Seg (width SAA)):] by MATRIX_1:25;
then A14: i in Seg (card P2) by A11, ZFMISC_1:87;
then card P2 <> 0 ;
then width SAA = card Q2 by Th1;
then j in dom (Sgm Q2) by A5, A12, FINSEQ_3:40;
then A15: (Sgm Q1) . Sq2j = (Sgm Q) . j by A9, FUNCT_1:13;
reconsider i9 = i, j9 = j as Element of NAT by ORDINAL1:def_12;
A16: [i9,j9] in Indices SAA by A10, MATRIX_1:26;
then A17: SAA * (i,j) = (Segm (A,P1,Q1)) * (Sp2i,Sq2j) by Def1;
i in dom (Sgm P2) by A7, A14, FINSEQ_3:40;
then A18: (Sgm P1) . Sp2i = (Sgm P) . i by A13, FUNCT_1:13;
[Sp2i,Sq2j] in Indices (Segm (A,P1,Q1)) by A8, A16, Th17;
then SAA * (i,j) = A * (((Sgm P1) . Sp2i),((Sgm Q1) . Sq2j)) by A17, Def1;
hence (Segm (A,P,Q)) * (i,j) = SAA * (i,j) by A10, A18, A15, Def1; ::_thesis: verum
end;
hence Segm ((Segm (A,P1,Q1)),P2,Q2) = Segm (A,P,Q) by MATRIX_1:27; ::_thesis: verum
end;
theorem Th57: :: MATRIX13:57
for D being non empty set
for A being Matrix of D
for P, Q, P1, Q1 being finite without_zero Subset of NAT holds
not ( ( P = {} implies Q = {} ) & ( Q = {} implies P = {} ) & [:P,Q:] c= Indices (Segm (A,P1,Q1)) & ( for P2, Q2 being finite without_zero Subset of NAT holds
( not P2 c= P1 or not Q2 c= Q1 or not P2 = (Sgm P1) .: P or not Q2 = (Sgm Q1) .: Q or not card P2 = card P or not card Q2 = card Q or not Segm ((Segm (A,P1,Q1)),P,Q) = Segm (A,P2,Q2) ) ) )
proof
let D be non empty set ; ::_thesis: for A being Matrix of D
for P, Q, P1, Q1 being finite without_zero Subset of NAT holds
not ( ( P = {} implies Q = {} ) & ( Q = {} implies P = {} ) & [:P,Q:] c= Indices (Segm (A,P1,Q1)) & ( for P2, Q2 being finite without_zero Subset of NAT holds
( not P2 c= P1 or not Q2 c= Q1 or not P2 = (Sgm P1) .: P or not Q2 = (Sgm Q1) .: Q or not card P2 = card P or not card Q2 = card Q or not Segm ((Segm (A,P1,Q1)),P,Q) = Segm (A,P2,Q2) ) ) )
let A be Matrix of D; ::_thesis: for P, Q, P1, Q1 being finite without_zero Subset of NAT holds
not ( ( P = {} implies Q = {} ) & ( Q = {} implies P = {} ) & [:P,Q:] c= Indices (Segm (A,P1,Q1)) & ( for P2, Q2 being finite without_zero Subset of NAT holds
( not P2 c= P1 or not Q2 c= Q1 or not P2 = (Sgm P1) .: P or not Q2 = (Sgm Q1) .: Q or not card P2 = card P or not card Q2 = card Q or not Segm ((Segm (A,P1,Q1)),P,Q) = Segm (A,P2,Q2) ) ) )
let P, Q, P1, Q1 be finite without_zero Subset of NAT; ::_thesis: not ( ( P = {} implies Q = {} ) & ( Q = {} implies P = {} ) & [:P,Q:] c= Indices (Segm (A,P1,Q1)) & ( for P2, Q2 being finite without_zero Subset of NAT holds
( not P2 c= P1 or not Q2 c= Q1 or not P2 = (Sgm P1) .: P or not Q2 = (Sgm Q1) .: Q or not card P2 = card P or not card Q2 = card Q or not Segm ((Segm (A,P1,Q1)),P,Q) = Segm (A,P2,Q2) ) ) )
assume that
A1: ( P = {} iff Q = {} ) and
A2: [:P,Q:] c= Indices (Segm (A,P1,Q1)) ; ::_thesis: ex P2, Q2 being finite without_zero Subset of NAT st
( P2 c= P1 & Q2 c= Q1 & P2 = (Sgm P1) .: P & Q2 = (Sgm Q1) .: Q & card P2 = card P & card Q2 = card Q & Segm ((Segm (A,P1,Q1)),P,Q) = Segm (A,P2,Q2) )
set S = Segm (A,P1,Q1);
A3: now__::_thesis:_(_P_c=_Seg_(card_P1)_&_Q_c=_Seg_(card_Q1)_)
percases ( P = {} or P <> {} ) ;
suppose P = {} ; ::_thesis: ( P c= Seg (card P1) & Q c= Seg (card Q1) )
hence ( P c= Seg (card P1) & Q c= Seg (card Q1) ) by A1, XBOOLE_1:2; ::_thesis: verum
end;
supposeA4: P <> {} ; ::_thesis: ( P c= Seg (card P1) & Q c= Seg (card Q1) )
then A5: Q c= Seg (width (Segm (A,P1,Q1))) by A1, A2, ZFMISC_1:114;
A6: len (Segm (A,P1,Q1)) = card P1 by MATRIX_1:def_2;
A7: Indices (Segm (A,P1,Q1)) = [:(Seg (len (Segm (A,P1,Q1)))),(Seg (width (Segm (A,P1,Q1)))):] by FINSEQ_1:def_3;
then len (Segm (A,P1,Q1)) <> 0 by A1, A2, A4;
hence ( P c= Seg (card P1) & Q c= Seg (card Q1) ) by A1, A2, A4, A7, A5, A6, Th1, ZFMISC_1:114; ::_thesis: verum
end;
end;
end;
set SQ = Sgm Q1;
set SP = Sgm P1;
A8: ex k being Nat st P1 c= Seg k by Th43;
then A9: Sgm P1 is one-to-one by FINSEQ_3:92;
A10: ex k being Nat st Q1 c= Seg k by Th43;
then A11: Sgm Q1 is one-to-one by FINSEQ_3:92;
A12: rng (Sgm Q1) = Q1 by A10, FINSEQ_1:def_13;
then A13: (Sgm Q1) .: Q c= Q1 by RELAT_1:111;
then A14: not 0 in (Sgm Q1) .: Q ;
rng (Sgm P1) = P1 by A8, FINSEQ_1:def_13;
then A15: (Sgm P1) .: P c= P1 by RELAT_1:111;
then not 0 in (Sgm P1) .: P ;
then reconsider P2 = (Sgm P1) .: P, Q2 = (Sgm Q1) .: Q as finite without_zero Subset of NAT by A15, A13, A14, MEASURE6:def_2, XBOOLE_1:1;
A16: dom (Sgm Q1) = Seg (card Q1) by A10, FINSEQ_3:40;
then A17: (Sgm Q1) " Q2 = Q by A3, A11, FUNCT_1:94;
A18: dom (Sgm P1) = Seg (card P1) by A8, FINSEQ_3:40;
then P,P2 are_equipotent by A3, A9, CARD_1:33;
then A19: card P = card P2 by CARD_1:5;
Q,Q2 are_equipotent by A3, A16, A11, CARD_1:33;
then A20: card Q = card Q2 by CARD_1:5;
(Sgm P1) " P2 = P by A3, A18, A9, FUNCT_1:94;
then Segm ((Segm (A,P1,Q1)),P,Q) = Segm (A,P2,Q2) by A15, A13, A17, Th56;
hence ex P2, Q2 being finite without_zero Subset of NAT st
( P2 c= P1 & Q2 c= Q1 & P2 = (Sgm P1) .: P & Q2 = (Sgm Q1) .: Q & card P2 = card P & card Q2 = card Q & Segm ((Segm (A,P1,Q1)),P,Q) = Segm (A,P2,Q2) ) by A12, A15, A19, A20, RELAT_1:111; ::_thesis: verum
end;
theorem Th58: :: MATRIX13:58
for n, i, j being Nat
for K being Field
for M being Matrix of n,K holds Segm (M,((Seg n) \ {i}),((Seg n) \ {j})) = Deleting (M,i,j)
proof
let n, i, j be Nat; ::_thesis: for K being Field
for M being Matrix of n,K holds Segm (M,((Seg n) \ {i}),((Seg n) \ {j})) = Deleting (M,i,j)
let K be Field; ::_thesis: for M being Matrix of n,K holds Segm (M,((Seg n) \ {i}),((Seg n) \ {j})) = Deleting (M,i,j)
let M be Matrix of n,K; ::_thesis: Segm (M,((Seg n) \ {i}),((Seg n) \ {j})) = Deleting (M,i,j)
A1: width M = n by MATRIX_1:24;
A2: len M = n by MATRIX_1:24;
then A3: dom M = Seg n by FINSEQ_1:def_3;
percases ( not i in Seg n or i in Seg n ) ;
supposeA4: not i in Seg n ; ::_thesis: Segm (M,((Seg n) \ {i}),((Seg n) \ {j})) = Deleting (M,i,j)
then A5: Seg n = (Seg n) \ {i} by ZFMISC_1:57;
Del (M,i) = M by A3, A4, FINSEQ_3:104;
hence Segm (M,((Seg n) \ {i}),((Seg n) \ {j})) = Deleting (M,i,j) by A2, A1, A5, Th52; ::_thesis: verum
end;
supposeA6: i in Seg n ; ::_thesis: Segm (M,((Seg n) \ {i}),((Seg n) \ {j})) = Deleting (M,i,j)
set Q1 = Seg n;
set Q = (Seg n) \ {j};
set P = (Seg n) \ {i};
set SS = Segm (M,((Seg n) \ {i}),(Seg n));
consider m being Nat such that
A7: len M = m + 1 and
A8: len (Del (M,i)) = m by A3, A6, FINSEQ_3:104;
percases ( m = 0 or m > 0 ) ;
supposeA9: m = 0 ; ::_thesis: Segm (M,((Seg n) \ {i}),((Seg n) \ {j})) = Deleting (M,i,j)
then len (Deleting (M,i,j)) = 0 by A8, MATRIX_2:def_5;
then A10: Deleting (M,i,j) = {} ;
A11: (Seg n) \ {1} = {} by A2, A7, A9, FINSEQ_1:2, XBOOLE_1:37;
i = 1 by A2, A6, A7, A9, FINSEQ_1:2, TARSKI:def_1;
then len (Segm (M,((Seg n) \ {i}),((Seg n) \ {j}))) = 0 by A11, MATRIX_1:def_2;
hence Segm (M,((Seg n) \ {i}),((Seg n) \ {j})) = Deleting (M,i,j) by A10; ::_thesis: verum
end;
suppose m > 0 ; ::_thesis: Segm (M,((Seg n) \ {i}),((Seg n) \ {j})) = Deleting (M,i,j)
then n > 1 + 0 by A2, A7, XREAL_1:8;
then A12: n = width (DelLine (M,i)) by A2, A1, LAPLACE:4;
A13: (Seg n) \ {j} c= Seg n by XBOOLE_1:36;
(Seg n) \ {i} c= Seg n by XBOOLE_1:36;
then A14: rng (Sgm ((Seg n) \ {i})) = (Seg n) \ {i} by FINSEQ_1:def_13;
dom (Sgm ((Seg n) \ {i})) = Seg (card ((Seg n) \ {i})) by FINSEQ_3:40, XBOOLE_1:36;
then A15: (Sgm ((Seg n) \ {i})) " ((Seg n) \ {i}) = Seg (card ((Seg n) \ {i})) by A14, RELAT_1:134
.= Seg (len (Segm (M,((Seg n) \ {i}),(Seg n)))) by MATRIX_1:def_2 ;
A16: Segm (M,((Seg n) \ {i}),(Seg n)) = Del (M,i) by A2, A1, Th51;
then A17: Deleting (M,i,j) = Segm ((Segm (M,((Seg n) \ {i}),(Seg n))),(Seg (len (Segm (M,((Seg n) \ {i}),(Seg n))))),((Seg (width (Segm (M,((Seg n) \ {i}),(Seg n))))) \ {j})) by Th52;
Sgm (Seg n) = idseq n by FINSEQ_3:48;
then (Sgm (Seg n)) " ((Seg n) \ {j}) = (Seg (width (Segm (M,((Seg n) \ {i}),(Seg n))))) \ {j} by A13, A12, A16, FUNCT_2:94;
hence Segm (M,((Seg n) \ {i}),((Seg n) \ {j})) = Deleting (M,i,j) by A13, A15, A17, Th56; ::_thesis: verum
end;
end;
end;
end;
end;
theorem Th59: :: MATRIX13:59
for D being non empty set
for n9, m9, i being Nat
for A9 being Matrix of n9,m9,D
for Q, P being finite without_zero Subset of NAT
for F, FQ being FinSequence of D st len F = width A9 & FQ = F * (Sgm Q) & [:P,Q:] c= Indices A9 holds
RLine ((Segm (A9,P,Q)),i,FQ) = Segm ((RLine (A9,((Sgm P) . i),F)),P,Q)
proof
let D be non empty set ; ::_thesis: for n9, m9, i being Nat
for A9 being Matrix of n9,m9,D
for Q, P being finite without_zero Subset of NAT
for F, FQ being FinSequence of D st len F = width A9 & FQ = F * (Sgm Q) & [:P,Q:] c= Indices A9 holds
RLine ((Segm (A9,P,Q)),i,FQ) = Segm ((RLine (A9,((Sgm P) . i),F)),P,Q)
let n9, m9, i be Nat; ::_thesis: for A9 being Matrix of n9,m9,D
for Q, P being finite without_zero Subset of NAT
for F, FQ being FinSequence of D st len F = width A9 & FQ = F * (Sgm Q) & [:P,Q:] c= Indices A9 holds
RLine ((Segm (A9,P,Q)),i,FQ) = Segm ((RLine (A9,((Sgm P) . i),F)),P,Q)
let A9 be Matrix of n9,m9,D; ::_thesis: for Q, P being finite without_zero Subset of NAT
for F, FQ being FinSequence of D st len F = width A9 & FQ = F * (Sgm Q) & [:P,Q:] c= Indices A9 holds
RLine ((Segm (A9,P,Q)),i,FQ) = Segm ((RLine (A9,((Sgm P) . i),F)),P,Q)
let Q, P be finite without_zero Subset of NAT; ::_thesis: for F, FQ being FinSequence of D st len F = width A9 & FQ = F * (Sgm Q) & [:P,Q:] c= Indices A9 holds
RLine ((Segm (A9,P,Q)),i,FQ) = Segm ((RLine (A9,((Sgm P) . i),F)),P,Q)
let F, FQ be FinSequence of D; ::_thesis: ( len F = width A9 & FQ = F * (Sgm Q) & [:P,Q:] c= Indices A9 implies RLine ((Segm (A9,P,Q)),i,FQ) = Segm ((RLine (A9,((Sgm P) . i),F)),P,Q) )
assume that
A1: len F = width A9 and
A2: FQ = F * (Sgm Q) and
A3: [:P,Q:] c= Indices A9 ; ::_thesis: RLine ((Segm (A9,P,Q)),i,FQ) = Segm ((RLine (A9,((Sgm P) . i),F)),P,Q)
set SQ = Sgm Q;
set SP = Sgm P;
A4: card P = len (Segm (A9,P,Q)) by MATRIX_1:def_2;
ex m being Nat st Q c= Seg m by Th43;
then A5: rng (Sgm Q) = Q by FINSEQ_1:def_13;
A6: ex n being Nat st P c= Seg n by Th43;
then A7: rng (Sgm P) = P by FINSEQ_1:def_13;
A8: Sgm P is one-to-one by A6, FINSEQ_3:92;
A9: dom (Sgm P) = Seg (card P) by A6, FINSEQ_3:40;
percases ( i in dom (Sgm P) or not i in dom (Sgm P) ) ;
suppose i in dom (Sgm P) ; ::_thesis: RLine ((Segm (A9,P,Q)),i,FQ) = Segm ((RLine (A9,((Sgm P) . i),F)),P,Q)
then (Sgm P) " {((Sgm P) . i)} = {i} by A8, FINSEQ_5:4;
hence RLine ((Segm (A9,P,Q)),i,FQ) = Segm ((RLine (A9,((Sgm P) . i),F)),P,Q) by A1, A2, A3, A7, A5, Th37; ::_thesis: verum
end;
supposeA10: not i in dom (Sgm P) ; ::_thesis: RLine ((Segm (A9,P,Q)),i,FQ) = Segm ((RLine (A9,((Sgm P) . i),F)),P,Q)
A11: not 0 in Seg (len A9) ;
(Sgm P) . i = 0 by A10, FUNCT_1:def_2;
hence Segm ((RLine (A9,((Sgm P) . i),F)),P,Q) = Segm (A9,P,Q) by A11, Th40
.= RLine ((Segm (A9,P,Q)),i,FQ) by A9, A4, A10, Th40 ;
::_thesis: verum
end;
end;
end;
theorem Th60: :: MATRIX13:60
for D being non empty set
for n9, m9 being Nat
for A9 being Matrix of n9,m9,D
for Q being finite without_zero Subset of NAT
for F being FinSequence of D
for i being Nat
for P being finite without_zero Subset of NAT st not i in P & [:P,Q:] c= Indices A9 holds
Segm (A9,P,Q) = Segm ((RLine (A9,i,F)),P,Q)
proof
let D be non empty set ; ::_thesis: for n9, m9 being Nat
for A9 being Matrix of n9,m9,D
for Q being finite without_zero Subset of NAT
for F being FinSequence of D
for i being Nat
for P being finite without_zero Subset of NAT st not i in P & [:P,Q:] c= Indices A9 holds
Segm (A9,P,Q) = Segm ((RLine (A9,i,F)),P,Q)
let n9, m9 be Nat; ::_thesis: for A9 being Matrix of n9,m9,D
for Q being finite without_zero Subset of NAT
for F being FinSequence of D
for i being Nat
for P being finite without_zero Subset of NAT st not i in P & [:P,Q:] c= Indices A9 holds
Segm (A9,P,Q) = Segm ((RLine (A9,i,F)),P,Q)
let A9 be Matrix of n9,m9,D; ::_thesis: for Q being finite without_zero Subset of NAT
for F being FinSequence of D
for i being Nat
for P being finite without_zero Subset of NAT st not i in P & [:P,Q:] c= Indices A9 holds
Segm (A9,P,Q) = Segm ((RLine (A9,i,F)),P,Q)
let Q be finite without_zero Subset of NAT; ::_thesis: for F being FinSequence of D
for i being Nat
for P being finite without_zero Subset of NAT st not i in P & [:P,Q:] c= Indices A9 holds
Segm (A9,P,Q) = Segm ((RLine (A9,i,F)),P,Q)
let F be FinSequence of D; ::_thesis: for i being Nat
for P being finite without_zero Subset of NAT st not i in P & [:P,Q:] c= Indices A9 holds
Segm (A9,P,Q) = Segm ((RLine (A9,i,F)),P,Q)
let i be Nat; ::_thesis: for P being finite without_zero Subset of NAT st not i in P & [:P,Q:] c= Indices A9 holds
Segm (A9,P,Q) = Segm ((RLine (A9,i,F)),P,Q)
let P be finite without_zero Subset of NAT; ::_thesis: ( not i in P & [:P,Q:] c= Indices A9 implies Segm (A9,P,Q) = Segm ((RLine (A9,i,F)),P,Q) )
assume that
A1: not i in P and
A2: [:P,Q:] c= Indices A9 ; ::_thesis: Segm (A9,P,Q) = Segm ((RLine (A9,i,F)),P,Q)
ex m being Nat st Q c= Seg m by Th43;
then A3: rng (Sgm Q) = Q by FINSEQ_1:def_13;
ex n being Nat st P c= Seg n by Th43;
then rng (Sgm P) = P by FINSEQ_1:def_13;
hence Segm (A9,P,Q) = Segm ((RLine (A9,i,F)),P,Q) by A1, A2, A3, Th38; ::_thesis: verum
end;
theorem :: MATRIX13:61
for D being non empty set
for A being Matrix of D
for P, Q being finite without_zero Subset of NAT holds
not ( [:P,Q:] c= Indices A & ( card P = 0 implies card Q = 0 ) & ( card Q = 0 implies card P = 0 ) & not (Segm (A,P,Q)) @ = Segm ((A @),Q,P) )
proof
let D be non empty set ; ::_thesis: for A being Matrix of D
for P, Q being finite without_zero Subset of NAT holds
not ( [:P,Q:] c= Indices A & ( card P = 0 implies card Q = 0 ) & ( card Q = 0 implies card P = 0 ) & not (Segm (A,P,Q)) @ = Segm ((A @),Q,P) )
let A be Matrix of D; ::_thesis: for P, Q being finite without_zero Subset of NAT holds
not ( [:P,Q:] c= Indices A & ( card P = 0 implies card Q = 0 ) & ( card Q = 0 implies card P = 0 ) & not (Segm (A,P,Q)) @ = Segm ((A @),Q,P) )
let P, Q be finite without_zero Subset of NAT; ::_thesis: not ( [:P,Q:] c= Indices A & ( card P = 0 implies card Q = 0 ) & ( card Q = 0 implies card P = 0 ) & not (Segm (A,P,Q)) @ = Segm ((A @),Q,P) )
assume that
A1: [:P,Q:] c= Indices A and
A2: ( card P = 0 iff card Q = 0 ) ; ::_thesis: (Segm (A,P,Q)) @ = Segm ((A @),Q,P)
ex m being Nat st Q c= Seg m by Th43;
then A3: rng (Sgm Q) = Q by FINSEQ_1:def_13;
ex n being Nat st P c= Seg n by Th43;
then rng (Sgm P) = P by FINSEQ_1:def_13;
hence (Segm (A,P,Q)) @ = Segm ((A @),Q,P) by A1, A2, A3, Th18; ::_thesis: verum
end;
theorem Th62: :: MATRIX13:62
for D being non empty set
for A being Matrix of D
for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices A & ( card Q = 0 implies card P = 0 ) holds
Segm (A,P,Q) = (Segm ((A @),Q,P)) @
proof
let D be non empty set ; ::_thesis: for A being Matrix of D
for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices A & ( card Q = 0 implies card P = 0 ) holds
Segm (A,P,Q) = (Segm ((A @),Q,P)) @
let A be Matrix of D; ::_thesis: for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices A & ( card Q = 0 implies card P = 0 ) holds
Segm (A,P,Q) = (Segm ((A @),Q,P)) @
let P, Q be finite without_zero Subset of NAT; ::_thesis: ( [:P,Q:] c= Indices A & ( card Q = 0 implies card P = 0 ) implies Segm (A,P,Q) = (Segm ((A @),Q,P)) @ )
assume that
A1: [:P,Q:] c= Indices A and
A2: ( card Q = 0 implies card P = 0 ) ; ::_thesis: Segm (A,P,Q) = (Segm ((A @),Q,P)) @
ex m being Nat st Q c= Seg m by Th43;
then A3: rng (Sgm Q) = Q by FINSEQ_1:def_13;
ex n being Nat st P c= Seg n by Th43;
then rng (Sgm P) = P by FINSEQ_1:def_13;
hence Segm (A,P,Q) = (Segm ((A @),Q,P)) @ by A1, A2, A3, Th19; ::_thesis: verum
end;
theorem Th63: :: MATRIX13:63
for K being Field
for a being Element of K
for M being Matrix of K
for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M holds
a * (Segm (M,P,Q)) = Segm ((a * M),P,Q)
proof
let K be Field; ::_thesis: for a being Element of K
for M being Matrix of K
for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M holds
a * (Segm (M,P,Q)) = Segm ((a * M),P,Q)
let a be Element of K; ::_thesis: for M being Matrix of K
for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M holds
a * (Segm (M,P,Q)) = Segm ((a * M),P,Q)
let M be Matrix of K; ::_thesis: for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M holds
a * (Segm (M,P,Q)) = Segm ((a * M),P,Q)
let P, Q be finite without_zero Subset of NAT; ::_thesis: ( [:P,Q:] c= Indices M implies a * (Segm (M,P,Q)) = Segm ((a * M),P,Q) )
ex n being Nat st P c= Seg n by Th43;
then A1: rng (Sgm P) = P by FINSEQ_1:def_13;
ex k being Nat st Q c= Seg k by Th43;
then A2: rng (Sgm Q) = Q by FINSEQ_1:def_13;
assume [:P,Q:] c= Indices M ; ::_thesis: a * (Segm (M,P,Q)) = Segm ((a * M),P,Q)
hence a * (Segm (M,P,Q)) = Segm ((a * M),P,Q) by A1, A2, Th41; ::_thesis: verum
end;
definition
let D be non empty set ;
let A be Matrix of D;
let P, Q be finite without_zero Subset of NAT;
assume A1: card P = card Q ;
func EqSegm (A,P,Q) -> Matrix of card P,D equals :Def3: :: MATRIX13:def 3
Segm (A,P,Q);
coherence
Segm (A,P,Q) is Matrix of card P,D by A1;
end;
:: deftheorem Def3 defines EqSegm MATRIX13:def_3_:_
for D being non empty set
for A being Matrix of D
for P, Q being finite without_zero Subset of NAT st card P = card Q holds
EqSegm (A,P,Q) = Segm (A,P,Q);
theorem Th64: :: MATRIX13:64
for K being Field
for M being Matrix of K
for P, Q being finite without_zero Subset of NAT
for i, j being Nat st i in Seg (card P) & j in Seg (card P) & card P = card Q holds
( Delete ((EqSegm (M,P,Q)),i,j) = EqSegm (M,(P \ {((Sgm P) . i)}),(Q \ {((Sgm Q) . j)})) & card (P \ {((Sgm P) . i)}) = card (Q \ {((Sgm Q) . j)}) )
proof
let K be Field; ::_thesis: for M being Matrix of K
for P, Q being finite without_zero Subset of NAT
for i, j being Nat st i in Seg (card P) & j in Seg (card P) & card P = card Q holds
( Delete ((EqSegm (M,P,Q)),i,j) = EqSegm (M,(P \ {((Sgm P) . i)}),(Q \ {((Sgm Q) . j)})) & card (P \ {((Sgm P) . i)}) = card (Q \ {((Sgm Q) . j)}) )
let M be Matrix of K; ::_thesis: for P, Q being finite without_zero Subset of NAT
for i, j being Nat st i in Seg (card P) & j in Seg (card P) & card P = card Q holds
( Delete ((EqSegm (M,P,Q)),i,j) = EqSegm (M,(P \ {((Sgm P) . i)}),(Q \ {((Sgm Q) . j)})) & card (P \ {((Sgm P) . i)}) = card (Q \ {((Sgm Q) . j)}) )
let P1, Q1 be finite without_zero Subset of NAT; ::_thesis: for i, j being Nat st i in Seg (card P1) & j in Seg (card P1) & card P1 = card Q1 holds
( Delete ((EqSegm (M,P1,Q1)),i,j) = EqSegm (M,(P1 \ {((Sgm P1) . i)}),(Q1 \ {((Sgm Q1) . j)})) & card (P1 \ {((Sgm P1) . i)}) = card (Q1 \ {((Sgm Q1) . j)}) )
let i, j be Nat; ::_thesis: ( i in Seg (card P1) & j in Seg (card P1) & card P1 = card Q1 implies ( Delete ((EqSegm (M,P1,Q1)),i,j) = EqSegm (M,(P1 \ {((Sgm P1) . i)}),(Q1 \ {((Sgm Q1) . j)})) & card (P1 \ {((Sgm P1) . i)}) = card (Q1 \ {((Sgm Q1) . j)}) ) )
assume that
A1: i in Seg (card P1) and
A2: j in Seg (card P1) and
A3: card P1 = card Q1 ; ::_thesis: ( Delete ((EqSegm (M,P1,Q1)),i,j) = EqSegm (M,(P1 \ {((Sgm P1) . i)}),(Q1 \ {((Sgm Q1) . j)})) & card (P1 \ {((Sgm P1) . i)}) = card (Q1 \ {((Sgm Q1) . j)}) )
set SQ1 = Sgm Q1;
A4: ex m being Nat st Q1 c= Seg m by Th43;
then A5: dom (Sgm Q1) = Seg (card Q1) by FINSEQ_3:40;
A6: rng (Sgm Q1) = Q1 by A4, FINSEQ_1:def_13;
then A7: (Sgm Q1) " Q1 = Seg (card P1) by A3, A5, RELAT_1:134;
set Q = Q1 \ {((Sgm Q1) . j)};
set Q2 = (Seg (card Q1)) \ {j};
A8: Q1 \ {((Sgm Q1) . j)} c= Q1 by XBOOLE_1:36;
set SP1 = Sgm P1;
A9: ex n being Nat st P1 c= Seg n by Th43;
then A10: dom (Sgm P1) = Seg (card P1) by FINSEQ_3:40;
A11: rng (Sgm P1) = P1 by A9, FINSEQ_1:def_13;
then A12: (Sgm P1) " P1 = Seg (card P1) by A10, RELAT_1:134;
Sgm Q1 is one-to-one by A4, FINSEQ_3:92;
then (Sgm Q1) " {((Sgm Q1) . j)} = {j} by A2, A3, A5, FINSEQ_5:4;
then A13: (Seg (card Q1)) \ {j} = (Sgm Q1) " (Q1 \ {((Sgm Q1) . j)}) by A3, A7, FUNCT_1:69;
A14: (Sgm P1) . i in P1 by A1, A10, A11, FUNCT_1:def_3;
set P2 = (Seg (card P1)) \ {i};
set P = P1 \ {((Sgm P1) . i)};
Sgm P1 is one-to-one by A9, FINSEQ_3:92;
then (Sgm P1) " {((Sgm P1) . i)} = {i} by A1, A10, FINSEQ_5:4;
then A15: (Seg (card P1)) \ {i} = (Sgm P1) " (P1 \ {((Sgm P1) . i)}) by A12, FUNCT_1:69;
set E = EqSegm (M,P1,Q1);
A16: P1 \ {((Sgm P1) . i)} c= P1 by XBOOLE_1:36;
card P1 <> 0 by A1;
then reconsider C = (card P1) - 1 as Element of NAT by NAT_1:20;
A17: card P1 = C + 1 ;
(Sgm Q1) . j in Q1 by A2, A3, A5, A6, FUNCT_1:def_3;
then A18: card (Q1 \ {((Sgm Q1) . j)}) = C by A3, A17, STIRL2_1:55;
Delete ((EqSegm (M,P1,Q1)),i,j) = Deleting ((EqSegm (M,P1,Q1)),i,j) by A1, A2, LAPLACE:def_1
.= Segm ((EqSegm (M,P1,Q1)),((Seg (card P1)) \ {i}),((Seg (card Q1)) \ {j})) by A3, Th58
.= Segm ((Segm (M,P1,Q1)),((Seg (card P1)) \ {i}),((Seg (card Q1)) \ {j})) by A3, Def3
.= Segm (M,(P1 \ {((Sgm P1) . i)}),(Q1 \ {((Sgm Q1) . j)})) by A16, A8, A15, A13, Th56
.= EqSegm (M,(P1 \ {((Sgm P1) . i)}),(Q1 \ {((Sgm Q1) . j)})) by A14, A17, A18, Def3, STIRL2_1:55 ;
hence ( Delete ((EqSegm (M,P1,Q1)),i,j) = EqSegm (M,(P1 \ {((Sgm P1) . i)}),(Q1 \ {((Sgm Q1) . j)})) & card (P1 \ {((Sgm P1) . i)}) = card (Q1 \ {((Sgm Q1) . j)}) ) by A14, A17, A18, STIRL2_1:55; ::_thesis: verum
end;
Lm3: for K being Field
for M being Matrix of K
for P, Q being finite without_zero Subset of NAT
for i being Nat st i in Seg (card P) & Det (EqSegm (M,P,Q)) <> 0. K holds
ex j being Nat st
( j in Seg (card P) & Det (Delete ((EqSegm (M,P,Q)),i,j)) <> 0. K )
proof
let K be Field; ::_thesis: for M being Matrix of K
for P, Q being finite without_zero Subset of NAT
for i being Nat st i in Seg (card P) & Det (EqSegm (M,P,Q)) <> 0. K holds
ex j being Nat st
( j in Seg (card P) & Det (Delete ((EqSegm (M,P,Q)),i,j)) <> 0. K )
let M be Matrix of K; ::_thesis: for P, Q being finite without_zero Subset of NAT
for i being Nat st i in Seg (card P) & Det (EqSegm (M,P,Q)) <> 0. K holds
ex j being Nat st
( j in Seg (card P) & Det (Delete ((EqSegm (M,P,Q)),i,j)) <> 0. K )
let P, Q be finite without_zero Subset of NAT; ::_thesis: for i being Nat st i in Seg (card P) & Det (EqSegm (M,P,Q)) <> 0. K holds
ex j being Nat st
( j in Seg (card P) & Det (Delete ((EqSegm (M,P,Q)),i,j)) <> 0. K )
let i be Nat; ::_thesis: ( i in Seg (card P) & Det (EqSegm (M,P,Q)) <> 0. K implies ex j being Nat st
( j in Seg (card P) & Det (Delete ((EqSegm (M,P,Q)),i,j)) <> 0. K ) )
assume that
A1: i in Seg (card P) and
A2: Det (EqSegm (M,P,Q)) <> 0. K ; ::_thesis: ex j being Nat st
( j in Seg (card P) & Det (Delete ((EqSegm (M,P,Q)),i,j)) <> 0. K )
set C = card P;
set E = EqSegm (M,P,Q);
set LL = LaplaceExpL ((EqSegm (M,P,Q)),i);
set CC = (card P) |-> (0. K);
Sum ((card P) |-> (0. K)) = 0. K by MATRIX_3:11;
then A3: LaplaceExpL ((EqSegm (M,P,Q)),i) <> (card P) |-> (0. K) by A1, A2, LAPLACE:25;
len (LaplaceExpL ((EqSegm (M,P,Q)),i)) = card P by LAPLACE:def_7;
then A4: dom (LaplaceExpL ((EqSegm (M,P,Q)),i)) = Seg (card P) by FINSEQ_1:def_3;
dom ((card P) |-> (0. K)) = Seg (card P) by FUNCOP_1:13;
then consider j being Nat such that
A5: j in dom (LaplaceExpL ((EqSegm (M,P,Q)),i)) and
A6: (LaplaceExpL ((EqSegm (M,P,Q)),i)) . j <> ((card P) |-> (0. K)) . j by A3, A4, FINSEQ_1:13;
A7: (LaplaceExpL ((EqSegm (M,P,Q)),i)) . j = ((EqSegm (M,P,Q)) * (i,j)) * (Cofactor ((EqSegm (M,P,Q)),i,j)) by A5, LAPLACE:def_7;
((card P) |-> (0. K)) . j = 0. K by A4, A5, FINSEQ_2:57;
then Cofactor ((EqSegm (M,P,Q)),i,j) <> 0. K by A6, A7, VECTSP_1:12;
then Minor ((EqSegm (M,P,Q)),i,j) <> 0. K by VECTSP_1:12;
hence ex j being Nat st
( j in Seg (card P) & Det (Delete ((EqSegm (M,P,Q)),i,j)) <> 0. K ) by A4, A5; ::_thesis: verum
end;
theorem Th65: :: MATRIX13:65
for K being Field
for M being Matrix of K
for P, P1, Q1 being finite without_zero Subset of NAT st card P1 = card Q1 & P c= P1 & Det (EqSegm (M,P1,Q1)) <> 0. K holds
ex Q being finite without_zero Subset of NAT st
( Q c= Q1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K )
proof
let K be Field; ::_thesis: for M being Matrix of K
for P, P1, Q1 being finite without_zero Subset of NAT st card P1 = card Q1 & P c= P1 & Det (EqSegm (M,P1,Q1)) <> 0. K holds
ex Q being finite without_zero Subset of NAT st
( Q c= Q1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K )
let M be Matrix of K; ::_thesis: for P, P1, Q1 being finite without_zero Subset of NAT st card P1 = card Q1 & P c= P1 & Det (EqSegm (M,P1,Q1)) <> 0. K holds
ex Q being finite without_zero Subset of NAT st
( Q c= Q1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K )
let P, P1, Q1 be finite without_zero Subset of NAT; ::_thesis: ( card P1 = card Q1 & P c= P1 & Det (EqSegm (M,P1,Q1)) <> 0. K implies ex Q being finite without_zero Subset of NAT st
( Q c= Q1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K ) )
assume that
A1: card P1 = card Q1 and
A2: P c= P1 and
A3: Det (EqSegm (M,P1,Q1)) <> 0. K ; ::_thesis: ex Q being finite without_zero Subset of NAT st
( Q c= Q1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K )
defpred S1[ Nat] means for P being finite without_zero Subset of NAT st P c= P1 & card P = $1 holds
ex Q being finite without_zero Subset of NAT st
( Q c= Q1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K );
A4: for k being Element of NAT st k < card P1 & S1[k + 1] holds
S1[k]
proof
let k be Element of NAT ; ::_thesis: ( k < card P1 & S1[k + 1] implies S1[k] )
assume that
A5: k < card P1 and
A6: S1[k + 1] ; ::_thesis: S1[k]
let P be finite without_zero Subset of NAT; ::_thesis: ( P c= P1 & card P = k implies ex Q being finite without_zero Subset of NAT st
( Q c= Q1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K ) )
assume that
A7: P c= P1 and
A8: card P = k ; ::_thesis: ex Q being finite without_zero Subset of NAT st
( Q c= Q1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K )
P c< P1 by A5, A7, A8, XBOOLE_0:def_8;
then P1 \ P <> {} by XBOOLE_1:105;
then consider x being set such that
A9: x in P1 \ P by XBOOLE_0:def_1;
reconsider x = x as non zero Element of NAT by A9;
reconsider Px = P \/ {x} as finite without_zero Subset of NAT ;
A10: not x in P by A9, XBOOLE_0:def_5;
then A11: card Px = k + 1 by A8, CARD_2:41;
x in P1 by A9, XBOOLE_0:def_5;
then {x} c= P1 by ZFMISC_1:31;
then Px c= P1 by A7, XBOOLE_1:8;
then consider Q2 being finite without_zero Subset of NAT such that
A12: Q2 c= Q1 and
A13: card Px = card Q2 and
A14: Det (EqSegm (M,Px,Q2)) <> 0. K by A6, A8, A10, CARD_2:41;
set E = EqSegm (M,Px,Q2);
A15: Px \ {x} = P by A10, ZFMISC_1:117;
x in {x} by TARSKI:def_1;
then A16: x in Px by XBOOLE_0:def_3;
A17: ex n being Nat st Px c= Seg n by Th43;
then A18: dom (Sgm Px) = Seg (card Px) by FINSEQ_3:40;
rng (Sgm Px) = Px by A17, FINSEQ_1:def_13;
then consider i being set such that
A19: i in Seg (card Px) and
A20: (Sgm Px) . i = x by A18, A16, FUNCT_1:def_3;
A21: (k + 1) -' 1 = (k + 1) - 1 by XREAL_0:def_2;
reconsider i = i as Element of NAT by A19;
consider j being Nat such that
A22: j in Seg (card Px) and
A23: Det (Delete ((EqSegm (M,Px,Q2)),i,j)) <> 0. K by A14, A19, Lm3;
take Q = Q2 \ {((Sgm Q2) . j)}; ::_thesis: ( Q c= Q1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K )
Q c= Q2 by XBOOLE_1:36;
hence ( Q c= Q1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K ) by A8, A11, A12, A13, A19, A20, A22, A23, A15, A21, Th64, XBOOLE_1:1; ::_thesis: verum
end;
A24: S1[ card P1]
proof
let P be finite without_zero Subset of NAT; ::_thesis: ( P c= P1 & card P = card P1 implies ex Q being finite without_zero Subset of NAT st
( Q c= Q1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K ) )
assume that
A25: P c= P1 and
A26: card P = card P1 ; ::_thesis: ex Q being finite without_zero Subset of NAT st
( Q c= Q1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K )
P = P1 by A25, A26, CARD_FIN:1;
hence ex Q being finite without_zero Subset of NAT st
( Q c= Q1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K ) by A1, A3; ::_thesis: verum
end;
for k being Element of NAT st k <= card P1 holds
S1[k] from PRE_POLY:sch_1(A24, A4);
hence ex Q being finite without_zero Subset of NAT st
( Q c= Q1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K ) by A2, NAT_1:43; ::_thesis: verum
end;
Lm4: for j being Nat
for K being Field
for M being Matrix of K
for P, Q being finite without_zero Subset of NAT
for i being Nat st j in Seg (card P) & Det (EqSegm (M,P,Q)) <> 0. K holds
ex i being Nat st
( i in Seg (card P) & Det (Delete ((EqSegm (M,P,Q)),i,j)) <> 0. K )
proof
let j be Nat; ::_thesis: for K being Field
for M being Matrix of K
for P, Q being finite without_zero Subset of NAT
for i being Nat st j in Seg (card P) & Det (EqSegm (M,P,Q)) <> 0. K holds
ex i being Nat st
( i in Seg (card P) & Det (Delete ((EqSegm (M,P,Q)),i,j)) <> 0. K )
let K be Field; ::_thesis: for M being Matrix of K
for P, Q being finite without_zero Subset of NAT
for i being Nat st j in Seg (card P) & Det (EqSegm (M,P,Q)) <> 0. K holds
ex i being Nat st
( i in Seg (card P) & Det (Delete ((EqSegm (M,P,Q)),i,j)) <> 0. K )
let M be Matrix of K; ::_thesis: for P, Q being finite without_zero Subset of NAT
for i being Nat st j in Seg (card P) & Det (EqSegm (M,P,Q)) <> 0. K holds
ex i being Nat st
( i in Seg (card P) & Det (Delete ((EqSegm (M,P,Q)),i,j)) <> 0. K )
let P, Q be finite without_zero Subset of NAT; ::_thesis: for i being Nat st j in Seg (card P) & Det (EqSegm (M,P,Q)) <> 0. K holds
ex i being Nat st
( i in Seg (card P) & Det (Delete ((EqSegm (M,P,Q)),i,j)) <> 0. K )
let i be Nat; ::_thesis: ( j in Seg (card P) & Det (EqSegm (M,P,Q)) <> 0. K implies ex i being Nat st
( i in Seg (card P) & Det (Delete ((EqSegm (M,P,Q)),i,j)) <> 0. K ) )
assume that
A1: j in Seg (card P) and
A2: Det (EqSegm (M,P,Q)) <> 0. K ; ::_thesis: ex i being Nat st
( i in Seg (card P) & Det (Delete ((EqSegm (M,P,Q)),i,j)) <> 0. K )
set C = card P;
set E = EqSegm (M,P,Q);
set LC = LaplaceExpC ((EqSegm (M,P,Q)),j);
set CC = (card P) |-> (0. K);
Sum ((card P) |-> (0. K)) = 0. K by MATRIX_3:11;
then A3: LaplaceExpC ((EqSegm (M,P,Q)),j) <> (card P) |-> (0. K) by A1, A2, LAPLACE:27;
len (LaplaceExpC ((EqSegm (M,P,Q)),j)) = card P by LAPLACE:def_8;
then A4: dom (LaplaceExpC ((EqSegm (M,P,Q)),j)) = Seg (card P) by FINSEQ_1:def_3;
dom ((card P) |-> (0. K)) = Seg (card P) by FUNCOP_1:13;
then consider i being Nat such that
A5: i in dom (LaplaceExpC ((EqSegm (M,P,Q)),j)) and
A6: (LaplaceExpC ((EqSegm (M,P,Q)),j)) . i <> ((card P) |-> (0. K)) . i by A3, A4, FINSEQ_1:13;
A7: (LaplaceExpC ((EqSegm (M,P,Q)),j)) . i = ((EqSegm (M,P,Q)) * (i,j)) * (Cofactor ((EqSegm (M,P,Q)),i,j)) by A5, LAPLACE:def_8;
((card P) |-> (0. K)) . i = 0. K by A4, A5, FINSEQ_2:57;
then Cofactor ((EqSegm (M,P,Q)),i,j) <> 0. K by A6, A7, VECTSP_1:12;
then Minor ((EqSegm (M,P,Q)),i,j) <> 0. K by VECTSP_1:12;
hence ex i being Nat st
( i in Seg (card P) & Det (Delete ((EqSegm (M,P,Q)),i,j)) <> 0. K ) by A4, A5; ::_thesis: verum
end;
theorem :: MATRIX13:66
for K being Field
for M being Matrix of K
for P1, Q, Q1 being finite without_zero Subset of NAT st card P1 = card Q1 & Q c= Q1 & Det (EqSegm (M,P1,Q1)) <> 0. K holds
ex P being finite without_zero Subset of NAT st
( P c= P1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K )
proof
let K be Field; ::_thesis: for M being Matrix of K
for P1, Q, Q1 being finite without_zero Subset of NAT st card P1 = card Q1 & Q c= Q1 & Det (EqSegm (M,P1,Q1)) <> 0. K holds
ex P being finite without_zero Subset of NAT st
( P c= P1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K )
let M be Matrix of K; ::_thesis: for P1, Q, Q1 being finite without_zero Subset of NAT st card P1 = card Q1 & Q c= Q1 & Det (EqSegm (M,P1,Q1)) <> 0. K holds
ex P being finite without_zero Subset of NAT st
( P c= P1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K )
let P1, Q, Q1 be finite without_zero Subset of NAT; ::_thesis: ( card P1 = card Q1 & Q c= Q1 & Det (EqSegm (M,P1,Q1)) <> 0. K implies ex P being finite without_zero Subset of NAT st
( P c= P1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K ) )
assume that
A1: card P1 = card Q1 and
A2: Q c= Q1 and
A3: Det (EqSegm (M,P1,Q1)) <> 0. K ; ::_thesis: ex P being finite without_zero Subset of NAT st
( P c= P1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K )
defpred S1[ Nat] means for Q being finite without_zero Subset of NAT st Q c= Q1 & card Q = $1 holds
ex P being finite without_zero Subset of NAT st
( P c= P1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K );
A4: for k being Element of NAT st k < card Q1 & S1[k + 1] holds
S1[k]
proof
let k be Element of NAT ; ::_thesis: ( k < card Q1 & S1[k + 1] implies S1[k] )
assume that
A5: k < card Q1 and
A6: S1[k + 1] ; ::_thesis: S1[k]
let Q be finite without_zero Subset of NAT; ::_thesis: ( Q c= Q1 & card Q = k implies ex P being finite without_zero Subset of NAT st
( P c= P1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K ) )
assume that
A7: Q c= Q1 and
A8: card Q = k ; ::_thesis: ex P being finite without_zero Subset of NAT st
( P c= P1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K )
Q c< Q1 by A5, A7, A8, XBOOLE_0:def_8;
then Q1 \ Q <> {} by XBOOLE_1:105;
then consider x being set such that
A9: x in Q1 \ Q by XBOOLE_0:def_1;
reconsider x = x as non zero Element of NAT by A9;
reconsider Qx = Q \/ {x} as finite without_zero Subset of NAT ;
A10: not x in Q by A9, XBOOLE_0:def_5;
then A11: card Qx = k + 1 by A8, CARD_2:41;
x in Q1 by A9, XBOOLE_0:def_5;
then {x} c= Q1 by ZFMISC_1:31;
then Qx c= Q1 by A7, XBOOLE_1:8;
then consider P2 being finite without_zero Subset of NAT such that
A12: P2 c= P1 and
A13: card Qx = card P2 and
A14: Det (EqSegm (M,P2,Qx)) <> 0. K by A6, A8, A10, CARD_2:41;
A15: (k + 1) -' 1 = (k + 1) - 1 by XREAL_0:def_2;
x in {x} by TARSKI:def_1;
then A16: x in Qx by XBOOLE_0:def_3;
A17: ex n being Nat st Qx c= Seg n by Th43;
then A18: dom (Sgm Qx) = Seg (card Qx) by FINSEQ_3:40;
rng (Sgm Qx) = Qx by A17, FINSEQ_1:def_13;
then consider j being set such that
A19: j in Seg (card Qx) and
A20: (Sgm Qx) . j = x by A18, A16, FUNCT_1:def_3;
set E = EqSegm (M,P2,Qx);
reconsider j = j as Element of NAT by A19;
consider i being Nat such that
A21: i in Seg (card Qx) and
A22: Det (Delete ((EqSegm (M,P2,Qx)),i,j)) <> 0. K by A13, A14, A19, Lm4;
take P = P2 \ {((Sgm P2) . i)}; ::_thesis: ( P c= P1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K )
A23: P c= P2 by XBOOLE_1:36;
A24: Qx \ {x} = Q by A10, ZFMISC_1:117;
then card P = card Q by A13, A19, A20, A21, Th64;
hence ( P c= P1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K ) by A8, A11, A12, A13, A19, A20, A21, A22, A24, A23, A15, Th64, XBOOLE_1:1; ::_thesis: verum
end;
A25: S1[ card Q1]
proof
let Q be finite without_zero Subset of NAT; ::_thesis: ( Q c= Q1 & card Q = card Q1 implies ex P being finite without_zero Subset of NAT st
( P c= P1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K ) )
assume that
A26: Q c= Q1 and
A27: card Q = card Q1 ; ::_thesis: ex P being finite without_zero Subset of NAT st
( P c= P1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K )
Q = Q1 by A26, A27, CARD_FIN:1;
hence ex P being finite without_zero Subset of NAT st
( P c= P1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K ) by A1, A3; ::_thesis: verum
end;
for k being Element of NAT st k <= card Q1 holds
S1[k] from PRE_POLY:sch_1(A25, A4);
hence ex P being finite without_zero Subset of NAT st
( P c= P1 & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K ) by A2, NAT_1:43; ::_thesis: verum
end;
theorem Th67: :: MATRIX13:67
for D being non empty set
for A being Matrix of D
for P, Q being finite without_zero Subset of NAT st card P = card Q holds
( [:P,Q:] c= Indices A iff ( P c= Seg (len A) & Q c= Seg (width A) ) )
proof
let D be non empty set ; ::_thesis: for A being Matrix of D
for P, Q being finite without_zero Subset of NAT st card P = card Q holds
( [:P,Q:] c= Indices A iff ( P c= Seg (len A) & Q c= Seg (width A) ) )
let A be Matrix of D; ::_thesis: for P, Q being finite without_zero Subset of NAT st card P = card Q holds
( [:P,Q:] c= Indices A iff ( P c= Seg (len A) & Q c= Seg (width A) ) )
let P, Q be finite without_zero Subset of NAT; ::_thesis: ( card P = card Q implies ( [:P,Q:] c= Indices A iff ( P c= Seg (len A) & Q c= Seg (width A) ) ) )
A1: Indices A = [:(Seg (len A)),(Seg (width A)):] by FINSEQ_1:def_3;
assume A2: card P = card Q ; ::_thesis: ( [:P,Q:] c= Indices A iff ( P c= Seg (len A) & Q c= Seg (width A) ) )
thus ( [:P,Q:] c= Indices A implies ( P c= Seg (len A) & Q c= Seg (width A) ) ) ::_thesis: ( P c= Seg (len A) & Q c= Seg (width A) implies [:P,Q:] c= Indices A )
proof
assume A3: [:P,Q:] c= Indices A ; ::_thesis: ( P c= Seg (len A) & Q c= Seg (width A) )
percases ( [:P,Q:] <> {} or [:P,Q:] = {} ) ;
suppose [:P,Q:] <> {} ; ::_thesis: ( P c= Seg (len A) & Q c= Seg (width A) )
hence ( P c= Seg (len A) & Q c= Seg (width A) ) by A1, A3, ZFMISC_1:114; ::_thesis: verum
end;
supposeA4: [:P,Q:] = {} ; ::_thesis: ( P c= Seg (len A) & Q c= Seg (width A) )
then A5: Q = {} by A2, CARD_1:27, ZFMISC_1:90;
P = {} by A2, A4, CARD_1:27, ZFMISC_1:90;
hence ( P c= Seg (len A) & Q c= Seg (width A) ) by A5, XBOOLE_1:2; ::_thesis: verum
end;
end;
end;
thus ( P c= Seg (len A) & Q c= Seg (width A) implies [:P,Q:] c= Indices A ) by A1, ZFMISC_1:96; ::_thesis: verum
end;
theorem Th68: :: MATRIX13:68
for m9, n9 being Nat
for K being Field
for M9 being Matrix of n9,m9,K
for P, Q being finite without_zero Subset of NAT
for i being Nat
for j0 being non zero Nat st j0 in Seg n9 & i in P & not j0 in P & card P = card Q & [:P,Q:] c= Indices M9 holds
( card P = card ((P \ {i}) \/ {j0}) & [:((P \ {i}) \/ {j0}),Q:] c= Indices M9 & ( Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q)) or Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = - (Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q))) ) )
proof
let m9, n9 be Nat; ::_thesis: for K being Field
for M9 being Matrix of n9,m9,K
for P, Q being finite without_zero Subset of NAT
for i being Nat
for j0 being non zero Nat st j0 in Seg n9 & i in P & not j0 in P & card P = card Q & [:P,Q:] c= Indices M9 holds
( card P = card ((P \ {i}) \/ {j0}) & [:((P \ {i}) \/ {j0}),Q:] c= Indices M9 & ( Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q)) or Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = - (Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q))) ) )
let K be Field; ::_thesis: for M9 being Matrix of n9,m9,K
for P, Q being finite without_zero Subset of NAT
for i being Nat
for j0 being non zero Nat st j0 in Seg n9 & i in P & not j0 in P & card P = card Q & [:P,Q:] c= Indices M9 holds
( card P = card ((P \ {i}) \/ {j0}) & [:((P \ {i}) \/ {j0}),Q:] c= Indices M9 & ( Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q)) or Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = - (Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q))) ) )
let M9 be Matrix of n9,m9,K; ::_thesis: for P, Q being finite without_zero Subset of NAT
for i being Nat
for j0 being non zero Nat st j0 in Seg n9 & i in P & not j0 in P & card P = card Q & [:P,Q:] c= Indices M9 holds
( card P = card ((P \ {i}) \/ {j0}) & [:((P \ {i}) \/ {j0}),Q:] c= Indices M9 & ( Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q)) or Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = - (Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q))) ) )
let P, Q be finite without_zero Subset of NAT; ::_thesis: for i being Nat
for j0 being non zero Nat st j0 in Seg n9 & i in P & not j0 in P & card P = card Q & [:P,Q:] c= Indices M9 holds
( card P = card ((P \ {i}) \/ {j0}) & [:((P \ {i}) \/ {j0}),Q:] c= Indices M9 & ( Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q)) or Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = - (Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q))) ) )
let i be Nat; ::_thesis: for j0 being non zero Nat st j0 in Seg n9 & i in P & not j0 in P & card P = card Q & [:P,Q:] c= Indices M9 holds
( card P = card ((P \ {i}) \/ {j0}) & [:((P \ {i}) \/ {j0}),Q:] c= Indices M9 & ( Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q)) or Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = - (Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q))) ) )
let j0 be non zero Nat; ::_thesis: ( j0 in Seg n9 & i in P & not j0 in P & card P = card Q & [:P,Q:] c= Indices M9 implies ( card P = card ((P \ {i}) \/ {j0}) & [:((P \ {i}) \/ {j0}),Q:] c= Indices M9 & ( Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q)) or Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = - (Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q))) ) ) )
assume that
A1: j0 in Seg n9 and
A2: i in P and
A3: not j0 in P and
A4: card P = card Q and
A5: [:P,Q:] c= Indices M9 ; ::_thesis: ( card P = card ((P \ {i}) \/ {j0}) & [:((P \ {i}) \/ {j0}),Q:] c= Indices M9 & ( Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q)) or Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = - (Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q))) ) )
set Pi = P \ {i};
A6: P \ {i} c= P by XBOOLE_1:36;
set SQ = Sgm Q;
set Pij = (P \ {i}) \/ {j0};
set SPij = Sgm ((P \ {i}) \/ {j0});
ex k being Nat st (P \ {i}) \/ {j0} c= Seg k by Th43;
then A7: rng (Sgm ((P \ {i}) \/ {j0})) = (P \ {i}) \/ {j0} by FINSEQ_1:def_13;
card P > 0 by A2;
then reconsider C = (card P) - 1 as Element of NAT by NAT_1:20;
card P = C + 1 ;
then A8: card (P \ {i}) = C by A2, STIRL2_1:55;
not j0 in P \ {i} by A3, XBOOLE_0:def_5;
hence A9: card ((P \ {i}) \/ {j0}) = C + 1 by A8, CARD_2:41
.= card P ;
::_thesis: ( [:((P \ {i}) \/ {j0}),Q:] c= Indices M9 & ( Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q)) or Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = - (Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q))) ) )
then reconsider SPij = Sgm ((P \ {i}) \/ {j0}), SQ9 = Sgm Q as Element of (card P) -tuples_on NAT by A4;
A10: Segm (M9,SPij,SQ9) = Segm (M9,((P \ {i}) \/ {j0}),Q) by A4, A9
.= EqSegm (M9,((P \ {i}) \/ {j0}),Q) by A4, A9, Def3 ;
P c= Seg (len M9) by A4, A5, Th67;
then A11: P \ {i} c= Seg (len M9) by A6, XBOOLE_1:1;
n9 = len M9 by MATRIX_1:def_2;
then {j0} c= Seg (len M9) by A1, ZFMISC_1:31;
then A12: (P \ {i}) \/ {j0} c= Seg (len M9) by A11, XBOOLE_1:8;
set SP = Sgm P;
ex m being Nat st Q c= Seg m by Th43;
then A13: rng (Sgm Q) = Q by FINSEQ_1:def_13;
ex n being Nat st P c= Seg n by Th43;
then rng (Sgm P) = P by FINSEQ_1:def_13;
then consider PT being Element of (card P) -tuples_on NAT such that
A14: rng PT = (P \ {i}) \/ {j0} and
A15: Segm ((RLine (M9,i,(Line (M9,j0)))),(Sgm P),(Sgm Q)) = Segm (M9,PT,(Sgm Q)) by A2, A5, A13, Th39;
Q c= Seg (width M9) by A4, A5, Th67;
hence [:((P \ {i}) \/ {j0}),Q:] c= Indices M9 by A4, A9, A12, Th67; ::_thesis: ( Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q)) or Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = - (Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q))) )
EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q) = Segm ((RLine (M9,i,(Line (M9,j0)))),P,Q) by A4, Def3
.= Segm (M9,PT,SQ9) by A4, A15 ;
hence ( Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q)) or Det (EqSegm ((RLine (M9,i,(Line (M9,j0)))),P,Q)) = - (Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q))) ) by A9, A7, A13, A14, A10, Th36; ::_thesis: verum
end;
theorem Th69: :: MATRIX13:69
for D being non empty set
for A being Matrix of D
for P, Q being finite without_zero Subset of NAT st card P = card Q holds
( [:P,Q:] c= Indices A iff [:Q,P:] c= Indices (A @) )
proof
let D be non empty set ; ::_thesis: for A being Matrix of D
for P, Q being finite without_zero Subset of NAT st card P = card Q holds
( [:P,Q:] c= Indices A iff [:Q,P:] c= Indices (A @) )
let A be Matrix of D; ::_thesis: for P, Q being finite without_zero Subset of NAT st card P = card Q holds
( [:P,Q:] c= Indices A iff [:Q,P:] c= Indices (A @) )
let P, Q be finite without_zero Subset of NAT; ::_thesis: ( card P = card Q implies ( [:P,Q:] c= Indices A iff [:Q,P:] c= Indices (A @) ) )
assume A1: card P = card Q ; ::_thesis: ( [:P,Q:] c= Indices A iff [:Q,P:] c= Indices (A @) )
percases ( Q = {} or Q <> {} ) ;
supposeA2: Q = {} ; ::_thesis: ( [:P,Q:] c= Indices A iff [:Q,P:] c= Indices (A @) )
then A3: [:Q,P:] = {} by ZFMISC_1:90;
[:P,Q:] = {} by A2, ZFMISC_1:90;
hence ( [:P,Q:] c= Indices A iff [:Q,P:] c= Indices (A @) ) by A3, XBOOLE_1:2; ::_thesis: verum
end;
supposeA4: Q <> {} ; ::_thesis: ( [:P,Q:] c= Indices A iff [:Q,P:] c= Indices (A @) )
thus ( [:P,Q:] c= Indices A implies [:Q,P:] c= Indices (A @) ) ::_thesis: ( [:Q,P:] c= Indices (A @) implies [:P,Q:] c= Indices A )
proof
assume A5: [:P,Q:] c= Indices A ; ::_thesis: [:Q,P:] c= Indices (A @)
then A6: P c= Seg (len A) by A1, Th67;
A7: Q c= Seg (width A) by A1, A5, Th67;
then A8: width A <> 0 by A4;
then A9: len (A @) = width A by MATRIX_2:10;
len A = width (A @) by A8, MATRIX_2:10;
hence [:Q,P:] c= Indices (A @) by A1, A7, A9, A6, Th67; ::_thesis: verum
end;
thus ( [:Q,P:] c= Indices (A @) implies [:P,Q:] c= Indices A ) ::_thesis: verum
proof
assume A10: [:Q,P:] c= Indices (A @) ; ::_thesis: [:P,Q:] c= Indices A
then A11: Q c= Seg (len (A @)) by A1, Th67;
then len (A @) <> 0 by A4;
then width A > 0 by MATRIX_1:def_6;
then A12: len A = width (A @) by MATRIX_2:10;
A13: len (A @) = width A by MATRIX_1:def_6;
P c= Seg (width (A @)) by A1, A10, Th67;
hence [:P,Q:] c= Indices A by A1, A11, A12, A13, Th67; ::_thesis: verum
end;
end;
end;
end;
theorem Th70: :: MATRIX13:70
for K being Field
for M being Matrix of K
for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M & card P = card Q holds
Det (EqSegm (M,P,Q)) = Det (EqSegm ((M @),Q,P))
proof
let K be Field; ::_thesis: for M being Matrix of K
for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M & card P = card Q holds
Det (EqSegm (M,P,Q)) = Det (EqSegm ((M @),Q,P))
let M be Matrix of K; ::_thesis: for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M & card P = card Q holds
Det (EqSegm (M,P,Q)) = Det (EqSegm ((M @),Q,P))
let P, Q be finite without_zero Subset of NAT; ::_thesis: ( [:P,Q:] c= Indices M & card P = card Q implies Det (EqSegm (M,P,Q)) = Det (EqSegm ((M @),Q,P)) )
assume that
A1: [:P,Q:] c= Indices M and
A2: card P = card Q ; ::_thesis: Det (EqSegm (M,P,Q)) = Det (EqSegm ((M @),Q,P))
EqSegm (M,P,Q) = Segm (M,P,Q) by A2, Def3
.= (Segm ((M @),Q,P)) @ by A1, A2, Th62
.= (EqSegm ((M @),Q,P)) @ by A2, Def3 ;
hence Det (EqSegm (M,P,Q)) = Det (EqSegm ((M @),Q,P)) by A2, MATRIXR2:43; ::_thesis: verum
end;
theorem Th71: :: MATRIX13:71
for n being Nat
for K being Field
for a being Element of K
for M being Matrix of n,K holds Det (a * M) = ((power K) . (a,n)) * (Det M)
proof
let n be Nat; ::_thesis: for K being Field
for a being Element of K
for M being Matrix of n,K holds Det (a * M) = ((power K) . (a,n)) * (Det M)
let K be Field; ::_thesis: for a being Element of K
for M being Matrix of n,K holds Det (a * M) = ((power K) . (a,n)) * (Det M)
let a be Element of K; ::_thesis: for M being Matrix of n,K holds Det (a * M) = ((power K) . (a,n)) * (Det M)
let M be Matrix of n,K; ::_thesis: Det (a * M) = ((power K) . (a,n)) * (Det M)
defpred S1[ Nat] means for k being Nat st k = $1 & k <= n holds
ex aM being Matrix of n,K st
( Det aM = ((power K) . (a,k)) * (Det M) & ( for i being Nat st i in Seg n holds
( ( i <= k implies Line (aM,i) = a * (Line (M,i)) ) & ( i > k implies Line (aM,i) = Line (M,i) ) ) ) );
A1: for m being Nat st S1[m] holds
S1[m + 1]
proof
let m be Nat; ::_thesis: ( S1[m] implies S1[m + 1] )
assume A2: S1[m] ; ::_thesis: S1[m + 1]
A3: m in NAT by ORDINAL1:def_12;
let k be Nat; ::_thesis: ( k = m + 1 & k <= n implies ex aM being Matrix of n,K st
( Det aM = ((power K) . (a,k)) * (Det M) & ( for i being Nat st i in Seg n holds
( ( i <= k implies Line (aM,i) = a * (Line (M,i)) ) & ( i > k implies Line (aM,i) = Line (M,i) ) ) ) ) )
assume that
A4: k = m + 1 and
A5: k <= n ; ::_thesis: ex aM being Matrix of n,K st
( Det aM = ((power K) . (a,k)) * (Det M) & ( for i being Nat st i in Seg n holds
( ( i <= k implies Line (aM,i) = a * (Line (M,i)) ) & ( i > k implies Line (aM,i) = Line (M,i) ) ) ) )
m <= n by A4, A5, NAT_1:13;
then consider aM being Matrix of n,K such that
A6: Det aM = ((power K) . (a,m)) * (Det M) and
A7: for i being Nat st i in Seg n holds
( ( i <= m implies Line (aM,i) = a * (Line (M,i)) ) & ( i > m implies Line (aM,i) = Line (M,i) ) ) by A2;
take R = RLine (aM,k,(a * (Line (aM,k)))); ::_thesis: ( Det R = ((power K) . (a,k)) * (Det M) & ( for i being Nat st i in Seg n holds
( ( i <= k implies Line (R,i) = a * (Line (M,i)) ) & ( i > k implies Line (R,i) = Line (M,i) ) ) ) )
0 + 1 <= k by A4, XREAL_1:7;
then k in Seg n by A4, A5;
hence Det R = a * (((power K) . (a,m)) * (Det M)) by A6, MATRIX11:35
.= (((power K) . (a,m)) * a) * (Det M) by GROUP_1:def_3
.= ((power K) . (a,k)) * (Det M) by A4, A3, GROUP_1:def_7 ;
::_thesis: for i being Nat st i in Seg n holds
( ( i <= k implies Line (R,i) = a * (Line (M,i)) ) & ( i > k implies Line (R,i) = Line (M,i) ) )
let i be Nat; ::_thesis: ( i in Seg n implies ( ( i <= k implies Line (R,i) = a * (Line (M,i)) ) & ( i > k implies Line (R,i) = Line (M,i) ) ) )
assume A8: i in Seg n ; ::_thesis: ( ( i <= k implies Line (R,i) = a * (Line (M,i)) ) & ( i > k implies Line (R,i) = Line (M,i) ) )
percases ( i < k or i > k or i = k ) by XXREAL_0:1;
supposeA9: ( i < k or i > k ) ; ::_thesis: ( ( i <= k implies Line (R,i) = a * (Line (M,i)) ) & ( i > k implies Line (R,i) = Line (M,i) ) )
then A10: ( ( i <= m & i < k ) or ( i > m & i > k ) ) by A4, NAT_1:13;
Line (R,i) = Line (aM,i) by A8, A9, MATRIX11:28;
hence ( ( i <= k implies Line (R,i) = a * (Line (M,i)) ) & ( i > k implies Line (R,i) = Line (M,i) ) ) by A7, A8, A10; ::_thesis: verum
end;
supposeA11: i = k ; ::_thesis: ( ( i <= k implies Line (R,i) = a * (Line (M,i)) ) & ( i > k implies Line (R,i) = Line (M,i) ) )
len (a * (Line (aM,k))) = len (Line (aM,k)) by MATRIXR1:16
.= width aM by MATRIX_1:def_7 ;
then A12: Line (R,i) = a * (Line (aM,k)) by A8, A11, MATRIX11:28;
i > m by A4, A11, NAT_1:13;
hence ( ( i <= k implies Line (R,i) = a * (Line (M,i)) ) & ( i > k implies Line (R,i) = Line (M,i) ) ) by A7, A8, A11, A12; ::_thesis: verum
end;
end;
end;
A13: S1[ 0 ]
proof
let k be Nat; ::_thesis: ( k = 0 & k <= n implies ex aM being Matrix of n,K st
( Det aM = ((power K) . (a,k)) * (Det M) & ( for i being Nat st i in Seg n holds
( ( i <= k implies Line (aM,i) = a * (Line (M,i)) ) & ( i > k implies Line (aM,i) = Line (M,i) ) ) ) ) )
assume that
A14: k = 0 and
k <= n ; ::_thesis: ex aM being Matrix of n,K st
( Det aM = ((power K) . (a,k)) * (Det M) & ( for i being Nat st i in Seg n holds
( ( i <= k implies Line (aM,i) = a * (Line (M,i)) ) & ( i > k implies Line (aM,i) = Line (M,i) ) ) ) )
take aM = M; ::_thesis: ( Det aM = ((power K) . (a,k)) * (Det M) & ( for i being Nat st i in Seg n holds
( ( i <= k implies Line (aM,i) = a * (Line (M,i)) ) & ( i > k implies Line (aM,i) = Line (M,i) ) ) ) )
(power K) . (a,0) = 1_ K by GROUP_1:def_7;
hence Det aM = ((power K) . (a,k)) * (Det M) by A14, VECTSP_1:def_8; ::_thesis: for i being Nat st i in Seg n holds
( ( i <= k implies Line (aM,i) = a * (Line (M,i)) ) & ( i > k implies Line (aM,i) = Line (M,i) ) )
let i be Nat; ::_thesis: ( i in Seg n implies ( ( i <= k implies Line (aM,i) = a * (Line (M,i)) ) & ( i > k implies Line (aM,i) = Line (M,i) ) ) )
assume i in Seg n ; ::_thesis: ( ( i <= k implies Line (aM,i) = a * (Line (M,i)) ) & ( i > k implies Line (aM,i) = Line (M,i) ) )
hence ( ( i <= k implies Line (aM,i) = a * (Line (M,i)) ) & ( i > k implies Line (aM,i) = Line (M,i) ) ) by A14; ::_thesis: verum
end;
for m being Nat holds S1[m] from NAT_1:sch_2(A13, A1);
then consider aM being Matrix of n,K such that
A15: Det aM = ((power K) . (a,n)) * (Det M) and
A16: for i being Nat st i in Seg n holds
( ( i <= n implies Line (aM,i) = a * (Line (M,i)) ) & ( i > n implies Line (aM,i) = Line (M,i) ) ) ;
set AM = a * M;
A17: len (a * M) = n by MATRIX_1:def_2;
A18: len M = n by MATRIX_1:def_2;
A19: now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_n_holds_
aM_._i_=_(a_*_M)_._i
let i be Nat; ::_thesis: ( 1 <= i & i <= n implies aM . i = (a * M) . i )
assume that
A20: 1 <= i and
A21: i <= n ; ::_thesis: aM . i = (a * M) . i
i in NAT by ORDINAL1:def_12;
then A22: i in Seg n by A20, A21;
hence aM . i = Line (aM,i) by MATRIX_2:8
.= a * (Line (M,i)) by A16, A21, A22
.= Line ((a * M),i) by A18, A20, A21, MATRIXR1:20
.= (a * M) . i by A22, MATRIX_2:8 ;
::_thesis: verum
end;
len aM = n by MATRIX_1:def_2;
hence Det (a * M) = ((power K) . (a,n)) * (Det M) by A15, A17, A19, FINSEQ_1:14; ::_thesis: verum
end;
theorem Th72: :: MATRIX13:72
for K being Field
for a being Element of K
for M being Matrix of K
for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M & card P = card Q holds
Det (EqSegm ((a * M),P,Q)) = ((power K) . (a,(card P))) * (Det (EqSegm (M,P,Q)))
proof
let K be Field; ::_thesis: for a being Element of K
for M being Matrix of K
for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M & card P = card Q holds
Det (EqSegm ((a * M),P,Q)) = ((power K) . (a,(card P))) * (Det (EqSegm (M,P,Q)))
let a be Element of K; ::_thesis: for M being Matrix of K
for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M & card P = card Q holds
Det (EqSegm ((a * M),P,Q)) = ((power K) . (a,(card P))) * (Det (EqSegm (M,P,Q)))
let M be Matrix of K; ::_thesis: for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M & card P = card Q holds
Det (EqSegm ((a * M),P,Q)) = ((power K) . (a,(card P))) * (Det (EqSegm (M,P,Q)))
let P, Q be finite without_zero Subset of NAT; ::_thesis: ( [:P,Q:] c= Indices M & card P = card Q implies Det (EqSegm ((a * M),P,Q)) = ((power K) . (a,(card P))) * (Det (EqSegm (M,P,Q))) )
assume that
A1: [:P,Q:] c= Indices M and
A2: card P = card Q ; ::_thesis: Det (EqSegm ((a * M),P,Q)) = ((power K) . (a,(card P))) * (Det (EqSegm (M,P,Q)))
EqSegm ((a * M),P,Q) = Segm ((a * M),P,Q) by A2, Def3
.= a * (Segm (M,P,Q)) by A1, Th63
.= a * (EqSegm (M,P,Q)) by A2, Def3 ;
hence Det (EqSegm ((a * M),P,Q)) = ((power K) . (a,(card P))) * (Det (EqSegm (M,P,Q))) by Th71; ::_thesis: verum
end;
definition
let K be Field;
let M be Matrix of K;
func the_rank_of M -> Element of NAT means :Def4: :: MATRIX13:def 4
( ex P, Q being finite without_zero Subset of NAT st
( [:P,Q:] c= Indices M & card P = card Q & card P = it & Det (EqSegm (M,P,Q)) <> 0. K ) & ( for P1, Q1 being finite without_zero Subset of NAT st [:P1,Q1:] c= Indices M & card P1 = card Q1 & Det (EqSegm (M,P1,Q1)) <> 0. K holds
card P1 <= it ) );
existence
ex b1 being Element of NAT st
( ex P, Q being finite without_zero Subset of NAT st
( [:P,Q:] c= Indices M & card P = card Q & card P = b1 & Det (EqSegm (M,P,Q)) <> 0. K ) & ( for P1, Q1 being finite without_zero Subset of NAT st [:P1,Q1:] c= Indices M & card P1 = card Q1 & Det (EqSegm (M,P1,Q1)) <> 0. K holds
card P1 <= b1 ) )
proof
defpred S1[ Nat] means ex P, Q being finite without_zero Subset of NAT st
( [:P,Q:] c= Indices M & card P = card Q & card Q = $1 & Det (EqSegm (M,P,Q)) <> 0. K );
A1: ex k being Nat st S1[k]
proof
set E = the empty finite without_zero Subset of NAT;
reconsider E = the empty finite without_zero Subset of NAT as finite without_zero Subset of NAT ;
take card E ; ::_thesis: S1[ card E]
take E ; ::_thesis: ex Q being finite without_zero Subset of NAT st
( [:E,Q:] c= Indices M & card E = card Q & card Q = card E & Det (EqSegm (M,E,Q)) <> 0. K )
take E ; ::_thesis: ( [:E,E:] c= Indices M & card E = card E & card E = card E & Det (EqSegm (M,E,E)) <> 0. K )
A2: E c= Seg (len M) by XBOOLE_1:2;
A3: E c= Seg (width M) by XBOOLE_1:2;
Det (EqSegm (M,E,E)) = 1_ K by CARD_1:27, MATRIXR2:41;
hence ( [:E,E:] c= Indices M & card E = card E & card E = card E & Det (EqSegm (M,E,E)) <> 0. K ) by A2, A3, Th67; ::_thesis: verum
end;
A4: for k being Nat st S1[k] holds
k <= len M
proof
let k be Nat; ::_thesis: ( S1[k] implies k <= len M )
A5: card (Seg (len M)) = len M by FINSEQ_1:57;
assume S1[k] ; ::_thesis: k <= len M
then consider P, Q being finite without_zero Subset of NAT such that
A6: [:P,Q:] c= Indices M and
A7: card P = card Q and
A8: card Q = k and
Det (EqSegm (M,P,Q)) <> 0. K ;
P c= Seg (len M) by A6, A7, Th67;
hence k <= len M by A7, A8, A5, NAT_1:43; ::_thesis: verum
end;
consider k being Nat such that
A9: S1[k] and
A10: for n being Nat st S1[n] holds
n <= k from NAT_1:sch_6(A4, A1);
take k ; ::_thesis: ( k is Element of NAT & ex P, Q being finite without_zero Subset of NAT st
( [:P,Q:] c= Indices M & card P = card Q & card P = k & Det (EqSegm (M,P,Q)) <> 0. K ) & ( for P1, Q1 being finite without_zero Subset of NAT st [:P1,Q1:] c= Indices M & card P1 = card Q1 & Det (EqSegm (M,P1,Q1)) <> 0. K holds
card P1 <= k ) )
thus ( k is Element of NAT & ex P, Q being finite without_zero Subset of NAT st
( [:P,Q:] c= Indices M & card P = card Q & card P = k & Det (EqSegm (M,P,Q)) <> 0. K ) & ( for P1, Q1 being finite without_zero Subset of NAT st [:P1,Q1:] c= Indices M & card P1 = card Q1 & Det (EqSegm (M,P1,Q1)) <> 0. K holds
card P1 <= k ) ) by A9, A10; ::_thesis: verum
end;
uniqueness
for b1, b2 being Element of NAT st ex P, Q being finite without_zero Subset of NAT st
( [:P,Q:] c= Indices M & card P = card Q & card P = b1 & Det (EqSegm (M,P,Q)) <> 0. K ) & ( for P1, Q1 being finite without_zero Subset of NAT st [:P1,Q1:] c= Indices M & card P1 = card Q1 & Det (EqSegm (M,P1,Q1)) <> 0. K holds
card P1 <= b1 ) & ex P, Q being finite without_zero Subset of NAT st
( [:P,Q:] c= Indices M & card P = card Q & card P = b2 & Det (EqSegm (M,P,Q)) <> 0. K ) & ( for P1, Q1 being finite without_zero Subset of NAT st [:P1,Q1:] c= Indices M & card P1 = card Q1 & Det (EqSegm (M,P1,Q1)) <> 0. K holds
card P1 <= b2 ) holds
b1 = b2
proof
let n1, n2 be Element of NAT ; ::_thesis: ( ex P, Q being finite without_zero Subset of NAT st
( [:P,Q:] c= Indices M & card P = card Q & card P = n1 & Det (EqSegm (M,P,Q)) <> 0. K ) & ( for P1, Q1 being finite without_zero Subset of NAT st [:P1,Q1:] c= Indices M & card P1 = card Q1 & Det (EqSegm (M,P1,Q1)) <> 0. K holds
card P1 <= n1 ) & ex P, Q being finite without_zero Subset of NAT st
( [:P,Q:] c= Indices M & card P = card Q & card P = n2 & Det (EqSegm (M,P,Q)) <> 0. K ) & ( for P1, Q1 being finite without_zero Subset of NAT st [:P1,Q1:] c= Indices M & card P1 = card Q1 & Det (EqSegm (M,P1,Q1)) <> 0. K holds
card P1 <= n2 ) implies n1 = n2 )
assume that
A11: ex P, Q being finite without_zero Subset of NAT st
( [:P,Q:] c= Indices M & card P = card Q & card P = n1 & Det (EqSegm (M,P,Q)) <> 0. K ) and
A12: for P1, Q1 being finite without_zero Subset of NAT st [:P1,Q1:] c= Indices M & card P1 = card Q1 & Det (EqSegm (M,P1,Q1)) <> 0. K holds
card P1 <= n1 and
A13: ex P, Q being finite without_zero Subset of NAT st
( [:P,Q:] c= Indices M & card P = card Q & card P = n2 & Det (EqSegm (M,P,Q)) <> 0. K ) and
A14: for P1, Q1 being finite without_zero Subset of NAT st [:P1,Q1:] c= Indices M & card P1 = card Q1 & Det (EqSegm (M,P1,Q1)) <> 0. K holds
card P1 <= n2 ; ::_thesis: n1 = n2
A15: n2 <= n1 by A12, A13;
n1 <= n2 by A11, A14;
hence n1 = n2 by A15, XXREAL_0:1; ::_thesis: verum
end;
end;
:: deftheorem Def4 defines the_rank_of MATRIX13:def_4_:_
for K being Field
for M being Matrix of K
for b3 being Element of NAT holds
( b3 = the_rank_of M iff ( ex P, Q being finite without_zero Subset of NAT st
( [:P,Q:] c= Indices M & card P = card Q & card P = b3 & Det (EqSegm (M,P,Q)) <> 0. K ) & ( for P1, Q1 being finite without_zero Subset of NAT st [:P1,Q1:] c= Indices M & card P1 = card Q1 & Det (EqSegm (M,P1,Q1)) <> 0. K holds
card P1 <= b3 ) ) );
theorem Th73: :: MATRIX13:73
for K being Field
for M being Matrix of K
for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M & card P = card Q holds
( card P <= len M & card Q <= width M )
proof
let K be Field; ::_thesis: for M being Matrix of K
for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M & card P = card Q holds
( card P <= len M & card Q <= width M )
let M be Matrix of K; ::_thesis: for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M & card P = card Q holds
( card P <= len M & card Q <= width M )
let P, Q be finite without_zero Subset of NAT; ::_thesis: ( [:P,Q:] c= Indices M & card P = card Q implies ( card P <= len M & card Q <= width M ) )
assume that
A1: [:P,Q:] c= Indices M and
A2: card P = card Q ; ::_thesis: ( card P <= len M & card Q <= width M )
Q c= Seg (width M) by A1, A2, Th67;
then A3: card Q <= card (Seg (width M)) by NAT_1:43;
P c= Seg (len M) by A1, A2, Th67;
then card P <= card (Seg (len M)) by NAT_1:43;
hence ( card P <= len M & card Q <= width M ) by A3, FINSEQ_1:57; ::_thesis: verum
end;
theorem Th74: :: MATRIX13:74
for K being Field
for M being Matrix of K holds
( the_rank_of M <= len M & the_rank_of M <= width M )
proof
let K be Field; ::_thesis: for M being Matrix of K holds
( the_rank_of M <= len M & the_rank_of M <= width M )
let M be Matrix of K; ::_thesis: ( the_rank_of M <= len M & the_rank_of M <= width M )
ex P, Q being finite without_zero Subset of NAT st
( [:P,Q:] c= Indices M & card P = card Q & card P = the_rank_of M & Det (EqSegm (M,P,Q)) <> 0. K ) by Def4;
hence ( the_rank_of M <= len M & the_rank_of M <= width M ) by Th73; ::_thesis: verum
end;
Lm5: for n, m being Nat
for K being Field
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT
for M being Matrix of K holds
not ( [:(rng nt),(rng mt):] c= Indices M & ( n = 0 implies m = 0 ) & ( m = 0 implies n = 0 ) & ( for P1, P2 being finite without_zero Subset of NAT holds
( not P1 = rng nt or not P2 = rng mt ) ) )
proof
let n, m be Nat; ::_thesis: for K being Field
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT
for M being Matrix of K holds
not ( [:(rng nt),(rng mt):] c= Indices M & ( n = 0 implies m = 0 ) & ( m = 0 implies n = 0 ) & ( for P1, P2 being finite without_zero Subset of NAT holds
( not P1 = rng nt or not P2 = rng mt ) ) )
let K be Field; ::_thesis: for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT
for M being Matrix of K holds
not ( [:(rng nt),(rng mt):] c= Indices M & ( n = 0 implies m = 0 ) & ( m = 0 implies n = 0 ) & ( for P1, P2 being finite without_zero Subset of NAT holds
( not P1 = rng nt or not P2 = rng mt ) ) )
let nt be Element of n -tuples_on NAT; ::_thesis: for mt being Element of m -tuples_on NAT
for M being Matrix of K holds
not ( [:(rng nt),(rng mt):] c= Indices M & ( n = 0 implies m = 0 ) & ( m = 0 implies n = 0 ) & ( for P1, P2 being finite without_zero Subset of NAT holds
( not P1 = rng nt or not P2 = rng mt ) ) )
let mt be Element of m -tuples_on NAT; ::_thesis: for M being Matrix of K holds
not ( [:(rng nt),(rng mt):] c= Indices M & ( n = 0 implies m = 0 ) & ( m = 0 implies n = 0 ) & ( for P1, P2 being finite without_zero Subset of NAT holds
( not P1 = rng nt or not P2 = rng mt ) ) )
let M be Matrix of K; ::_thesis: not ( [:(rng nt),(rng mt):] c= Indices M & ( n = 0 implies m = 0 ) & ( m = 0 implies n = 0 ) & ( for P1, P2 being finite without_zero Subset of NAT holds
( not P1 = rng nt or not P2 = rng mt ) ) )
assume that
A1: [:(rng nt),(rng mt):] c= Indices M and
A2: ( n = 0 iff m = 0 ) ; ::_thesis: ex P1, P2 being finite without_zero Subset of NAT st
( P1 = rng nt & P2 = rng mt )
rng nt is without_zero
proof
A3: Indices M = [:(Seg (len M)),(Seg (width M)):] by FINSEQ_1:def_3;
assume A4: 0 in rng nt ; :: according to MEASURE6:def_2 ::_thesis: contradiction
dom mt = Seg m by FINSEQ_2:124;
then rng mt <> {} by A2, A4, RELAT_1:42;
then rng nt c= Seg (len M) by A1, A4, A3, ZFMISC_1:114;
hence contradiction by A4; ::_thesis: verum
end;
then A5: rng nt is finite without_zero Subset of NAT by FINSEQ_1:def_4;
rng mt is without_zero
proof
assume A6: 0 in rng mt ; :: according to MEASURE6:def_2 ::_thesis: contradiction
dom nt = Seg n by FINSEQ_2:124;
then rng nt <> {} by A2, A6, RELAT_1:42;
then rng mt c= Seg (width M) by A1, A6, ZFMISC_1:114;
hence contradiction by A6; ::_thesis: verum
end;
then rng mt is finite without_zero Subset of NAT by FINSEQ_1:def_4;
hence ex P1, P2 being finite without_zero Subset of NAT st
( P1 = rng nt & P2 = rng mt ) by A5; ::_thesis: verum
end;
theorem Th75: :: MATRIX13:75
for n being Nat
for K being Field
for nt1, nt2 being Element of n -tuples_on NAT
for M being Matrix of K st [:(rng nt1),(rng nt2):] c= Indices M & Det (Segm (M,nt1,nt2)) <> 0. K holds
ex P1, P2 being finite without_zero Subset of NAT st
( P1 = rng nt1 & P2 = rng nt2 & card P1 = card P2 & card P1 = n & Det (EqSegm (M,P1,P2)) <> 0. K )
proof
let n be Nat; ::_thesis: for K being Field
for nt1, nt2 being Element of n -tuples_on NAT
for M being Matrix of K st [:(rng nt1),(rng nt2):] c= Indices M & Det (Segm (M,nt1,nt2)) <> 0. K holds
ex P1, P2 being finite without_zero Subset of NAT st
( P1 = rng nt1 & P2 = rng nt2 & card P1 = card P2 & card P1 = n & Det (EqSegm (M,P1,P2)) <> 0. K )
let K be Field; ::_thesis: for nt1, nt2 being Element of n -tuples_on NAT
for M being Matrix of K st [:(rng nt1),(rng nt2):] c= Indices M & Det (Segm (M,nt1,nt2)) <> 0. K holds
ex P1, P2 being finite without_zero Subset of NAT st
( P1 = rng nt1 & P2 = rng nt2 & card P1 = card P2 & card P1 = n & Det (EqSegm (M,P1,P2)) <> 0. K )
let nt1, nt2 be Element of n -tuples_on NAT; ::_thesis: for M being Matrix of K st [:(rng nt1),(rng nt2):] c= Indices M & Det (Segm (M,nt1,nt2)) <> 0. K holds
ex P1, P2 being finite without_zero Subset of NAT st
( P1 = rng nt1 & P2 = rng nt2 & card P1 = card P2 & card P1 = n & Det (EqSegm (M,P1,P2)) <> 0. K )
let M be Matrix of K; ::_thesis: ( [:(rng nt1),(rng nt2):] c= Indices M & Det (Segm (M,nt1,nt2)) <> 0. K implies ex P1, P2 being finite without_zero Subset of NAT st
( P1 = rng nt1 & P2 = rng nt2 & card P1 = card P2 & card P1 = n & Det (EqSegm (M,P1,P2)) <> 0. K ) )
assume that
A1: [:(rng nt1),(rng nt2):] c= Indices M and
A2: Det (Segm (M,nt1,nt2)) <> 0. K ; ::_thesis: ex P1, P2 being finite without_zero Subset of NAT st
( P1 = rng nt1 & P2 = rng nt2 & card P1 = card P2 & card P1 = n & Det (EqSegm (M,P1,P2)) <> 0. K )
( n = 0 iff n = 0 ) ;
then consider P1, P2 being finite without_zero Subset of NAT such that
A3: P1 = rng nt1 and
A4: P2 = rng nt2 by A1, Lm5;
nt2 is one-to-one by A2, Th31;
then A5: card P2 = len nt2 by A4, FINSEQ_4:62;
nt1 is one-to-one by A2, Th27;
then A6: card P1 = len nt1 by A3, FINSEQ_4:62;
then reconsider SP1 = Sgm P1, SP2 = Sgm P2 as Element of n -tuples_on NAT by A5, CARD_1:def_7;
ex m being Nat st P2 c= Seg m by Th43;
then A7: rng SP2 = P2 by FINSEQ_1:def_13;
ex k being Nat st P1 c= Seg k by Th43;
then rng SP1 = P1 by FINSEQ_1:def_13;
then A8: ( Det (Segm (M,nt1,nt2)) = Det (Segm (M,SP1,SP2)) or - (Det (Segm (M,nt1,nt2))) = Det (Segm (M,SP1,SP2)) ) by A3, A4, A7, Th36;
A9: len nt1 = n by CARD_1:def_7;
A10: len nt2 = n by CARD_1:def_7;
Segm (M,(Sgm P1),(Sgm P2)) = Segm (M,P1,P2)
.= EqSegm (M,P1,P2) by A6, A5, A9, A10, Def3 ;
hence ex P1, P2 being finite without_zero Subset of NAT st
( P1 = rng nt1 & P2 = rng nt2 & card P1 = card P2 & card P1 = n & Det (EqSegm (M,P1,P2)) <> 0. K ) by A2, A3, A4, A6, A5, A9, A10, A8, VECTSP_1:28; ::_thesis: verum
end;
theorem Th76: :: MATRIX13:76
for K being Field
for M being Matrix of K
for RANK being Element of NAT holds
( the_rank_of M = RANK iff ( ex rt1, rt2 being Element of RANK -tuples_on NAT st
( [:(rng rt1),(rng rt2):] c= Indices M & Det (Segm (M,rt1,rt2)) <> 0. K ) & ( for n being Nat
for nt1, nt2 being Element of n -tuples_on NAT st [:(rng nt1),(rng nt2):] c= Indices M & Det (Segm (M,nt1,nt2)) <> 0. K holds
n <= RANK ) ) )
proof
let K be Field; ::_thesis: for M being Matrix of K
for RANK being Element of NAT holds
( the_rank_of M = RANK iff ( ex rt1, rt2 being Element of RANK -tuples_on NAT st
( [:(rng rt1),(rng rt2):] c= Indices M & Det (Segm (M,rt1,rt2)) <> 0. K ) & ( for n being Nat
for nt1, nt2 being Element of n -tuples_on NAT st [:(rng nt1),(rng nt2):] c= Indices M & Det (Segm (M,nt1,nt2)) <> 0. K holds
n <= RANK ) ) )
let M be Matrix of K; ::_thesis: for RANK being Element of NAT holds
( the_rank_of M = RANK iff ( ex rt1, rt2 being Element of RANK -tuples_on NAT st
( [:(rng rt1),(rng rt2):] c= Indices M & Det (Segm (M,rt1,rt2)) <> 0. K ) & ( for n being Nat
for nt1, nt2 being Element of n -tuples_on NAT st [:(rng nt1),(rng nt2):] c= Indices M & Det (Segm (M,nt1,nt2)) <> 0. K holds
n <= RANK ) ) )
let RANK be Element of NAT ; ::_thesis: ( the_rank_of M = RANK iff ( ex rt1, rt2 being Element of RANK -tuples_on NAT st
( [:(rng rt1),(rng rt2):] c= Indices M & Det (Segm (M,rt1,rt2)) <> 0. K ) & ( for n being Nat
for nt1, nt2 being Element of n -tuples_on NAT st [:(rng nt1),(rng nt2):] c= Indices M & Det (Segm (M,nt1,nt2)) <> 0. K holds
n <= RANK ) ) )
thus ( the_rank_of M = RANK implies ( ex rt1, rt2 being Element of RANK -tuples_on NAT st
( [:(rng rt1),(rng rt2):] c= Indices M & Det (Segm (M,rt1,rt2)) <> 0. K ) & ( for n being Nat
for nt1, nt2 being Element of n -tuples_on NAT st [:(rng nt1),(rng nt2):] c= Indices M & Det (Segm (M,nt1,nt2)) <> 0. K holds
n <= RANK ) ) ) ::_thesis: ( ex rt1, rt2 being Element of RANK -tuples_on NAT st
( [:(rng rt1),(rng rt2):] c= Indices M & Det (Segm (M,rt1,rt2)) <> 0. K ) & ( for n being Nat
for nt1, nt2 being Element of n -tuples_on NAT st [:(rng nt1),(rng nt2):] c= Indices M & Det (Segm (M,nt1,nt2)) <> 0. K holds
n <= RANK ) implies the_rank_of M = RANK )
proof
assume A1: the_rank_of M = RANK ; ::_thesis: ( ex rt1, rt2 being Element of RANK -tuples_on NAT st
( [:(rng rt1),(rng rt2):] c= Indices M & Det (Segm (M,rt1,rt2)) <> 0. K ) & ( for n being Nat
for nt1, nt2 being Element of n -tuples_on NAT st [:(rng nt1),(rng nt2):] c= Indices M & Det (Segm (M,nt1,nt2)) <> 0. K holds
n <= RANK ) )
then consider P, Q being finite without_zero Subset of NAT such that
A2: [:P,Q:] c= Indices M and
A3: card P = card Q and
A4: card P = RANK and
A5: Det (EqSegm (M,P,Q)) <> 0. K by Def4;
reconsider Sp = Sgm P, Sq = Sgm Q as Element of RANK -tuples_on NAT by A3, A4;
ex k being Nat st P c= Seg k by Th43;
then A6: rng Sp = P by FINSEQ_1:def_13;
ex m being Nat st Q c= Seg m by Th43;
then A7: rng Sq = Q by FINSEQ_1:def_13;
EqSegm (M,P,Q) = Segm (M,P,Q) by A3, Def3
.= Segm (M,(Sgm P),(Sgm Q)) ;
hence ex rt1, rt2 being Element of RANK -tuples_on NAT st
( [:(rng rt1),(rng rt2):] c= Indices M & Det (Segm (M,rt1,rt2)) <> 0. K ) by A2, A3, A4, A5, A6, A7; ::_thesis: for n being Nat
for nt1, nt2 being Element of n -tuples_on NAT st [:(rng nt1),(rng nt2):] c= Indices M & Det (Segm (M,nt1,nt2)) <> 0. K holds
n <= RANK
let n be Nat; ::_thesis: for nt1, nt2 being Element of n -tuples_on NAT st [:(rng nt1),(rng nt2):] c= Indices M & Det (Segm (M,nt1,nt2)) <> 0. K holds
n <= RANK
let nt1, nt2 be Element of n -tuples_on NAT; ::_thesis: ( [:(rng nt1),(rng nt2):] c= Indices M & Det (Segm (M,nt1,nt2)) <> 0. K implies n <= RANK )
assume that
A8: [:(rng nt1),(rng nt2):] c= Indices M and
A9: Det (Segm (M,nt1,nt2)) <> 0. K ; ::_thesis: n <= RANK
ex P1, P2 being finite without_zero Subset of NAT st
( P1 = rng nt1 & P2 = rng nt2 & card P1 = card P2 & card P1 = n & Det (EqSegm (M,P1,P2)) <> 0. K ) by A8, A9, Th75;
hence n <= RANK by A1, A8, Def4; ::_thesis: verum
end;
assume that
A10: ex rt1, rt2 being Element of RANK -tuples_on NAT st
( [:(rng rt1),(rng rt2):] c= Indices M & Det (Segm (M,rt1,rt2)) <> 0. K ) and
A11: for n being Nat
for nt1, nt2 being Element of n -tuples_on NAT st [:(rng nt1),(rng nt2):] c= Indices M & Det (Segm (M,nt1,nt2)) <> 0. K holds
n <= RANK ; ::_thesis: the_rank_of M = RANK
consider rt1, rt2 being Element of RANK -tuples_on NAT such that
A12: [:(rng rt1),(rng rt2):] c= Indices M and
A13: Det (Segm (M,rt1,rt2)) <> 0. K by A10;
consider P, Q being finite without_zero Subset of NAT such that
A14: [:P,Q:] c= Indices M and
A15: card P = card Q and
A16: card P = the_rank_of M and
A17: Det (EqSegm (M,P,Q)) <> 0. K by Def4;
ex P1, P2 being finite without_zero Subset of NAT st
( P1 = rng rt1 & P2 = rng rt2 & card P1 = card P2 & card P1 = RANK & Det (EqSegm (M,P1,P2)) <> 0. K ) by A12, A13, Th75;
then A18: RANK <= card P by A12, A16, Def4;
reconsider SP = Sgm P, SQ = Sgm Q as Element of (card P) -tuples_on NAT by A15;
ex k being Nat st P c= Seg k by Th43;
then A19: rng SP = P by FINSEQ_1:def_13;
ex m being Nat st Q c= Seg m by Th43;
then A20: rng SQ = Q by FINSEQ_1:def_13;
EqSegm (M,P,Q) = Segm (M,P,Q) by A15, Def3
.= Segm (M,(Sgm P),(Sgm Q)) ;
then card P <= RANK by A11, A14, A15, A17, A19, A20;
hence the_rank_of M = RANK by A16, A18, XXREAL_0:1; ::_thesis: verum
end;
theorem Th77: :: MATRIX13:77
for n, m being Nat
for K being Field
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT
for M being Matrix of K st ( n = 0 or m = 0 ) holds
the_rank_of (Segm (M,nt,mt)) = 0
proof
let n, m be Nat; ::_thesis: for K being Field
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT
for M being Matrix of K st ( n = 0 or m = 0 ) holds
the_rank_of (Segm (M,nt,mt)) = 0
let K be Field; ::_thesis: for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT
for M being Matrix of K st ( n = 0 or m = 0 ) holds
the_rank_of (Segm (M,nt,mt)) = 0
let nt be Element of n -tuples_on NAT; ::_thesis: for mt being Element of m -tuples_on NAT
for M being Matrix of K st ( n = 0 or m = 0 ) holds
the_rank_of (Segm (M,nt,mt)) = 0
let mt be Element of m -tuples_on NAT; ::_thesis: for M being Matrix of K st ( n = 0 or m = 0 ) holds
the_rank_of (Segm (M,nt,mt)) = 0
let M be Matrix of K; ::_thesis: ( ( n = 0 or m = 0 ) implies the_rank_of (Segm (M,nt,mt)) = 0 )
set S = Segm (M,nt,mt);
assume ( n = 0 or m = 0 ) ; ::_thesis: the_rank_of (Segm (M,nt,mt)) = 0
then ( len (Segm (M,nt,mt)) = 0 or width (Segm (M,nt,mt)) = 0 ) by Th1, MATRIX_1:def_2;
hence the_rank_of (Segm (M,nt,mt)) = 0 by Th74; ::_thesis: verum
end;
theorem Th78: :: MATRIX13:78
for n, m being Nat
for K being Field
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT
for M being Matrix of K st [:(rng nt),(rng mt):] c= Indices M holds
the_rank_of M >= the_rank_of (Segm (M,nt,mt))
proof
let n, m be Nat; ::_thesis: for K being Field
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT
for M being Matrix of K st [:(rng nt),(rng mt):] c= Indices M holds
the_rank_of M >= the_rank_of (Segm (M,nt,mt))
let K be Field; ::_thesis: for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT
for M being Matrix of K st [:(rng nt),(rng mt):] c= Indices M holds
the_rank_of M >= the_rank_of (Segm (M,nt,mt))
let nt be Element of n -tuples_on NAT; ::_thesis: for mt being Element of m -tuples_on NAT
for M being Matrix of K st [:(rng nt),(rng mt):] c= Indices M holds
the_rank_of M >= the_rank_of (Segm (M,nt,mt))
let mt be Element of m -tuples_on NAT; ::_thesis: for M being Matrix of K st [:(rng nt),(rng mt):] c= Indices M holds
the_rank_of M >= the_rank_of (Segm (M,nt,mt))
let M be Matrix of K; ::_thesis: ( [:(rng nt),(rng mt):] c= Indices M implies the_rank_of M >= the_rank_of (Segm (M,nt,mt)) )
assume A1: [:(rng nt),(rng mt):] c= Indices M ; ::_thesis: the_rank_of M >= the_rank_of (Segm (M,nt,mt))
percases ( n = 0 or m = 0 or ( n > 0 & m > 0 ) ) ;
suppose ( n = 0 or m = 0 ) ; ::_thesis: the_rank_of M >= the_rank_of (Segm (M,nt,mt))
hence the_rank_of M >= the_rank_of (Segm (M,nt,mt)) by Th77; ::_thesis: verum
end;
supposeA2: ( n > 0 & m > 0 ) ; ::_thesis: the_rank_of M >= the_rank_of (Segm (M,nt,mt))
A3: dom mt = Seg m by FINSEQ_2:124;
A4: dom nt = Seg n by FINSEQ_2:124;
set S = Segm (M,nt,mt);
set RS = the_rank_of (Segm (M,nt,mt));
consider rt1, rt2 being Element of (the_rank_of (Segm (M,nt,mt))) -tuples_on NAT such that
A5: [:(rng rt1),(rng rt2):] c= Indices (Segm (M,nt,mt)) and
A6: Det (Segm ((Segm (M,nt,mt)),rt1,rt2)) <> 0. K by Th76;
set mr = mt * rt2;
A7: ex R1, R2 being finite without_zero Subset of NAT st
( R1 = rng rt1 & R2 = rng rt2 & card R1 = card R2 & card R1 = the_rank_of (Segm (M,nt,mt)) & Det (EqSegm ((Segm (M,nt,mt)),R1,R2)) <> 0. K ) by A5, A6, Th75;
set nr = nt * rt1;
A8: rng (mt * rt2) c= rng mt by RELAT_1:26;
len (Segm (M,nt,mt)) = n by A2, Th1;
then A9: rng rt1 c= dom nt by A5, A7, A4, Th67;
then dom (nt * rt1) = dom rt1 by RELAT_1:27;
then A10: dom (nt * rt1) = Seg (the_rank_of (Segm (M,nt,mt))) by FINSEQ_2:124;
width (Segm (M,nt,mt)) = m by A2, Th1;
then A11: rng rt2 c= dom mt by A5, A7, A3, Th67;
then dom (mt * rt2) = dom rt2 by RELAT_1:27;
then A12: dom (mt * rt2) = Seg (the_rank_of (Segm (M,nt,mt))) by FINSEQ_2:124;
rng mt c= NAT by FINSEQ_1:def_4;
then A13: rng (mt * rt2) c= NAT by A8, XBOOLE_1:1;
set SS = Segm ((Segm (M,nt,mt)),rt1,rt2);
A14: rng (nt * rt1) c= rng nt by RELAT_1:26;
rng nt c= NAT by FINSEQ_1:def_4;
then A15: rng (nt * rt1) c= NAT by A14, XBOOLE_1:1;
reconsider nr = nt * rt1, mr = mt * rt2 as FinSequence by A9, A11, FINSEQ_1:16;
reconsider nr = nr, mr = mr as FinSequence of NAT by A15, A13, FINSEQ_1:def_4;
A16: len nr = the_rank_of (Segm (M,nt,mt)) by A10, FINSEQ_1:def_3;
len mr = the_rank_of (Segm (M,nt,mt)) by A12, FINSEQ_1:def_3;
then reconsider nr = nr, mr = mr as Element of (the_rank_of (Segm (M,nt,mt))) -tuples_on NAT by A16, FINSEQ_2:92;
set MR = Segm (M,nr,mr);
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_(Segm_((Segm_(M,nt,mt)),rt1,rt2))_holds_
(Segm_(M,nr,mr))_*_(i,j)_=_(Segm_((Segm_(M,nt,mt)),rt1,rt2))_*_(i,j)
let i, j be Nat; ::_thesis: ( [i,j] in Indices (Segm ((Segm (M,nt,mt)),rt1,rt2)) implies (Segm (M,nr,mr)) * (i,j) = (Segm ((Segm (M,nt,mt)),rt1,rt2)) * (i,j) )
assume A17: [i,j] in Indices (Segm ((Segm (M,nt,mt)),rt1,rt2)) ; ::_thesis: (Segm (M,nr,mr)) * (i,j) = (Segm ((Segm (M,nt,mt)),rt1,rt2)) * (i,j)
reconsider I = i, J = j, rtI = rt1 . i, rtJ = rt2 . j as Element of NAT by ORDINAL1:def_12;
A18: [(rt1 . I),(rt2 . J)] in Indices (Segm (M,nt,mt)) by A5, A17, Th17;
A19: Indices (Segm ((Segm (M,nt,mt)),rt1,rt2)) = [:(dom nr),(dom mr):] by A10, A12, MATRIX_1:24;
then A20: i in dom nr by A17, ZFMISC_1:87;
A21: j in dom mr by A17, A19, ZFMISC_1:87;
[i,j] in Indices (Segm (M,nr,mr)) by A17, MATRIX_1:26;
hence (Segm (M,nr,mr)) * (i,j) = M * ((nr . I),(mr . J)) by Def1
.= M * ((nt . rtI),(mr . J)) by A20, FUNCT_1:12
.= M * ((nt . rtI),(mt . rtJ)) by A21, FUNCT_1:12
.= (Segm (M,nt,mt)) * ((rt1 . I),(rt2 . J)) by A18, Def1
.= (Segm ((Segm (M,nt,mt)),rt1,rt2)) * (i,j) by A17, Def1 ;
::_thesis: verum
end;
then A22: Segm ((Segm (M,nt,mt)),rt1,rt2) = Segm (M,nr,mr) by MATRIX_1:27;
[:(rng nr),(rng mr):] c= [:(rng nt),(rng mt):] by A14, A8, ZFMISC_1:96;
then [:(rng nr),(rng mr):] c= Indices M by A1, XBOOLE_1:1;
hence the_rank_of M >= the_rank_of (Segm (M,nt,mt)) by A6, A22, Th76; ::_thesis: verum
end;
end;
end;
theorem Th79: :: MATRIX13:79
for K being Field
for M being Matrix of K
for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M holds
the_rank_of M >= the_rank_of (Segm (M,P,Q))
proof
let K be Field; ::_thesis: for M being Matrix of K
for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M holds
the_rank_of M >= the_rank_of (Segm (M,P,Q))
let M be Matrix of K; ::_thesis: for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M holds
the_rank_of M >= the_rank_of (Segm (M,P,Q))
let P, Q be finite without_zero Subset of NAT; ::_thesis: ( [:P,Q:] c= Indices M implies the_rank_of M >= the_rank_of (Segm (M,P,Q)) )
ex k being Nat st P c= Seg k by Th43;
then A1: rng (Sgm P) = P by FINSEQ_1:def_13;
ex n being Nat st Q c= Seg n by Th43;
then A2: rng (Sgm Q) = Q by FINSEQ_1:def_13;
assume [:P,Q:] c= Indices M ; ::_thesis: the_rank_of M >= the_rank_of (Segm (M,P,Q))
hence the_rank_of M >= the_rank_of (Segm (M,P,Q)) by A1, A2, Th78; ::_thesis: verum
end;
theorem Th80: :: MATRIX13:80
for K being Field
for M being Matrix of K
for P, P1, Q, Q1 being finite without_zero Subset of NAT st P c= P1 & Q c= Q1 holds
the_rank_of (Segm (M,P,Q)) <= the_rank_of (Segm (M,P1,Q1))
proof
let K be Field; ::_thesis: for M being Matrix of K
for P, P1, Q, Q1 being finite without_zero Subset of NAT st P c= P1 & Q c= Q1 holds
the_rank_of (Segm (M,P,Q)) <= the_rank_of (Segm (M,P1,Q1))
let M be Matrix of K; ::_thesis: for P, P1, Q, Q1 being finite without_zero Subset of NAT st P c= P1 & Q c= Q1 holds
the_rank_of (Segm (M,P,Q)) <= the_rank_of (Segm (M,P1,Q1))
let P, P1, Q, Q1 be finite without_zero Subset of NAT; ::_thesis: ( P c= P1 & Q c= Q1 implies the_rank_of (Segm (M,P,Q)) <= the_rank_of (Segm (M,P1,Q1)) )
assume that
A1: P c= P1 and
A2: Q c= Q1 ; ::_thesis: the_rank_of (Segm (M,P,Q)) <= the_rank_of (Segm (M,P1,Q1))
set S1 = Segm (M,P1,Q1);
set S = Segm (M,P,Q);
consider P2, Q2 being finite without_zero Subset of NAT such that
A3: [:P2,Q2:] c= Indices (Segm (M,P,Q)) and
A4: card P2 = card Q2 and
A5: card P2 = the_rank_of (Segm (M,P,Q)) and
A6: Det (EqSegm ((Segm (M,P,Q)),P2,Q2)) <> 0. K by Def4;
( P2 = {} iff Q2 = {} ) by A4;
then consider P3, Q3 being finite without_zero Subset of NAT such that
A7: P3 c= P and
A8: Q3 c= Q and
P3 = (Sgm P) .: P2 and
Q3 = (Sgm Q) .: Q2 and
A9: card P3 = card P2 and
A10: card Q3 = card Q2 and
A11: Segm ((Segm (M,P,Q)),P2,Q2) = Segm (M,P3,Q3) by A3, Th57;
reconsider P4 = (Sgm P1) " P3, Q4 = (Sgm Q1) " Q3 as finite without_zero Subset of NAT by Th53;
A12: card Q4 = card P2 by A2, A4, A8, A10, Lm2, XBOOLE_1:1;
A13: card P4 = card P2 by A1, A7, A9, Lm2, XBOOLE_1:1;
ex k being Nat st Q4 c= Seg k by Th43;
then A14: rng (Sgm Q4) = Q4 by FINSEQ_1:def_13;
A15: Q3 c= Q1 by A2, A8, XBOOLE_1:1;
A16: P3 c= P1 by A1, A7, XBOOLE_1:1;
ex k being Nat st P4 c= Seg k by Th43;
then rng (Sgm P4) = P4 by FINSEQ_1:def_13;
then A17: [:P4,Q4:] c= Indices (Segm (M,P1,Q1)) by A16, A15, A14, Th56;
Segm ((Segm (M,P1,Q1)),P4,Q4) = Segm (M,P3,Q3) by A16, A15, Th56;
then EqSegm ((Segm (M,P,Q)),P2,Q2) = Segm ((Segm (M,P1,Q1)),P4,Q4) by A4, A11, Def3
.= EqSegm ((Segm (M,P1,Q1)),P4,Q4) by A4, A10, A15, A13, Def3, Lm2 ;
hence the_rank_of (Segm (M,P,Q)) <= the_rank_of (Segm (M,P1,Q1)) by A5, A6, A13, A12, A17, Def4; ::_thesis: verum
end;
theorem Th81: :: MATRIX13:81
for f, g being Function st rng f c= rng g holds
ex h being Function st
( dom h = dom f & rng h c= dom g & f = g * h )
proof
let f, g be Function; ::_thesis: ( rng f c= rng g implies ex h being Function st
( dom h = dom f & rng h c= dom g & f = g * h ) )
assume A1: rng f c= rng g ; ::_thesis: ex h being Function st
( dom h = dom f & rng h c= dom g & f = g * h )
defpred S1[ set , set ] means f . $1 = g . $2;
A2: for x being set st x in dom f holds
ex y being set st
( y in dom g & S1[x,y] ) by A1, GRAPH_5:1;
consider h being Function of (dom f),(dom g) such that
A3: for x being set st x in dom f holds
S1[x,h . x] from FUNCT_2:sch_1(A2);
percases ( dom g = {} or dom g <> {} ) ;
suppose dom g = {} ; ::_thesis: ex h being Function st
( dom h = dom f & rng h c= dom g & f = g * h )
then rng g = {} by RELAT_1:42;
then A4: f = g * {} by A1;
rng {} c= dom g by XBOOLE_1:2;
hence ex h being Function st
( dom h = dom f & rng h c= dom g & f = g * h ) by A4; ::_thesis: verum
end;
supposeA5: dom g <> {} ; ::_thesis: ex h being Function st
( dom h = dom f & rng h c= dom g & f = g * h )
A6: rng h c= dom g by RELAT_1:def_19;
A7: dom h = dom f by A5, FUNCT_2:def_1;
then A8: dom (g * h) = dom f by A6, RELAT_1:27;
now__::_thesis:_for_x_being_set_st_x_in_dom_f_holds_
f_._x_=_(g_*_h)_._x
let x be set ; ::_thesis: ( x in dom f implies f . x = (g * h) . x )
assume A9: x in dom f ; ::_thesis: f . x = (g * h) . x
thus f . x = g . (h . x) by A3, A9
.= (g * h) . x by A8, A9, FUNCT_1:12 ; ::_thesis: verum
end;
hence ex h being Function st
( dom h = dom f & rng h c= dom g & f = g * h ) by A7, A6, A8, FUNCT_1:2; ::_thesis: verum
end;
end;
end;
theorem Th82: :: MATRIX13:82
for n, m being Nat
for K being Field
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT
for M being Matrix of K st [:(rng nt),(rng mt):] = Indices M holds
the_rank_of M = the_rank_of (Segm (M,nt,mt))
proof
let n, m be Nat; ::_thesis: for K being Field
for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT
for M being Matrix of K st [:(rng nt),(rng mt):] = Indices M holds
the_rank_of M = the_rank_of (Segm (M,nt,mt))
let K be Field; ::_thesis: for nt being Element of n -tuples_on NAT
for mt being Element of m -tuples_on NAT
for M being Matrix of K st [:(rng nt),(rng mt):] = Indices M holds
the_rank_of M = the_rank_of (Segm (M,nt,mt))
let nt be Element of n -tuples_on NAT; ::_thesis: for mt being Element of m -tuples_on NAT
for M being Matrix of K st [:(rng nt),(rng mt):] = Indices M holds
the_rank_of M = the_rank_of (Segm (M,nt,mt))
let mt be Element of m -tuples_on NAT; ::_thesis: for M being Matrix of K st [:(rng nt),(rng mt):] = Indices M holds
the_rank_of M = the_rank_of (Segm (M,nt,mt))
let M be Matrix of K; ::_thesis: ( [:(rng nt),(rng mt):] = Indices M implies the_rank_of M = the_rank_of (Segm (M,nt,mt)) )
set RM = the_rank_of M;
set S = Segm (M,nt,mt);
consider rt1, rt2 being Element of (the_rank_of M) -tuples_on NAT such that
A1: [:(rng rt1),(rng rt2):] c= Indices M and
A2: Det (Segm (M,rt1,rt2)) <> 0. K by Th76;
assume A3: [:(rng nt),(rng mt):] = Indices M ; ::_thesis: the_rank_of M = the_rank_of (Segm (M,nt,mt))
A4: now__::_thesis:_the_rank_of_M_<=_the_rank_of_(Segm_(M,nt,mt))
percases ( the_rank_of M = 0 or the_rank_of M > 0 ) ;
suppose the_rank_of M = 0 ; ::_thesis: the_rank_of M <= the_rank_of (Segm (M,nt,mt))
hence the_rank_of M <= the_rank_of (Segm (M,nt,mt)) ; ::_thesis: verum
end;
supposeA5: the_rank_of M > 0 ; ::_thesis: the_rank_of M <= the_rank_of (Segm (M,nt,mt))
then len rt2 > 0 by CARD_1:def_7;
then A6: rt2 <> {} ;
len rt1 > 0 by A5, CARD_1:def_7;
then A7: rt1 <> {} ;
then rng nt <> {} by A3, A1, A6;
then dom nt <> {} by RELAT_1:42;
then A8: n <> 0 ;
then A9: width (Segm (M,nt,mt)) = m by Th1;
A10: dom mt = Seg m by FINSEQ_2:124;
set MR = Segm (M,rt1,rt2);
A11: dom rt2 = Seg (the_rank_of M) by FINSEQ_2:124;
rng rt1 c= rng nt by A3, A1, A7, A6, ZFMISC_1:114;
then consider R1 being Function such that
A12: dom R1 = dom rt1 and
A13: rng R1 c= dom nt and
A14: rt1 = nt * R1 by Th81;
rng rt2 c= rng mt by A3, A1, A7, A6, ZFMISC_1:114;
then consider R2 being Function such that
A15: dom R2 = dom rt2 and
A16: rng R2 c= dom mt and
A17: rt2 = mt * R2 by Th81;
A18: dom rt1 = Seg (the_rank_of M) by FINSEQ_2:124;
then reconsider R1 = R1, R2 = R2 as FinSequence by A12, A15, A11, FINSEQ_1:def_2;
A19: rng R1 c= NAT by A13, XBOOLE_1:1;
rng R2 c= NAT by A16, XBOOLE_1:1;
then reconsider R1 = R1, R2 = R2 as FinSequence of NAT by A19, FINSEQ_1:def_4;
A20: len R1 = the_rank_of M by A12, A18, FINSEQ_1:def_3;
len R2 = the_rank_of M by A15, A11, FINSEQ_1:def_3;
then reconsider R1 = R1, R2 = R2 as Element of (the_rank_of M) -tuples_on NAT by A20, FINSEQ_2:92;
set SR = Segm ((Segm (M,nt,mt)),R1,R2);
len (Segm (M,nt,mt)) = n by Th1, A8;
then A21: Indices (Segm (M,nt,mt)) = [:(Seg n),(Seg m):] by A9, FINSEQ_1:def_3;
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_(Segm_((Segm_(M,nt,mt)),R1,R2))_holds_
(Segm_(M,rt1,rt2))_*_(i,j)_=_(Segm_((Segm_(M,nt,mt)),R1,R2))_*_(i,j)
A22: dom mt = Seg m by FINSEQ_2:124;
let i, j be Nat; ::_thesis: ( [i,j] in Indices (Segm ((Segm (M,nt,mt)),R1,R2)) implies (Segm (M,rt1,rt2)) * (i,j) = (Segm ((Segm (M,nt,mt)),R1,R2)) * (i,j) )
assume A23: [i,j] in Indices (Segm ((Segm (M,nt,mt)),R1,R2)) ; ::_thesis: (Segm (M,rt1,rt2)) * (i,j) = (Segm ((Segm (M,nt,mt)),R1,R2)) * (i,j)
reconsider I = i, J = j, RI = R1 . i, RJ = R2 . j as Element of NAT by ORDINAL1:def_12;
A24: dom nt = Seg n by FINSEQ_2:124;
A25: Indices (Segm ((Segm (M,nt,mt)),R1,R2)) = [:(dom R1),(dom R2):] by A12, A15, A18, A11, MATRIX_1:24;
then A26: i in dom R1 by A23, ZFMISC_1:87;
A27: j in dom R2 by A23, A25, ZFMISC_1:87;
then A28: R2 . j in rng R2 by FUNCT_1:def_3;
R1 . i in rng R1 by A26, FUNCT_1:def_3;
then A29: [(R1 . I),(R2 . J)] in Indices (Segm (M,nt,mt)) by A13, A16, A21, A28, A24, A22, ZFMISC_1:87;
[i,j] in Indices (Segm (M,rt1,rt2)) by A23, MATRIX_1:26;
hence (Segm (M,rt1,rt2)) * (i,j) = M * ((rt1 . I),(rt2 . J)) by Def1
.= M * ((nt . RI),(rt2 . J)) by A12, A14, A26, FUNCT_1:12
.= M * ((nt . RI),(mt . RJ)) by A15, A17, A27, FUNCT_1:12
.= (Segm (M,nt,mt)) * ((R1 . I),(R2 . J)) by A29, Def1
.= (Segm ((Segm (M,nt,mt)),R1,R2)) * (i,j) by A23, Def1 ;
::_thesis: verum
end;
then A30: Det (Segm ((Segm (M,nt,mt)),R1,R2)) <> 0. K by A2, MATRIX_1:27;
dom nt = Seg n by FINSEQ_2:124;
then [:(rng R1),(rng R2):] c= Indices (Segm (M,nt,mt)) by A13, A16, A10, A21, ZFMISC_1:96;
hence the_rank_of M <= the_rank_of (Segm (M,nt,mt)) by A30, Th76; ::_thesis: verum
end;
end;
end;
the_rank_of M >= the_rank_of (Segm (M,nt,mt)) by A3, Th78;
hence the_rank_of M = the_rank_of (Segm (M,nt,mt)) by A4, XXREAL_0:1; ::_thesis: verum
end;
theorem Th83: :: MATRIX13:83
for n being Nat
for K being Field
for M being Matrix of n,K holds
( the_rank_of M = n iff Det M <> 0. K )
proof
let n be Nat; ::_thesis: for K being Field
for M being Matrix of n,K holds
( the_rank_of M = n iff Det M <> 0. K )
let K be Field; ::_thesis: for M being Matrix of n,K holds
( the_rank_of M = n iff Det M <> 0. K )
let M be Matrix of n,K; ::_thesis: ( the_rank_of M = n iff Det M <> 0. K )
A1: [:(Seg n),(Seg n):] c= Indices M by MATRIX_1:24;
A2: len M = n by MATRIX_1:24;
then A3: the_rank_of M <= n by Th74;
A4: width M = n by MATRIX_1:24;
then A5: M = Segm (M,(Seg n),(Seg n)) by A2, Th46
.= EqSegm (M,(Seg n),(Seg n)) by Def3 ;
A6: card (Seg n) = n by FINSEQ_1:57;
thus ( the_rank_of M = n implies Det M <> 0. K ) ::_thesis: ( Det M <> 0. K implies the_rank_of M = n )
proof
assume the_rank_of M = n ; ::_thesis: Det M <> 0. K
then consider P, Q being finite without_zero Subset of NAT such that
A7: [:P,Q:] c= Indices M and
A8: card P = card Q and
A9: card P = n and
A10: Det (EqSegm (M,P,Q)) <> 0. K by Def4;
P c= Seg n by A2, A7, A8, Th67;
then A11: P = Seg n by A6, A9, CARD_FIN:1;
Q c= Seg n by A4, A7, A8, Th67;
then Q = Seg n by A6, A8, A9, CARD_FIN:1;
then M = Segm (M,P,Q) by A2, A4, A11, Th46
.= EqSegm (M,P,Q) by A8, Def3 ;
hence Det M <> 0. K by A9, A10; ::_thesis: verum
end;
assume Det M <> 0. K ; ::_thesis: the_rank_of M = n
then the_rank_of M >= n by A6, A5, A1, Def4;
hence the_rank_of M = n by A3, XXREAL_0:1; ::_thesis: verum
end;
theorem :: MATRIX13:84
for K being Field
for M being Matrix of K holds the_rank_of M = the_rank_of (M @)
proof
let K be Field; ::_thesis: for M being Matrix of K holds the_rank_of M = the_rank_of (M @)
let M be Matrix of K; ::_thesis: the_rank_of M = the_rank_of (M @)
consider P, Q being finite without_zero Subset of NAT such that
A1: [:P,Q:] c= Indices M and
A2: card P = card Q and
A3: card P = the_rank_of M and
A4: Det (EqSegm (M,P,Q)) <> 0. K by Def4;
A5: [:Q,P:] c= Indices (M @) by A1, A2, Th69;
consider P1, Q1 being finite without_zero Subset of NAT such that
A6: [:P1,Q1:] c= Indices (M @) and
A7: card P1 = card Q1 and
A8: card P1 = the_rank_of (M @) and
A9: Det (EqSegm ((M @),P1,Q1)) <> 0. K by Def4;
A10: [:Q1,P1:] c= Indices M by A6, A7, Th69;
then Det (EqSegm (M,Q1,P1)) <> 0. K by A7, A9, Th70;
then A11: the_rank_of M >= the_rank_of (M @) by A7, A8, A10, Def4;
Det (EqSegm ((M @),Q,P)) <> 0. K by A1, A2, A4, Th70;
then the_rank_of (M @) >= the_rank_of M by A2, A3, A5, Def4;
hence the_rank_of M = the_rank_of (M @) by A11, XXREAL_0:1; ::_thesis: verum
end;
Lm6: for n being Nat
for K being Field
for a being Element of K st a <> 0. K holds
(power K) . (a,n) <> 0. K
proof
let n be Nat; ::_thesis: for K being Field
for a being Element of K st a <> 0. K holds
(power K) . (a,n) <> 0. K
let K be Field; ::_thesis: for a being Element of K st a <> 0. K holds
(power K) . (a,n) <> 0. K
let a be Element of K; ::_thesis: ( a <> 0. K implies (power K) . (a,n) <> 0. K )
defpred S1[ Nat] means for n being Nat st n = $1 holds
(power K) . (a,n) <> 0. K;
assume A1: a <> 0. K ; ::_thesis: (power K) . (a,n) <> 0. K
A2: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume S1[k] ; ::_thesis: S1[k + 1]
then A3: (power K) . (a,k) <> 0. K ;
A4: k in NAT by ORDINAL1:def_12;
let n be Nat; ::_thesis: ( n = k + 1 implies (power K) . (a,n) <> 0. K )
assume n = k + 1 ; ::_thesis: (power K) . (a,n) <> 0. K
then (power K) . (a,n) = ((power K) . (a,k)) * a by A4, GROUP_1:def_7;
hence (power K) . (a,n) <> 0. K by A1, A3, VECTSP_1:12; ::_thesis: verum
end;
A5: S1[ 0 ]
proof
A6: 1_ K <> 0. K ;
let n be Nat; ::_thesis: ( n = 0 implies (power K) . (a,n) <> 0. K )
assume n = 0 ; ::_thesis: (power K) . (a,n) <> 0. K
hence (power K) . (a,n) <> 0. K by A6, GROUP_1:def_7; ::_thesis: verum
end;
for k being Nat holds S1[k] from NAT_1:sch_2(A5, A2);
hence (power K) . (a,n) <> 0. K ; ::_thesis: verum
end;
theorem :: MATRIX13:85
for n, m being Nat
for K being Field
for M being Matrix of n,m,K
for F being Permutation of (Seg n) holds the_rank_of M = the_rank_of (M * F)
proof
let n, m be Nat; ::_thesis: for K being Field
for M being Matrix of n,m,K
for F being Permutation of (Seg n) holds the_rank_of M = the_rank_of (M * F)
let K be Field; ::_thesis: for M being Matrix of n,m,K
for F being Permutation of (Seg n) holds the_rank_of M = the_rank_of (M * F)
let M be Matrix of n,m,K; ::_thesis: for F being Permutation of (Seg n) holds the_rank_of M = the_rank_of (M * F)
let F be Permutation of (Seg n); ::_thesis: the_rank_of M = the_rank_of (M * F)
set P = Seg (len M);
set Q = Seg (width M);
set SP = Sgm (Seg (len M));
set SQ = Sgm (Seg (width M));
A1: card (Seg (len M)) = len M by FINSEQ_1:57;
A2: len M = n by MATRIX_1:def_2;
then reconsider F9 = F as Permutation of (Seg (card (Seg (len M)))) by A1;
A3: rng F = Seg n by FUNCT_2:def_3;
A4: dom F = Seg n by FUNCT_2:52;
A5: dom (Sgm (Seg (len M))) = Seg (card (Seg (len M))) by FINSEQ_3:40;
then A6: dom ((Sgm (Seg (len M))) * F) = dom F by A2, A1, A3, RELAT_1:27;
then reconsider SPF = (Sgm (Seg (len M))) * F as FinSequence by A4, FINSEQ_1:def_2;
A7: rng ((Sgm (Seg (len M))) * F) = rng (Sgm (Seg (len M))) by A2, A1, A5, A3, RELAT_1:28;
then reconsider SPF = SPF as FinSequence of NAT by FINSEQ_1:def_4;
len SPF = card (Seg (len M)) by A2, A1, A6, A4, FINSEQ_1:def_3;
then reconsider SPF = SPF as Element of (card (Seg (len M))) -tuples_on NAT by FINSEQ_2:92;
A8: Indices M = [:(Seg (len M)),(Seg (width M)):] by FINSEQ_1:def_3;
A9: rng (Sgm (Seg (width M))) = Seg (width M) by FINSEQ_1:def_13;
A10: rng (Sgm (Seg (len M))) = Seg (len M) by FINSEQ_1:def_13;
Segm (M,SPF,(Sgm (Seg (width M)))) = (Segm (M,(Seg (len M)),(Seg (width M)))) * F9 by Th33
.= M * F by Th46 ;
hence the_rank_of M = the_rank_of (M * F) by A7, A8, A10, A9, Th82; ::_thesis: verum
end;
theorem :: MATRIX13:86
for K being Field
for a being Element of K
for M being Matrix of K st a <> 0. K holds
the_rank_of M = the_rank_of (a * M)
proof
let K be Field; ::_thesis: for a being Element of K
for M being Matrix of K st a <> 0. K holds
the_rank_of M = the_rank_of (a * M)
let a be Element of K; ::_thesis: for M being Matrix of K st a <> 0. K holds
the_rank_of M = the_rank_of (a * M)
let M be Matrix of K; ::_thesis: ( a <> 0. K implies the_rank_of M = the_rank_of (a * M) )
consider P, Q being finite without_zero Subset of NAT such that
A1: [:P,Q:] c= Indices M and
A2: card P = card Q and
A3: card P = the_rank_of M and
A4: Det (EqSegm (M,P,Q)) <> 0. K by Def4;
consider P1, Q1 being finite without_zero Subset of NAT such that
A5: [:P1,Q1:] c= Indices (a * M) and
A6: card P1 = card Q1 and
A7: card P1 = the_rank_of (a * M) and
A8: Det (EqSegm ((a * M),P1,Q1)) <> 0. K by Def4;
A9: Indices M = Indices (a * M) by MATRIXR1:18;
then Det (EqSegm ((a * M),P1,Q1)) = ((power K) . (a,(card P1))) * (Det (EqSegm (M,P1,Q1))) by A5, A6, Th72;
then Det (EqSegm (M,P1,Q1)) <> 0. K by A8, VECTSP_1:12;
then A10: the_rank_of M >= the_rank_of (a * M) by A9, A5, A6, A7, Def4;
assume a <> 0. K ; ::_thesis: the_rank_of M = the_rank_of (a * M)
then A11: (power K) . (a,(card P)) <> 0. K by Lm6;
Det (EqSegm ((a * M),P,Q)) = ((power K) . (a,(card P))) * (Det (EqSegm (M,P,Q))) by A1, A2, Th72;
then Det (EqSegm ((a * M),P,Q)) <> 0. K by A4, A11, VECTSP_1:12;
then the_rank_of (a * M) >= the_rank_of M by A9, A1, A2, A3, Def4;
hence the_rank_of M = the_rank_of (a * M) by A10, XXREAL_0:1; ::_thesis: verum
end;
theorem Th87: :: MATRIX13:87
for K being Field
for a being Element of K
for p, pf being FinSequence of K
for f being Function st pf = p * f & rng f c= dom p holds
(a * p) * f = a * pf
proof
let K be Field; ::_thesis: for a being Element of K
for p, pf being FinSequence of K
for f being Function st pf = p * f & rng f c= dom p holds
(a * p) * f = a * pf
let a be Element of K; ::_thesis: for p, pf being FinSequence of K
for f being Function st pf = p * f & rng f c= dom p holds
(a * p) * f = a * pf
let p, pf be FinSequence of K; ::_thesis: for f being Function st pf = p * f & rng f c= dom p holds
(a * p) * f = a * pf
let f be Function; ::_thesis: ( pf = p * f & rng f c= dom p implies (a * p) * f = a * pf )
assume that
A1: pf = p * f and
A2: rng f c= dom p ; ::_thesis: (a * p) * f = a * pf
len (a * p) = len p by MATRIXR1:16;
then A3: dom (a * p) = Seg (len p) by FINSEQ_1:def_3;
A4: Seg (len p) = dom p by FINSEQ_1:def_3;
then A5: dom ((a * p) * f) = dom f by A2, A3, RELAT_1:27;
len (a * pf) = len pf by MATRIXR1:16;
then A6: dom (a * pf) = Seg (len pf) by FINSEQ_1:def_3;
A7: Seg (len pf) = dom pf by FINSEQ_1:def_3;
then A8: dom (a * pf) = dom f by A1, A2, A6, RELAT_1:27;
now__::_thesis:_for_x_being_set_st_x_in_dom_(a_*_pf)_holds_
(a_*_pf)_._x_=_((a_*_p)_*_f)_._x
set KK = the carrier of K;
A9: rng pf c= the carrier of K by FINSEQ_1:def_4;
let x be set ; ::_thesis: ( x in dom (a * pf) implies (a * pf) . x = ((a * p) * f) . x )
assume A10: x in dom (a * pf) ; ::_thesis: (a * pf) . x = ((a * p) * f) . x
A11: f . x in rng f by A8, A10, FUNCT_1:def_3;
then A12: f . x in dom p by A2;
pf . x in rng pf by A6, A7, A10, FUNCT_1:def_3;
then reconsider pf9x = pf . x as Element of K by A9;
A13: p . (f . x) in rng p by A2, A11, FUNCT_1:def_3;
thus (a * pf) . x = a * pf9x by A10, FVSUM_1:50
.= (a * p) . (f . x) by A1, A3, A4, A6, A7, A10, A12, A13, FUNCT_1:12, FVSUM_1:50
.= ((a * p) * f) . x by A5, A8, A10, FUNCT_1:12 ; ::_thesis: verum
end;
hence (a * p) * f = a * pf by A1, A2, A6, A7, A5, FUNCT_1:2, RELAT_1:27; ::_thesis: verum
end;
theorem Th88: :: MATRIX13:88
for K being Field
for p, pf, q, qf being FinSequence of K
for f being Function st pf = p * f & rng f c= dom p & qf = q * f & rng f c= dom q holds
(p + q) * f = pf + qf
proof
let K be Field; ::_thesis: for p, pf, q, qf being FinSequence of K
for f being Function st pf = p * f & rng f c= dom p & qf = q * f & rng f c= dom q holds
(p + q) * f = pf + qf
let p, pf, q, qf be FinSequence of K; ::_thesis: for f being Function st pf = p * f & rng f c= dom p & qf = q * f & rng f c= dom q holds
(p + q) * f = pf + qf
let f be Function; ::_thesis: ( pf = p * f & rng f c= dom p & qf = q * f & rng f c= dom q implies (p + q) * f = pf + qf )
assume that
A1: pf = p * f and
A2: rng f c= dom p and
A3: qf = q * f and
A4: rng f c= dom q ; ::_thesis: (p + q) * f = pf + qf
A5: dom pf = dom f by A1, A2, RELAT_1:27;
set KK = the carrier of K;
A6: dom pf = Seg (len pf) by FINSEQ_1:def_3;
A7: dom qf = dom f by A3, A4, RELAT_1:27;
then len qf = len pf by A5, A6, FINSEQ_1:def_3;
then reconsider pf9 = pf, qf9 = qf as Element of (len pf) -tuples_on the carrier of K by FINSEQ_2:92;
A8: dom (pf9 + qf9) = dom f by A5, A6, FINSEQ_2:124;
set pq = p + q;
A9: rng q c= the carrier of K by FINSEQ_1:def_4;
rng p c= the carrier of K by FINSEQ_1:def_4;
then [:(rng p),(rng q):] c= [: the carrier of K, the carrier of K:] by A9, ZFMISC_1:96;
then [:(rng p),(rng q):] c= dom the addF of K by FUNCT_2:def_1;
then A10: dom (p + q) = (dom p) /\ (dom q) by FUNCOP_1:69;
then A11: rng f c= dom (p + q) by A2, A4, XBOOLE_1:19;
A12: now__::_thesis:_for_x_being_set_st_x_in_dom_f_holds_
((p_+_q)_*_f)_._x_=_(pf9_+_qf9)_._x
A13: rng qf c= the carrier of K by FINSEQ_1:def_4;
A14: rng pf c= the carrier of K by FINSEQ_1:def_4;
let x be set ; ::_thesis: ( x in dom f implies ((p + q) * f) . x = (pf9 + qf9) . x )
assume A15: x in dom f ; ::_thesis: ((p + q) * f) . x = (pf9 + qf9) . x
A16: f . x in rng f by A15, FUNCT_1:def_3;
dom p = Seg (len p) by FINSEQ_1:def_3;
then f . x in Seg (len p) by A2, A16;
then reconsider n = x, fx = f . x as Element of NAT by A5, A15;
A17: qf . x in rng qf by A7, A15, FUNCT_1:def_3;
pf . x in rng pf by A5, A15, FUNCT_1:def_3;
then reconsider pfn = pf . n, qfn = qf . n as Element of K by A14, A17, A13;
A18: pfn = p . fx by A1, A15, FUNCT_1:13;
A19: qfn = q . fx by A3, A15, FUNCT_1:13;
thus ((p + q) * f) . x = (p + q) . fx by A15, FUNCT_1:13
.= pfn + qfn by A11, A16, A18, A19, FVSUM_1:17
.= (pf9 + qf9) . x by A5, A6, A15, FVSUM_1:18 ; ::_thesis: verum
end;
dom ((p + q) * f) = dom f by A2, A4, A10, RELAT_1:27, XBOOLE_1:19;
hence (p + q) * f = pf + qf by A8, A12, FUNCT_1:2; ::_thesis: verum
end;
theorem Th89: :: MATRIX13:89
for n9, m9, i being Nat
for K being Field
for a being Element of K
for M9 being Matrix of n9,m9,K st a <> 0. K holds
the_rank_of M9 = the_rank_of (RLine (M9,i,(a * (Line (M9,i)))))
proof
let n9, m9, i be Nat; ::_thesis: for K being Field
for a being Element of K
for M9 being Matrix of n9,m9,K st a <> 0. K holds
the_rank_of M9 = the_rank_of (RLine (M9,i,(a * (Line (M9,i)))))
let K be Field; ::_thesis: for a being Element of K
for M9 being Matrix of n9,m9,K st a <> 0. K holds
the_rank_of M9 = the_rank_of (RLine (M9,i,(a * (Line (M9,i)))))
let a be Element of K; ::_thesis: for M9 being Matrix of n9,m9,K st a <> 0. K holds
the_rank_of M9 = the_rank_of (RLine (M9,i,(a * (Line (M9,i)))))
let M9 be Matrix of n9,m9,K; ::_thesis: ( a <> 0. K implies the_rank_of M9 = the_rank_of (RLine (M9,i,(a * (Line (M9,i))))) )
set L = Line (M9,i);
set aL = a * (Line (M9,i));
set R = RLine (M9,i,(a * (Line (M9,i))));
A1: Indices M9 = Indices (RLine (M9,i,(a * (Line (M9,i))))) by MATRIX_1:26;
consider P, Q being finite without_zero Subset of NAT such that
A2: [:P,Q:] c= Indices M9 and
A3: card P = card Q and
A4: card P = the_rank_of M9 and
A5: Det (EqSegm (M9,P,Q)) <> 0. K by Def4;
assume A6: a <> 0. K ; ::_thesis: the_rank_of M9 = the_rank_of (RLine (M9,i,(a * (Line (M9,i)))))
A7: now__::_thesis:_the_rank_of_(RLine_(M9,i,(a_*_(Line_(M9,i)))))_>=_the_rank_of_M9
percases ( i in P or not i in P ) ;
supposeA8: i in P ; ::_thesis: the_rank_of (RLine (M9,i,(a * (Line (M9,i))))) >= the_rank_of M9
A9: len (Line (M9,i)) = width M9 by MATRIX_1:def_7;
then A10: dom (Line (M9,i)) = Seg (width M9) by FINSEQ_1:def_3;
ex n being Nat st Q c= Seg n by Th43;
then A11: rng (Sgm Q) = Q by FINSEQ_1:def_13;
A12: ex k being Nat st P c= Seg k by Th43;
then A13: dom (Sgm P) = Seg (card P) by FINSEQ_3:40;
rng (Sgm P) = P by A12, FINSEQ_1:def_13;
then consider x being set such that
A14: x in dom (Sgm P) and
A15: (Sgm P) . x = i by A8, FUNCT_1:def_3;
reconsider x = x as Element of NAT by A14;
A16: Q c= Seg (width M9) by A2, A3, Th67;
then Line ((Segm (M9,P,Q)),x) = (Line (M9,i)) * (Sgm Q) by A14, A15, A13, Th47;
then A17: a * (Line ((Segm (M9,P,Q)),x)) = (a * (Line (M9,i))) * (Sgm Q) by A11, A16, A10, Th87;
A18: len (a * (Line (M9,i))) = len (Line (M9,i)) by MATRIXR1:16;
RLine ((EqSegm (M9,P,Q)),x,(a * (Line ((EqSegm (M9,P,Q)),x)))) = RLine ((Segm (M9,P,Q)),x,(a * (Line ((EqSegm (M9,P,Q)),x)))) by A3, Def3
.= RLine ((Segm (M9,P,Q)),x,(a * (Line ((Segm (M9,P,Q)),x)))) by A3, Def3
.= Segm ((RLine (M9,i,(a * (Line (M9,i))))),P,Q) by A2, A15, A9, A17, A18, Th59
.= EqSegm ((RLine (M9,i,(a * (Line (M9,i))))),P,Q) by A3, Def3 ;
then Det (EqSegm ((RLine (M9,i,(a * (Line (M9,i))))),P,Q)) = a * (Det (EqSegm (M9,P,Q))) by A14, A13, MATRIX11:35;
then Det (EqSegm ((RLine (M9,i,(a * (Line (M9,i))))),P,Q)) <> 0. K by A6, A5, VECTSP_1:12;
hence the_rank_of (RLine (M9,i,(a * (Line (M9,i))))) >= the_rank_of M9 by A2, A3, A4, A1, Def4; ::_thesis: verum
end;
supposeA19: not i in P ; ::_thesis: the_rank_of (RLine (M9,i,(a * (Line (M9,i))))) >= the_rank_of M9
EqSegm (M9,P,Q) = Segm (M9,P,Q) by A3, Def3
.= Segm ((RLine (M9,i,(a * (Line (M9,i))))),P,Q) by A2, A19, Th60
.= EqSegm ((RLine (M9,i,(a * (Line (M9,i))))),P,Q) by A3, Def3 ;
hence the_rank_of (RLine (M9,i,(a * (Line (M9,i))))) >= the_rank_of M9 by A2, A3, A4, A5, A1, Def4; ::_thesis: verum
end;
end;
end;
consider P1, Q1 being finite without_zero Subset of NAT such that
A20: [:P1,Q1:] c= Indices (RLine (M9,i,(a * (Line (M9,i))))) and
A21: card P1 = card Q1 and
A22: card P1 = the_rank_of (RLine (M9,i,(a * (Line (M9,i))))) and
A23: Det (EqSegm ((RLine (M9,i,(a * (Line (M9,i))))),P1,Q1)) <> 0. K by Def4;
now__::_thesis:_the_rank_of_M9_>=_the_rank_of_(RLine_(M9,i,(a_*_(Line_(M9,i)))))
percases ( i in P1 or not i in P1 ) ;
supposeA24: i in P1 ; ::_thesis: the_rank_of M9 >= the_rank_of (RLine (M9,i,(a * (Line (M9,i)))))
A25: len (Line (M9,i)) = width M9 by MATRIX_1:def_7;
then A26: dom (Line (M9,i)) = Seg (width M9) by FINSEQ_1:def_3;
ex n being Nat st Q1 c= Seg n by Th43;
then A27: rng (Sgm Q1) = Q1 by FINSEQ_1:def_13;
A28: ex k being Nat st P1 c= Seg k by Th43;
then A29: dom (Sgm P1) = Seg (card P1) by FINSEQ_3:40;
rng (Sgm P1) = P1 by A28, FINSEQ_1:def_13;
then consider x being set such that
A30: x in dom (Sgm P1) and
A31: (Sgm P1) . x = i by A24, FUNCT_1:def_3;
reconsider x = x as Element of NAT by A30;
A32: Q1 c= Seg (width M9) by A1, A20, A21, Th67;
then Line ((Segm (M9,P1,Q1)),x) = (Line (M9,i)) * (Sgm Q1) by A30, A31, A29, Th47;
then A33: a * (Line ((Segm (M9,P1,Q1)),x)) = (a * (Line (M9,i))) * (Sgm Q1) by A27, A32, A26, Th87;
A34: len (a * (Line (M9,i))) = len (Line (M9,i)) by MATRIXR1:16;
RLine ((EqSegm (M9,P1,Q1)),x,(a * (Line ((EqSegm (M9,P1,Q1)),x)))) = RLine ((Segm (M9,P1,Q1)),x,(a * (Line ((EqSegm (M9,P1,Q1)),x)))) by A21, Def3
.= RLine ((Segm (M9,P1,Q1)),x,(a * (Line ((Segm (M9,P1,Q1)),x)))) by A21, Def3
.= Segm ((RLine (M9,i,(a * (Line (M9,i))))),P1,Q1) by A1, A20, A31, A25, A33, A34, Th59
.= EqSegm ((RLine (M9,i,(a * (Line (M9,i))))),P1,Q1) by A21, Def3 ;
then Det (EqSegm ((RLine (M9,i,(a * (Line (M9,i))))),P1,Q1)) = a * (Det (EqSegm (M9,P1,Q1))) by A30, A29, MATRIX11:35;
then Det (EqSegm (M9,P1,Q1)) <> 0. K by A23, VECTSP_1:12;
hence the_rank_of M9 >= the_rank_of (RLine (M9,i,(a * (Line (M9,i))))) by A1, A20, A21, A22, Def4; ::_thesis: verum
end;
supposeA35: not i in P1 ; ::_thesis: the_rank_of M9 >= the_rank_of (RLine (M9,i,(a * (Line (M9,i)))))
EqSegm (M9,P1,Q1) = Segm (M9,P1,Q1) by A21, Def3
.= Segm ((RLine (M9,i,(a * (Line (M9,i))))),P1,Q1) by A1, A20, A35, Th60
.= EqSegm ((RLine (M9,i,(a * (Line (M9,i))))),P1,Q1) by A21, Def3 ;
hence the_rank_of M9 >= the_rank_of (RLine (M9,i,(a * (Line (M9,i))))) by A1, A20, A21, A22, A23, Def4; ::_thesis: verum
end;
end;
end;
hence the_rank_of M9 = the_rank_of (RLine (M9,i,(a * (Line (M9,i))))) by A7, XXREAL_0:1; ::_thesis: verum
end;
theorem Th90: :: MATRIX13:90
for i being Nat
for K being Field
for M being Matrix of K st Line (M,i) = (width M) |-> (0. K) holds
the_rank_of (DelLine (M,i)) = the_rank_of M
proof
let i be Nat; ::_thesis: for K being Field
for M being Matrix of K st Line (M,i) = (width M) |-> (0. K) holds
the_rank_of (DelLine (M,i)) = the_rank_of M
let K be Field; ::_thesis: for M being Matrix of K st Line (M,i) = (width M) |-> (0. K) holds
the_rank_of (DelLine (M,i)) = the_rank_of M
let M be Matrix of K; ::_thesis: ( Line (M,i) = (width M) |-> (0. K) implies the_rank_of (DelLine (M,i)) = the_rank_of M )
set D = DelLine (M,i);
A1: Indices M = [:(Seg (len M)),(Seg (width M)):] by FINSEQ_1:def_3;
A2: Segm (M,((Seg (len M)) \ {i}),(Seg (width M))) = DelLine (M,i) by Th51;
consider P, Q being finite without_zero Subset of NAT such that
A3: [:P,Q:] c= Indices (DelLine (M,i)) and
A4: card P = card Q and
A5: card P = the_rank_of (DelLine (M,i)) and
A6: Det (EqSegm ((DelLine (M,i)),P,Q)) <> 0. K by Def4;
EqSegm ((DelLine (M,i)),P,Q) = Segm ((DelLine (M,i)),P,Q) by A4, Def3;
then A7: the_rank_of (Segm ((DelLine (M,i)),P,Q)) = card P by A6, Th83;
( P = {} iff Q = {} ) by A4;
then consider P2, Q2 being finite without_zero Subset of NAT such that
A8: P2 c= (Seg (len M)) \ {i} and
A9: Q2 c= Seg (width M) and
P2 = (Sgm ((Seg (len M)) \ {i})) .: P and
Q2 = (Sgm (Seg (width M))) .: Q and
card P2 = card P and
card Q2 = card Q and
A10: Segm ((DelLine (M,i)),P,Q) = Segm (M,P2,Q2) by A3, A2, Th57;
(Seg (len M)) \ {i} c= Seg (len M) by XBOOLE_1:36;
then P2 c= Seg (len M) by A8, XBOOLE_1:1;
then [:P2,Q2:] c= Indices M by A9, A1, ZFMISC_1:96;
then A11: the_rank_of (DelLine (M,i)) <= the_rank_of M by A5, A10, A7, Th79;
consider p, q being finite without_zero Subset of NAT such that
A12: [:p,q:] c= Indices M and
A13: card p = card q and
A14: card p = the_rank_of M and
A15: Det (EqSegm (M,p,q)) <> 0. K by Def4;
EqSegm (M,p,q) = Segm (M,p,q) by A13, Def3;
then A16: the_rank_of (Segm (M,p,q)) = card p by A15, Th83;
assume A17: Line (M,i) = (width M) |-> (0. K) ; ::_thesis: the_rank_of (DelLine (M,i)) = the_rank_of M
not i in p
proof
assume A18: i in p ; ::_thesis: contradiction
then reconsider i0 = i as non zero Element of NAT ;
{i} c= p by A18, ZFMISC_1:31;
then consider q1 being finite without_zero Subset of NAT such that
A19: q1 c= q and
A20: card {i} = card q1 and
A21: Det (EqSegm (M,{i0},q1)) <> 0. K by A13, A15, Th65;
consider y being set such that
A22: {y} = q1 by A20, CARD_2:42;
A23: card {i} = 1 by CARD_1:30;
A24: q c= Seg (width M) by A12, A13, Th67;
y in {y} by TARSKI:def_1;
then reconsider y = y as non zero Element of NAT by A22;
y in q1 by A22, TARSKI:def_1;
then A25: y in q by A19;
then A26: M * (i0,y) = (Line (M,i)) . y by A24, MATRIX_1:def_7;
A27: (Line (M,i)) . y = 0. K by A17, A25, A24, FINSEQ_2:57;
EqSegm (M,{i0},q1) = Segm (M,{i0},{y}) by A20, A22, Def3
.= <*<*(0. K)*>*> by A26, A27, Th44 ;
hence contradiction by A21, A23, MATRIX_3:34; ::_thesis: verum
end;
then A28: p \ {i} = p by ZFMISC_1:57;
p c= Seg (len M) by A12, A13, Th67;
then A29: p c= (Seg (len M)) \ {i} by A28, XBOOLE_1:33;
q c= Seg (width M) by A12, A13, Th67;
then the_rank_of (Segm (M,p,q)) <= the_rank_of (DelLine (M,i)) by A2, A29, Th80;
hence the_rank_of (DelLine (M,i)) = the_rank_of M by A11, A14, A16, XXREAL_0:1; ::_thesis: verum
end;
theorem Th91: :: MATRIX13:91
for n9, m9, i being Nat
for K being Field
for M9 being Matrix of n9,m9,K
for p being FinSequence of K st len p = width M9 holds
the_rank_of (DelLine (M9,i)) = the_rank_of (RLine (M9,i,((0. K) * p)))
proof
let n9, m9, i be Nat; ::_thesis: for K being Field
for M9 being Matrix of n9,m9,K
for p being FinSequence of K st len p = width M9 holds
the_rank_of (DelLine (M9,i)) = the_rank_of (RLine (M9,i,((0. K) * p)))
let K be Field; ::_thesis: for M9 being Matrix of n9,m9,K
for p being FinSequence of K st len p = width M9 holds
the_rank_of (DelLine (M9,i)) = the_rank_of (RLine (M9,i,((0. K) * p)))
let M9 be Matrix of n9,m9,K; ::_thesis: for p being FinSequence of K st len p = width M9 holds
the_rank_of (DelLine (M9,i)) = the_rank_of (RLine (M9,i,((0. K) * p)))
let p be FinSequence of K; ::_thesis: ( len p = width M9 implies the_rank_of (DelLine (M9,i)) = the_rank_of (RLine (M9,i,((0. K) * p))) )
assume A1: len p = width M9 ; ::_thesis: the_rank_of (DelLine (M9,i)) = the_rank_of (RLine (M9,i,((0. K) * p)))
set R = RLine (M9,i,((0. K) * p));
A2: Seg (len M9) = dom M9 by FINSEQ_1:def_3;
A3: len M9 = n9 by MATRIX_1:def_2;
percases ( not i in dom M9 or i in dom M9 ) ;
supposeA4: not i in dom M9 ; ::_thesis: the_rank_of (DelLine (M9,i)) = the_rank_of (RLine (M9,i,((0. K) * p)))
then RLine (M9,i,((0. K) * p)) = M9 by A2, Th40;
hence the_rank_of (DelLine (M9,i)) = the_rank_of (RLine (M9,i,((0. K) * p))) by A4, FINSEQ_3:104; ::_thesis: verum
end;
supposeA5: i in dom M9 ; ::_thesis: the_rank_of (DelLine (M9,i)) = the_rank_of (RLine (M9,i,((0. K) * p)))
then A6: n9 <> 0 by A2, A3;
set KK = the carrier of K;
A7: p is Element of (len p) -tuples_on the carrier of K by FINSEQ_2:92;
A8: len ((0. K) * p) = len p by MATRIXR1:16;
then Line ((RLine (M9,i,((0. K) * p))),i) = (0. K) * p by A1, A2, A3, A5, MATRIX11:28;
then A9: Line ((RLine (M9,i,((0. K) * p))),i) = (len p) |-> (0. K) by A7, FVSUM_1:58;
reconsider 0p = (0. K) * p as Element of the carrier of K * by FINSEQ_1:def_11;
A10: i in NAT by ORDINAL1:def_12;
RLine (M9,i,((0. K) * p)) = Replace (M9,i,0p) by A1, A8, MATRIX11:29;
then A11: Replace ((RLine (M9,i,((0. K) * p))),i,0p) = Replace (M9,i,0p) by FUNCT_7:34;
A12: width (RLine (M9,i,((0. K) * p))) = m9 by Th1, A6;
width M9 = m9 by Th1, A6;
then the_rank_of (RLine (M9,i,((0. K) * p))) = the_rank_of (DelLine ((RLine (M9,i,((0. K) * p))),i)) by A1, A12, A9, Th90;
hence the_rank_of (DelLine (M9,i)) = the_rank_of (RLine (M9,i,((0. K) * p))) by A11, A10, COMPUT_1:4; ::_thesis: verum
end;
end;
end;
theorem Th92: :: MATRIX13:92
for n9, m9, j, i being Nat
for K being Field
for a being Element of K
for M9 being Matrix of n9,m9,K st j in Seg (len M9) & ( i = j implies a <> - (1_ K) ) holds
the_rank_of M9 = the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j))))))
proof
let n9, m9, j, i be Nat; ::_thesis: for K being Field
for a being Element of K
for M9 being Matrix of n9,m9,K st j in Seg (len M9) & ( i = j implies a <> - (1_ K) ) holds
the_rank_of M9 = the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j))))))
let K be Field; ::_thesis: for a being Element of K
for M9 being Matrix of n9,m9,K st j in Seg (len M9) & ( i = j implies a <> - (1_ K) ) holds
the_rank_of M9 = the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j))))))
let a be Element of K; ::_thesis: for M9 being Matrix of n9,m9,K st j in Seg (len M9) & ( i = j implies a <> - (1_ K) ) holds
the_rank_of M9 = the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j))))))
let M9 be Matrix of n9,m9,K; ::_thesis: ( j in Seg (len M9) & ( i = j implies a <> - (1_ K) ) implies the_rank_of M9 = the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) )
assume that
A1: j in Seg (len M9) and
A2: ( i = j implies a <> - (1_ K) ) ; ::_thesis: the_rank_of M9 = the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j))))))
percases ( not i in Seg (len M9) or i in Seg (len M9) ) ;
suppose not i in Seg (len M9) ; ::_thesis: the_rank_of M9 = the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j))))))
hence the_rank_of M9 = the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) by Th40; ::_thesis: verum
end;
supposeA3: i in Seg (len M9) ; ::_thesis: the_rank_of M9 = the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j))))))
set KK = the carrier of K;
set W = width M9;
set Lj = Line (M9,j);
set Li = Line (M9,i);
set R = RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))));
reconsider Li9 = Line (M9,i), Lj9 = Line (M9,j), LL = (Line (M9,i)) + (a * (Line (M9,j))) as Element of the carrier of K * by FINSEQ_1:def_11;
A4: len (Line (M9,i)) = width M9 by CARD_1:def_7;
then A5: dom (Line (M9,i)) = Seg (width M9) by FINSEQ_1:def_3;
A6: len ((Line (M9,i)) + (a * (Line (M9,j)))) = width M9 by CARD_1:def_7;
then A7: width M9 = width (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) by MATRIX11:def_3;
then A8: RLine ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),i,(Line (M9,i))) = Replace ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),i,Li9) by A4, MATRIX11:29
.= Replace ((Replace (M9,i,LL)),i,Li9) by A6, MATRIX11:29
.= Replace (M9,i,Li9) by FUNCT_7:34
.= RLine (M9,i,(Line (M9,i))) by A4, MATRIX11:29
.= M9 by MATRIX11:30 ;
A9: len M9 = n9 by MATRIX_1:def_2;
then A10: Line ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),i) = (Line (M9,i)) + (a * (Line (M9,j))) by A3, A6, MATRIX11:28;
A11: len (Line (M9,j)) = width M9 by CARD_1:def_7;
then A12: dom (Line (M9,j)) = Seg (width M9) by FINSEQ_1:def_3;
len (a * (Line (M9,j))) = width M9 by CARD_1:def_7;
then A13: dom (a * (Line (M9,j))) = Seg (width M9) by FINSEQ_1:def_3;
A14: Indices (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) = Indices M9 by MATRIX_1:26;
consider P1, Q1 being finite without_zero Subset of NAT such that
A15: [:P1,Q1:] c= Indices M9 and
A16: card P1 = card Q1 and
A17: card P1 = the_rank_of M9 and
A18: Det (EqSegm (M9,P1,Q1)) <> 0. K by Def4;
A19: EqSegm (M9,P1,Q1) = Segm (M9,P1,Q1) by A16, Def3;
A20: EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P1,Q1) = Segm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P1,Q1) by A16, Def3;
A21: RLine ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),i,(Line (M9,j))) = Replace ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),i,Lj9) by A11, A7, MATRIX11:29
.= Replace ((Replace (M9,i,LL)),i,Lj9) by A6, MATRIX11:29
.= Replace (M9,i,Lj9) by FUNCT_7:34
.= RLine (M9,i,(Line (M9,j))) by A11, MATRIX11:29 ;
A22: now__::_thesis:_the_rank_of_(RLine_(M9,i,((Line_(M9,i))_+_(a_*_(Line_(M9,j))))))_>=_the_rank_of_M9
percases ( not i in P1 or i in P1 ) ;
suppose not i in P1 ; ::_thesis: the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) >= the_rank_of M9
then Segm (M9,P1,Q1) = Segm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P1,Q1) by A15, Th60;
hence the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) >= the_rank_of M9 by A14, A15, A16, A17, A18, A20, A19, Def4; ::_thesis: verum
end;
supposeA23: i in P1 ; ::_thesis: the_rank_of M9 <= the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j))))))
set SM = EqSegm (M9,P1,Q1);
set SR = EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P1,Q1);
A24: rng (Line (M9,j)) c= the carrier of K by FINSEQ_1:def_4;
A25: ex k being Nat st P1 c= Seg k by Th43;
then A26: dom (Sgm P1) = Seg (card P1) by FINSEQ_3:40;
A27: Q1 c= Seg (width M9) by A15, A16, Th67;
then A28: dom (Sgm Q1) = Seg (card Q1) by FINSEQ_3:40;
A29: rng (Sgm Q1) c= Seg (width M9) by A27, FINSEQ_1:def_13;
then A30: dom ((Line (M9,j)) * (Sgm Q1)) = dom (Sgm Q1) by A12, RELAT_1:27;
then reconsider LjQ = (Line (M9,j)) * (Sgm Q1) as FinSequence by A28, FINSEQ_1:def_2;
rng LjQ c= rng (Line (M9,j)) by RELAT_1:26;
then rng LjQ c= the carrier of K by A24, XBOOLE_1:1;
then reconsider LjQ = LjQ as FinSequence of K by FINSEQ_1:def_4;
A31: len LjQ = card P1 by A16, A30, A28, FINSEQ_1:def_3;
rng (Sgm P1) = P1 by A25, FINSEQ_1:def_13;
then consider m being set such that
A32: m in dom (Sgm P1) and
A33: (Sgm P1) . m = i by A23, FUNCT_1:def_3;
reconsider m = m as Element of NAT by A32;
A34: len (Line ((EqSegm (M9,P1,Q1)),m)) = width (EqSegm (M9,P1,Q1)) by MATRIX_1:def_7;
A35: Line ((EqSegm (M9,P1,Q1)),m) = (Line (M9,i)) * (Sgm Q1) by A19, A32, A33, A26, A27, Th47;
then A36: RLine ((EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P1,Q1)),m,(Line ((EqSegm (M9,P1,Q1)),m))) = Segm (M9,P1,Q1) by A4, A7, A8, A14, A15, A16, A20, A33, Th59;
Q1 c= Seg (width (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j))))))) by A14, A15, A16, Th67;
then A37: Line ((EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P1,Q1)),m) = (Line ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),i)) * (Sgm Q1) by A20, A32, A33, A26, Th47;
A38: RLine ((EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P1,Q1)),m,LjQ) = Segm ((RLine (M9,i,(Line (M9,j)))),P1,Q1) by A11, A7, A21, A14, A15, A16, A20, A33, Th59;
A39: a * LjQ = (a * (Line (M9,j))) * (Sgm Q1) by A12, A29, Th87;
A40: len LjQ = len (a * LjQ) by MATRIXR1:16;
A41: width (EqSegm (M9,P1,Q1)) = card P1 by MATRIX_1:24;
A42: Segm ((RLine (M9,i,(Line (M9,j)))),P1,Q1) = EqSegm ((RLine (M9,i,(Line (M9,j)))),P1,Q1) by A16, Def3;
EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P1,Q1) = RLine ((EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P1,Q1)),m,(Line ((EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P1,Q1)),m))) by MATRIX11:30
.= RLine ((EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P1,Q1)),m,((Line ((EqSegm (M9,P1,Q1)),m)) + (a * LjQ))) by A10, A5, A13, A29, A37, A35, A39, Th88 ;
then A43: Det (EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P1,Q1)) = (Det (RLine ((EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P1,Q1)),m,(Line ((EqSegm (M9,P1,Q1)),m))))) + (Det (RLine ((EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P1,Q1)),m,(a * LjQ)))) by A32, A26, A31, A40, A34, A41, MATRIX11:36
.= (Det (RLine ((EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P1,Q1)),m,(Line ((EqSegm (M9,P1,Q1)),m))))) + (a * (Det (RLine ((EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P1,Q1)),m,LjQ)))) by A32, A26, A31, MATRIX11:34
.= (Det (EqSegm (M9,P1,Q1))) + (a * (Det (EqSegm ((RLine (M9,i,(Line (M9,j)))),P1,Q1)))) by A16, A36, A38, A42, Def3 ;
percases ( Det (EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P1,Q1)) <> 0. K or Det (EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P1,Q1)) = 0. K ) ;
suppose Det (EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P1,Q1)) <> 0. K ; ::_thesis: the_rank_of M9 <= the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j))))))
hence the_rank_of M9 <= the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) by A14, A15, A16, A17, Def4; ::_thesis: verum
end;
supposeA44: Det (EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P1,Q1)) = 0. K ; ::_thesis: the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) >= the_rank_of M9
reconsider j0 = j as non zero Element of NAT by A1;
percases ( i = j or ( i <> j & j in P1 ) or ( i <> j & not j in P1 ) ) ;
supposeA45: i = j ; ::_thesis: the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) >= the_rank_of M9
then Det (EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P1,Q1)) = (Det (EqSegm (M9,P1,Q1))) + (a * (Det (EqSegm (M9,P1,Q1)))) by A43, MATRIX11:30
.= ((1_ K) * (Det (EqSegm (M9,P1,Q1)))) + (a * (Det (EqSegm (M9,P1,Q1)))) by VECTSP_1:def_4
.= ((1_ K) + a) * (Det (EqSegm (M9,P1,Q1))) by VECTSP_1:def_7 ;
then (1_ K) + a = 0. K by A18, A44, VECTSP_1:12;
then a = (0. K) - (1_ K) by VECTSP_2:2
.= (0. K) + (- (1_ K))
.= - (1_ K) by RLVECT_1:def_4 ;
hence the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) >= the_rank_of M9 by A2, A45; ::_thesis: verum
end;
supposeA46: ( i <> j & j in P1 ) ; ::_thesis: the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) >= the_rank_of M9
rng (Sgm P1) = P1 by A25, FINSEQ_1:def_13;
then consider l being set such that
A47: l in dom (Sgm P1) and
A48: (Sgm P1) . l = j0 by A46, FUNCT_1:def_3;
reconsider l = l as Element of NAT by A47;
0. K = Det (RLine ((EqSegm (M9,P1,Q1)),m,(Line ((EqSegm (M9,P1,Q1)),l)))) by A32, A33, A26, A46, A47, A48, MATRIX11:51;
then A49: 0. K = a * (Det (RLine ((EqSegm (M9,P1,Q1)),m,(Line ((EqSegm (M9,P1,Q1)),l))))) by VECTSP_1:12;
RLine ((EqSegm (M9,P1,Q1)),m,(Line ((EqSegm (M9,P1,Q1)),l))) = EqSegm ((RLine (M9,i,(Line (M9,j)))),P1,Q1) by A11, A15, A16, A19, A33, A26, A27, A38, A42, A47, A48, Th47, Th59;
hence the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) >= the_rank_of M9 by A18, A43, A44, A49, RLVECT_1:def_4; ::_thesis: verum
end;
supposeA50: ( i <> j & not j in P1 ) ; ::_thesis: the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) >= the_rank_of M9
set Pij = (P1 \ {i}) \/ {j0};
A51: not i in P1 \ {i} by ZFMISC_1:56;
A52: [:((P1 \ {i}) \/ {j0}),Q1:] c= Indices (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) by A1, A9, A14, A15, A16, A23, A50, Th68;
a * (Det (EqSegm ((RLine (M9,i,(Line (M9,j)))),P1,Q1))) <> 0. K by A18, A43, A44, RLVECT_1:def_4;
then A53: Det (EqSegm ((RLine (M9,i,(Line (M9,j)))),P1,Q1)) <> 0. K by VECTSP_1:12;
( Det (EqSegm ((RLine (M9,i,(Line (M9,j)))),P1,Q1)) = Det (EqSegm (M9,((P1 \ {i}) \/ {j0}),Q1)) or Det (EqSegm ((RLine (M9,i,(Line (M9,j)))),P1,Q1)) = - (Det (EqSegm (M9,((P1 \ {i}) \/ {j0}),Q1))) ) by A1, A9, A15, A16, A23, A50, Th68;
then A54: Det (EqSegm (M9,((P1 \ {i}) \/ {j0}),Q1)) <> 0. K by A53, VECTSP_2:3;
not i in {j} by A50, TARSKI:def_1;
then A55: not i in (P1 \ {i}) \/ {j0} by A51, XBOOLE_0:def_3;
A56: card P1 = card ((P1 \ {i}) \/ {j0}) by A1, A9, A15, A16, A23, A50, Th68;
then EqSegm (M9,((P1 \ {i}) \/ {j0}),Q1) = Segm (M9,((P1 \ {i}) \/ {j0}),Q1) by A16, Def3
.= Segm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),((P1 \ {i}) \/ {j0}),Q1) by A14, A52, A55, Th60
.= EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),((P1 \ {i}) \/ {j0}),Q1) by A16, A56, Def3 ;
hence the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) >= the_rank_of M9 by A16, A17, A54, A56, A52, Def4; ::_thesis: verum
end;
end;
end;
end;
end;
end;
end;
consider P, Q being finite without_zero Subset of NAT such that
A57: [:P,Q:] c= Indices (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) and
A58: card P = card Q and
A59: card P = the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) and
A60: Det (EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P,Q)) <> 0. K by Def4;
A61: EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P,Q) = Segm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P,Q) by A58, Def3;
A62: EqSegm (M9,P,Q) = Segm (M9,P,Q) by A58, Def3;
now__::_thesis:_the_rank_of_(RLine_(M9,i,((Line_(M9,i))_+_(a_*_(Line_(M9,j))))))_<=_the_rank_of_M9
percases ( not i in P or i in P ) ;
suppose not i in P ; ::_thesis: the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) <= the_rank_of M9
then Segm (M9,P,Q) = Segm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P,Q) by A57, A14, Th60;
hence the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) <= the_rank_of M9 by A57, A58, A59, A60, A61, A62, A14, Def4; ::_thesis: verum
end;
supposeA63: i in P ; ::_thesis: the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) <= the_rank_of M9
set SM = EqSegm (M9,P,Q);
set SR = EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P,Q);
A64: rng (Line (M9,j)) c= the carrier of K by FINSEQ_1:def_4;
A65: ex k being Nat st P c= Seg k by Th43;
then A66: dom (Sgm P) = Seg (card P) by FINSEQ_3:40;
A67: Q c= Seg (width M9) by A57, A58, A14, Th67;
then A68: dom (Sgm Q) = Seg (card Q) by FINSEQ_3:40;
A69: rng (Sgm Q) c= Seg (width M9) by A67, FINSEQ_1:def_13;
then A70: dom ((Line (M9,j)) * (Sgm Q)) = dom (Sgm Q) by A12, RELAT_1:27;
then reconsider LjQ = (Line (M9,j)) * (Sgm Q) as FinSequence by A68, FINSEQ_1:def_2;
rng LjQ c= rng (Line (M9,j)) by RELAT_1:26;
then rng LjQ c= the carrier of K by A64, XBOOLE_1:1;
then reconsider LjQ = LjQ as FinSequence of K by FINSEQ_1:def_4;
A71: len LjQ = card P by A58, A70, A68, FINSEQ_1:def_3;
rng (Sgm P) = P by A65, FINSEQ_1:def_13;
then consider m being set such that
A72: m in dom (Sgm P) and
A73: (Sgm P) . m = i by A63, FUNCT_1:def_3;
reconsider m = m as Element of NAT by A72;
A74: len (Line ((EqSegm (M9,P,Q)),m)) = width (EqSegm (M9,P,Q)) by MATRIX_1:def_7;
A75: Line ((EqSegm (M9,P,Q)),m) = (Line (M9,i)) * (Sgm Q) by A62, A72, A73, A66, A67, Th47;
then A76: RLine ((EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P,Q)),m,(Line ((EqSegm (M9,P,Q)),m))) = Segm (M9,P,Q) by A4, A7, A8, A57, A58, A61, A73, Th59;
Q c= Seg (width (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j))))))) by A57, A58, Th67;
then A77: Line ((EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P,Q)),m) = (Line ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),i)) * (Sgm Q) by A61, A72, A73, A66, Th47;
A78: RLine ((EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P,Q)),m,LjQ) = Segm ((RLine (M9,i,(Line (M9,j)))),P,Q) by A11, A7, A21, A57, A58, A61, A73, Th59;
A79: a * LjQ = (a * (Line (M9,j))) * (Sgm Q) by A12, A69, Th87;
A80: len LjQ = len (a * LjQ) by MATRIXR1:16;
A81: width (EqSegm (M9,P,Q)) = card P by MATRIX_1:24;
A82: Segm ((RLine (M9,i,(Line (M9,j)))),P,Q) = EqSegm ((RLine (M9,i,(Line (M9,j)))),P,Q) by A58, Def3;
EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P,Q) = RLine ((EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P,Q)),m,(Line ((EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P,Q)),m))) by MATRIX11:30
.= RLine ((EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P,Q)),m,((Line ((EqSegm (M9,P,Q)),m)) + (a * LjQ))) by A10, A5, A13, A69, A77, A75, A79, Th88 ;
then A83: Det (EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P,Q)) = (Det (RLine ((EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P,Q)),m,(Line ((EqSegm (M9,P,Q)),m))))) + (Det (RLine ((EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P,Q)),m,(a * LjQ)))) by A72, A66, A71, A80, A74, A81, MATRIX11:36
.= (Det (RLine ((EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P,Q)),m,(Line ((EqSegm (M9,P,Q)),m))))) + (a * (Det (RLine ((EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P,Q)),m,LjQ)))) by A72, A66, A71, MATRIX11:34
.= (Det (EqSegm (M9,P,Q))) + (a * (Det (EqSegm ((RLine (M9,i,(Line (M9,j)))),P,Q)))) by A58, A76, A78, A82, Def3 ;
percases ( Det (EqSegm (M9,P,Q)) <> 0. K or Det (EqSegm (M9,P,Q)) = 0. K ) ;
suppose Det (EqSegm (M9,P,Q)) <> 0. K ; ::_thesis: the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) <= the_rank_of M9
hence the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) <= the_rank_of M9 by A57, A58, A59, A14, Def4; ::_thesis: verum
end;
supposeA84: Det (EqSegm (M9,P,Q)) = 0. K ; ::_thesis: the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) <= the_rank_of M9
reconsider j0 = j as non zero Element of NAT by A1;
percases ( i = j or ( i <> j & j in P ) or ( i <> j & not j in P ) ) ;
suppose i = j ; ::_thesis: the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) <= the_rank_of M9
then Det (EqSegm ((RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))),P,Q)) = (Det (EqSegm (M9,P,Q))) + (a * (Det (EqSegm (M9,P,Q)))) by A83, MATRIX11:30
.= ((1_ K) * (Det (EqSegm (M9,P,Q)))) + (a * (Det (EqSegm (M9,P,Q)))) by VECTSP_1:def_4
.= ((1_ K) + a) * (Det (EqSegm (M9,P,Q))) by VECTSP_1:def_7 ;
hence the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) <= the_rank_of M9 by A60, A84, VECTSP_1:12; ::_thesis: verum
end;
supposeA85: ( i <> j & j in P ) ; ::_thesis: the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) <= the_rank_of M9
rng (Sgm P) = P by A65, FINSEQ_1:def_13;
then consider l being set such that
A86: l in dom (Sgm P) and
A87: (Sgm P) . l = j0 by A85, FUNCT_1:def_3;
reconsider l = l as Element of NAT by A86;
A88: RLine ((EqSegm (M9,P,Q)),m,(Line ((EqSegm (M9,P,Q)),l))) = EqSegm ((RLine (M9,i,(Line (M9,j)))),P,Q) by A11, A57, A58, A62, A14, A73, A66, A67, A78, A82, A86, A87, Th47, Th59;
0. K = Det (RLine ((EqSegm (M9,P,Q)),m,(Line ((EqSegm (M9,P,Q)),l)))) by A72, A73, A66, A85, A86, A87, MATRIX11:51;
then 0. K = a * (Det (RLine ((EqSegm (M9,P,Q)),m,(Line ((EqSegm (M9,P,Q)),l))))) by VECTSP_1:12
.= (Det (EqSegm (M9,P,Q))) + (a * (Det (EqSegm ((RLine (M9,i,(Line (M9,j)))),P,Q)))) by A84, A88, RLVECT_1:def_4 ;
hence the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) <= the_rank_of M9 by A60, A83; ::_thesis: verum
end;
supposeA89: ( i <> j & not j in P ) ; ::_thesis: the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) <= the_rank_of M9
set Pij = (P \ {i}) \/ {j0};
A90: card P = card ((P \ {i}) \/ {j0}) by A1, A9, A57, A58, A63, A89, Th68;
a * (Det (EqSegm ((RLine (M9,i,(Line (M9,j)))),P,Q))) <> 0. K by A60, A83, A84, RLVECT_1:def_4;
then A91: Det (EqSegm ((RLine (M9,i,(Line (M9,j)))),P,Q)) <> 0. K by VECTSP_1:12;
( Det (EqSegm ((RLine (M9,i,(Line (M9,j)))),P,Q)) = Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q)) or Det (EqSegm ((RLine (M9,i,(Line (M9,j)))),P,Q)) = - (Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q))) ) by A1, A9, A57, A58, A14, A63, A89, Th68;
then A92: Det (EqSegm (M9,((P \ {i}) \/ {j0}),Q)) <> 0. K by A91, VECTSP_2:3;
[:((P \ {i}) \/ {j0}),Q:] c= Indices M9 by A1, A9, A57, A58, A14, A63, A89, Th68;
hence the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) <= the_rank_of M9 by A58, A59, A92, A90, Def4; ::_thesis: verum
end;
end;
end;
end;
end;
end;
end;
hence the_rank_of M9 = the_rank_of (RLine (M9,i,((Line (M9,i)) + (a * (Line (M9,j)))))) by A22, XXREAL_0:1; ::_thesis: verum
end;
end;
end;
theorem Th93: :: MATRIX13:93
for n9, m9, j, i being Nat
for K being Field
for a being Element of K
for M9 being Matrix of n9,m9,K st j in Seg (len M9) & j <> i holds
the_rank_of (DelLine (M9,i)) = the_rank_of (RLine (M9,i,(a * (Line (M9,j)))))
proof
let n9, m9, j, i be Nat; ::_thesis: for K being Field
for a being Element of K
for M9 being Matrix of n9,m9,K st j in Seg (len M9) & j <> i holds
the_rank_of (DelLine (M9,i)) = the_rank_of (RLine (M9,i,(a * (Line (M9,j)))))
let K be Field; ::_thesis: for a being Element of K
for M9 being Matrix of n9,m9,K st j in Seg (len M9) & j <> i holds
the_rank_of (DelLine (M9,i)) = the_rank_of (RLine (M9,i,(a * (Line (M9,j)))))
let a be Element of K; ::_thesis: for M9 being Matrix of n9,m9,K st j in Seg (len M9) & j <> i holds
the_rank_of (DelLine (M9,i)) = the_rank_of (RLine (M9,i,(a * (Line (M9,j)))))
let M9 be Matrix of n9,m9,K; ::_thesis: ( j in Seg (len M9) & j <> i implies the_rank_of (DelLine (M9,i)) = the_rank_of (RLine (M9,i,(a * (Line (M9,j))))) )
assume that
A1: j in Seg (len M9) and
A2: i <> j ; ::_thesis: the_rank_of (DelLine (M9,i)) = the_rank_of (RLine (M9,i,(a * (Line (M9,j)))))
percases ( i in Seg (len M9) or not i in Seg (len M9) ) ;
supposeA3: i in Seg (len M9) ; ::_thesis: the_rank_of (DelLine (M9,i)) = the_rank_of (RLine (M9,i,(a * (Line (M9,j)))))
set Li = Line (M9,i);
set W = width M9;
set R = RLine (M9,i,((0. K) * (Line (M9,i))));
A4: width M9 = len ((0. K) * (Line (M9,i))) by CARD_1:def_7;
then A5: len (RLine (M9,i,((0. K) * (Line (M9,i))))) = len M9 by MATRIX11:def_3;
set Lj = Line (M9,j);
A6: width M9 = len (a * (Line (M9,j))) by CARD_1:def_7;
reconsider 0Li = (0. K) * (Line (M9,i)), aLj = a * (Line (M9,j)) as Element of the carrier of K * by FINSEQ_1:def_11;
width (RLine (M9,i,((0. K) * (Line (M9,i))))) = width M9 by A4, MATRIX11:def_3;
then A7: RLine ((RLine (M9,i,((0. K) * (Line (M9,i))))),i,aLj) = Replace ((RLine (M9,i,((0. K) * (Line (M9,i))))),i,aLj) by A6, MATRIX11:29
.= Replace ((Replace (M9,i,0Li)),i,aLj) by A4, MATRIX11:29
.= Replace (M9,i,aLj) by FUNCT_7:34
.= RLine (M9,i,aLj) by A6, MATRIX11:29 ;
A8: len M9 = n9 by MATRIX_1:def_2;
then A9: Line ((RLine (M9,i,((0. K) * (Line (M9,i))))),j) = Line (M9,j) by A1, A2, MATRIX11:28;
Line ((RLine (M9,i,((0. K) * (Line (M9,i))))),i) = (0. K) * (Line (M9,i)) by A3, A4, A8, MATRIX11:28;
then A10: (Line ((RLine (M9,i,((0. K) * (Line (M9,i))))),i)) + (a * (Line ((RLine (M9,i,((0. K) * (Line (M9,i))))),j))) = ((width M9) |-> (0. K)) + (a * (Line (M9,j))) by A9, FVSUM_1:58
.= a * (Line (M9,j)) by FVSUM_1:21 ;
width M9 = len (Line (M9,i)) by CARD_1:def_7;
hence the_rank_of (DelLine (M9,i)) = the_rank_of (RLine (M9,i,((0. K) * (Line (M9,i))))) by Th91
.= the_rank_of (RLine (M9,i,(a * (Line (M9,j))))) by A1, A2, A5, A10, A7, Th92 ;
::_thesis: verum
end;
supposeA11: not i in Seg (len M9) ; ::_thesis: the_rank_of (DelLine (M9,i)) = the_rank_of (RLine (M9,i,(a * (Line (M9,j)))))
then not i in dom M9 by FINSEQ_1:def_3;
then DelLine (M9,i) = M9 by FINSEQ_3:104;
hence the_rank_of (DelLine (M9,i)) = the_rank_of (RLine (M9,i,(a * (Line (M9,j))))) by A11, Th40; ::_thesis: verum
end;
end;
end;
theorem Th94: :: MATRIX13:94
for K being Field
for M being Matrix of K holds
( the_rank_of M > 0 iff ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K ) )
proof
let K be Field; ::_thesis: for M being Matrix of K holds
( the_rank_of M > 0 iff ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K ) )
let M be Matrix of K; ::_thesis: ( the_rank_of M > 0 iff ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K ) )
set r = the_rank_of M;
thus ( the_rank_of M > 0 implies ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K ) ) ::_thesis: ( ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K ) implies the_rank_of M > 0 )
proof
consider P, Q being finite without_zero Subset of NAT such that
A1: [:P,Q:] c= Indices M and
A2: card P = card Q and
A3: card P = the_rank_of M and
A4: Det (EqSegm (M,P,Q)) <> 0. K by Def4;
assume the_rank_of M > 0 ; ::_thesis: ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K )
then consider x being set such that
A5: x in P by A3, CARD_1:27, XBOOLE_0:def_1;
reconsider x = x as non zero Element of NAT by A5;
{x} c= P by A5, ZFMISC_1:31;
then consider Q1 being finite without_zero Subset of NAT such that
A6: Q1 c= Q and
A7: card {x} = card Q1 and
A8: Det (EqSegm (M,{x},Q1)) <> 0. K by A2, A4, Th65;
consider y being set such that
A9: {y} = Q1 by A7, CARD_2:42;
y in {y} by TARSKI:def_1;
then reconsider y = y as non zero Element of NAT by A9;
take x ; ::_thesis: ex j being Nat st
( [x,j] in Indices M & M * (x,j) <> 0. K )
take y ; ::_thesis: ( [x,y] in Indices M & M * (x,y) <> 0. K )
y in Q1 by A9, TARSKI:def_1;
then [x,y] in [:P,Q:] by A5, A6, ZFMISC_1:87;
hence [x,y] in Indices M by A1; ::_thesis: M * (x,y) <> 0. K
A10: card {x} = 1 by CARD_1:30;
EqSegm (M,{x},Q1) = Segm (M,{x},{y}) by A7, A9, Def3
.= <*<*(M * (x,y))*>*> by Th44 ;
hence M * (x,y) <> 0. K by A8, A10, MATRIX_3:34; ::_thesis: verum
end;
given i, j being Nat such that A11: [i,j] in Indices M and
A12: M * (i,j) <> 0. K ; ::_thesis: the_rank_of M > 0
A13: j in Seg (width M) by A11, ZFMISC_1:87;
Indices M = [:(Seg (len M)),(Seg (width M)):] by FINSEQ_1:def_3;
then A14: i in Seg (len M) by A11, ZFMISC_1:87;
then reconsider i = i, j = j as non zero Element of NAT by A13;
A15: card {i} = 1 by CARD_1:30;
A16: card {j} = 1 by CARD_1:30;
then EqSegm (M,{i},{j}) = Segm (M,{i},{j}) by Def3, CARD_1:30
.= <*<*(M * (i,j))*>*> by Th44 ;
then A17: Det (EqSegm (M,{i},{j})) <> 0. K by A12, A15, MATRIX_3:34;
A18: {j} c= Seg (width M) by A13, ZFMISC_1:31;
{i} c= Seg (len M) by A14, ZFMISC_1:31;
then [:{i},{j}:] c= Indices M by A15, A16, A18, Th67;
hence the_rank_of M > 0 by A15, A16, A17, Def4; ::_thesis: verum
end;
theorem :: MATRIX13:95
for K being Field
for M being Matrix of K holds
( the_rank_of M = 0 iff M = 0. (K,(len M),(width M)) )
proof
let K be Field; ::_thesis: for M being Matrix of K holds
( the_rank_of M = 0 iff M = 0. (K,(len M),(width M)) )
let M be Matrix of K; ::_thesis: ( the_rank_of M = 0 iff M = 0. (K,(len M),(width M)) )
set NULL = 0. (K,(len M),(width M));
reconsider M9 = M as Matrix of len M, width M,K by MATRIX_2:7;
thus ( the_rank_of M = 0 implies M = 0. (K,(len M),(width M)) ) ::_thesis: ( M = 0. (K,(len M),(width M)) implies the_rank_of M = 0 )
proof
assume A1: the_rank_of M = 0 ; ::_thesis: M = 0. (K,(len M),(width M))
now__::_thesis:_for_i,_j_being_Nat_st_[i,j]_in_Indices_M9_holds_
M9_*_(i,j)_=_(0._(K,(len_M),(width_M)))_*_(i,j)
A2: Indices M9 = Indices (0. (K,(len M),(width M))) by MATRIX_1:26;
let i, j be Nat; ::_thesis: ( [i,j] in Indices M9 implies M9 * (i,j) = (0. (K,(len M),(width M))) * (i,j) )
assume A3: [i,j] in Indices M9 ; ::_thesis: M9 * (i,j) = (0. (K,(len M),(width M))) * (i,j)
reconsider i9 = i, j9 = j as Element of NAT by ORDINAL1:def_12;
M * (i9,j9) = 0. K by A1, A3, Th94;
hence M9 * (i,j) = (0. (K,(len M),(width M))) * (i,j) by A3, A2, MATRIX_3:1; ::_thesis: verum
end;
hence M = 0. (K,(len M),(width M)) by MATRIX_1:27; ::_thesis: verum
end;
assume A4: M = 0. (K,(len M),(width M)) ; ::_thesis: the_rank_of M = 0
assume the_rank_of M <> 0 ; ::_thesis: contradiction
then ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K ) by Th94;
hence contradiction by A4, MATRIX_3:1; ::_thesis: verum
end;
theorem Th96: :: MATRIX13:96
for K being Field
for M being Matrix of K holds
( the_rank_of M = 1 iff ( ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K ) & ( for i0, j0, n0, m0 being non zero Nat st i0 <> j0 & n0 <> m0 & [:{i0,j0},{n0,m0}:] c= Indices M holds
Det (EqSegm (M,{i0,j0},{n0,m0})) = 0. K ) ) )
proof
let K be Field; ::_thesis: for M being Matrix of K holds
( the_rank_of M = 1 iff ( ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K ) & ( for i0, j0, n0, m0 being non zero Nat st i0 <> j0 & n0 <> m0 & [:{i0,j0},{n0,m0}:] c= Indices M holds
Det (EqSegm (M,{i0,j0},{n0,m0})) = 0. K ) ) )
let M be Matrix of K; ::_thesis: ( the_rank_of M = 1 iff ( ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K ) & ( for i0, j0, n0, m0 being non zero Nat st i0 <> j0 & n0 <> m0 & [:{i0,j0},{n0,m0}:] c= Indices M holds
Det (EqSegm (M,{i0,j0},{n0,m0})) = 0. K ) ) )
consider P, Q being finite without_zero Subset of NAT such that
A1: [:P,Q:] c= Indices M and
A2: card P = card Q and
A3: card P = the_rank_of M and
A4: Det (EqSegm (M,P,Q)) <> 0. K by Def4;
thus ( the_rank_of M = 1 implies ( ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K ) & ( for i0, j0, n0, m0 being non zero Nat st i0 <> j0 & n0 <> m0 & [:{i0,j0},{n0,m0}:] c= Indices M holds
Det (EqSegm (M,{i0,j0},{n0,m0})) = 0. K ) ) ) ::_thesis: ( ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K ) & ( for i0, j0, n0, m0 being non zero Nat st i0 <> j0 & n0 <> m0 & [:{i0,j0},{n0,m0}:] c= Indices M holds
Det (EqSegm (M,{i0,j0},{n0,m0})) = 0. K ) implies the_rank_of M = 1 )
proof
assume A5: the_rank_of M = 1 ; ::_thesis: ( ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K ) & ( for i0, j0, n0, m0 being non zero Nat st i0 <> j0 & n0 <> m0 & [:{i0,j0},{n0,m0}:] c= Indices M holds
Det (EqSegm (M,{i0,j0},{n0,m0})) = 0. K ) )
hence ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K ) by Th94; ::_thesis: for i0, j0, n0, m0 being non zero Nat st i0 <> j0 & n0 <> m0 & [:{i0,j0},{n0,m0}:] c= Indices M holds
Det (EqSegm (M,{i0,j0},{n0,m0})) = 0. K
let i0, j0, n0, m0 be non zero Nat; ::_thesis: ( i0 <> j0 & n0 <> m0 & [:{i0,j0},{n0,m0}:] c= Indices M implies Det (EqSegm (M,{i0,j0},{n0,m0})) = 0. K )
assume that
A6: i0 <> j0 and
A7: n0 <> m0 and
A8: [:{i0,j0},{n0,m0}:] c= Indices M ; ::_thesis: Det (EqSegm (M,{i0,j0},{n0,m0})) = 0. K
A9: card {n0,m0} = 2 by A7, CARD_2:57;
assume A10: Det (EqSegm (M,{i0,j0},{n0,m0})) <> 0. K ; ::_thesis: contradiction
card {i0,j0} = 2 by A6, CARD_2:57;
hence contradiction by A5, A8, A10, A9, Def4; ::_thesis: verum
end;
assume ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K ) ; ::_thesis: ( ex i0, j0, n0, m0 being non zero Nat st
( i0 <> j0 & n0 <> m0 & [:{i0,j0},{n0,m0}:] c= Indices M & not Det (EqSegm (M,{i0,j0},{n0,m0})) = 0. K ) or the_rank_of M = 1 )
then A11: the_rank_of M > 0 by Th94;
assume A12: for i0, j0, n0, m0 being non zero Nat st i0 <> j0 & n0 <> m0 & [:{i0,j0},{n0,m0}:] c= Indices M holds
Det (EqSegm (M,{i0,j0},{n0,m0})) = 0. K ; ::_thesis: the_rank_of M = 1
assume the_rank_of M <> 1 ; ::_thesis: contradiction
then card P > 1 by A11, A3, NAT_1:25;
then card P >= 1 + 1 by NAT_1:13;
then consider P1 being finite Subset of P such that
A13: card P1 = 2 by FINSEQ_4:72;
not 0 in P1 ;
then reconsider P1 = P1 as finite without_zero Subset of NAT by MEASURE6:def_2, XBOOLE_1:1;
consider Q1 being finite without_zero Subset of NAT such that
A14: Q1 c= Q and
A15: card P1 = card Q1 and
A16: Det (EqSegm (M,P1,Q1)) <> 0. K by A2, A4, Th65;
consider n, m being set such that
A17: n <> m and
A18: Q1 = {n,m} by A13, A15, CARD_2:60;
A19: n in Q1 by A18, TARSKI:def_2;
m in Q1 by A18, TARSKI:def_2;
then reconsider n = n, m = m as non zero Element of NAT by A19;
consider i, j being set such that
A20: i <> j and
A21: P1 = {i,j} by A13, CARD_2:60;
A22: i in P1 by A21, TARSKI:def_2;
j in P1 by A21, TARSKI:def_2;
then reconsider i = i, j = j as non zero Element of NAT by A22;
[:P1,Q1:] c= [:P,Q:] by A14, ZFMISC_1:96;
then [:{i,j},{n,m}:] c= Indices M by A1, A21, A18, XBOOLE_1:1;
hence contradiction by A12, A20, A21, A16, A17, A18; ::_thesis: verum
end;
theorem Th97: :: MATRIX13:97
for K being Field
for M being Matrix of K holds
( the_rank_of M = 1 iff ( ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K ) & ( for i, j, n, m being Nat st [:{i,j},{n,m}:] c= Indices M holds
(M * (i,n)) * (M * (j,m)) = (M * (i,m)) * (M * (j,n)) ) ) )
proof
let K be Field; ::_thesis: for M being Matrix of K holds
( the_rank_of M = 1 iff ( ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K ) & ( for i, j, n, m being Nat st [:{i,j},{n,m}:] c= Indices M holds
(M * (i,n)) * (M * (j,m)) = (M * (i,m)) * (M * (j,n)) ) ) )
let M be Matrix of K; ::_thesis: ( the_rank_of M = 1 iff ( ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K ) & ( for i, j, n, m being Nat st [:{i,j},{n,m}:] c= Indices M holds
(M * (i,n)) * (M * (j,m)) = (M * (i,m)) * (M * (j,n)) ) ) )
thus ( the_rank_of M = 1 implies ( ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K ) & ( for i, j, n, m being Nat st [:{i,j},{n,m}:] c= Indices M holds
(M * (i,n)) * (M * (j,m)) = (M * (i,m)) * (M * (j,n)) ) ) ) ::_thesis: ( ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K ) & ( for i, j, n, m being Nat st [:{i,j},{n,m}:] c= Indices M holds
(M * (i,n)) * (M * (j,m)) = (M * (i,m)) * (M * (j,n)) ) implies the_rank_of M = 1 )
proof
assume A1: the_rank_of M = 1 ; ::_thesis: ( ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K ) & ( for i, j, n, m being Nat st [:{i,j},{n,m}:] c= Indices M holds
(M * (i,n)) * (M * (j,m)) = (M * (i,m)) * (M * (j,n)) ) )
hence ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K ) by Th96; ::_thesis: for i, j, n, m being Nat st [:{i,j},{n,m}:] c= Indices M holds
(M * (i,n)) * (M * (j,m)) = (M * (i,m)) * (M * (j,n))
let i, j, n, p be Nat; ::_thesis: ( [:{i,j},{n,p}:] c= Indices M implies (M * (i,n)) * (M * (j,p)) = (M * (i,p)) * (M * (j,n)) )
assume A2: [:{i,j},{n,p}:] c= Indices M ; ::_thesis: (M * (i,n)) * (M * (j,p)) = (M * (i,p)) * (M * (j,n))
percases ( i = j or n = p or ( i <> j & n <> p ) ) ;
suppose ( i = j or n = p ) ; ::_thesis: (M * (i,n)) * (M * (j,p)) = (M * (i,p)) * (M * (j,n))
hence (M * (i,n)) * (M * (j,p)) = (M * (i,p)) * (M * (j,n)) ; ::_thesis: verum
end;
supposeA3: ( i <> j & n <> p ) ; ::_thesis: (M * (i,n)) * (M * (j,p)) = (M * (i,p)) * (M * (j,n))
Indices M = [:(Seg (len M)),(Seg (width M)):] by FINSEQ_1:def_3;
then A4: {i,j} c= Seg (len M) by A2, ZFMISC_1:114;
A5: i in {i,j} by TARSKI:def_2;
A6: n in {n,p} by TARSKI:def_2;
A7: p in {n,p} by TARSKI:def_2;
A8: j in {i,j} by TARSKI:def_2;
{n,p} c= Seg (width M) by A2, ZFMISC_1:114;
then reconsider I = i, J = j, P = p, N = n as non zero Element of NAT by A4, A5, A8, A7, A6;
A9: card {I,J} = 2 by A3, CARD_2:57;
set JP = M * (J,P);
set JN = M * (J,N);
set IP = M * (I,P);
set IN = M * (I,N);
A10: Det (EqSegm (M,{I,J},{N,P})) = 0. K by A1, A2, A3, Th96;
card {N,P} = 2 by A3, CARD_2:57;
then A11: EqSegm (M,{I,J},{N,P}) = Segm (M,{I,J},{N,P}) by A9, Def3;
percases ( ( I < J & N < P ) or ( I < J & N > P ) or ( I > J & N < P ) or ( I > J & N > P ) ) by A3, XXREAL_0:1;
suppose ( I < J & N < P ) ; ::_thesis: (M * (i,n)) * (M * (j,p)) = (M * (i,p)) * (M * (j,n))
then 0. K = Det (((M * (I,N)),(M * (I,P))) ][ ((M * (J,N)),(M * (J,P)))) by A9, A11, A10, Th45
.= ((M * (I,N)) * (M * (J,P))) - ((M * (I,P)) * (M * (J,N))) by MATRIX_9:13 ;
hence (M * (i,n)) * (M * (j,p)) = (M * (i,p)) * (M * (j,n)) by VECTSP_1:19; ::_thesis: verum
end;
suppose ( I < J & N > P ) ; ::_thesis: (M * (i,n)) * (M * (j,p)) = (M * (i,p)) * (M * (j,n))
then 0. K = Det (((M * (I,P)),(M * (I,N))) ][ ((M * (J,P)),(M * (J,N)))) by A9, A11, A10, Th45
.= ((M * (I,P)) * (M * (J,N))) - ((M * (I,N)) * (M * (J,P))) by MATRIX_9:13 ;
hence (M * (i,n)) * (M * (j,p)) = (M * (i,p)) * (M * (j,n)) by VECTSP_1:19; ::_thesis: verum
end;
suppose ( I > J & N < P ) ; ::_thesis: (M * (i,n)) * (M * (j,p)) = (M * (i,p)) * (M * (j,n))
then 0. K = Det (((M * (J,N)),(M * (J,P))) ][ ((M * (I,N)),(M * (I,P)))) by A9, A11, A10, Th45
.= ((M * (J,N)) * (M * (I,P))) - ((M * (J,P)) * (M * (I,N))) by MATRIX_9:13 ;
hence (M * (i,n)) * (M * (j,p)) = (M * (i,p)) * (M * (j,n)) by VECTSP_1:19; ::_thesis: verum
end;
suppose ( I > J & N > P ) ; ::_thesis: (M * (i,n)) * (M * (j,p)) = (M * (i,p)) * (M * (j,n))
then 0. K = Det (((M * (J,P)),(M * (J,N))) ][ ((M * (I,P)),(M * (I,N)))) by A9, A11, A10, Th45
.= ((M * (J,P)) * (M * (I,N))) - ((M * (J,N)) * (M * (I,P))) by MATRIX_9:13 ;
hence (M * (i,n)) * (M * (j,p)) = (M * (i,p)) * (M * (j,n)) by VECTSP_1:19; ::_thesis: verum
end;
end;
end;
end;
end;
assume that
A12: ex i, j being Nat st
( [i,j] in Indices M & M * (i,j) <> 0. K ) and
A13: for i, j, n, m being Nat st [:{i,j},{n,m}:] c= Indices M holds
(M * (i,n)) * (M * (j,m)) = (M * (i,m)) * (M * (j,n)) ; ::_thesis: the_rank_of M = 1
now__::_thesis:_for_i0,_j0,_n0,_m0_being_non_zero_Nat_st_i0_<>_j0_&_n0_<>_m0_&_[:{i0,j0},{n0,m0}:]_c=_Indices_M_holds_
Det_(EqSegm_(M,{i0,j0},{n0,m0}))_=_0._K
let i0, j0, n0, m0 be non zero Nat; ::_thesis: ( i0 <> j0 & n0 <> m0 & [:{i0,j0},{n0,m0}:] c= Indices M implies Det (EqSegm (M,{b1,b2},{b3,b4})) = 0. K )
assume that
A14: i0 <> j0 and
A15: n0 <> m0 and
A16: [:{i0,j0},{n0,m0}:] c= Indices M ; ::_thesis: Det (EqSegm (M,{b1,b2},{b3,b4})) = 0. K
A17: card {i0,j0} = 2 by A14, CARD_2:57;
set JM = M * (j0,m0);
set JN = M * (j0,n0);
set IM = M * (i0,m0);
set IN = M * (i0,n0);
A18: (M * (i0,n0)) * (M * (j0,m0)) = (M * (i0,m0)) * (M * (j0,n0)) by A13, A16;
card {n0,m0} = 2 by A15, CARD_2:57;
then A19: EqSegm (M,{i0,j0},{n0,m0}) = Segm (M,{i0,j0},{n0,m0}) by A17, Def3;
percases ( ( i0 < j0 & n0 < m0 ) or ( i0 < j0 & n0 > m0 ) or ( i0 > j0 & n0 < m0 ) or ( i0 > j0 & n0 > m0 ) ) by A14, A15, XXREAL_0:1;
suppose ( i0 < j0 & n0 < m0 ) ; ::_thesis: Det (EqSegm (M,{b1,b2},{b3,b4})) = 0. K
then EqSegm (M,{i0,j0},{n0,m0}) = ((M * (i0,n0)),(M * (i0,m0))) ][ ((M * (j0,n0)),(M * (j0,m0))) by A19, Th45;
hence Det (EqSegm (M,{i0,j0},{n0,m0})) = ((M * (i0,n0)) * (M * (j0,m0))) - ((M * (i0,m0)) * (M * (j0,n0))) by A17, MATRIX_9:13
.= 0. K by A18, VECTSP_1:19 ;
::_thesis: verum
end;
suppose ( i0 < j0 & n0 > m0 ) ; ::_thesis: Det (EqSegm (M,{b1,b2},{b3,b4})) = 0. K
then EqSegm (M,{i0,j0},{n0,m0}) = ((M * (i0,m0)),(M * (i0,n0))) ][ ((M * (j0,m0)),(M * (j0,n0))) by A19, Th45;
hence Det (EqSegm (M,{i0,j0},{n0,m0})) = ((M * (i0,m0)) * (M * (j0,n0))) - ((M * (i0,n0)) * (M * (j0,m0))) by A17, MATRIX_9:13
.= 0. K by A18, VECTSP_1:19 ;
::_thesis: verum
end;
suppose ( i0 > j0 & n0 < m0 ) ; ::_thesis: Det (EqSegm (M,{b1,b2},{b3,b4})) = 0. K
then EqSegm (M,{i0,j0},{n0,m0}) = ((M * (j0,n0)),(M * (j0,m0))) ][ ((M * (i0,n0)),(M * (i0,m0))) by A19, Th45;
hence Det (EqSegm (M,{i0,j0},{n0,m0})) = ((M * (j0,n0)) * (M * (i0,m0))) - ((M * (j0,m0)) * (M * (i0,n0))) by A17, MATRIX_9:13
.= 0. K by A18, VECTSP_1:19 ;
::_thesis: verum
end;
suppose ( i0 > j0 & n0 > m0 ) ; ::_thesis: Det (EqSegm (M,{b1,b2},{b3,b4})) = 0. K
then EqSegm (M,{i0,j0},{n0,m0}) = ((M * (j0,m0)),(M * (j0,n0))) ][ ((M * (i0,m0)),(M * (i0,n0))) by A19, Th45;
hence Det (EqSegm (M,{i0,j0},{n0,m0})) = ((M * (j0,m0)) * (M * (i0,n0))) - ((M * (j0,n0)) * (M * (i0,m0))) by A17, MATRIX_9:13
.= 0. K by A18, VECTSP_1:19 ;
::_thesis: verum
end;
end;
end;
hence the_rank_of M = 1 by A12, Th96; ::_thesis: verum
end;
theorem :: MATRIX13:98
for K being Field
for M being Matrix of K holds
( the_rank_of M = 1 iff ex i being Nat st
( i in Seg (len M) & ex j being Nat st
( j in Seg (width M) & M * (i,j) <> 0. K ) & ( for k being Nat st k in Seg (len M) holds
ex a being Element of K st Line (M,k) = a * (Line (M,i)) ) ) )
proof
let K be Field; ::_thesis: for M being Matrix of K holds
( the_rank_of M = 1 iff ex i being Nat st
( i in Seg (len M) & ex j being Nat st
( j in Seg (width M) & M * (i,j) <> 0. K ) & ( for k being Nat st k in Seg (len M) holds
ex a being Element of K st Line (M,k) = a * (Line (M,i)) ) ) )
let M be Matrix of K; ::_thesis: ( the_rank_of M = 1 iff ex i being Nat st
( i in Seg (len M) & ex j being Nat st
( j in Seg (width M) & M * (i,j) <> 0. K ) & ( for k being Nat st k in Seg (len M) holds
ex a being Element of K st Line (M,k) = a * (Line (M,i)) ) ) )
A1: Indices M = [:(Seg (len M)),(Seg (width M)):] by FINSEQ_1:def_3;
thus ( the_rank_of M = 1 implies ex i being Nat st
( i in Seg (len M) & ex j being Nat st
( j in Seg (width M) & M * (i,j) <> 0. K ) & ( for k being Nat st k in Seg (len M) holds
ex a being Element of K st Line (M,k) = a * (Line (M,i)) ) ) ) ::_thesis: ( ex i being Nat st
( i in Seg (len M) & ex j being Nat st
( j in Seg (width M) & M * (i,j) <> 0. K ) & ( for k being Nat st k in Seg (len M) holds
ex a being Element of K st Line (M,k) = a * (Line (M,i)) ) ) implies the_rank_of M = 1 )
proof
assume A2: the_rank_of M = 1 ; ::_thesis: ex i being Nat st
( i in Seg (len M) & ex j being Nat st
( j in Seg (width M) & M * (i,j) <> 0. K ) & ( for k being Nat st k in Seg (len M) holds
ex a being Element of K st Line (M,k) = a * (Line (M,i)) ) )
then consider i, j being Nat such that
A3: [i,j] in Indices M and
A4: M * (i,j) <> 0. K by Th97;
take i ; ::_thesis: ( i in Seg (len M) & ex j being Nat st
( j in Seg (width M) & M * (i,j) <> 0. K ) & ( for k being Nat st k in Seg (len M) holds
ex a being Element of K st Line (M,k) = a * (Line (M,i)) ) )
A5: j in Seg (width M) by A3, ZFMISC_1:87;
hence ( i in Seg (len M) & ex j being Nat st
( j in Seg (width M) & M * (i,j) <> 0. K ) ) by A1, A3, A4, ZFMISC_1:87; ::_thesis: for k being Nat st k in Seg (len M) holds
ex a being Element of K st Line (M,k) = a * (Line (M,i))
set Li = Line (M,i);
set ij = M * (i,j);
let k be Nat; ::_thesis: ( k in Seg (len M) implies ex a being Element of K st Line (M,k) = a * (Line (M,i)) )
assume A6: k in Seg (len M) ; ::_thesis: ex a being Element of K st Line (M,k) = a * (Line (M,i))
set Lk = Line (M,k);
set kj = M * (k,j);
take a = (M * (k,j)) * ((M * (i,j)) "); ::_thesis: Line (M,k) = a * (Line (M,i))
A7: i in Seg (len M) by A1, A3, ZFMISC_1:87;
A8: now__::_thesis:_for_n_being_Nat_st_1_<=_n_&_n_<=_width_M_holds_
(Line_(M,k))_._n_=_(a_*_(Line_(M,i)))_._n
let n be Nat; ::_thesis: ( 1 <= n & n <= width M implies (Line (M,k)) . n = (a * (Line (M,i))) . n )
assume that
A9: 1 <= n and
A10: n <= width M ; ::_thesis: (Line (M,k)) . n = (a * (Line (M,i))) . n
n in NAT by ORDINAL1:def_12;
then A11: n in Seg (width M) by A9, A10;
then A12: {j,n} c= Seg (width M) by A5, ZFMISC_1:32;
(Line (M,i)) . n = M * (i,n) by A11, MATRIX_1:def_7;
then A13: (a * (Line (M,i))) . n = a * (M * (i,n)) by A11, FVSUM_1:51;
A14: M * (k,j) = (M * (k,j)) * (1_ K) by VECTSP_1:def_4
.= (M * (k,j)) * ((M * (i,j)) * ((M * (i,j)) ")) by A4, VECTSP_1:def_10
.= a * (M * (i,j)) by GROUP_1:def_3 ;
{i,k} c= Seg (len M) by A7, A6, ZFMISC_1:32;
then [:{i,k},{j,n}:] c= Indices M by A1, A12, ZFMISC_1:96;
then A15: (M * (i,j)) * (M * (k,n)) = (a * (M * (i,j))) * (M * (i,n)) by A2, A14, Th97
.= (M * (i,j)) * (a * (M * (i,n))) by GROUP_1:def_3 ;
(Line (M,k)) . n = M * (k,n) by A11, MATRIX_1:def_7;
hence (Line (M,k)) . n = (a * (Line (M,i))) . n by A4, A15, A13, VECTSP_2:8; ::_thesis: verum
end;
A16: len (a * (Line (M,i))) = width M by CARD_1:def_7;
len (Line (M,k)) = width M by CARD_1:def_7;
hence Line (M,k) = a * (Line (M,i)) by A16, A8, FINSEQ_1:14; ::_thesis: verum
end;
given i being Nat such that A17: i in Seg (len M) and
A18: ex j being Nat st
( j in Seg (width M) & M * (i,j) <> 0. K ) and
A19: for k being Nat st k in Seg (len M) holds
ex a being Element of K st Line (M,k) = a * (Line (M,i)) ; ::_thesis: the_rank_of M = 1
A20: now__::_thesis:_for_I,_J,_n,_m_being_Nat_st_[:{I,J},{n,m}:]_c=_Indices_M_holds_
(M_*_(I,n))_*_(M_*_(J,m))_=_(M_*_(I,m))_*_(M_*_(J,n))
set Li = Line (M,i);
let I, J, n, m be Nat; ::_thesis: ( [:{I,J},{n,m}:] c= Indices M implies (M * (I,n)) * (M * (J,m)) = (M * (I,m)) * (M * (J,n)) )
assume A21: [:{I,J},{n,m}:] c= Indices M ; ::_thesis: (M * (I,n)) * (M * (J,m)) = (M * (I,m)) * (M * (J,n))
A22: {n,m} c= Seg (width M) by A21, ZFMISC_1:114;
then A23: n in Seg (width M) by ZFMISC_1:32;
then A24: (Line (M,i)) . n = M * (i,n) by MATRIX_1:def_7;
set LJ = Line (M,J);
set LI = Line (M,I);
A25: {I,J} c= Seg (len M) by A1, A21, ZFMISC_1:114;
then I in Seg (len M) by ZFMISC_1:32;
then consider a being Element of K such that
A26: Line (M,I) = a * (Line (M,i)) by A19;
J in Seg (len M) by A25, ZFMISC_1:32;
then consider b being Element of K such that
A27: Line (M,J) = b * (Line (M,i)) by A19;
(Line (M,J)) . n = M * (J,n) by A23, MATRIX_1:def_7;
then A28: M * (J,n) = b * (M * (i,n)) by A27, A23, A24, FVSUM_1:51;
A29: m in Seg (width M) by A22, ZFMISC_1:32;
then A30: (Line (M,i)) . m = M * (i,m) by MATRIX_1:def_7;
(Line (M,J)) . m = M * (J,m) by A29, MATRIX_1:def_7;
then A31: M * (J,m) = b * (M * (i,m)) by A27, A29, A30, FVSUM_1:51;
(Line (M,I)) . m = M * (I,m) by A29, MATRIX_1:def_7;
then A32: M * (I,m) = a * (M * (i,m)) by A26, A29, A30, FVSUM_1:51;
(Line (M,I)) . n = M * (I,n) by A23, MATRIX_1:def_7;
then M * (I,n) = a * (M * (i,n)) by A26, A23, A24, FVSUM_1:51;
hence (M * (I,n)) * (M * (J,m)) = ((a * (M * (i,n))) * b) * (M * (i,m)) by A31, GROUP_1:def_3
.= ((b * (M * (i,n))) * a) * (M * (i,m)) by GROUP_1:def_3
.= (M * (I,m)) * (M * (J,n)) by A28, A32, GROUP_1:def_3 ;
::_thesis: verum
end;
consider j being Nat such that
A33: j in Seg (width M) and
A34: M * (i,j) <> 0. K by A18;
[i,j] in Indices M by A1, A17, A33, ZFMISC_1:87;
hence the_rank_of M = 1 by A34, A20, Th97; ::_thesis: verum
end;
registration
let K be Field;
cluster Relation-like NAT -defined the carrier of K * -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V147() tabular diagonal for FinSequence of the carrier of K * ;
existence
ex b1 being Matrix of K st b1 is diagonal
proof
set E = the V162(K) Matrix of 1,K;
take the V162(K) Matrix of 1,K ; ::_thesis: the V162(K) Matrix of 1,K is diagonal
thus the V162(K) Matrix of 1,K is diagonal ; ::_thesis: verum
end;
end;
theorem Th99: :: MATRIX13:99
for K being Field
for M being diagonal Matrix of K
for NonZero1 being set st NonZero1 = { i where i is Element of NAT : ( [i,i] in Indices M & M * (i,i) <> 0. K ) } holds
for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K holds
( P c= NonZero1 & Q c= NonZero1 )
proof
let K be Field; ::_thesis: for M being diagonal Matrix of K
for NonZero1 being set st NonZero1 = { i where i is Element of NAT : ( [i,i] in Indices M & M * (i,i) <> 0. K ) } holds
for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K holds
( P c= NonZero1 & Q c= NonZero1 )
let M be diagonal Matrix of K; ::_thesis: for NonZero1 being set st NonZero1 = { i where i is Element of NAT : ( [i,i] in Indices M & M * (i,i) <> 0. K ) } holds
for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K holds
( P c= NonZero1 & Q c= NonZero1 )
let NonZero1 be set ; ::_thesis: ( NonZero1 = { i where i is Element of NAT : ( [i,i] in Indices M & M * (i,i) <> 0. K ) } implies for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K holds
( P c= NonZero1 & Q c= NonZero1 ) )
assume A1: NonZero1 = { i where i is Element of NAT : ( [i,i] in Indices M & M * (i,i) <> 0. K ) } ; ::_thesis: for P, Q being finite without_zero Subset of NAT st [:P,Q:] c= Indices M & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K holds
( P c= NonZero1 & Q c= NonZero1 )
let P, Q be finite without_zero Subset of NAT; ::_thesis: ( [:P,Q:] c= Indices M & card P = card Q & Det (EqSegm (M,P,Q)) <> 0. K implies ( P c= NonZero1 & Q c= NonZero1 ) )
assume that
A2: [:P,Q:] c= Indices M and
A3: card P = card Q and
A4: Det (EqSegm (M,P,Q)) <> 0. K ; ::_thesis: ( P c= NonZero1 & Q c= NonZero1 )
set S = Segm (M,P,Q);
set SQ = Sgm Q;
set SP = Sgm P;
set ES = EqSegm (M,P,Q);
A5: Indices (Segm (M,P,Q)) = [:(Seg (len (Segm (M,P,Q)))),(Seg (width (Segm (M,P,Q)))):] by FINSEQ_1:def_3;
A6: EqSegm (M,P,Q) = Segm (M,P,Q) by A3, Def3;
thus P c= NonZero1 ::_thesis: Q c= NonZero1
proof
assume not P c= NonZero1 ; ::_thesis: contradiction
then consider x being set such that
A7: x in P and
A8: not x in NonZero1 by TARSKI:def_3;
A9: P c= Seg (len M) by A2, A3, Th67;
then A10: rng (Sgm P) = P by FINSEQ_1:def_13;
then consider y being set such that
A11: y in dom (Sgm P) and
A12: (Sgm P) . y = x by A7, FUNCT_1:def_3;
reconsider x = x, y = y as Element of NAT by A7, A11;
set L = Line ((Segm (M,P,Q)),y);
A13: dom (Sgm P) = Seg (card P) by A9, FINSEQ_3:40;
Q c= Seg (width M) by A2, A3, Th67;
then A14: rng (Sgm Q) = Q by FINSEQ_1:def_13;
A15: now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_width_(Segm_(M,P,Q))_holds_
(Line_((Segm_(M,P,Q)),y))_._i_=_((0._K)_*_(Line_((Segm_(M,P,Q)),y)))_._i
let i be Nat; ::_thesis: ( 1 <= i & i <= width (Segm (M,P,Q)) implies (Line ((Segm (M,P,Q)),y)) . i = ((0. K) * (Line ((Segm (M,P,Q)),y))) . i )
assume that
A16: 1 <= i and
A17: i <= width (Segm (M,P,Q)) ; ::_thesis: (Line ((Segm (M,P,Q)),y)) . i = ((0. K) * (Line ((Segm (M,P,Q)),y))) . i
i in NAT by ORDINAL1:def_12;
then A18: i in Seg (width (Segm (M,P,Q))) by A16, A17;
then A19: (Line ((Segm (M,P,Q)),y)) . i = (Segm (M,P,Q)) * (y,i) by MATRIX_1:def_7;
y in Seg (len (Segm (M,P,Q))) by A3, A13, A11, MATRIX_1:24;
then A20: [y,i] in Indices (Segm (M,P,Q)) by A5, A18, ZFMISC_1:87;
then A21: (Segm (M,P,Q)) * (y,i) = M * (x,((Sgm Q) . i)) by A12, Def1;
A22: (0. K) * (0. K) = 0. K by VECTSP_1:7;
A23: ( (Sgm Q) . i <> x or (Sgm Q) . i = x ) ;
[x,((Sgm Q) . i)] in Indices M by A2, A10, A14, A12, A20, Th17;
then (Line ((Segm (M,P,Q)),y)) . i = 0. K by A1, A8, A21, A19, A23, MATRIX_1:def_14;
hence (Line ((Segm (M,P,Q)),y)) . i = ((0. K) * (Line ((Segm (M,P,Q)),y))) . i by A18, A22, FVSUM_1:51; ::_thesis: verum
end;
A24: len (Line ((Segm (M,P,Q)),y)) = width (Segm (M,P,Q)) by MATRIX_1:def_7;
len (Line ((Segm (M,P,Q)),y)) = len ((0. K) * (Line ((Segm (M,P,Q)),y))) by MATRIXR1:16;
then Line ((Segm (M,P,Q)),y) = (0. K) * (Line ((Segm (M,P,Q)),y)) by A24, A15, FINSEQ_1:14;
then A25: Det (RLine ((EqSegm (M,P,Q)),y,(Line ((EqSegm (M,P,Q)),y)))) = (0. K) * (Det (EqSegm (M,P,Q))) by A6, A13, A11, MATRIX11:35;
RLine ((EqSegm (M,P,Q)),y,(Line ((EqSegm (M,P,Q)),y))) = EqSegm (M,P,Q) by MATRIX11:30;
hence contradiction by A4, A25, VECTSP_1:7; ::_thesis: verum
end;
A26: dom (Segm (M,P,Q)) = Seg (len (Segm (M,P,Q))) by FINSEQ_1:def_3;
A27: len (Segm (M,P,Q)) = card P by A3, MATRIX_1:24;
A28: width (Segm (M,P,Q)) = card P by A3, MATRIX_1:24;
thus Q c= NonZero1 ::_thesis: verum
proof
A29: Q c= Seg (width M) by A2, A3, Th67;
then A30: dom (Sgm Q) = Seg (card Q) by FINSEQ_3:40;
assume not Q c= NonZero1 ; ::_thesis: contradiction
then consider x being set such that
A31: x in Q and
A32: not x in NonZero1 by TARSKI:def_3;
A33: rng (Sgm Q) = Q by A29, FINSEQ_1:def_13;
then consider y being set such that
A34: y in dom (Sgm Q) and
A35: (Sgm Q) . y = x by A31, FUNCT_1:def_3;
reconsider x = x, y = y as Element of NAT by A31, A34;
set C = Col ((EqSegm (M,P,Q)),y);
P c= Seg (len M) by A2, A3, Th67;
then A36: rng (Sgm P) = P by FINSEQ_1:def_13;
now__::_thesis:_for_k_being_Element_of_NAT_st_k_in_Seg_(card_P)_holds_
(Col_((EqSegm_(M,P,Q)),y))_._k_=_0._K
let k be Element of NAT ; ::_thesis: ( k in Seg (card P) implies (Col ((EqSegm (M,P,Q)),y)) . k = 0. K )
assume A37: k in Seg (card P) ; ::_thesis: (Col ((EqSegm (M,P,Q)),y)) . k = 0. K
A38: (Segm (M,P,Q)) * (k,y) = (Col ((EqSegm (M,P,Q)),y)) . k by A6, A27, A26, A37, MATRIX_1:def_8;
A39: [k,y] in Indices (Segm (M,P,Q)) by A3, A5, A27, A28, A30, A34, A37, ZFMISC_1:87;
then A40: (Segm (M,P,Q)) * (k,y) = M * (((Sgm P) . k),x) by A35, Def1;
A41: ( (Sgm P) . k <> x or (Sgm P) . k = x ) ;
[((Sgm P) . k),x] in Indices M by A2, A36, A33, A35, A39, Th17;
hence (Col ((EqSegm (M,P,Q)),y)) . k = 0. K by A1, A32, A40, A38, A41, MATRIX_1:def_14; ::_thesis: verum
end;
hence contradiction by A3, A4, A30, A34, MATRIX_9:53; ::_thesis: verum
end;
end;
theorem Th100: :: MATRIX13:100
for K being Field
for M being diagonal Matrix of K
for P being finite without_zero Subset of NAT st [:P,P:] c= Indices M holds
Segm (M,P,P) is V162(b1)
proof
let K be Field; ::_thesis: for M being diagonal Matrix of K
for P being finite without_zero Subset of NAT st [:P,P:] c= Indices M holds
Segm (M,P,P) is V162(K)
let M be diagonal Matrix of K; ::_thesis: for P being finite without_zero Subset of NAT st [:P,P:] c= Indices M holds
Segm (M,P,P) is V162(K)
let P be finite without_zero Subset of NAT; ::_thesis: ( [:P,P:] c= Indices M implies Segm (M,P,P) is V162(K) )
assume A1: [:P,P:] c= Indices M ; ::_thesis: Segm (M,P,P) is V162(K)
set S = Segm (M,P,P);
set SP = Sgm P;
let i, j be Element of NAT ; :: according to MATRIX_7:def_2 ::_thesis: ( not i in Seg (card P) or not j in Seg (card P) or i = j or (Segm (M,P,P)) * (i,j) = 0. K )
assume that
A2: i in Seg (card P) and
A3: j in Seg (card P) and
A4: i <> j ; ::_thesis: (Segm (M,P,P)) * (i,j) = 0. K
A5: P c= Seg (len M) by A1, A2, Th67;
then A6: Sgm P is one-to-one by FINSEQ_3:92;
[i,j] in [:(Seg (card P)),(Seg (card P)):] by A2, A3, ZFMISC_1:87;
then A7: [i,j] in Indices (Segm (M,P,P)) by MATRIX_1:24;
dom (Sgm P) = Seg (card P) by A5, FINSEQ_3:40;
then A8: (Sgm P) . i <> (Sgm P) . j by A2, A3, A4, A6, FUNCT_1:def_4;
rng (Sgm P) = P by A5, FINSEQ_1:def_13;
then A9: [((Sgm P) . i),((Sgm P) . j)] in Indices M by A1, A7, Th17;
(Segm (M,P,P)) * (i,j) = M * (((Sgm P) . i),((Sgm P) . j)) by A7, Def1;
hence (Segm (M,P,P)) * (i,j) = 0. K by A9, A8, MATRIX_1:def_14; ::_thesis: verum
end;
theorem :: MATRIX13:101
for K being Field
for M being diagonal Matrix of K
for NonZero1 being set st NonZero1 = { i where i is Element of NAT : ( [i,i] in Indices M & M * (i,i) <> 0. K ) } holds
the_rank_of M = card NonZero1
proof
let K be Field; ::_thesis: for M being diagonal Matrix of K
for NonZero1 being set st NonZero1 = { i where i is Element of NAT : ( [i,i] in Indices M & M * (i,i) <> 0. K ) } holds
the_rank_of M = card NonZero1
let M be diagonal Matrix of K; ::_thesis: for NonZero1 being set st NonZero1 = { i where i is Element of NAT : ( [i,i] in Indices M & M * (i,i) <> 0. K ) } holds
the_rank_of M = card NonZero1
consider P, Q being finite without_zero Subset of NAT such that
A1: [:P,Q:] c= Indices M and
A2: card P = card Q and
A3: card P = the_rank_of M and
A4: Det (EqSegm (M,P,Q)) <> 0. K by Def4;
let NZ be set ; ::_thesis: ( NZ = { i where i is Element of NAT : ( [i,i] in Indices M & M * (i,i) <> 0. K ) } implies the_rank_of M = card NZ )
assume A5: NZ = { i where i is Element of NAT : ( [i,i] in Indices M & M * (i,i) <> 0. K ) } ; ::_thesis: the_rank_of M = card NZ
A6: NZ c= Seg (width M)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in NZ or x in Seg (width M) )
assume x in NZ ; ::_thesis: x in Seg (width M)
then ex i being Element of NAT st
( x = i & [i,i] in Indices M & M * (i,i) <> 0. K ) by A5;
hence x in Seg (width M) by ZFMISC_1:87; ::_thesis: verum
end;
then not 0 in NZ ;
then reconsider nz = NZ as finite without_zero Subset of NAT by A6, MEASURE6:def_2, XBOOLE_1:1;
set S = Segm (M,nz,nz);
NZ c= dom M
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in NZ or x in dom M )
assume x in NZ ; ::_thesis: x in dom M
then ex i being Element of NAT st
( x = i & [i,i] in Indices M & M * (i,i) <> 0. K ) by A5;
hence x in dom M by ZFMISC_1:87; ::_thesis: verum
end;
then A7: [:nz,nz:] c= Indices M by A6, ZFMISC_1:96;
then reconsider S = Segm (M,nz,nz) as V162(K) Matrix of card nz,K by Th100;
set d = diagonal_of_Matrix S;
now__::_thesis:_Det_S_<>_0._K
percases ( card nz = 0 or card nz >= 1 ) by NAT_1:14;
suppose card nz = 0 ; ::_thesis: Det S <> 0. K
then Det S = 1_ K by MATRIXR2:41;
hence Det S <> 0. K ; ::_thesis: verum
end;
supposeA8: card nz >= 1 ; ::_thesis: Det S <> 0. K
set Sn = Sgm nz;
A9: now__::_thesis:_for_k_being_Element_of_NAT_st_k_in_dom_(diagonal_of_Matrix_S)_holds_
(diagonal_of_Matrix_S)_._k_<>_0._K
A10: rng (Sgm nz) = nz by A6, FINSEQ_1:def_13;
A11: dom (diagonal_of_Matrix S) = Seg (len (diagonal_of_Matrix S)) by FINSEQ_1:def_3;
A12: len (diagonal_of_Matrix S) = card nz by MATRIX_3:def_10;
let k be Element of NAT ; ::_thesis: ( k in dom (diagonal_of_Matrix S) implies (diagonal_of_Matrix S) . k <> 0. K )
assume A13: k in dom (diagonal_of_Matrix S) ; ::_thesis: (diagonal_of_Matrix S) . k <> 0. K
A14: (diagonal_of_Matrix S) . k = S * (k,k) by A13, A11, A12, MATRIX_3:def_10;
dom (Sgm nz) = dom (diagonal_of_Matrix S) by A6, A11, A12, FINSEQ_3:40;
then (Sgm nz) . k in nz by A13, A10, FUNCT_1:def_3;
then A15: ex i being Element of NAT st
( i = (Sgm nz) . k & [i,i] in Indices M & M * (i,i) <> 0. K ) by A5;
Indices S = [:(Seg (card nz)),(Seg (card nz)):] by MATRIX_1:24;
then [k,k] in Indices S by A13, A11, A12, ZFMISC_1:87;
hence (diagonal_of_Matrix S) . k <> 0. K by A14, A15, Def1; ::_thesis: verum
end;
Det S = Product (diagonal_of_Matrix S) by A8, MATRIX_7:17;
hence Det S <> 0. K by A9, FVSUM_1:82; ::_thesis: verum
end;
end;
end;
then Det (EqSegm (M,nz,nz)) <> 0. K by Def3;
then A16: the_rank_of M >= card nz by A7, Def4;
P c= nz by A5, A1, A2, A4, Th99;
then card P <= card nz by NAT_1:43;
hence the_rank_of M = card NZ by A16, A3, XXREAL_0:1; ::_thesis: verum
end;
theorem Th102: :: MATRIX13:102
for n being Nat
for K being Field
for a being Element of K
for v1, v2, v being Vector of (n -VectSp_over K)
for t1, t2, t being Element of n -tuples_on the carrier of K holds
( the carrier of (n -VectSp_over K) = n -tuples_on the carrier of K & 0. (n -VectSp_over K) = n |-> (0. K) & ( t1 = v1 & t2 = v2 implies t1 + t2 = v1 + v2 ) & ( t = v implies a * t = a * v ) )
proof
let n be Nat; ::_thesis: for K being Field
for a being Element of K
for v1, v2, v being Vector of (n -VectSp_over K)
for t1, t2, t being Element of n -tuples_on the carrier of K holds
( the carrier of (n -VectSp_over K) = n -tuples_on the carrier of K & 0. (n -VectSp_over K) = n |-> (0. K) & ( t1 = v1 & t2 = v2 implies t1 + t2 = v1 + v2 ) & ( t = v implies a * t = a * v ) )
let K be Field; ::_thesis: for a being Element of K
for v1, v2, v being Vector of (n -VectSp_over K)
for t1, t2, t being Element of n -tuples_on the carrier of K holds
( the carrier of (n -VectSp_over K) = n -tuples_on the carrier of K & 0. (n -VectSp_over K) = n |-> (0. K) & ( t1 = v1 & t2 = v2 implies t1 + t2 = v1 + v2 ) & ( t = v implies a * t = a * v ) )
let a be Element of K; ::_thesis: for v1, v2, v being Vector of (n -VectSp_over K)
for t1, t2, t being Element of n -tuples_on the carrier of K holds
( the carrier of (n -VectSp_over K) = n -tuples_on the carrier of K & 0. (n -VectSp_over K) = n |-> (0. K) & ( t1 = v1 & t2 = v2 implies t1 + t2 = v1 + v2 ) & ( t = v implies a * t = a * v ) )
let v1, v2, v be Vector of (n -VectSp_over K); ::_thesis: for t1, t2, t being Element of n -tuples_on the carrier of K holds
( the carrier of (n -VectSp_over K) = n -tuples_on the carrier of K & 0. (n -VectSp_over K) = n |-> (0. K) & ( t1 = v1 & t2 = v2 implies t1 + t2 = v1 + v2 ) & ( t = v implies a * t = a * v ) )
let t1, t2, t be Element of n -tuples_on the carrier of K; ::_thesis: ( the carrier of (n -VectSp_over K) = n -tuples_on the carrier of K & 0. (n -VectSp_over K) = n |-> (0. K) & ( t1 = v1 & t2 = v2 implies t1 + t2 = v1 + v2 ) & ( t = v implies a * t = a * v ) )
A1: addLoopStr(# the carrier of (n -VectSp_over K), the addF of (n -VectSp_over K), the ZeroF of (n -VectSp_over K) #) = n -Group_over K by PRVECT_1:def_5;
A2: n -Group_over K = addLoopStr(# (n -tuples_on the carrier of K),(product ( the addF of K,n)),(n |-> (0. K)) #) by PRVECT_1:def_3;
hence ( the carrier of (n -VectSp_over K) = n -tuples_on the carrier of K & 0. (n -VectSp_over K) = n |-> (0. K) ) by A1; ::_thesis: ( ( t1 = v1 & t2 = v2 implies t1 + t2 = v1 + v2 ) & ( t = v implies a * t = a * v ) )
thus ( t1 = v1 & t2 = v2 implies t1 + t2 = v1 + v2 ) by A2, A1, PRVECT_1:def_1; ::_thesis: ( t = v implies a * t = a * v )
assume A3: t = v ; ::_thesis: a * t = a * v
rng t c= the carrier of K by RELAT_1:def_19;
then A4: (id the carrier of K) * t = t by RELAT_1:53;
thus a * v = (n -Mult_over K) . (a,v) by PRVECT_1:def_5
.= the multF of K [;] (a,t) by A3, PRVECT_1:def_4
.= a * t by A4, FUNCOP_1:34 ; ::_thesis: verum
end;
registration
let K be Field;
let n be Nat;
clustern -VectSp_over K -> right_complementable Abelian add-associative right_zeroed ;
coherence
( n -VectSp_over K is right_complementable & n -VectSp_over K is Abelian & n -VectSp_over K is add-associative & n -VectSp_over K is right_zeroed )
proof
set nV = n -VectSp_over K;
A1: now__::_thesis:_for_v,_u_being_Vector_of_(n_-VectSp_over_K)_holds_v_+_u_=_u_+_v
let v, u be Vector of (n -VectSp_over K); ::_thesis: v + u = u + v
reconsider V = v, U = u as Element of n -tuples_on the carrier of K by Th102;
thus v + u = V + U by Th102
.= U + V by FINSEQOP:33
.= u + v by Th102 ; ::_thesis: verum
end;
A2: n in NAT by ORDINAL1:def_12;
A3: now__::_thesis:_for_v_being_Vector_of_(n_-VectSp_over_K)_holds_v_+_(0._(n_-VectSp_over_K))_=_v
let v be Vector of (n -VectSp_over K); ::_thesis: v + (0. (n -VectSp_over K)) = v
reconsider V = v, n0 = 0. (n -VectSp_over K) as Element of n -tuples_on the carrier of K by Th102;
thus v + (0. (n -VectSp_over K)) = V + n0 by Th102
.= V + (n |-> (0. K)) by Th102
.= v by A2, FVSUM_1:21 ; ::_thesis: verum
end;
A4: n -VectSp_over K is right_complementable
proof
reconsider N = n as Element of NAT by ORDINAL1:def_12;
let v be Vector of (n -VectSp_over K); :: according to ALGSTR_0:def_16 ::_thesis: v is right_complementable
reconsider V = v as Element of N -tuples_on the carrier of K by Th102;
reconsider u = - V as Element of (n -VectSp_over K) by Th102;
v + u = V + (- V) by Th102
.= n |-> (0. K) by FVSUM_1:26
.= 0. (n -VectSp_over K) by Th102 ;
hence ex u being Vector of (n -VectSp_over K) st v + u = 0. (n -VectSp_over K) ; :: according to ALGSTR_0:def_11 ::_thesis: verum
end;
now__::_thesis:_for_u,_v,_w_being_Vector_of_(n_-VectSp_over_K)_holds_(u_+_v)_+_w_=_u_+_(v_+_w)
let u, v, w be Vector of (n -VectSp_over K); ::_thesis: (u + v) + w = u + (v + w)
reconsider V = v, U = u, W = w, UV = u + v, VW = v + w as Element of n -tuples_on the carrier of K by Th102;
thus (u + v) + w = UV + W by Th102
.= (U + V) + W by Th102
.= U + (V + W) by FINSEQOP:28
.= U + VW by Th102
.= u + (v + w) by Th102 ; ::_thesis: verum
end;
hence ( n -VectSp_over K is right_complementable & n -VectSp_over K is Abelian & n -VectSp_over K is add-associative & n -VectSp_over K is right_zeroed ) by A3, A4, A1, RLVECT_1:def_2, RLVECT_1:def_3, RLVECT_1:def_4; ::_thesis: verum
end;
end;
registration
let K be Field;
let n be Nat;
cluster -> Relation-like Function-like for Element of the carrier of (n -VectSp_over K);
correctness
coherence
for b1 being Vector of (n -VectSp_over K) holds
( b1 is Function-like & b1 is Relation-like );
proof
let v be Element of (n -VectSp_over K); ::_thesis: ( v is Function-like & v is Relation-like )
v is Element of n -tuples_on the carrier of K by Th102;
hence ( v is Function-like & v is Relation-like ) ; ::_thesis: verum
end;
end;
Lm7: for m, n being Nat
for K being Field
for M being Matrix of m,n,K holds rng M is Subset of (n -VectSp_over K)
proof
let m, n be Nat; ::_thesis: for K being Field
for M being Matrix of m,n,K holds rng M is Subset of (n -VectSp_over K)
let K be Field; ::_thesis: for M being Matrix of m,n,K holds rng M is Subset of (n -VectSp_over K)
let M be Matrix of m,n,K; ::_thesis: rng M is Subset of (n -VectSp_over K)
rng M c= the carrier of (n -VectSp_over K)
proof
consider m being Nat such that
A1: for x being set st x in rng M holds
ex p being FinSequence of K st
( x = p & len p = m ) by MATRIX_1:9;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng M or x in the carrier of (n -VectSp_over K) )
assume A2: x in rng M ; ::_thesis: x in the carrier of (n -VectSp_over K)
consider p being FinSequence of K such that
A3: x = p and
len p = m by A2, A1;
len p = n by A2, A3, MATRIX_1:def_2;
then p is Element of n -tuples_on the carrier of K by FINSEQ_2:92;
then p in n -tuples_on the carrier of K ;
hence x in the carrier of (n -VectSp_over K) by A3, Th102; ::_thesis: verum
end;
hence rng M is Subset of (n -VectSp_over K) ; ::_thesis: verum
end;
notation
let K be Field;
let m, n be Nat;
let M be Matrix of m,n,K;
synonym lines M for rng K;
synonym without_repeated_line M for one-to-one ;
end;
definition
let K be Field;
let m, n be Nat;
let M be Matrix of m,n,K;
:: original: lines
redefine func lines M -> Subset of (n -VectSp_over K);
coherence
lines is Subset of (n -VectSp_over K) by Lm7;
end;
theorem Th103: :: MATRIX13:103
for x being set
for n, m being Nat
for K being Field
for M being Matrix of m,n,K holds
( x in lines M iff ex i being Nat st
( i in Seg m & x = Line (M,i) ) )
proof
let x be set ; ::_thesis: for n, m being Nat
for K being Field
for M being Matrix of m,n,K holds
( x in lines M iff ex i being Nat st
( i in Seg m & x = Line (M,i) ) )
let n, m be Nat; ::_thesis: for K being Field
for M being Matrix of m,n,K holds
( x in lines M iff ex i being Nat st
( i in Seg m & x = Line (M,i) ) )
let K be Field; ::_thesis: for M being Matrix of m,n,K holds
( x in lines M iff ex i being Nat st
( i in Seg m & x = Line (M,i) ) )
let M be Matrix of m,n,K; ::_thesis: ( x in lines M iff ex i being Nat st
( i in Seg m & x = Line (M,i) ) )
thus ( x in lines M implies ex i being Nat st
( i in Seg m & x = Line (M,i) ) ) ::_thesis: ( ex i being Nat st
( i in Seg m & x = Line (M,i) ) implies x in lines M )
proof
assume x in lines M ; ::_thesis: ex i being Nat st
( i in Seg m & x = Line (M,i) )
then consider i being set such that
A1: i in dom M and
A2: M . i = x by FUNCT_1:def_3;
A3: len M = m by MATRIX_1:def_2;
reconsider i = i as Element of NAT by A1;
A4: dom M = Seg (len M) by FINSEQ_1:def_3;
then M . i = Line (M,i) by A1, A3, MATRIX_2:8;
hence ex i being Nat st
( i in Seg m & x = Line (M,i) ) by A1, A2, A4, A3; ::_thesis: verum
end;
given i being Nat such that A5: i in Seg m and
A6: x = Line (M,i) ; ::_thesis: x in lines M
A7: len M = m by MATRIX_1:def_2;
dom M = Seg (len M) by FINSEQ_1:def_3;
then M . i in rng M by A5, A7, FUNCT_1:def_3;
hence x in lines M by A5, A6, MATRIX_2:8; ::_thesis: verum
end;
theorem :: MATRIX13:104
for n being Nat
for K being Field
for V being finite Subset of (n -VectSp_over K) ex M being Matrix of card V,n,K st
( M is without_repeated_line & lines M = V )
proof
let n be Nat; ::_thesis: for K being Field
for V being finite Subset of (n -VectSp_over K) ex M being Matrix of card V,n,K st
( M is without_repeated_line & lines M = V )
let K be Field; ::_thesis: for V being finite Subset of (n -VectSp_over K) ex M being Matrix of card V,n,K st
( M is without_repeated_line & lines M = V )
let V be finite Subset of (n -VectSp_over K); ::_thesis: ex M being Matrix of card V,n,K st
( M is without_repeated_line & lines M = V )
set cV = card V;
card (Seg (card V)) = card V by FINSEQ_1:57;
then Seg (card V),V are_equipotent by CARD_1:5;
then consider m being Function such that
A1: m is one-to-one and
A2: dom m = Seg (card V) and
A3: rng m = V by WELLORD2:def_4;
reconsider M = m as FinSequence by A2, FINSEQ_1:def_2;
now__::_thesis:_for_x_being_set_st_x_in_rng_M_holds_
ex_p_being_FinSequence_of_K_st_
(_x_=_p_&_len_p_=_n_)
let x be set ; ::_thesis: ( x in rng M implies ex p being FinSequence of K st
( x = p & len p = n ) )
assume x in rng M ; ::_thesis: ex p being FinSequence of K st
( x = p & len p = n )
then reconsider p = x as Element of n -tuples_on the carrier of K by A3, Th102;
len p = n by CARD_1:def_7;
hence ex p being FinSequence of K st
( x = p & len p = n ) ; ::_thesis: verum
end;
then reconsider M = M as Matrix of K by MATRIX_1:9;
A4: len M = card V by A2, FINSEQ_1:def_3;
the carrier of (n -VectSp_over K) = n -tuples_on the carrier of K by Th102;
then for p being FinSequence of K st p in rng M holds
len p = n by A3, CARD_1:def_7;
then M is Matrix of card V,n,K by A4, MATRIX_1:def_2;
hence ex M being Matrix of card V,n,K st
( M is without_repeated_line & lines M = V ) by A1, A3; ::_thesis: verum
end;
definition
let K be Field;
let n be Nat;
let F be FinSequence of (n -VectSp_over K);
func FinS2MX F -> Matrix of len F,n,K equals :: MATRIX13:def 5
F;
coherence
F is Matrix of len F,n,K
proof
A1: F is FinSequence of n -tuples_on the carrier of K by Th102;
now__::_thesis:_for_x_being_set_st_x_in_rng_F_holds_
ex_p_being_FinSequence_of_K_st_
(_x_=_p_&_len_p_=_n_)
A2: rng F c= n -tuples_on the carrier of K by A1, FINSEQ_1:def_4;
let x be set ; ::_thesis: ( x in rng F implies ex p being FinSequence of K st
( x = p & len p = n ) )
assume x in rng F ; ::_thesis: ex p being FinSequence of K st
( x = p & len p = n )
then reconsider p = x as Element of n -tuples_on the carrier of K by A2;
len p = n by CARD_1:def_7;
hence ex p being FinSequence of K st
( x = p & len p = n ) ; ::_thesis: verum
end;
then reconsider F9 = F as Matrix of K by MATRIX_1:9;
now__::_thesis:_for_p_being_FinSequence_of_K_st_p_in_rng_F9_holds_
len_p_=_n
A3: rng F9 c= n -tuples_on the carrier of K by A1, FINSEQ_1:def_4;
let p be FinSequence of K; ::_thesis: ( p in rng F9 implies len p = n )
assume p in rng F9 ; ::_thesis: len p = n
hence len p = n by A3, CARD_1:def_7; ::_thesis: verum
end;
hence F is Matrix of len F,n,K by MATRIX_1:def_2; ::_thesis: verum
end;
end;
:: deftheorem defines FinS2MX MATRIX13:def_5_:_
for K being Field
for n being Nat
for F being FinSequence of (n -VectSp_over K) holds FinS2MX F = F;
definition
let K be Field;
let m, n be Nat;
let M be Matrix of m,n,K;
func MX2FinS M -> FinSequence of (n -VectSp_over K) equals :: MATRIX13:def 6
M;
coherence
M is FinSequence of (n -VectSp_over K)
proof
lines M is Subset of (n -VectSp_over K) ;
hence M is FinSequence of (n -VectSp_over K) by FINSEQ_1:def_4; ::_thesis: verum
end;
end;
:: deftheorem defines MX2FinS MATRIX13:def_6_:_
for K being Field
for m, n being Nat
for M being Matrix of m,n,K holds MX2FinS M = M;
theorem Th105: :: MATRIX13:105
for n, m being Nat
for K being Field
for M being Matrix of m,n,K st the_rank_of M = m holds
M is without_repeated_line
proof
let n, m be Nat; ::_thesis: for K being Field
for M being Matrix of m,n,K st the_rank_of M = m holds
M is without_repeated_line
let K be Field; ::_thesis: for M being Matrix of m,n,K st the_rank_of M = m holds
M is without_repeated_line
let M be Matrix of m,n,K; ::_thesis: ( the_rank_of M = m implies M is without_repeated_line )
assume A1: the_rank_of M = m ; ::_thesis: M is without_repeated_line
A2: len M = m by MATRIX_1:def_2;
assume not M is without_repeated_line ; ::_thesis: contradiction
then consider x1, x2 being set such that
A3: x1 in dom M and
A4: x2 in dom M and
A5: M . x1 = M . x2 and
A6: x1 <> x2 by FUNCT_1:def_4;
reconsider x1 = x1, x2 = x2 as Element of NAT by A3, A4;
consider k being Nat such that
A7: len M = k + 1 and
A8: len (Del (M,x1)) = k by A3, FINSEQ_3:104;
A9: dom M = Seg (len M) by FINSEQ_1:def_3;
then A10: M . x2 = Line (M,x2) by A4, A2, MATRIX_2:8;
M . x1 = Line (M,x1) by A3, A9, A2, MATRIX_2:8;
then M = RLine (M,x1,(Line (M,x2))) by A5, A10, MATRIX11:30
.= RLine (M,x1,((1_ K) * (Line (M,x2)))) by FVSUM_1:57 ;
then m = the_rank_of (DelLine (M,x1)) by A1, A4, A6, A9, Th93;
then m <= k by A8, Th74;
hence contradiction by A2, A7, NAT_1:13; ::_thesis: verum
end;
theorem Th106: :: MATRIX13:106
for m, n, i being Nat
for K being Field
for a being Element of K
for L being Linear_Combination of n -VectSp_over K
for M being Matrix of m,n,K st i in Seg (len M) & a = L . (M . i) holds
Line ((FinS2MX (L (#) (MX2FinS M))),i) = a * (Line (M,i))
proof
let m, n, i be Nat; ::_thesis: for K being Field
for a being Element of K
for L being Linear_Combination of n -VectSp_over K
for M being Matrix of m,n,K st i in Seg (len M) & a = L . (M . i) holds
Line ((FinS2MX (L (#) (MX2FinS M))),i) = a * (Line (M,i))
let K be Field; ::_thesis: for a being Element of K
for L being Linear_Combination of n -VectSp_over K
for M being Matrix of m,n,K st i in Seg (len M) & a = L . (M . i) holds
Line ((FinS2MX (L (#) (MX2FinS M))),i) = a * (Line (M,i))
let a be Element of K; ::_thesis: for L being Linear_Combination of n -VectSp_over K
for M being Matrix of m,n,K st i in Seg (len M) & a = L . (M . i) holds
Line ((FinS2MX (L (#) (MX2FinS M))),i) = a * (Line (M,i))
let L be Linear_Combination of n -VectSp_over K; ::_thesis: for M being Matrix of m,n,K st i in Seg (len M) & a = L . (M . i) holds
Line ((FinS2MX (L (#) (MX2FinS M))),i) = a * (Line (M,i))
let M be Matrix of m,n,K; ::_thesis: ( i in Seg (len M) & a = L . (M . i) implies Line ((FinS2MX (L (#) (MX2FinS M))),i) = a * (Line (M,i)) )
assume that
A1: i in Seg (len M) and
A2: a = L . (M . i) ; ::_thesis: Line ((FinS2MX (L (#) (MX2FinS M))),i) = a * (Line (M,i))
set MX = MX2FinS M;
set LM = L (#) (MX2FinS M);
i in dom M by A1, FINSEQ_1:def_3;
then A3: M . i = (MX2FinS M) /. i by PARTFUN1:def_6;
len M = m by MATRIX_1:def_2;
then A4: Line (M,i) = M . i by A1, MATRIX_2:8;
then reconsider L = Line (M,i) as Element of n -tuples_on the carrier of K by A3, Th102;
set FLM = FinS2MX (L (#) (MX2FinS M));
A5: len (L (#) (MX2FinS M)) = len M by VECTSP_6:def_5;
then A6: i in dom (FinS2MX (L (#) (MX2FinS M))) by A1, FINSEQ_1:def_3;
Line ((FinS2MX (L (#) (MX2FinS M))),i) = (FinS2MX (L (#) (MX2FinS M))) . i by A1, A5, MATRIX_2:8;
hence Line ((FinS2MX (L (#) (MX2FinS M))),i) = a * ((MX2FinS M) /. i) by A2, A3, A6, VECTSP_6:def_5
.= a * L by A3, A4, Th102
.= a * (Line (M,i)) ;
::_thesis: verum
end;
theorem Th107: :: MATRIX13:107
for m, i, n being Nat
for K being Field
for L being Linear_Combination of n -VectSp_over K
for M being Matrix of m,n,K st M is without_repeated_line & Carrier L c= lines M & i in Seg n holds
(Sum L) . i = Sum (Col ((FinS2MX (L (#) (MX2FinS M))),i))
proof
let m, i, n be Nat; ::_thesis: for K being Field
for L being Linear_Combination of n -VectSp_over K
for M being Matrix of m,n,K st M is without_repeated_line & Carrier L c= lines M & i in Seg n holds
(Sum L) . i = Sum (Col ((FinS2MX (L (#) (MX2FinS M))),i))
let K be Field; ::_thesis: for L being Linear_Combination of n -VectSp_over K
for M being Matrix of m,n,K st M is without_repeated_line & Carrier L c= lines M & i in Seg n holds
(Sum L) . i = Sum (Col ((FinS2MX (L (#) (MX2FinS M))),i))
let L be Linear_Combination of n -VectSp_over K; ::_thesis: for M being Matrix of m,n,K st M is without_repeated_line & Carrier L c= lines M & i in Seg n holds
(Sum L) . i = Sum (Col ((FinS2MX (L (#) (MX2FinS M))),i))
let M be Matrix of m,n,K; ::_thesis: ( M is without_repeated_line & Carrier L c= lines M & i in Seg n implies (Sum L) . i = Sum (Col ((FinS2MX (L (#) (MX2FinS M))),i)) )
assume that
A1: M is without_repeated_line and
A2: Carrier L c= lines M and
A3: i in Seg n ; ::_thesis: (Sum L) . i = Sum (Col ((FinS2MX (L (#) (MX2FinS M))),i))
set MX = MX2FinS M;
set V = n -VectSp_over K;
set LM = L (#) (MX2FinS M);
A4: len (L (#) (MX2FinS M)) = len M by VECTSP_6:def_5;
set FLM = FinS2MX (L (#) (MX2FinS M));
set C = Col ((FinS2MX (L (#) (MX2FinS M))),i);
len (L (#) (MX2FinS M)) = len (Col ((FinS2MX (L (#) (MX2FinS M))),i)) by MATRIX_1:def_8;
then consider g being Function of NAT, the carrier of K such that
A5: Sum (Col ((FinS2MX (L (#) (MX2FinS M))),i)) = g . (len M) and
A6: g . 0 = 0. K and
A7: for j being Element of NAT
for a being Element of K st j < len M & a = (Col ((FinS2MX (L (#) (MX2FinS M))),i)) . (j + 1) holds
g . (j + 1) = (g . j) + a by A4, RLVECT_1:def_12;
Sum L = Sum (L (#) (MX2FinS M)) by A1, A2, VECTSP_9:3;
then consider f being Function of NAT, the carrier of (n -VectSp_over K) such that
A8: Sum L = f . (len M) and
A9: f . 0 = 0. (n -VectSp_over K) and
A10: for j being Element of NAT
for v being Vector of (n -VectSp_over K) st j < len M & v = (L (#) (MX2FinS M)) . (j + 1) holds
f . (j + 1) = (f . j) + v by A4, RLVECT_1:def_12;
defpred S1[ Nat] means ( $1 <= len M implies for v being Element of (n -VectSp_over K) st v = f . $1 holds
v . i = g . $1 );
A11: len M = m by MATRIX_1:def_2;
A12: for k being Nat st S1[k] holds
S1[k + 1]
proof
reconsider N = n as Element of NAT by ORDINAL1:def_12;
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A13: S1[k] ; ::_thesis: S1[k + 1]
set k1 = k + 1;
reconsider kk = k as Element of NAT by ORDINAL1:def_12;
assume A14: k + 1 <= len M ; ::_thesis: for v being Element of (n -VectSp_over K) st v = f . (k + 1) holds
v . i = g . (k + 1)
then A15: k < len M by NAT_1:13;
A16: width (FinS2MX (L (#) (MX2FinS M))) = n by A4, A14, Th1;
1 <= k + 1 by NAT_1:14;
then A17: k + 1 in Seg (len M) by A14;
then A18: k + 1 in dom (FinS2MX (L (#) (MX2FinS M))) by A4, FINSEQ_1:def_3;
A19: (MX2FinS M) . (k + 1) = Line (M,(k + 1)) by A11, A17, MATRIX_2:8;
then (MX2FinS M) . (k + 1) in lines M by A11, A17, Th103;
then reconsider MXK1 = (MX2FinS M) . (k + 1) as Element of (n -VectSp_over K) ;
k + 1 in dom (MX2FinS M) by A17, FINSEQ_1:def_3;
then (MX2FinS M) /. (k + 1) = (MX2FinS M) . (k + 1) by PARTFUN1:def_6;
then A20: (L (#) (MX2FinS M)) . (k + 1) = (L . MXK1) * MXK1 by A18, VECTSP_6:def_5;
then reconsider LMK1 = (L (#) (MX2FinS M)) . (k + 1) as Element of (n -VectSp_over K) ;
let v be Vector of (n -VectSp_over K); ::_thesis: ( v = f . (k + 1) implies v . i = g . (k + 1) )
assume v = f . (k + 1) ; ::_thesis: v . i = g . (k + 1)
then A21: v = LMK1 + (f . kk) by A10, A15;
reconsider lmk1 = LMK1, mxk1 = MXK1, fk = f . kk as Element of N -tuples_on the carrier of K by Th102;
LMK1 = (L . MXK1) * mxk1 by A20, Th102
.= Line ((FinS2MX (L (#) (MX2FinS M))),(k + 1)) by A17, A19, Th106 ;
then A22: LMK1 . i = (FinS2MX (L (#) (MX2FinS M))) * ((k + 1),i) by A3, A16, MATRIX_1:def_7;
dom lmk1 = Seg n by FINSEQ_2:124;
then A23: lmk1 . i in rng lmk1 by A3, FUNCT_1:def_3;
rng lmk1 c= the carrier of K by FINSEQ_1:def_4;
then reconsider lmk1i = lmk1 . i as Element of K by A23;
(Col ((FinS2MX (L (#) (MX2FinS M))),i)) . (k + 1) = (FinS2MX (L (#) (MX2FinS M))) * ((k + 1),i) by A18, MATRIX_1:def_8;
then A24: g . (k + 1) = lmk1i + (g . kk) by A7, A22, A15;
A25: LMK1 + (f . kk) = lmk1 + fk by Th102;
fk . i = g . kk by A13, A14, NAT_1:13;
hence v . i = g . (k + 1) by A3, A24, A21, A25, FVSUM_1:18; ::_thesis: verum
end;
A26: S1[ 0 ]
proof
assume 0 <= len M ; ::_thesis: for v being Element of (n -VectSp_over K) st v = f . 0 holds
v . i = g . 0
A27: 0. (n -VectSp_over K) = n |-> (0. K) by Th102;
let v be Vector of (n -VectSp_over K); ::_thesis: ( v = f . 0 implies v . i = g . 0 )
assume v = f . 0 ; ::_thesis: v . i = g . 0
hence v . i = g . 0 by A3, A9, A6, A27, FINSEQ_2:57; ::_thesis: verum
end;
for k being Nat holds S1[k] from NAT_1:sch_2(A26, A12);
hence (Sum L) . i = Sum (Col ((FinS2MX (L (#) (MX2FinS M))),i)) by A8, A5; ::_thesis: verum
end;
theorem Th108: :: MATRIX13:108
for n, m being Nat
for K being Field
for M, M1 being Matrix of m,n,K st M is without_repeated_line & ( for i being Nat st i in Seg m holds
ex a being Element of K st Line (M1,i) = a * (Line (M,i)) ) holds
ex L being Linear_Combination of lines M st L (#) (MX2FinS M) = M1
proof
let n, m be Nat; ::_thesis: for K being Field
for M, M1 being Matrix of m,n,K st M is without_repeated_line & ( for i being Nat st i in Seg m holds
ex a being Element of K st Line (M1,i) = a * (Line (M,i)) ) holds
ex L being Linear_Combination of lines M st L (#) (MX2FinS M) = M1
let K be Field; ::_thesis: for M, M1 being Matrix of m,n,K st M is without_repeated_line & ( for i being Nat st i in Seg m holds
ex a being Element of K st Line (M1,i) = a * (Line (M,i)) ) holds
ex L being Linear_Combination of lines M st L (#) (MX2FinS M) = M1
let M, M1 be Matrix of m,n,K; ::_thesis: ( M is without_repeated_line & ( for i being Nat st i in Seg m holds
ex a being Element of K st Line (M1,i) = a * (Line (M,i)) ) implies ex L being Linear_Combination of lines M st L (#) (MX2FinS M) = M1 )
assume that
A1: M is without_repeated_line and
A2: for i being Nat st i in Seg m holds
ex a being Element of K st Line (M1,i) = a * (Line (M,i)) ; ::_thesis: ex L being Linear_Combination of lines M st L (#) (MX2FinS M) = M1
set V = n -VectSp_over K;
defpred S1[ set , set ] means for i being Nat st $1 = i holds
ex a being Element of K st
( a = $2 & Line (M1,i) = a * (Line (M,i)) );
A3: for k being Nat st k in Seg (len M) holds
ex x being Element of K st S1[k,x]
proof
A4: len M = m by MATRIX_1:def_2;
let k be Nat; ::_thesis: ( k in Seg (len M) implies ex x being Element of K st S1[k,x] )
assume k in Seg (len M) ; ::_thesis: ex x being Element of K st S1[k,x]
then consider a being Element of K such that
A5: Line (M1,k) = a * (Line (M,k)) by A2, A4;
take a ; ::_thesis: S1[k,a]
thus S1[k,a] by A5; ::_thesis: verum
end;
consider p being FinSequence of K such that
A6: dom p = Seg (len M) and
A7: for k being Nat st k in Seg (len M) holds
S1[k,p . k] from FINSEQ_1:sch_5(A3);
defpred S2[ set , set ] means for v being Vector of (n -VectSp_over K) st $1 = v holds
( ( not v in lines M implies $2 = 0. K ) & ( v in lines M implies for k being Nat st k in Seg m & v = Line (M,k) holds
$2 = p . k ) );
A8: for x being set st x in the carrier of (n -VectSp_over K) holds
ex y being set st
( y in the carrier of K & S2[x,y] )
proof
let x be set ; ::_thesis: ( x in the carrier of (n -VectSp_over K) implies ex y being set st
( y in the carrier of K & S2[x,y] ) )
assume x in the carrier of (n -VectSp_over K) ; ::_thesis: ex y being set st
( y in the carrier of K & S2[x,y] )
then reconsider v = x as Element of (n -VectSp_over K) ;
percases ( v in lines M or not v in lines M ) ;
supposeA9: v in lines M ; ::_thesis: ex y being set st
( y in the carrier of K & S2[x,y] )
A10: rng p c= the carrier of K by FINSEQ_1:def_4;
consider j being Nat such that
A11: j in Seg m and
A12: v = Line (M,j) by A9, Th103;
len M = m by MATRIX_1:def_2;
then p . j in rng p by A6, A11, FUNCT_1:def_3;
then reconsider pj = p . j as Element of the carrier of K by A10;
take pj ; ::_thesis: ( pj in the carrier of K & S2[x,pj] )
thus pj in the carrier of K ; ::_thesis: S2[x,pj]
let w be Vector of (n -VectSp_over K); ::_thesis: ( x = w implies ( ( not w in lines M implies pj = 0. K ) & ( w in lines M implies for k being Nat st k in Seg m & w = Line (M,k) holds
pj = p . k ) ) )
assume A13: w = x ; ::_thesis: ( ( not w in lines M implies pj = 0. K ) & ( w in lines M implies for k being Nat st k in Seg m & w = Line (M,k) holds
pj = p . k ) )
thus ( not w in lines M implies pj = 0. K ) by A9, A13; ::_thesis: ( w in lines M implies for k being Nat st k in Seg m & w = Line (M,k) holds
pj = p . k )
thus ( w in lines M implies for k being Nat st k in Seg m & w = Line (M,k) holds
pj = p . k ) ::_thesis: verum
proof
len M = m by MATRIX_1:def_2;
then A14: dom M = Seg m by FINSEQ_1:def_3;
A15: M . j = Line (M,j) by A11, MATRIX_2:8;
assume w in lines M ; ::_thesis: for k being Nat st k in Seg m & w = Line (M,k) holds
pj = p . k
let k be Nat; ::_thesis: ( k in Seg m & w = Line (M,k) implies pj = p . k )
assume that
A16: k in Seg m and
A17: w = Line (M,k) ; ::_thesis: pj = p . k
M . k = Line (M,k) by A16, MATRIX_2:8;
hence pj = p . k by A1, A11, A12, A13, A16, A17, A14, A15, FUNCT_1:def_4; ::_thesis: verum
end;
end;
supposeA18: not v in lines M ; ::_thesis: ex y being set st
( y in the carrier of K & S2[x,y] )
take 0K = 0. K; ::_thesis: ( 0K in the carrier of K & S2[x,0K] )
thus 0K in the carrier of K ; ::_thesis: S2[x,0K]
let w be Vector of (n -VectSp_over K); ::_thesis: ( x = w implies ( ( not w in lines M implies 0K = 0. K ) & ( w in lines M implies for k being Nat st k in Seg m & w = Line (M,k) holds
0K = p . k ) ) )
assume A19: w = x ; ::_thesis: ( ( not w in lines M implies 0K = 0. K ) & ( w in lines M implies for k being Nat st k in Seg m & w = Line (M,k) holds
0K = p . k ) )
thus ( not w in lines M implies 0K = 0. K ) ; ::_thesis: ( w in lines M implies for k being Nat st k in Seg m & w = Line (M,k) holds
0K = p . k )
thus ( w in lines M implies for k being Nat st k in Seg m & w = Line (M,k) holds
0K = p . k ) by A18, A19; ::_thesis: verum
end;
end;
end;
consider l being Function of the carrier of (n -VectSp_over K), the carrier of K such that
A20: for x being set st x in the carrier of (n -VectSp_over K) holds
S2[x,l . x] from FUNCT_2:sch_1(A8);
reconsider L = l as Element of Funcs ( the carrier of (n -VectSp_over K), the carrier of K) by FUNCT_2:8;
for v being Vector of (n -VectSp_over K) st not v in lines M holds
L . v = 0. K by A20;
then reconsider L = L as Linear_Combination of n -VectSp_over K by VECTSP_6:def_1;
A21: Carrier L c= lines M
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Carrier L or x in lines M )
assume x in Carrier L ; ::_thesis: x in lines M
then ex v being Vector of (n -VectSp_over K) st
( x = v & L . v <> 0. K ) by VECTSP_6:1;
hence x in lines M by A20; ::_thesis: verum
end;
set MX = MX2FinS M;
A22: len M = m by MATRIX_1:def_2;
reconsider L = L as Linear_Combination of lines M by A21, VECTSP_6:def_4;
set LM = L (#) (MX2FinS M);
A23: len (L (#) (MX2FinS M)) = len M by VECTSP_6:def_5;
A24: now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_k_<=_m_holds_
M1_._k_=_(L_(#)_(MX2FinS_M))_._k
let k be Nat; ::_thesis: ( 1 <= k & k <= m implies M1 . k = (L (#) (MX2FinS M)) . k )
assume that
A25: 1 <= k and
A26: k <= m ; ::_thesis: M1 . k = (L (#) (MX2FinS M)) . k
k in NAT by ORDINAL1:def_12;
then A27: k in Seg m by A25, A26;
then consider a being Element of K such that
A28: p . k = a and
A29: Line (M1,k) = a * (Line (M,k)) by A7, A22;
dom (MX2FinS M) = Seg m by A22, FINSEQ_1:def_3;
then A30: (MX2FinS M) /. k = M . k by A27, PARTFUN1:def_6;
A31: Line (M,k) in lines M by A27, Th103;
then reconsider LMk = Line (M,k) as Element of n -tuples_on the carrier of K by Th102;
A32: LMk = M . k by A27, MATRIX_2:8;
dom (L (#) (MX2FinS M)) = Seg m by A22, A23, FINSEQ_1:def_3;
then A33: (L (#) (MX2FinS M)) . k = (L . ((MX2FinS M) /. k)) * ((MX2FinS M) /. k) by A27, VECTSP_6:def_5;
L . LMk = p . k by A20, A27, A31;
then (L (#) (MX2FinS M)) . k = a * LMk by A28, A33, A30, A32, Th102;
hence M1 . k = (L (#) (MX2FinS M)) . k by A27, A29, MATRIX_2:8; ::_thesis: verum
end;
len M1 = m by MATRIX_1:def_2;
hence ex L being Linear_Combination of lines M st L (#) (MX2FinS M) = M1 by A22, A23, A24, FINSEQ_1:14; ::_thesis: verum
end;
theorem Th109: :: MATRIX13:109
for n, m being Nat
for K being Field
for M being Matrix of m,n,K st M is without_repeated_line holds
( ( ( for i being Nat st i in Seg m holds
Line (M,i) <> n |-> (0. K) ) & ( for M1 being Matrix of m,n,K st ( for i being Nat st i in Seg m holds
ex a being Element of K st Line (M1,i) = a * (Line (M,i)) ) & ( for j being Nat st j in Seg n holds
Sum (Col (M1,j)) = 0. K ) holds
M1 = 0. (K,m,n) ) ) iff lines M is linearly-independent )
proof
let n, m be Nat; ::_thesis: for K being Field
for M being Matrix of m,n,K st M is without_repeated_line holds
( ( ( for i being Nat st i in Seg m holds
Line (M,i) <> n |-> (0. K) ) & ( for M1 being Matrix of m,n,K st ( for i being Nat st i in Seg m holds
ex a being Element of K st Line (M1,i) = a * (Line (M,i)) ) & ( for j being Nat st j in Seg n holds
Sum (Col (M1,j)) = 0. K ) holds
M1 = 0. (K,m,n) ) ) iff lines M is linearly-independent )
let K be Field; ::_thesis: for M being Matrix of m,n,K st M is without_repeated_line holds
( ( ( for i being Nat st i in Seg m holds
Line (M,i) <> n |-> (0. K) ) & ( for M1 being Matrix of m,n,K st ( for i being Nat st i in Seg m holds
ex a being Element of K st Line (M1,i) = a * (Line (M,i)) ) & ( for j being Nat st j in Seg n holds
Sum (Col (M1,j)) = 0. K ) holds
M1 = 0. (K,m,n) ) ) iff lines M is linearly-independent )
set V = n -VectSp_over K;
set n0 = n |-> (0. K);
A1: len (n |-> (0. K)) = n by CARD_1:def_7;
let M be Matrix of m,n,K; ::_thesis: ( M is without_repeated_line implies ( ( ( for i being Nat st i in Seg m holds
Line (M,i) <> n |-> (0. K) ) & ( for M1 being Matrix of m,n,K st ( for i being Nat st i in Seg m holds
ex a being Element of K st Line (M1,i) = a * (Line (M,i)) ) & ( for j being Nat st j in Seg n holds
Sum (Col (M1,j)) = 0. K ) holds
M1 = 0. (K,m,n) ) ) iff lines M is linearly-independent ) )
assume A2: M is without_repeated_line ; ::_thesis: ( ( ( for i being Nat st i in Seg m holds
Line (M,i) <> n |-> (0. K) ) & ( for M1 being Matrix of m,n,K st ( for i being Nat st i in Seg m holds
ex a being Element of K st Line (M1,i) = a * (Line (M,i)) ) & ( for j being Nat st j in Seg n holds
Sum (Col (M1,j)) = 0. K ) holds
M1 = 0. (K,m,n) ) ) iff lines M is linearly-independent )
thus ( ( for i being Nat st i in Seg m holds
Line (M,i) <> n |-> (0. K) ) & ( for M1 being Matrix of m,n,K st ( for i being Nat st i in Seg m holds
ex a being Element of K st Line (M1,i) = a * (Line (M,i)) ) & ( for j being Nat st j in Seg n holds
Sum (Col (M1,j)) = 0. K ) holds
M1 = 0. (K,m,n) ) implies lines M is linearly-independent ) ::_thesis: ( lines M is linearly-independent implies ( ( for i being Nat st i in Seg m holds
Line (M,i) <> n |-> (0. K) ) & ( for M1 being Matrix of m,n,K st ( for i being Nat st i in Seg m holds
ex a being Element of K st Line (M1,i) = a * (Line (M,i)) ) & ( for j being Nat st j in Seg n holds
Sum (Col (M1,j)) = 0. K ) holds
M1 = 0. (K,m,n) ) ) )
proof
set MX = MX2FinS M;
set V = n -VectSp_over K;
assume that
A3: for i being Nat st i in Seg m holds
Line (M,i) <> n |-> (0. K) and
A4: for M1 being Matrix of m,n,K st ( for i being Nat st i in Seg m holds
ex a being Element of K st Line (M1,i) = a * (Line (M,i)) ) & ( for j being Nat st j in Seg n holds
Sum (Col (M1,j)) = 0. K ) holds
M1 = 0. (K,m,n) ; ::_thesis: lines M is linearly-independent
let L be Linear_Combination of lines M; :: according to VECTSP_7:def_1 ::_thesis: ( not Sum L = 0. (n -VectSp_over K) or Carrier L = {} )
assume A5: Sum L = 0. (n -VectSp_over K) ; ::_thesis: Carrier L = {}
set LM = L (#) (MX2FinS M);
set FLM = FinS2MX (L (#) (MX2FinS M));
A6: len M = m by MATRIX_1:def_2;
then reconsider flm = FinS2MX (L (#) (MX2FinS M)) as Matrix of m,n,K by VECTSP_6:def_5;
A7: for i being Nat st i in Seg m holds
ex a being Element of K st Line (flm,i) = a * (Line (M,i))
proof
let i be Nat; ::_thesis: ( i in Seg m implies ex a being Element of K st Line (flm,i) = a * (Line (M,i)) )
assume A8: i in Seg m ; ::_thesis: ex a being Element of K st Line (flm,i) = a * (Line (M,i))
Line (M,i) in lines M by A8, Th103;
then reconsider LM = Line (M,i) as Element of (n -VectSp_over K) ;
reconsider LLM = L . LM as Element of K ;
Line (M,i) = M . i by A8, MATRIX_2:8;
then Line (flm,i) = LLM * (Line (M,i)) by A6, A8, Th106;
hence ex a being Element of K st Line (flm,i) = a * (Line (M,i)) ; ::_thesis: verum
end;
A9: len (n |-> (0. K)) = n by CARD_1:def_7;
assume Carrier L <> {} ; ::_thesis: contradiction
then consider x being set such that
A10: x in Carrier L by XBOOLE_0:def_1;
Carrier L c= lines M by VECTSP_6:def_4;
then consider i being Nat such that
A11: i in Seg m and
A12: x = Line (M,i) by A10, Th103;
consider v being Vector of (n -VectSp_over K) such that
A13: x = v and
A14: L . v <> 0. K by A10, VECTSP_6:1;
reconsider LM = Line (M,i) as Element of n -tuples_on the carrier of K by A12, A13, Th102;
Line (M,i) = M . i by A11, MATRIX_2:8;
then A15: Line (flm,i) = (L . v) * LM by A6, A11, A12, A13, Th106;
now__::_thesis:_for_j_being_Nat_st_j_in_Seg_n_holds_
Sum_(Col_(flm,j))_=_0._K
let j be Nat; ::_thesis: ( j in Seg n implies Sum (Col (flm,j)) = 0. K )
assume A16: j in Seg n ; ::_thesis: Sum (Col (flm,j)) = 0. K
A17: (n |-> (0. K)) . j = 0. K by A16, FINSEQ_2:57;
Carrier L c= lines M by VECTSP_6:def_4;
then (Sum L) . j = Sum (Col (flm,j)) by A2, A16, Th107;
hence Sum (Col (flm,j)) = 0. K by A5, A17, Th102; ::_thesis: verum
end;
then flm = 0. (K,m,n) by A4, A7;
then A18: flm . i = n |-> (0. K) by A11, FUNCOP_1:7;
A19: dom LM = Seg n by FINSEQ_2:124;
len LM = n by CARD_1:def_7;
then consider j being Nat such that
A20: 1 <= j and
A21: j <= n and
A22: LM . j <> (n |-> (0. K)) . j by A3, A11, A9, FINSEQ_1:14;
A23: j in NAT by ORDINAL1:def_12;
then A24: j in Seg n by A20, A21;
then A25: LM . j <> 0. K by A22, FINSEQ_2:57;
j in Seg n by A20, A21, A23;
then A26: LM . j in rng LM by A19, FUNCT_1:def_3;
rng LM c= the carrier of K by RELAT_1:def_19;
then reconsider LMj = LM . j as Element of K by A26;
n in NAT by ORDINAL1:def_12;
then A27: ((L . v) * LM) . j = (L . v) * LMj by A24, FVSUM_1:51;
flm . i = Line (flm,i) by A11, MATRIX_2:8;
then (Line (flm,i)) . j = 0. K by A18, A24, FINSEQ_2:57;
hence contradiction by A14, A25, A27, A15, VECTSP_1:12; ::_thesis: verum
end;
assume A28: lines M is linearly-independent ; ::_thesis: ( ( for i being Nat st i in Seg m holds
Line (M,i) <> n |-> (0. K) ) & ( for M1 being Matrix of m,n,K st ( for i being Nat st i in Seg m holds
ex a being Element of K st Line (M1,i) = a * (Line (M,i)) ) & ( for j being Nat st j in Seg n holds
Sum (Col (M1,j)) = 0. K ) holds
M1 = 0. (K,m,n) ) )
hereby ::_thesis: for M1 being Matrix of m,n,K st ( for i being Nat st i in Seg m holds
ex a being Element of K st Line (M1,i) = a * (Line (M,i)) ) & ( for j being Nat st j in Seg n holds
Sum (Col (M1,j)) = 0. K ) holds
M1 = 0. (K,m,n)
let i be Nat; ::_thesis: ( i in Seg m implies Line (M,i) <> n |-> (0. K) )
assume i in Seg m ; ::_thesis: Line (M,i) <> n |-> (0. K)
then Line (M,i) in lines M by Th103;
then Line (M,i) <> 0. (n -VectSp_over K) by A28, VECTSP_7:2;
hence Line (M,i) <> n |-> (0. K) by Th102; ::_thesis: verum
end;
let M1 be Matrix of m,n,K; ::_thesis: ( ( for i being Nat st i in Seg m holds
ex a being Element of K st Line (M1,i) = a * (Line (M,i)) ) & ( for j being Nat st j in Seg n holds
Sum (Col (M1,j)) = 0. K ) implies M1 = 0. (K,m,n) )
assume that
A29: for i being Nat st i in Seg m holds
ex a being Element of K st Line (M1,i) = a * (Line (M,i)) and
A30: for j being Nat st j in Seg n holds
Sum (Col (M1,j)) = 0. K ; ::_thesis: M1 = 0. (K,m,n)
consider L being Linear_Combination of lines M such that
A31: L (#) (MX2FinS M) = M1 by A2, A29, Th108;
A32: Carrier L c= lines M by VECTSP_6:def_4;
A33: now__::_thesis:_for_j_being_Nat_st_1_<=_j_&_j_<=_n_holds_
(Sum_L)_._j_=_(n_|->_(0._K))_._j
let j be Nat; ::_thesis: ( 1 <= j & j <= n implies (Sum L) . j = (n |-> (0. K)) . j )
assume that
A34: 1 <= j and
A35: j <= n ; ::_thesis: (Sum L) . j = (n |-> (0. K)) . j
j in NAT by ORDINAL1:def_12;
then A36: j in Seg n by A34, A35;
hence (Sum L) . j = Sum (Col ((FinS2MX (L (#) (MX2FinS M))),j)) by A2, A32, Th107
.= 0. K by A30, A31, A36
.= (n |-> (0. K)) . j by A36, FINSEQ_2:57 ;
::_thesis: verum
end;
reconsider SumL = Sum L as Element of n -tuples_on the carrier of K by Th102;
len SumL = n by CARD_1:def_7;
then SumL = n |-> (0. K) by A1, A33, FINSEQ_1:14
.= 0. (n -VectSp_over K) by Th102 ;
then A37: Carrier L = {} by A28, VECTSP_7:def_1;
assume M1 <> 0. (K,m,n) ; ::_thesis: contradiction
then consider I, J being Nat such that
A38: [I,J] in Indices M1 and
A39: M1 * (I,J) <> (0. (K,m,n)) * (I,J) by MATRIX_1:27;
[I,J] in Indices (0. (K,m,n)) by A38, MATRIX_1:26;
then A40: M1 * (I,J) <> 0. K by A39, MATRIX_3:1;
reconsider ii = I, jj = J as Element of NAT by ORDINAL1:def_12;
A41: Indices M1 = Indices M by MATRIX_1:26;
then Indices M1 = [:(Seg (len M)),(Seg (width M)):] by FINSEQ_1:def_3;
then A42: ii in Seg (len M) by A38, ZFMISC_1:87;
A43: len M = m by MATRIX_1:def_2;
then Line (M,ii) in lines M by A42, Th103;
then reconsider Mii = M . ii as Element of (n -VectSp_over K) by A42, A43, MATRIX_2:8;
A44: jj in Seg (width M) by A38, A41, ZFMISC_1:87;
then A45: (Line (M,ii)) . jj = M * (ii,jj) by MATRIX_1:def_7;
jj in Seg (width M1) by A38, ZFMISC_1:87;
then M1 * (I,J) = (Line ((FinS2MX (L (#) (MX2FinS M))),ii)) . jj by A31, MATRIX_1:def_7
.= ((L . Mii) * (Line (M,ii))) . jj by A42, Th106
.= (L . Mii) * (M * (ii,jj)) by A44, A45, FVSUM_1:51 ;
then L . Mii <> 0. K by A40, VECTSP_1:12;
hence contradiction by A37, VECTSP_6:1; ::_thesis: verum
end;
theorem Th110: :: MATRIX13:110
for n, m being Nat
for K being Field
for M being Matrix of m,n,K st the_rank_of M = m holds
lines M is linearly-independent
proof
let n, m be Nat; ::_thesis: for K being Field
for M being Matrix of m,n,K st the_rank_of M = m holds
lines M is linearly-independent
let K be Field; ::_thesis: for M being Matrix of m,n,K st the_rank_of M = m holds
lines M is linearly-independent
let M be Matrix of m,n,K; ::_thesis: ( the_rank_of M = m implies lines M is linearly-independent )
assume A1: the_rank_of M = m ; ::_thesis: lines M is linearly-independent
reconsider N = n as Element of NAT by ORDINAL1:def_12;
set V = n -VectSp_over K;
percases ( m = 0 or m <> 0 ) ;
suppose m = 0 ; ::_thesis: lines M is linearly-independent
then len M = 0 by MATRIX_1:def_2;
then M = {} ;
hence lines M is linearly-independent ; ::_thesis: verum
end;
supposeA2: m <> 0 ; ::_thesis: lines M is linearly-independent
then A3: width M = n by Th1;
A4: M is without_repeated_line by A1, Th105;
A5: now__::_thesis:_for_M1_being_Matrix_of_m,n,K_st_(_for_i_being_Nat_st_i_in_Seg_m_holds_
ex_a_being_Element_of_K_st_Line_(M1,i)_=_a_*_(Line_(M,i))_)_&_(_for_j_being_Nat_st_j_in_Seg_n_holds_
Sum_(Col_(M1,j))_=_0._K_)_holds_
not_M1_<>_0._(K,m,n)
set n0 = n |-> (0. K);
set NULL = 0. (K,m,n);
let M1 be Matrix of m,n,K; ::_thesis: ( ( for i being Nat st i in Seg m holds
ex a being Element of K st Line (M1,i) = a * (Line (M,i)) ) & ( for j being Nat st j in Seg n holds
Sum (Col (M1,j)) = 0. K ) implies not M1 <> 0. (K,m,n) )
assume that
A6: for i being Nat st i in Seg m holds
ex a being Element of K st Line (M1,i) = a * (Line (M,i)) and
A7: for j being Nat st j in Seg n holds
Sum (Col (M1,j)) = 0. K ; ::_thesis: not M1 <> 0. (K,m,n)
assume M1 <> 0. (K,m,n) ; ::_thesis: contradiction
then consider i, j being Nat such that
A8: [i,j] in Indices M1 and
A9: M1 * (i,j) <> (0. (K,m,n)) * (i,j) by MATRIX_1:27;
reconsider i = i, j = j as Element of NAT by ORDINAL1:def_12;
A10: len M = m by MATRIX_1:def_2;
Indices M1 = Indices (0. (K,m,n)) by MATRIX_1:26;
then A11: M1 * (i,j) <> 0. K by A8, A9, MATRIX_3:1;
A12: Indices M = [:(Seg m),(Seg n):] by A3, MATRIX_1:25;
Indices M1 = Indices M by MATRIX_1:26;
then A13: i in Seg m by A12, A8, ZFMISC_1:87;
then consider a being Element of K such that
A14: Line (M1,i) = a * (Line (M,i)) by A6;
A15: width M1 = n by A2, Th1;
then A16: j in Seg n by A8, ZFMISC_1:87;
then A17: (Line (M,i)) . j = M * (i,j) by A3, MATRIX_1:def_7;
set R = RLine (M,i,(a * (Line (M,i))));
consider L being Linear_Combination of lines M such that
A18: L (#) (MX2FinS M) = M1 by A1, A6, Th105, Th108;
set LM = L (#) (MX2FinS M);
len M1 = len M by A18, VECTSP_6:def_5;
then consider f being Function of NAT, the carrier of (n -VectSp_over K) such that
A19: Sum (L (#) (MX2FinS M)) = f . m and
A20: f . 0 = 0. (n -VectSp_over K) and
A21: for j being Element of NAT
for v being Vector of (n -VectSp_over K) st j < m & v = (L (#) (MX2FinS M)) . (j + 1) holds
f . (j + 1) = (f . j) + v by A18, A10, RLVECT_1:def_12;
set RR = RLine ((RLine (M,i,(a * (Line (M,i))))),i,(n |-> (0. K)));
A22: len (RLine ((RLine (M,i,(a * (Line (M,i))))),i,(n |-> (0. K)))) = m by MATRIX_1:def_2;
defpred S1[ Nat] means ( $1 < i implies for t being Element of n -tuples_on the carrier of K st t = f . $1 holds
the_rank_of (RLine (M,i,(a * (Line (M,i))))) = the_rank_of (RLine ((RLine (M,i,(a * (Line (M,i))))),i,((Line ((RLine (M,i,(a * (Line (M,i))))),i)) + t))) );
width M = len (Line (M,i)) by MATRIX_1:def_7
.= len (a * (Line (M,i))) by MATRIXR1:16 ;
then A23: width (RLine (M,i,(a * (Line (M,i))))) = width M by MATRIX11:def_3;
A24: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A25: S1[k] ; ::_thesis: S1[k + 1]
reconsider kk = k as Element of NAT by ORDINAL1:def_12;
set k1 = k + 1;
A26: 1 <= k + 1 by NAT_1:14;
A27: i <= m by A13, FINSEQ_1:1;
reconsider LR = Line ((RLine (M,i,(a * (Line (M,i))))),i), LM1 = Line (M1,(k + 1)) as Element of N -tuples_on the carrier of K by A2, Th1;
assume A28: k + 1 < i ; ::_thesis: for t being Element of n -tuples_on the carrier of K st t = f . (k + 1) holds
the_rank_of (RLine (M,i,(a * (Line (M,i))))) = the_rank_of (RLine ((RLine (M,i,(a * (Line (M,i))))),i,((Line ((RLine (M,i,(a * (Line (M,i))))),i)) + t)))
let t be Element of n -tuples_on the carrier of K; ::_thesis: ( t = f . (k + 1) implies the_rank_of (RLine (M,i,(a * (Line (M,i))))) = the_rank_of (RLine ((RLine (M,i,(a * (Line (M,i))))),i,((Line ((RLine (M,i,(a * (Line (M,i))))),i)) + t))) )
assume A29: t = f . (k + 1) ; ::_thesis: the_rank_of (RLine (M,i,(a * (Line (M,i))))) = the_rank_of (RLine ((RLine (M,i,(a * (Line (M,i))))),i,((Line ((RLine (M,i,(a * (Line (M,i))))),i)) + t)))
reconsider t1 = f . kk, T = t as Element of N -tuples_on the carrier of K by Th102;
set RR = RLine ((RLine (M,i,(a * (Line (M,i))))),i,((Line ((RLine (M,i,(a * (Line (M,i))))),i)) + t1));
reconsider LRt = LR + t, LRt1 = LR + t1 as Element of the carrier of K * by FINSEQ_1:def_11;
A30: len (LR + T) = n by CARD_1:def_7;
A31: len (LR + t1) = width (RLine (M,i,(a * (Line (M,i))))) by A3, A23, CARD_1:def_7;
then width (RLine ((RLine (M,i,(a * (Line (M,i))))),i,((Line ((RLine (M,i,(a * (Line (M,i))))),i)) + t1))) = width (RLine (M,i,(a * (Line (M,i))))) by MATRIX11:def_3;
then A32: RLine ((RLine ((RLine (M,i,(a * (Line (M,i))))),i,((Line ((RLine (M,i,(a * (Line (M,i))))),i)) + t1))),i,(LR + t)) = Replace ((RLine ((RLine (M,i,(a * (Line (M,i))))),i,((Line ((RLine (M,i,(a * (Line (M,i))))),i)) + t1))),i,LRt) by A3, A23, A30, MATRIX11:29
.= Replace ((Replace ((RLine (M,i,(a * (Line (M,i))))),i,LRt1)),i,LRt) by A31, MATRIX11:29
.= Replace ((RLine (M,i,(a * (Line (M,i))))),i,LRt) by FUNCT_7:34
.= RLine ((RLine (M,i,(a * (Line (M,i))))),i,(LR + t)) by A3, A23, A30, MATRIX11:29 ;
i <= m by A13, FINSEQ_1:1;
then k + 1 < m by A28, XXREAL_0:2;
then A33: k + 1 in Seg m by A26;
then A34: Line (M1,(k + 1)) = M1 . (k + 1) by MATRIX_2:8;
Line (M1,(k + 1)) in lines M1 by A33, Th103;
then reconsider LMk1 = (L (#) (MX2FinS M)) . (k + 1) as Element of (n -VectSp_over K) by A18, A33, MATRIX_2:8;
consider a being Element of K such that
A35: Line (M1,(k + 1)) = a * (Line (M,(k + 1))) by A6, A33;
A36: Line (M,(k + 1)) = Line ((RLine (M,i,(a * (Line (M,i))))),(k + 1)) by A28, A33, MATRIX11:28
.= Line ((RLine ((RLine (M,i,(a * (Line (M,i))))),i,((Line ((RLine (M,i,(a * (Line (M,i))))),i)) + t1))),(k + 1)) by A28, A33, MATRIX11:28 ;
k < i by A28, NAT_1:13;
then k < m by A27, XXREAL_0:2;
then t = (f . kk) + LMk1 by A21, A29
.= t1 + (Line (M1,(k + 1))) by A15, A18, A34, Th102 ;
then A37: LR + t = (LR + t1) + LM1 by FINSEQOP:28
.= (Line ((RLine ((RLine (M,i,(a * (Line (M,i))))),i,((Line ((RLine (M,i,(a * (Line (M,i))))),i)) + t1))),i)) + (a * (Line ((RLine ((RLine (M,i,(a * (Line (M,i))))),i,((Line ((RLine (M,i,(a * (Line (M,i))))),i)) + t1))),(k + 1)))) by A13, A35, A31, A36, MATRIX11:28 ;
A38: len (RLine ((RLine (M,i,(a * (Line (M,i))))),i,((Line ((RLine (M,i,(a * (Line (M,i))))),i)) + t1))) = m by MATRIX_1:def_2;
the_rank_of (RLine (M,i,(a * (Line (M,i))))) = the_rank_of (RLine ((RLine (M,i,(a * (Line (M,i))))),i,((Line ((RLine (M,i,(a * (Line (M,i))))),i)) + t1))) by A25, A28, NAT_1:13;
hence the_rank_of (RLine (M,i,(a * (Line (M,i))))) = the_rank_of (RLine ((RLine (M,i,(a * (Line (M,i))))),i,((Line ((RLine (M,i,(a * (Line (M,i))))),i)) + t))) by A28, A33, A38, A32, A37, Th92; ::_thesis: verum
end;
defpred S2[ Nat] means ( i <= $1 & $1 <= m implies for t being Element of n -tuples_on the carrier of K st t = f . $1 holds
the_rank_of (RLine (M,i,(a * (Line (M,i))))) = the_rank_of (RLine ((RLine (M,i,(a * (Line (M,i))))),i,t)) );
A39: S1[ 0 ]
proof
assume 0 < i ; ::_thesis: for t being Element of n -tuples_on the carrier of K st t = f . 0 holds
the_rank_of (RLine (M,i,(a * (Line (M,i))))) = the_rank_of (RLine ((RLine (M,i,(a * (Line (M,i))))),i,((Line ((RLine (M,i,(a * (Line (M,i))))),i)) + t)))
let t be Element of n -tuples_on the carrier of K; ::_thesis: ( t = f . 0 implies the_rank_of (RLine (M,i,(a * (Line (M,i))))) = the_rank_of (RLine ((RLine (M,i,(a * (Line (M,i))))),i,((Line ((RLine (M,i,(a * (Line (M,i))))),i)) + t))) )
assume t = f . 0 ; ::_thesis: the_rank_of (RLine (M,i,(a * (Line (M,i))))) = the_rank_of (RLine ((RLine (M,i,(a * (Line (M,i))))),i,((Line ((RLine (M,i,(a * (Line (M,i))))),i)) + t)))
then t = n |-> (0. K) by A20, Th102;
then (Line ((RLine (M,i,(a * (Line (M,i))))),i)) + t = Line ((RLine (M,i,(a * (Line (M,i))))),i) by A3, A23, FVSUM_1:21;
hence the_rank_of (RLine (M,i,(a * (Line (M,i))))) = the_rank_of (RLine ((RLine (M,i,(a * (Line (M,i))))),i,((Line ((RLine (M,i,(a * (Line (M,i))))),i)) + t))) by MATRIX11:30; ::_thesis: verum
end;
A40: for k being Nat holds S1[k] from NAT_1:sch_2(A39, A24);
A41: for k being Nat st S2[k] holds
S2[k + 1]
proof
let k be Nat; ::_thesis: ( S2[k] implies S2[k + 1] )
assume A42: S2[k] ; ::_thesis: S2[k + 1]
reconsider kk = k as Element of NAT by ORDINAL1:def_12;
reconsider t1 = f . kk as Element of N -tuples_on the carrier of K by Th102;
set k1 = k + 1;
reconsider LR = Line ((RLine (M,i,(a * (Line (M,i))))),i), LM1 = Line (M1,(k + 1)) as Element of n -tuples_on the carrier of K by A2, Th1;
assume that
A43: i <= k + 1 and
A44: k + 1 <= m ; ::_thesis: for t being Element of n -tuples_on the carrier of K st t = f . (k + 1) holds
the_rank_of (RLine (M,i,(a * (Line (M,i))))) = the_rank_of (RLine ((RLine (M,i,(a * (Line (M,i))))),i,t))
A45: k < m by A44, NAT_1:13;
1 <= k + 1 by NAT_1:14;
then A46: k + 1 in Seg m by A44;
then A47: Line (M1,(k + 1)) = M1 . (k + 1) by MATRIX_2:8;
Line (M1,(k + 1)) in lines M1 by A46, Th103;
then reconsider LMk1 = (L (#) (MX2FinS M)) . (k + 1) as Element of (n -VectSp_over K) by A18, A46, MATRIX_2:8;
let t be Element of n -tuples_on the carrier of K; ::_thesis: ( t = f . (k + 1) implies the_rank_of (RLine (M,i,(a * (Line (M,i))))) = the_rank_of (RLine ((RLine (M,i,(a * (Line (M,i))))),i,t)) )
assume t = f . (k + 1) ; ::_thesis: the_rank_of (RLine (M,i,(a * (Line (M,i))))) = the_rank_of (RLine ((RLine (M,i,(a * (Line (M,i))))),i,t))
then A48: t = (f . kk) + LMk1 by A21, A45
.= t1 + LM1 by A18, A47, Th102 ;
consider b being Element of K such that
A49: Line (M1,(k + 1)) = b * (Line (M,(k + 1))) by A6, A46;
reconsider T = t, T1 = t1 as Element of the carrier of K * by FINSEQ_1:def_11;
percases ( i = k + 1 or i < k + 1 ) by A43, XXREAL_0:1;
supposeA50: i = k + 1 ; ::_thesis: the_rank_of (RLine (M,i,(a * (Line (M,i))))) = the_rank_of (RLine ((RLine (M,i,(a * (Line (M,i))))),i,t))
len LM1 = n by CARD_1:def_7;
then LR = LM1 by A3, A14, A46, A50, MATRIX11:28;
then A51: LR + t1 = t by A48, FINSEQOP:33;
k < i by A50, NAT_1:13;
hence the_rank_of (RLine (M,i,(a * (Line (M,i))))) = the_rank_of (RLine ((RLine (M,i,(a * (Line (M,i))))),i,t)) by A40, A51; ::_thesis: verum
end;
supposeA52: i < k + 1 ; ::_thesis: the_rank_of (RLine (M,i,(a * (Line (M,i))))) = the_rank_of (RLine ((RLine (M,i,(a * (Line (M,i))))),i,t))
set RR = RLine ((RLine (M,i,(a * (Line (M,i))))),i,t1);
A53: len t1 = width (RLine (M,i,(a * (Line (M,i))))) by A3, A23, CARD_1:def_7;
then Line ((RLine ((RLine (M,i,(a * (Line (M,i))))),i,t1)),i) = t1 by A13, MATRIX11:28;
then A54: t = (Line ((RLine ((RLine (M,i,(a * (Line (M,i))))),i,t1)),i)) + (b * (Line ((RLine (M,i,(a * (Line (M,i))))),(k + 1)))) by A46, A49, A48, A52, MATRIX11:28
.= (Line ((RLine ((RLine (M,i,(a * (Line (M,i))))),i,t1)),i)) + (b * (Line ((RLine ((RLine (M,i,(a * (Line (M,i))))),i,t1)),(k + 1)))) by A46, A52, MATRIX11:28 ;
A55: len t = n by CARD_1:def_7;
width (RLine ((RLine (M,i,(a * (Line (M,i))))),i,t1)) = width (RLine (M,i,(a * (Line (M,i))))) by A53, MATRIX11:def_3;
then A56: RLine ((RLine ((RLine (M,i,(a * (Line (M,i))))),i,t1)),i,t) = Replace ((RLine ((RLine (M,i,(a * (Line (M,i))))),i,t1)),i,T) by A3, A23, A55, MATRIX11:29
.= Replace ((Replace ((RLine (M,i,(a * (Line (M,i))))),i,T1)),i,T) by A53, MATRIX11:29
.= Replace ((RLine (M,i,(a * (Line (M,i))))),i,T) by FUNCT_7:34
.= RLine ((RLine (M,i,(a * (Line (M,i))))),i,t) by A3, A23, A55, MATRIX11:29 ;
A57: len (RLine ((RLine (M,i,(a * (Line (M,i))))),i,t1)) = m by MATRIX_1:def_2;
the_rank_of (RLine (M,i,(a * (Line (M,i))))) = the_rank_of (RLine ((RLine (M,i,(a * (Line (M,i))))),i,t1)) by A42, A44, A52, NAT_1:13;
hence the_rank_of (RLine (M,i,(a * (Line (M,i))))) = the_rank_of (RLine ((RLine (M,i,(a * (Line (M,i))))),i,t)) by A46, A52, A54, A56, A57, Th92; ::_thesis: verum
end;
end;
end;
A58: S2[ 0 ] by A13;
A59: for k being Nat holds S2[k] from NAT_1:sch_2(A58, A41);
reconsider SumLM = Sum (L (#) (MX2FinS M)) as Element of n -tuples_on the carrier of K by Th102;
A60: now__::_thesis:_for_j_being_Nat_st_1_<=_j_&_j_<=_n_holds_
SumLM_._j_=_(n_|->_(0._K))_._j
let j be Nat; ::_thesis: ( 1 <= j & j <= n implies SumLM . j = (n |-> (0. K)) . j )
assume that
A61: 1 <= j and
A62: j <= n ; ::_thesis: SumLM . j = (n |-> (0. K)) . j
j in NAT by ORDINAL1:def_12;
then A63: j in Seg n by A61, A62;
A64: Carrier L c= lines M by VECTSP_6:def_4;
M1 = FinS2MX (L (#) (MX2FinS M)) by A18;
then Sum (Col (M1,j)) = (Sum L) . j by A1, A63, A64, Th105, Th107
.= SumLM . j by A4, A64, VECTSP_9:3 ;
hence SumLM . j = 0. K by A7, A63
.= (n |-> (0. K)) . j by A63, FINSEQ_2:57 ;
::_thesis: verum
end;
dom (RLine ((RLine (M,i,(a * (Line (M,i))))),i,(n |-> (0. K)))) = Seg (len (RLine ((RLine (M,i,(a * (Line (M,i))))),i,(n |-> (0. K))))) by FINSEQ_1:def_3;
then consider k being Nat such that
A65: m = k + 1 and
A66: len (Del ((RLine ((RLine (M,i,(a * (Line (M,i))))),i,(n |-> (0. K)))),i)) = k by A13, A22, FINSEQ_3:104;
A67: len SumLM = n by CARD_1:def_7;
M1 * (i,j) = (Line (M1,i)) . j by A15, A16, MATRIX_1:def_7
.= a * (M * (i,j)) by A3, A16, A14, A17, FVSUM_1:51 ;
then a <> 0. K by A11, VECTSP_1:12;
then A68: m = the_rank_of (RLine (M,i,(a * (Line (M,i))))) by A1, Th89;
A69: len (n |-> (0. K)) = n by CARD_1:def_7;
then A70: width (RLine ((RLine (M,i,(a * (Line (M,i))))),i,(n |-> (0. K)))) = width (RLine (M,i,(a * (Line (M,i))))) by A3, A23, MATRIX11:def_3;
A71: Line ((RLine ((RLine (M,i,(a * (Line (M,i))))),i,(n |-> (0. K)))),i) = n |-> (0. K) by A3, A13, A23, A69, MATRIX11:28;
i <= m by A13, FINSEQ_1:1;
then m = the_rank_of (RLine ((RLine (M,i,(a * (Line (M,i))))),i,(n |-> (0. K)))) by A68, A19, A59, A67, A69, A60, FINSEQ_1:14;
then m = the_rank_of (DelLine ((RLine ((RLine (M,i,(a * (Line (M,i))))),i,(n |-> (0. K)))),i)) by A3, A23, A71, A70, Th90;
then m <= k by A66, Th74;
hence contradiction by A65, NAT_1:13; ::_thesis: verum
end;
now__::_thesis:_for_i_being_Nat_st_i_in_Seg_m_holds_
not_Line_(M,i)_=_n_|->_(0._K)
A72: len M = m by MATRIX_1:def_2;
A73: dom M = Seg (len M) by FINSEQ_1:def_3;
let i be Nat; ::_thesis: ( i in Seg m implies not Line (M,i) = n |-> (0. K) )
assume i in Seg m ; ::_thesis: not Line (M,i) = n |-> (0. K)
then consider k being Nat such that
A74: m = k + 1 and
A75: len (Del (M,i)) = k by A73, A72, FINSEQ_3:104;
assume Line (M,i) = n |-> (0. K) ; ::_thesis: contradiction
then the_rank_of (DelLine (M,i)) = m by A1, A3, Th90;
then m <= k by A75, Th74;
hence contradiction by A74, NAT_1:13; ::_thesis: verum
end;
hence lines M is linearly-independent by A4, A5, Th109; ::_thesis: verum
end;
end;
end;
theorem Th111: :: MATRIX13:111
for n being Nat
for K being Field
for M being Diagonal of n,K st the_rank_of M = n holds
lines M is Basis of n -VectSp_over K
proof
let n be Nat; ::_thesis: for K being Field
for M being Diagonal of n,K st the_rank_of M = n holds
lines M is Basis of n -VectSp_over K
let K be Field; ::_thesis: for M being Diagonal of n,K st the_rank_of M = n holds
lines M is Basis of n -VectSp_over K
let M be Diagonal of n,K; ::_thesis: ( the_rank_of M = n implies lines M is Basis of n -VectSp_over K )
assume A1: the_rank_of M = n ; ::_thesis: lines M is Basis of n -VectSp_over K
set lM = lines M;
set V = n -VectSp_over K;
reconsider V9 = n -VectSp_over K as Subspace of n -VectSp_over K by VECTSP_4:24;
now__::_thesis:_for_v_being_Vector_of_(n_-VectSp_over_K)_holds_
(_(_v_in_Lin_(lines_M)_implies_v_in_V9_)_&_(_v_in_V9_implies_v_in_Lin_(lines_M)_)_)
let v be Vector of (n -VectSp_over K); ::_thesis: ( ( v in Lin (lines M) implies v in V9 ) & ( v in V9 implies v in Lin (lines M) ) )
thus ( v in Lin (lines M) implies v in V9 ) by STRUCT_0:def_5; ::_thesis: ( v in V9 implies v in Lin (lines M) )
thus ( v in V9 implies v in Lin (lines M) ) ::_thesis: verum
proof
reconsider t = v as Element of n -tuples_on the carrier of K by Th102;
assume v in V9 ; ::_thesis: v in Lin (lines M)
deffunc H1( Nat) -> Element of (width M) -tuples_on the carrier of K = ((t /. $1) * ((M * ($1,$1)) ")) * (Line (M,$1));
consider f being FinSequence of (width M) -tuples_on the carrier of K such that
A2: len f = n and
A3: for j being Nat st j in dom f holds
f . j = H1(j) from FINSEQ_2:sch_1();
A4: dom f = Seg n by A2, FINSEQ_1:def_3;
width M = n by MATRIX_1:24;
then reconsider f = f as FinSequence of the carrier of (n -VectSp_over K) by Th102;
reconsider M1 = FinS2MX f as Matrix of n,K by A2;
now__::_thesis:_for_i_being_Nat_st_i_in_Seg_n_holds_
ex_a_being_Element_of_K_st_Line_(M1,i)_=_a_*_(Line_(M,i))
let i be Nat; ::_thesis: ( i in Seg n implies ex a being Element of K st Line (M1,i) = a * (Line (M,i)) )
assume A5: i in Seg n ; ::_thesis: ex a being Element of K st Line (M1,i) = a * (Line (M,i))
Line (M1,i) = M1 . i by A5, MATRIX_2:8
.= H1(i) by A3, A4, A5 ;
hence ex a being Element of K st Line (M1,i) = a * (Line (M,i)) ; ::_thesis: verum
end;
then consider L being Linear_Combination of lines M such that
A6: L (#) (MX2FinS M) = M1 by A1, Th105, Th108;
set MX = MX2FinS M;
A7: len t = n by CARD_1:def_7;
reconsider SumL = Sum L as Element of n -tuples_on the carrier of K by Th102;
A8: Carrier L c= lines M by VECTSP_6:def_4;
A9: now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_n_holds_
SumL_._i_=_t_._i
set diag = diagonal_of_Matrix M;
let i be Nat; ::_thesis: ( 1 <= i & i <= n implies SumL . i = t . i )
assume that
A10: 1 <= i and
A11: i <= n ; ::_thesis: SumL . i = t . i
i in NAT by ORDINAL1:def_12;
then A12: i in Seg n by A10, A11;
then A13: (diagonal_of_Matrix M) . i = M * (i,i) by MATRIX_3:def_10;
A14: len (diagonal_of_Matrix M) = n by MATRIX_3:def_10;
then A15: dom (diagonal_of_Matrix M) = Seg n by FINSEQ_1:def_3;
A16: width M = n by MATRIX_1:24;
then A17: (Line (M,i)) . i = M * (i,i) by A12, MATRIX_1:def_7;
set C = Col (M1,i);
A18: dom t = Seg n by FINSEQ_2:124;
len (Col (M1,i)) = len M1 by MATRIX_1:def_8;
then A19: dom (Col (M1,i)) = Seg (len M1) by FINSEQ_1:def_3;
len M = n by MATRIX_1:24;
then A20: dom M = Seg n by FINSEQ_1:def_3;
A21: Det M <> 0. K by A1, Th83;
A22: len M1 = n by MATRIX_1:24;
then A23: dom M1 = Seg n by FINSEQ_1:def_3;
Det M = Product (diagonal_of_Matrix M) by A10, A11, A14, MATRIX_7:17, NAT_1:14;
then A24: (diagonal_of_Matrix M) . i <> 0. K by A12, A21, A15, FVSUM_1:82;
A25: Line (M1,i) = M1 . i by A12, MATRIX_2:8
.= ((t /. i) * ((M * (i,i)) ")) * (Line (M,i)) by A3, A12, A23 ;
A26: width M1 = n by MATRIX_1:24;
now__::_thesis:_for_k_being_Nat_st_k_in_dom_(Col_(M1,i))_&_k_<>_i_holds_
0._K_=_(Col_(M1,i))_._k
let k be Nat; ::_thesis: ( k in dom (Col (M1,i)) & k <> i implies 0. K = (Col (M1,i)) . k )
assume that
A27: k in dom (Col (M1,i)) and
A28: k <> i ; ::_thesis: 0. K = (Col (M1,i)) . k
A29: [k,i] in Indices M by A22, A16, A12, A19, A20, A27, ZFMISC_1:87;
A30: (Line (M,k)) . i = M * (k,i) by A16, A12, MATRIX_1:def_7
.= 0. K by A28, A29, MATRIX_1:def_14 ;
A31: (MX2FinS M) /. k = M . k by A22, A19, A20, A27, PARTFUN1:def_6
.= Line (M,k) by A22, A19, A27, MATRIX_2:8 ;
Line (M1,k) = M1 . k by A19, A27, MATRIX_2:8
.= (L . ((MX2FinS M) /. k)) * ((MX2FinS M) /. k) by A6, A22, A19, A23, A27, VECTSP_6:def_5
.= (L . ((MX2FinS M) /. k)) * (Line (M,k)) by A16, A31, Th102 ;
then (Line (M1,k)) . i = (L . ((MX2FinS M) /. k)) * (0. K) by A16, A12, A30, FVSUM_1:51
.= 0. K by VECTSP_1:7 ;
hence 0. K = M1 * (k,i) by A26, A12, MATRIX_1:def_7
.= (Col (M1,i)) . k by A22, A19, A23, A27, MATRIX_1:def_8 ;
::_thesis: verum
end;
then (Col (M1,i)) . i = Sum (Col (M1,i)) by A22, A12, A19, MATRIX_3:12
.= SumL . i by A1, A6, A8, A12, Th105, Th107 ;
hence SumL . i = M1 * (i,i) by A12, A23, MATRIX_1:def_8
.= (Line (M1,i)) . i by A26, A12, MATRIX_1:def_7
.= ((t /. i) * ((M * (i,i)) ")) * (M * (i,i)) by A16, A12, A25, A17, FVSUM_1:51
.= (t /. i) * (((M * (i,i)) ") * (M * (i,i))) by GROUP_1:def_3
.= (t /. i) * (1_ K) by A24, A13, VECTSP_1:def_10
.= t /. i by VECTSP_1:def_4
.= t . i by A12, A18, PARTFUN1:def_6 ;
::_thesis: verum
end;
len SumL = n by CARD_1:def_7;
then SumL = t by A7, A9, FINSEQ_1:14;
hence v in Lin (lines M) by VECTSP_7:7; ::_thesis: verum
end;
end;
then A32: Lin (lines M) = VectSpStr(# the carrier of (n -VectSp_over K), the addF of (n -VectSp_over K), the ZeroF of (n -VectSp_over K), the lmult of (n -VectSp_over K) #) by VECTSP_4:30;
lines M is linearly-independent by A1, Th110;
hence lines M is Basis of n -VectSp_over K by A32, VECTSP_7:def_3; ::_thesis: verum
end;
Lm8: for n being Nat
for K being Field holds
( lines (1. (K,n)) is Basis of n -VectSp_over K & the_rank_of (1. (K,n)) = n )
proof
let n be Nat; ::_thesis: for K being Field holds
( lines (1. (K,n)) is Basis of n -VectSp_over K & the_rank_of (1. (K,n)) = n )
let K be Field; ::_thesis: ( lines (1. (K,n)) is Basis of n -VectSp_over K & the_rank_of (1. (K,n)) = n )
set ONE = 1. (K,n);
A1: n in NAT by ORDINAL1:def_12;
( n = 0 or n >= 1 ) by NAT_1:14;
then A2: Det (1. (K,n)) = 1_ K by A1, MATRIXR2:41, MATRIX_7:16;
for i, j being Nat st [i,j] in Indices (1. (K,n)) & (1. (K,n)) * (i,j) <> 0. K holds
i = j by MATRIX_1:def_11;
then A3: 1. (K,n) is V162(K) by MATRIX_1:def_14;
1_ K <> 0. K ;
then the_rank_of (1. (K,n)) = n by A2, Th83;
hence ( lines (1. (K,n)) is Basis of n -VectSp_over K & the_rank_of (1. (K,n)) = n ) by A3, Th111; ::_thesis: verum
end;
registration
let K be Field;
let n be Nat;
clustern -VectSp_over K -> finite-dimensional ;
coherence
n -VectSp_over K is finite-dimensional
proof
lines (1. (K,n)) is Basis of n -VectSp_over K by Lm8;
hence n -VectSp_over K is finite-dimensional by MATRLIN:def_1; ::_thesis: verum
end;
end;
theorem :: MATRIX13:112
for n being Nat
for K being Field holds dim (n -VectSp_over K) = n
proof
let n be Nat; ::_thesis: for K being Field holds dim (n -VectSp_over K) = n
let K be Field; ::_thesis: dim (n -VectSp_over K) = n
set ONE = 1. (K,n);
len (1. (K,n)) = n by MATRIX_1:24;
then A1: dom (1. (K,n)) = Seg n by FINSEQ_1:def_3;
then A2: (1. (K,n)) .: (Seg n) = lines (1. (K,n)) by RELAT_1:113;
the_rank_of (1. (K,n)) = n by Lm8;
then 1. (K,n) is without_repeated_line by Th105;
then Seg n,(1. (K,n)) .: (Seg n) are_equipotent by A1, CARD_1:33;
then card (Seg n) = card (lines (1. (K,n))) by A2, CARD_1:5;
then A3: card (lines (1. (K,n))) = n by FINSEQ_1:57;
lines (1. (K,n)) is Basis of n -VectSp_over K by Lm8;
hence dim (n -VectSp_over K) = n by A3, VECTSP_9:def_1; ::_thesis: verum
end;
theorem Th113: :: MATRIX13:113
for n, m being Nat
for K being Field
for M being Matrix of m,n,K
for i being Nat
for a being Element of K st ( for j being Nat st j in Seg m holds
M * (j,i) = a ) holds
( M is without_repeated_line iff Segm (M,(Seg (len M)),((Seg (width M)) \ {i})) is without_repeated_line )
proof
let n, m be Nat; ::_thesis: for K being Field
for M being Matrix of m,n,K
for i being Nat
for a being Element of K st ( for j being Nat st j in Seg m holds
M * (j,i) = a ) holds
( M is without_repeated_line iff Segm (M,(Seg (len M)),((Seg (width M)) \ {i})) is without_repeated_line )
let K be Field; ::_thesis: for M being Matrix of m,n,K
for i being Nat
for a being Element of K st ( for j being Nat st j in Seg m holds
M * (j,i) = a ) holds
( M is without_repeated_line iff Segm (M,(Seg (len M)),((Seg (width M)) \ {i})) is without_repeated_line )
let M be Matrix of m,n,K; ::_thesis: for i being Nat
for a being Element of K st ( for j being Nat st j in Seg m holds
M * (j,i) = a ) holds
( M is without_repeated_line iff Segm (M,(Seg (len M)),((Seg (width M)) \ {i})) is without_repeated_line )
let i be Nat; ::_thesis: for a being Element of K st ( for j being Nat st j in Seg m holds
M * (j,i) = a ) holds
( M is without_repeated_line iff Segm (M,(Seg (len M)),((Seg (width M)) \ {i})) is without_repeated_line )
let a be Element of K; ::_thesis: ( ( for j being Nat st j in Seg m holds
M * (j,i) = a ) implies ( M is without_repeated_line iff Segm (M,(Seg (len M)),((Seg (width M)) \ {i})) is without_repeated_line ) )
assume A1: for j being Nat st j in Seg m holds
M * (j,i) = a ; ::_thesis: ( M is without_repeated_line iff Segm (M,(Seg (len M)),((Seg (width M)) \ {i})) is without_repeated_line )
set Sl = Sgm (Seg (len M));
set SMi = (Seg (width M)) \ {i};
set S = Segm (M,(Seg (len M)),((Seg (width M)) \ {i}));
set Si = Sgm ((Seg (width M)) \ {i});
A2: len M = m by MATRIX_1:def_2;
A3: card (Seg (len M)) = len M by FINSEQ_1:57;
len (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) = card (Seg (len M)) by MATRIX_1:def_2;
then A4: dom (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) = Seg m by A2, A3, FINSEQ_1:def_3;
A5: dom M = Seg m by A2, FINSEQ_1:def_3;
thus ( M is without_repeated_line implies Segm (M,(Seg (len M)),((Seg (width M)) \ {i})) is without_repeated_line ) ::_thesis: ( Segm (M,(Seg (len M)),((Seg (width M)) \ {i})) is without_repeated_line implies M is without_repeated_line )
proof
A6: (Seg (width M)) \ {i} c= Seg (width M) by XBOOLE_1:36;
assume A7: M is without_repeated_line ; ::_thesis: Segm (M,(Seg (len M)),((Seg (width M)) \ {i})) is without_repeated_line
let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in dom (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) or not x2 in dom (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) or not (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) . x1 = (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) . x2 or x1 = x2 )
assume that
A8: x1 in dom (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) and
A9: x2 in dom (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) and
A10: (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) . x1 = (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) . x2 ; ::_thesis: x1 = x2
reconsider i1 = x1, i2 = x2 as Element of NAT by A8, A9;
A11: Sgm (Seg (len M)) = idseq m by A2, FINSEQ_3:48;
then A12: (Line (M,i1)) * (Sgm ((Seg (width M)) \ {i})) = (Line (M,((Sgm (Seg (len M))) . i1))) * (Sgm ((Seg (width M)) \ {i})) by A4, A8, FINSEQ_2:49
.= Line ((Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))),i1) by A2, A3, A4, A8, Th47, XBOOLE_1:36
.= (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) . i1 by A2, A3, A4, A8, MATRIX_2:8
.= Line ((Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))),i2) by A2, A3, A4, A9, A10, MATRIX_2:8
.= (Line (M,((Sgm (Seg (len M))) . i2))) * (Sgm ((Seg (width M)) \ {i})) by A2, A3, A4, A9, Th47, XBOOLE_1:36
.= (Line (M,i2)) * (Sgm ((Seg (width M)) \ {i})) by A4, A9, A11, FINSEQ_2:49 ;
A13: now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_k_<=_width_M_holds_
(Line_(M,i1))_._k_=_(Line_(M,i2))_._k
let k be Nat; ::_thesis: ( 1 <= k & k <= width M implies (Line (M,i1)) . b1 = (Line (M,i2)) . b1 )
assume that
A14: 1 <= k and
A15: k <= width M ; ::_thesis: (Line (M,i1)) . b1 = (Line (M,i2)) . b1
k in NAT by ORDINAL1:def_12;
then A16: k in Seg (width M) by A14, A15;
percases ( k = i or k <> i ) ;
supposeA17: k = i ; ::_thesis: (Line (M,i1)) . b1 = (Line (M,i2)) . b1
then A18: M * (i2,k) = a by A1, A4, A9;
A19: M * (i1,k) = (Line (M,i1)) . k by A16, MATRIX_1:def_7;
M * (i1,k) = a by A1, A4, A8, A17;
hence (Line (M,i1)) . k = (Line (M,i2)) . k by A16, A18, A19, MATRIX_1:def_7; ::_thesis: verum
end;
supposeA20: k <> i ; ::_thesis: (Line (M,i1)) . b1 = (Line (M,i2)) . b1
A21: rng (Sgm ((Seg (width M)) \ {i})) = (Seg (width M)) \ {i} by A6, FINSEQ_1:def_13;
k in (Seg (width M)) \ {i} by A16, A20, ZFMISC_1:56;
then consider x being set such that
A22: x in dom (Sgm ((Seg (width M)) \ {i})) and
A23: (Sgm ((Seg (width M)) \ {i})) . x = k by A21, FUNCT_1:def_3;
thus (Line (M,i1)) . k = ((Line (M,i1)) * (Sgm ((Seg (width M)) \ {i}))) . x by A22, A23, FUNCT_1:13
.= (Line (M,i2)) . k by A12, A22, A23, FUNCT_1:13 ; ::_thesis: verum
end;
end;
end;
A24: len (Line (M,i2)) = width M by CARD_1:def_7;
len (Line (M,i1)) = width M by CARD_1:def_7;
then Line (M,i1) = Line (M,i2) by A24, A13, FINSEQ_1:14
.= M . i2 by A4, A9, MATRIX_2:8 ;
then M . i1 = M . i2 by A4, A8, MATRIX_2:8;
hence x1 = x2 by A5, A4, A7, A8, A9, FUNCT_1:def_4; ::_thesis: verum
end;
thus ( Segm (M,(Seg (len M)),((Seg (width M)) \ {i})) is without_repeated_line implies M is without_repeated_line ) ::_thesis: verum
proof
A25: Sgm (Seg (len M)) = idseq m by A2, FINSEQ_3:48;
assume A26: Segm (M,(Seg (len M)),((Seg (width M)) \ {i})) is without_repeated_line ; ::_thesis: M is without_repeated_line
let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in dom M or not x2 in dom M or not M . x1 = M . x2 or x1 = x2 )
assume that
A27: x1 in dom M and
A28: x2 in dom M and
A29: M . x1 = M . x2 ; ::_thesis: x1 = x2
reconsider i1 = x1, i2 = x2 as Element of NAT by A27, A28;
A30: Line (M,i1) = M . i1 by A5, A27, MATRIX_2:8;
A31: Line (M,i2) = M . i2 by A5, A28, MATRIX_2:8;
(Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) . x1 = Line ((Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))),i1) by A2, A3, A5, A27, MATRIX_2:8
.= (Line (M,((Sgm (Seg (len M))) . i1))) * (Sgm ((Seg (width M)) \ {i})) by A2, A3, A5, A27, Th47, XBOOLE_1:36
.= (Line (M,i2)) * (Sgm ((Seg (width M)) \ {i})) by A5, A27, A29, A25, A30, A31, FINSEQ_2:49
.= (Line (M,((Sgm (Seg (len M))) . i2))) * (Sgm ((Seg (width M)) \ {i})) by A5, A28, A25, FINSEQ_2:49
.= Line ((Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))),i2) by A2, A3, A5, A28, Th47, XBOOLE_1:36
.= (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) . x2 by A2, A3, A5, A28, MATRIX_2:8 ;
hence x1 = x2 by A5, A4, A26, A27, A28, FUNCT_1:def_4; ::_thesis: verum
end;
end;
theorem Th114: :: MATRIX13:114
for n, m being Nat
for K being Field
for M being Matrix of m,n,K
for i being Nat st M is without_repeated_line & lines M is linearly-independent & ( for j being Nat st j in Seg m holds
M * (j,i) = 0. K ) holds
lines (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) is linearly-independent
proof
let n, m be Nat; ::_thesis: for K being Field
for M being Matrix of m,n,K
for i being Nat st M is without_repeated_line & lines M is linearly-independent & ( for j being Nat st j in Seg m holds
M * (j,i) = 0. K ) holds
lines (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) is linearly-independent
let K be Field; ::_thesis: for M being Matrix of m,n,K
for i being Nat st M is without_repeated_line & lines M is linearly-independent & ( for j being Nat st j in Seg m holds
M * (j,i) = 0. K ) holds
lines (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) is linearly-independent
let M be Matrix of m,n,K; ::_thesis: for i being Nat st M is without_repeated_line & lines M is linearly-independent & ( for j being Nat st j in Seg m holds
M * (j,i) = 0. K ) holds
lines (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) is linearly-independent
let i be Nat; ::_thesis: ( M is without_repeated_line & lines M is linearly-independent & ( for j being Nat st j in Seg m holds
M * (j,i) = 0. K ) implies lines (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) is linearly-independent )
assume that
A1: M is without_repeated_line and
A2: lines M is linearly-independent and
A3: for j being Nat st j in Seg m holds
M * (j,i) = 0. K ; ::_thesis: lines (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) is linearly-independent
set SMi = (Seg (width M)) \ {i};
set Sl = Seg (len M);
set S = Segm (M,(Seg (len M)),((Seg (width M)) \ {i}));
A4: (Seg (width M)) \ {i} c= Seg (width M) by XBOOLE_1:36;
A5: card (Seg (len M)) = len M by FINSEQ_1:57;
A6: len M = m by MATRIX_1:def_2;
percases ( m = 0 or m <> 0 ) ;
suppose m = 0 ; ::_thesis: lines (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) is linearly-independent
then len (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) = 0 by A6, MATRIX_1:def_2;
then Segm (M,(Seg (len M)),((Seg (width M)) \ {i})) = {} ;
hence lines (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) is linearly-independent ; ::_thesis: verum
end;
suppose m <> 0 ; ::_thesis: lines (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) is linearly-independent
then A7: width M = n by Th1;
A8: now__::_thesis:_for_k_being_Nat_st_k_in_Seg_(card_(Seg_(len_M)))_holds_
not_Line_((Segm_(M,(Seg_(len_M)),((Seg_(width_M))_\_{i}))),k)_=_(card_((Seg_(width_M))_\_{i}))_|->_(0._K)
set n0 = n |-> (0. K);
A9: len (n |-> (0. K)) = n by CARD_1:def_7;
A10: dom (Sgm ((Seg (width M)) \ {i})) = Seg (card ((Seg (width M)) \ {i})) by FINSEQ_3:40, XBOOLE_1:36;
let k be Nat; ::_thesis: ( k in Seg (card (Seg (len M))) implies not Line ((Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))),k) = (card ((Seg (width M)) \ {i})) |-> (0. K) )
assume A11: k in Seg (card (Seg (len M))) ; ::_thesis: not Line ((Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))),k) = (card ((Seg (width M)) \ {i})) |-> (0. K)
Line (M,k) in lines M by A5, A6, A11, Th103;
then reconsider LM = Line (M,k) as Element of n -tuples_on the carrier of K by Th102;
A12: len LM = n by CARD_1:def_7;
LM <> n |-> (0. K) by A1, A2, A5, A6, A11, Th109;
then consider n9 being Nat such that
A13: 1 <= n9 and
A14: n9 <= n and
A15: LM . n9 <> (n |-> (0. K)) . n9 by A12, A9, FINSEQ_1:14;
n9 in NAT by ORDINAL1:def_12;
then A16: n9 in Seg n by A13, A14;
then A17: (n |-> (0. K)) . n9 = 0. K by FINSEQ_2:57;
Sgm (Seg (len M)) = idseq m by A6, FINSEQ_3:48;
then A18: (Sgm (Seg (len M))) . k = k by A5, A6, A11, FINSEQ_2:49;
A19: rng (Sgm ((Seg (width M)) \ {i})) = (Seg (width M)) \ {i} by A4, FINSEQ_1:def_13;
LM . n9 = M * (k,n9) by A7, A16, MATRIX_1:def_7;
then n9 <> i by A3, A5, A6, A11, A15, A17;
then n9 in (Seg (width M)) \ {i} by A7, A16, ZFMISC_1:56;
then consider x being set such that
A20: x in dom (Sgm ((Seg (width M)) \ {i})) and
A21: (Sgm ((Seg (width M)) \ {i})) . x = n9 by A19, FUNCT_1:def_3;
assume A22: Line ((Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))),k) = (card ((Seg (width M)) \ {i})) |-> (0. K) ; ::_thesis: contradiction
reconsider x = x as Element of NAT by A20;
Line ((Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))),k) = (Line (M,((Sgm (Seg (len M))) . k))) * (Sgm ((Seg (width M)) \ {i})) by A11, Th47, XBOOLE_1:36;
then (Line ((Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))),k)) . x = (Line (M,((Sgm (Seg (len M))) . k))) . n9 by A20, A21, FUNCT_1:13;
hence contradiction by A22, A15, A17, A20, A10, A18, FINSEQ_2:57; ::_thesis: verum
end;
A23: now__::_thesis:_for_M1_being_Matrix_of_card_(Seg_(len_M)),_card_((Seg_(width_M))_\_{i}),K_st_(_for_i_being_Nat_st_i_in_Seg_(card_(Seg_(len_M)))_holds_
ex_a_being_Element_of_K_st_Line_(M1,i)_=_a_*_(Line_((Segm_(M,(Seg_(len_M)),((Seg_(width_M))_\_{i}))),i))_)_&_(_for_j_being_Nat_st_j_in_Seg_(card_((Seg_(width_M))_\_{i}))_holds_
Sum_(Col_(M1,j))_=_0._K_)_holds_
M1_=_0._(K,(card_(Seg_(len_M))),(card_((Seg_(width_M))_\_{i})))
set NULL = 0. (K,(card (Seg (len M))),(card ((Seg (width M)) \ {i})));
let M1 be Matrix of card (Seg (len M)), card ((Seg (width M)) \ {i}),K; ::_thesis: ( ( for i being Nat st i in Seg (card (Seg (len M))) holds
ex a being Element of K st Line (M1,i) = a * (Line ((Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))),i)) ) & ( for j being Nat st j in Seg (card ((Seg (width M)) \ {i})) holds
Sum (Col (M1,j)) = 0. K ) implies M1 = 0. (K,(card (Seg (len M))),(card ((Seg (width M)) \ {i}))) )
assume that
A24: for i being Nat st i in Seg (card (Seg (len M))) holds
ex a being Element of K st Line (M1,i) = a * (Line ((Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))),i)) and
A25: for j being Nat st j in Seg (card ((Seg (width M)) \ {i})) holds
Sum (Col (M1,j)) = 0. K ; ::_thesis: M1 = 0. (K,(card (Seg (len M))),(card ((Seg (width M)) \ {i})))
defpred S1[ set , set ] means for i being Nat st $1 = i holds
ex a being Element of K st
( a = $2 & Line (M1,i) = a * (Line ((Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))),i)) );
A26: for k being Nat st k in Seg m holds
ex x being Element of K st S1[k,x]
proof
let k be Nat; ::_thesis: ( k in Seg m implies ex x being Element of K st S1[k,x] )
assume k in Seg m ; ::_thesis: ex x being Element of K st S1[k,x]
then consider a being Element of K such that
A27: Line (M1,k) = a * (Line ((Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))),k)) by A5, A6, A24;
take a ; ::_thesis: S1[k,a]
thus S1[k,a] by A27; ::_thesis: verum
end;
consider p being FinSequence of K such that
A28: dom p = Seg m and
A29: for k being Nat st k in Seg m holds
S1[k,p . k] from FINSEQ_1:sch_5(A26);
deffunc H1( Nat) -> Element of (width M) -tuples_on the carrier of K = (p /. $1) * (Line (M,$1));
consider f being FinSequence of (width M) -tuples_on the carrier of K such that
A30: len f = m and
A31: for j being Nat st j in dom f holds
f . j = H1(j) from FINSEQ_2:sch_1();
reconsider f9 = f as FinSequence of the carrier of (n -VectSp_over K) by A7, Th102;
FinS2MX f9 is Matrix of m,n,K by A30;
then reconsider Mf = f as Matrix of m,n,K ;
A32: dom f = Seg m by A30, FINSEQ_1:def_3;
len Mf = m by MATRIX_1:def_2;
then A33: dom Mf = Seg m by FINSEQ_1:def_3;
A34: now__::_thesis:_for_j_being_Nat_st_j_in_Seg_n_holds_
Sum_(Col_(Mf,j))_=_0._K
A35: len M1 = m by A5, A6, MATRIX_1:def_2;
A36: len Mf = m by MATRIX_1:def_2;
A37: dom Mf = Seg (len Mf) by FINSEQ_1:def_3;
A38: dom M1 = Seg (len M1) by FINSEQ_1:def_3;
let j be Nat; ::_thesis: ( j in Seg n implies Sum (Col (Mf,b1)) = 0. K )
assume A39: j in Seg n ; ::_thesis: Sum (Col (Mf,b1)) = 0. K
set C = Col (Mf,j);
A40: len (Col (Mf,j)) = len Mf by MATRIX_1:def_8
.= m by MATRIX_1:def_2 ;
percases ( j = i or j <> i ) ;
supposeA41: j = i ; ::_thesis: Sum (Col (Mf,b1)) = 0. K
set m0 = m |-> (0. K);
A42: now__::_thesis:_for_n9_being_Nat_st_1_<=_n9_&_n9_<=_m_holds_
(Col_(Mf,j))_._n9_=_(m_|->_(0._K))_._n9
let n9 be Nat; ::_thesis: ( 1 <= n9 & n9 <= m implies (Col (Mf,j)) . n9 = (m |-> (0. K)) . n9 )
assume that
A43: 1 <= n9 and
A44: n9 <= m ; ::_thesis: (Col (Mf,j)) . n9 = (m |-> (0. K)) . n9
A45: width M = n by A43, A44, Th1;
A46: width Mf = n by A43, A44, Th1;
n9 in NAT by ORDINAL1:def_12;
then A47: n9 in Seg m by A43, A44;
then A48: Mf . n9 = Mf /. n9 by A33, PARTFUN1:def_6;
0. K = M * (n9,i) by A3, A47
.= (Line (M,n9)) . i by A39, A41, A45, MATRIX_1:def_7 ;
then (p /. n9) * (0. K) = ((p /. n9) * (Line (M,n9))) . i by A39, A41, A45, FVSUM_1:51
.= (Mf /. n9) . i by A31, A32, A47, A48
.= (Line (Mf,n9)) . i by A47, A48, MATRIX_2:8
.= Mf * (n9,i) by A39, A41, A46, MATRIX_1:def_7
.= (Col (Mf,i)) . n9 by A36, A37, A47, MATRIX_1:def_8 ;
hence (Col (Mf,j)) . n9 = 0. K by A41, VECTSP_1:7
.= (m |-> (0. K)) . n9 by A47, FINSEQ_2:57 ;
::_thesis: verum
end;
len (m |-> (0. K)) = m by CARD_1:def_7;
then Col (Mf,j) = m |-> (0. K) by A40, A42, FINSEQ_1:14;
hence Sum (Col (Mf,j)) = 0. K by A40, MATRIX_3:11; ::_thesis: verum
end;
supposeA49: j <> i ; ::_thesis: Sum (Col (Mf,b1)) = 0. K
A50: rng (Sgm ((Seg (width M)) \ {i})) = (Seg (width M)) \ {i} by A4, FINSEQ_1:def_13;
j in (Seg (width M)) \ {i} by A7, A39, A49, ZFMISC_1:56;
then consider x being set such that
A51: x in dom (Sgm ((Seg (width M)) \ {i})) and
A52: (Sgm ((Seg (width M)) \ {i})) . x = j by A50, FUNCT_1:def_3;
reconsider x = x as Element of NAT by A51;
set C1 = Col (M1,x);
A53: dom (Sgm ((Seg (width M)) \ {i})) = Seg (card ((Seg (width M)) \ {i})) by FINSEQ_3:40, XBOOLE_1:36;
A54: now__::_thesis:_for_n9_being_Nat_st_1_<=_n9_&_n9_<=_m_holds_
(Col_(Mf,j))_._n9_=_(Col_(M1,x))_._n9
let n9 be Nat; ::_thesis: ( 1 <= n9 & n9 <= m implies (Col (Mf,j)) . n9 = (Col (M1,x)) . n9 )
assume that
A55: 1 <= n9 and
A56: n9 <= m ; ::_thesis: (Col (Mf,j)) . n9 = (Col (M1,x)) . n9
A57: width Mf = n by A55, A56, Th1;
A58: width (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) = card ((Seg (width M)) \ {i}) by A6, A55, A56, Th1;
A59: Sgm (Seg (len M)) = idseq m by A6, FINSEQ_3:48;
A60: width M1 = card ((Seg (width M)) \ {i}) by A6, A55, A56, Th1;
A61: (Line (M,n9)) . j = M * (n9,j) by A7, A39, MATRIX_1:def_7;
n9 in NAT by ORDINAL1:def_12;
then A62: n9 in Seg m by A55, A56;
then consider a being Element of K such that
A63: a = p . n9 and
A64: Line (M1,n9) = a * (Line ((Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))),n9)) by A29;
A65: Mf . n9 = Mf /. n9 by A33, A62, PARTFUN1:def_6;
(idseq m) . n9 = n9 by A62, FINSEQ_2:49;
then (Line (M,n9)) * (Sgm ((Seg (width M)) \ {i})) = Line ((Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))),n9) by A5, A6, A62, A59, Th47, XBOOLE_1:36;
then A66: (Line ((Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))),n9)) . x = (Line (M,n9)) . j by A51, A52, FUNCT_1:13;
thus (Col (Mf,j)) . n9 = Mf * (n9,j) by A36, A37, A62, MATRIX_1:def_8
.= (Line (Mf,n9)) . j by A39, A57, MATRIX_1:def_7
.= (Mf /. n9) . j by A62, A65, MATRIX_2:8
.= ((p /. n9) * (Line (M,n9))) . j by A31, A32, A62, A65
.= (a * (Line (M,n9))) . j by A28, A62, A63, PARTFUN1:def_6
.= a * (M * (n9,j)) by A7, A39, A61, FVSUM_1:51
.= (a * (Line ((Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))),n9))) . x by A51, A53, A58, A66, A61, FVSUM_1:51
.= M1 * (n9,x) by A51, A53, A60, A64, MATRIX_1:def_7
.= (Col (M1,x)) . n9 by A35, A38, A62, MATRIX_1:def_8 ; ::_thesis: verum
end;
A67: len (Col (Mf,j)) = len Mf by MATRIX_1:def_8;
len (Col (M1,x)) = len M1 by MATRIX_1:def_8;
then Col (Mf,j) = Col (M1,x) by A36, A35, A67, A54, FINSEQ_1:14;
hence Sum (Col (Mf,j)) = 0. K by A25, A51, A53; ::_thesis: verum
end;
end;
end;
now__::_thesis:_for_j_being_Nat_st_j_in_Seg_m_holds_
ex_pj_being_Element_of_the_carrier_of_K_st_Line_(Mf,j)_=_pj_*_(Line_(M,j))
let j be Nat; ::_thesis: ( j in Seg m implies ex pj being Element of the carrier of K st Line (Mf,j) = pj * (Line (M,j)) )
assume A68: j in Seg m ; ::_thesis: ex pj being Element of the carrier of K st Line (Mf,j) = pj * (Line (M,j))
take pj = p /. j; ::_thesis: Line (Mf,j) = pj * (Line (M,j))
thus Line (Mf,j) = Mf . j by A68, MATRIX_2:8
.= pj * (Line (M,j)) by A31, A32, A68 ; ::_thesis: verum
end;
then A69: Mf = 0. (K,m,n) by A1, A2, A34, Th109;
A70: now__::_thesis:_for_j_being_Nat_st_1_<=_j_&_j_<=_m_holds_
M1_._j_=_(0._(K,(card_(Seg_(len_M))),(card_((Seg_(width_M))_\_{i}))))_._j
let j be Nat; ::_thesis: ( 1 <= j & j <= m implies M1 . j = (0. (K,(card (Seg (len M))),(card ((Seg (width M)) \ {i})))) . j )
assume that
A71: 1 <= j and
A72: j <= m ; ::_thesis: M1 . j = (0. (K,(card (Seg (len M))),(card ((Seg (width M)) \ {i})))) . j
j in NAT by ORDINAL1:def_12;
then A73: j in Seg m by A71, A72;
then consider a being Element of K such that
A74: a = p . j and
A75: Line (M1,j) = a * (Line ((Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))),j)) by A29;
A76: Line ((0. (K,m,n)),j) = (0. (K,m,n)) . j by A73, MATRIX_2:8
.= n |-> (0. K) by A73, FUNCOP_1:7 ;
p . j = p /. j by A28, A73, PARTFUN1:def_6;
then A77: a * (Line (M,j)) = Mf . j by A31, A32, A73, A74
.= Line (Mf,j) by A73, MATRIX_2:8 ;
A78: rng (Sgm ((Seg (width M)) \ {i})) = (Seg (width M)) \ {i} by A4, FINSEQ_1:def_13;
Sgm (Seg (len M)) = idseq m by A6, FINSEQ_3:48;
then (Sgm (Seg (len M))) . j = j by A73, FINSEQ_2:49;
then A79: Line ((Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))),j) = (Line (M,j)) * (Sgm ((Seg (width M)) \ {i})) by A5, A6, A73, Th47, XBOOLE_1:36;
(Seg n) /\ ((Seg (width M)) \ {i}) = (Seg (width M)) \ {i} by A7, XBOOLE_1:28, XBOOLE_1:36;
then A80: (Sgm ((Seg (width M)) \ {i})) " (Seg n) = (Sgm ((Seg (width M)) \ {i})) " (rng (Sgm ((Seg (width M)) \ {i}))) by A78, RELAT_1:133
.= dom (Sgm ((Seg (width M)) \ {i})) by RELAT_1:134
.= Seg (card ((Seg (width M)) \ {i})) by FINSEQ_3:40, XBOOLE_1:36 ;
dom (Line (M,j)) = Seg (len (Line (M,j))) by FINSEQ_1:def_3
.= Seg (width M) by CARD_1:def_7 ;
then Line (M1,j) = (Line ((0. (K,m,n)),j)) * (Sgm ((Seg (width M)) \ {i})) by A69, A75, A77, A78, A79, Th87, XBOOLE_1:36
.= (card ((Seg (width M)) \ {i})) |-> (0. K) by A80, A76, FUNCOP_1:19
.= (0. (K,(card (Seg (len M))),(card ((Seg (width M)) \ {i})))) . j by A5, A6, A73, FUNCOP_1:7 ;
hence M1 . j = (0. (K,(card (Seg (len M))),(card ((Seg (width M)) \ {i})))) . j by A5, A6, A73, MATRIX_2:8; ::_thesis: verum
end;
A81: len (0. (K,(card (Seg (len M))),(card ((Seg (width M)) \ {i})))) = m by A5, A6, MATRIX_1:def_2;
len M1 = m by A5, A6, MATRIX_1:def_2;
hence M1 = 0. (K,(card (Seg (len M))),(card ((Seg (width M)) \ {i}))) by A81, A70, FINSEQ_1:14; ::_thesis: verum
end;
Segm (M,(Seg (len M)),((Seg (width M)) \ {i})) is without_repeated_line by A1, A3, Th113;
hence lines (Segm (M,(Seg (len M)),((Seg (width M)) \ {i}))) is linearly-independent by A8, A23, Th109; ::_thesis: verum
end;
end;
end;
theorem Th115: :: MATRIX13:115
for K being Field
for a being Element of K
for V being VectSp of K
for U being finite Subset of V st U is linearly-independent holds
for u, v being Vector of V st u in U & v in U & u <> v holds
(U \ {u}) \/ {(u + (a * v))} is linearly-independent
proof
let K be Field; ::_thesis: for a being Element of K
for V being VectSp of K
for U being finite Subset of V st U is linearly-independent holds
for u, v being Vector of V st u in U & v in U & u <> v holds
(U \ {u}) \/ {(u + (a * v))} is linearly-independent
let a be Element of K; ::_thesis: for V being VectSp of K
for U being finite Subset of V st U is linearly-independent holds
for u, v being Vector of V st u in U & v in U & u <> v holds
(U \ {u}) \/ {(u + (a * v))} is linearly-independent
let V be VectSp of K; ::_thesis: for U being finite Subset of V st U is linearly-independent holds
for u, v being Vector of V st u in U & v in U & u <> v holds
(U \ {u}) \/ {(u + (a * v))} is linearly-independent
let U be finite Subset of V; ::_thesis: ( U is linearly-independent implies for u, v being Vector of V st u in U & v in U & u <> v holds
(U \ {u}) \/ {(u + (a * v))} is linearly-independent )
assume A1: U is linearly-independent ; ::_thesis: for u, v being Vector of V st u in U & v in U & u <> v holds
(U \ {u}) \/ {(u + (a * v))} is linearly-independent
let u, v be Vector of V; ::_thesis: ( u in U & v in U & u <> v implies (U \ {u}) \/ {(u + (a * v))} is linearly-independent )
assume that
A2: u in U and
A3: v in U and
A4: u <> v ; ::_thesis: (U \ {u}) \/ {(u + (a * v))} is linearly-independent
set ua = u + (a * v);
set Uu = U \ {u};
set Uua = (U \ {u}) \/ {(u + (a * v))};
percases ( u = u + (a * v) or u <> u + (a * v) ) ;
suppose u = u + (a * v) ; ::_thesis: (U \ {u}) \/ {(u + (a * v))} is linearly-independent
hence (U \ {u}) \/ {(u + (a * v))} is linearly-independent by A1, A2, ZFMISC_1:116; ::_thesis: verum
end;
supposeA5: u <> u + (a * v) ; ::_thesis: (U \ {u}) \/ {(u + (a * v))} is linearly-independent
now__::_thesis:_for_L_being_Linear_Combination_of_(U_\_{u})_\/_{(u_+_(a_*_v))}_st_Sum_L_=_0._V_holds_
Carrier_L_=_{}
let L be Linear_Combination of (U \ {u}) \/ {(u + (a * v))}; ::_thesis: ( Sum L = 0. V implies Carrier b1 = {} )
assume A6: Sum L = 0. V ; ::_thesis: Carrier b1 = {}
percases ( L . (u + (a * v)) = 0. K or L . (u + (a * v)) <> 0. K ) ;
supposeA7: L . (u + (a * v)) = 0. K ; ::_thesis: Carrier b1 = {}
Carrier L c= U \ {u}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Carrier L or x in U \ {u} )
assume A8: x in Carrier L ; ::_thesis: x in U \ {u}
then consider v being Vector of V such that
A9: x = v and
A10: L . v <> 0. K by VECTSP_6:1;
Carrier L c= (U \ {u}) \/ {(u + (a * v))} by VECTSP_6:def_4;
then ( v in U \ {u} or ( v in {(u + (a * v))} & not v in {(u + (a * v))} ) ) by A7, A8, A9, A10, TARSKI:def_1, XBOOLE_0:def_3;
hence x in U \ {u} by A9; ::_thesis: verum
end;
then reconsider L9 = L as Linear_Combination of U \ {u} by VECTSP_6:def_4;
A11: Sum L9 = 0. V by A6;
U \ {u} is linearly-independent by A1, VECTSP_7:1, XBOOLE_1:36;
hence Carrier L = {} by A11, VECTSP_7:def_1; ::_thesis: verum
end;
supposeA12: L . (u + (a * v)) <> 0. K ; ::_thesis: Carrier b1 = {}
A13: Carrier L c= (U \ {u}) \/ {(u + (a * v))} by VECTSP_6:def_4;
U \ {u} c= U by XBOOLE_1:36;
then (U \ {u}) \/ {(u + (a * v))} c= U \/ {(u + (a * v))} by XBOOLE_1:13;
then A14: Carrier L c= U \/ {(u + (a * v))} by A13, XBOOLE_1:1;
u + (a * v) in {(u + (a * v))} by TARSKI:def_1;
then u + (a * v) in Lin {(u + (a * v))} by VECTSP_7:8;
then consider Lua being Linear_Combination of {(u + (a * v))} such that
A15: u + (a * v) = Sum Lua by VECTSP_7:7;
reconsider LLua = (L . (u + (a * v))) * Lua as Linear_Combination of {(u + (a * v))} by VECTSP_6:31;
A16: Carrier LLua c= {(u + (a * v))} by VECTSP_6:def_4;
then not u in Carrier LLua by A5, TARSKI:def_1;
then A17: LLua . u = 0. K by VECTSP_6:2;
v in {v} by TARSKI:def_1;
then v in Lin {v} by VECTSP_7:8;
then consider Lv being Linear_Combination of {v} such that
A18: v = Sum Lv by VECTSP_7:7;
reconsider LLv = ((L . (u + (a * v))) * a) * Lv as Linear_Combination of {v} by VECTSP_6:31;
A19: Carrier LLv c= {v} by VECTSP_6:def_4;
then not u in Carrier LLv by A4, TARSKI:def_1;
then A20: LLv . u = 0. K by VECTSP_6:2;
v <> u + (a * v)
proof
assume v = u + (a * v) ; ::_thesis: contradiction
then v - (a * v) = u + ((a * v) - (a * v)) by RLVECT_1:def_3
.= u + (0. V) by VECTSP_1:16
.= u by RLVECT_1:def_4 ;
then u = ((1_ K) * v) + (- (a * v)) by VECTSP_1:def_17
.= ((1_ K) * v) + ((- a) * v) by VECTSP_1:21
.= ((1_ K) - a) * v by VECTSP_1:def_15 ;
then A21: {v,u} is linearly-dependent by A4, VECTSP_7:5;
{v,u} c= U by A2, A3, ZFMISC_1:32;
hence contradiction by A1, A21, VECTSP_7:1; ::_thesis: verum
end;
then not u + (a * v) in Carrier LLv by A19, TARSKI:def_1;
then A22: LLv . (u + (a * v)) = 0. K by VECTSP_6:2;
A23: u + (a * v) <> 0. V
proof
{v,u} c= U by A2, A3, ZFMISC_1:32;
then A24: {v,u} is linearly-independent by A1, VECTSP_7:1;
A25: (1_ K) * u = u by VECTSP_1:def_17;
assume 0. V = u + (a * v) ; ::_thesis: contradiction
then 1_ K = 0. K by A4, A24, A25, VECTSP_7:6;
hence contradiction ; ::_thesis: verum
end;
A26: u <> 0. V by A1, A2, VECTSP_7:2;
(Lua . (u + (a * v))) * (u + (a * v)) = u + (a * v) by A15, VECTSP_6:17
.= (1_ K) * (u + (a * v)) by VECTSP_1:def_17 ;
then A27: Lua . (u + (a * v)) = 1_ K by A23, VECTSP10:4;
u in {u} by TARSKI:def_1;
then u in Lin {u} by VECTSP_7:8;
then consider Lu being Linear_Combination of {u} such that
A28: u = Sum Lu by VECTSP_7:7;
reconsider LLu = (L . (u + (a * v))) * Lu as Linear_Combination of {u} by VECTSP_6:31;
A29: Carrier LLu c= {u} by VECTSP_6:def_4;
then not u + (a * v) in Carrier LLu by A5, TARSKI:def_1;
then A30: LLu . (u + (a * v)) = 0. K by VECTSP_6:2;
{u} c= U by A2, ZFMISC_1:31;
then A31: Carrier LLu c= U by A29, XBOOLE_1:1;
((L + LLv) + LLu) - LLua = (L + (LLv + LLu)) - LLua by VECTSP_6:26
.= (L + (LLv + LLu)) + (- LLua) by VECTSP_6:def_11
.= L + ((LLv + LLu) + (- LLua)) by VECTSP_6:26
.= L + ((LLv + LLu) - LLua) by VECTSP_6:def_11 ;
then A32: Carrier (((L + LLv) + LLu) - LLua) c= (Carrier L) \/ (Carrier ((LLv + LLu) - LLua)) by VECTSP_6:23;
A33: Carrier ((LLv + LLu) - LLua) c= (Carrier (LLv + LLu)) \/ (Carrier LLua) by VECTSP_6:41;
A34: Carrier (LLv + LLu) c= (Carrier LLv) \/ (Carrier LLu) by VECTSP_6:23;
{v} c= U by A3, ZFMISC_1:31;
then Carrier LLv c= U by A19, XBOOLE_1:1;
then (Carrier LLv) \/ (Carrier LLu) c= U by A31, XBOOLE_1:8;
then Carrier (LLv + LLu) c= U by A34, XBOOLE_1:1;
then (Carrier (LLv + LLu)) \/ (Carrier LLua) c= U \/ {(u + (a * v))} by A16, XBOOLE_1:13;
then Carrier ((LLv + LLu) - LLua) c= U \/ {(u + (a * v))} by A33, XBOOLE_1:1;
then (Carrier L) \/ (Carrier ((LLv + LLu) - LLua)) c= U \/ {(u + (a * v))} by A14, XBOOLE_1:8;
then A35: Carrier (((L + LLv) + LLu) - LLua) c= U \/ {(u + (a * v))} by A32, XBOOLE_1:1;
A36: (((L + LLv) + LLu) - LLua) . (u + (a * v)) = (((L + LLv) + LLu) + (- LLua)) . (u + (a * v)) by VECTSP_6:def_11
.= (((L + LLv) + LLu) . (u + (a * v))) + ((- LLua) . (u + (a * v))) by VECTSP_6:22
.= (((L + LLv) . (u + (a * v))) + (LLu . (u + (a * v)))) + ((- LLua) . (u + (a * v))) by VECTSP_6:22
.= (((L . (u + (a * v))) + (0. K)) + (0. K)) + ((- LLua) . (u + (a * v))) by A22, A30, VECTSP_6:22
.= ((L . (u + (a * v))) + (0. K)) + ((- LLua) . (u + (a * v))) by RLVECT_1:def_4
.= (L . (u + (a * v))) + ((- LLua) . (u + (a * v))) by RLVECT_1:def_4
.= (L . (u + (a * v))) - (LLua . (u + (a * v))) by VECTSP_6:36
.= (L . (u + (a * v))) - ((L . (u + (a * v))) * (1_ K)) by A27, VECTSP_6:def_9
.= (L . (u + (a * v))) - (L . (u + (a * v))) by VECTSP_1:def_4
.= 0. K by VECTSP_1:19 ;
Carrier (((L + LLv) + LLu) - LLua) c= U
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Carrier (((L + LLv) + LLu) - LLua) or x in U )
assume A37: x in Carrier (((L + LLv) + LLu) - LLua) ; ::_thesis: x in U
assume not x in U ; ::_thesis: contradiction
then A38: x in {(u + (a * v))} by A35, A37, XBOOLE_0:def_3;
ex v being Element of V st
( x = v & (((L + LLv) + LLu) - LLua) . v <> 0. K ) by A37, VECTSP_6:1;
hence contradiction by A36, A38, TARSKI:def_1; ::_thesis: verum
end;
then reconsider LLL = ((L + LLv) + LLu) - LLua as Linear_Combination of U by VECTSP_6:def_4;
A39: not u in U \ {u} by ZFMISC_1:56;
not u in {(u + (a * v))} by A5, TARSKI:def_1;
then not u in Carrier L by A13, A39, XBOOLE_0:def_3;
then A40: L . u = 0. K by VECTSP_6:2;
(Lu . u) * u = u by A28, VECTSP_6:17
.= (1_ K) * u by VECTSP_1:def_17 ;
then A41: Lu . u = 1_ K by A26, VECTSP10:4;
LLL . u = (((L + LLv) + LLu) + (- LLua)) . u by VECTSP_6:def_11
.= (((L + LLv) + LLu) . u) + ((- LLua) . u) by VECTSP_6:22
.= (((L + LLv) . u) + (LLu . u)) + ((- LLua) . u) by VECTSP_6:22
.= (((L . u) + (LLv . u)) + (LLu . u)) + ((- LLua) . u) by VECTSP_6:22
.= (((0. K) + (0. K)) + (LLu . u)) - (0. K) by A20, A17, A40, VECTSP_6:36
.= ((0. K) + (LLu . u)) - (0. K) by RLVECT_1:def_4
.= (LLu . u) - (0. K) by RLVECT_1:def_4
.= LLu . u by VECTSP_1:18
.= (L . (u + (a * v))) * (1_ K) by A41, VECTSP_6:def_9
.= L . (u + (a * v)) by VECTSP_1:def_4 ;
then A42: u in Carrier LLL by A12, VECTSP_6:1;
Sum (((L + LLv) + LLu) - LLua) = (Sum ((L + LLv) + LLu)) - (Sum LLua) by VECTSP_6:47
.= ((Sum (L + LLv)) + (Sum LLu)) - (Sum LLua) by VECTSP_6:44
.= (((Sum L) + (Sum LLv)) + (Sum LLu)) - (Sum LLua) by VECTSP_6:44
.= (((Sum L) + (Sum LLv)) + (Sum LLu)) - ((L . (u + (a * v))) * (u + (a * v))) by A15, VECTSP_6:45
.= (((Sum L) + (Sum LLv)) + ((L . (u + (a * v))) * u)) - ((L . (u + (a * v))) * (u + (a * v))) by A28, VECTSP_6:45
.= (((Sum L) + ((a * (L . (u + (a * v)))) * v)) + ((L . (u + (a * v))) * u)) - ((L . (u + (a * v))) * (u + (a * v))) by A18, VECTSP_6:45
.= (((Sum L) + ((L . (u + (a * v))) * (a * v))) + ((L . (u + (a * v))) * u)) - ((L . (u + (a * v))) * (u + (a * v))) by VECTSP_1:def_16
.= ((Sum L) + (((L . (u + (a * v))) * (a * v)) + ((L . (u + (a * v))) * u))) - ((L . (u + (a * v))) * (u + (a * v))) by RLVECT_1:def_3
.= ((Sum L) + ((L . (u + (a * v))) * ((a * v) + u))) - ((L . (u + (a * v))) * (u + (a * v))) by VECTSP_1:def_14
.= (Sum L) + (((L . (u + (a * v))) * (u + (a * v))) - ((L . (u + (a * v))) * (u + (a * v)))) by RLVECT_1:def_3
.= (0. V) + (0. V) by A6, VECTSP_1:16
.= 0. V by RLVECT_1:def_4 ;
hence Carrier L = {} by A1, A42, VECTSP_7:def_1; ::_thesis: verum
end;
end;
end;
hence (U \ {u}) \/ {(u + (a * v))} is linearly-independent by VECTSP_7:def_1; ::_thesis: verum
end;
end;
end;
theorem Th116: :: MATRIX13:116
for x being set
for K being Field
for V being VectSp of K
for u, v being Vector of V holds
( x in Lin {u,v} iff ex a, b being Element of K st x = (a * u) + (b * v) )
proof
let x be set ; ::_thesis: for K being Field
for V being VectSp of K
for u, v being Vector of V holds
( x in Lin {u,v} iff ex a, b being Element of K st x = (a * u) + (b * v) )
let K be Field; ::_thesis: for V being VectSp of K
for u, v being Vector of V holds
( x in Lin {u,v} iff ex a, b being Element of K st x = (a * u) + (b * v) )
let V be VectSp of K; ::_thesis: for u, v being Vector of V holds
( x in Lin {u,v} iff ex a, b being Element of K st x = (a * u) + (b * v) )
let u, v be Vector of V; ::_thesis: ( x in Lin {u,v} iff ex a, b being Element of K st x = (a * u) + (b * v) )
percases ( u = v or u <> v ) ;
supposeA1: u = v ; ::_thesis: ( x in Lin {u,v} iff ex a, b being Element of K st x = (a * u) + (b * v) )
then A2: {u,v} = {u} by ENUMSET1:29;
thus ( x in Lin {u,v} implies ex a, b being Element of K st x = (a * u) + (b * v) ) ::_thesis: ( ex a, b being Element of K st x = (a * u) + (b * v) implies x in Lin {u,v} )
proof
assume x in Lin {u,v} ; ::_thesis: ex a, b being Element of K st x = (a * u) + (b * v)
then consider a being Element of K such that
A3: x = a * u by A2, VECTSP10:3;
x = (a * u) + (0. V) by A3, RLVECT_1:def_4
.= (a * u) + ((0. K) * v) by VECTSP10:1 ;
hence ex a, b being Element of K st x = (a * u) + (b * v) ; ::_thesis: verum
end;
given a, b being Element of K such that A4: x = (a * u) + (b * v) ; ::_thesis: x in Lin {u,v}
x = (a + b) * u by A1, A4, VECTSP_1:def_15;
hence x in Lin {u,v} by A2, VECTSP10:3; ::_thesis: verum
end;
supposeA5: u <> v ; ::_thesis: ( x in Lin {u,v} iff ex a, b being Element of K st x = (a * u) + (b * v) )
thus ( x in Lin {u,v} implies ex a, b being Element of K st x = (a * u) + (b * v) ) ::_thesis: ( ex a, b being Element of K st x = (a * u) + (b * v) implies x in Lin {u,v} )
proof
assume x in Lin {u,v} ; ::_thesis: ex a, b being Element of K st x = (a * u) + (b * v)
then consider L being Linear_Combination of {u,v} such that
A6: x = Sum L by VECTSP_7:7;
x = ((L . u) * u) + ((L . v) * v) by A5, A6, VECTSP_6:18;
hence ex a, b being Element of K st x = (a * u) + (b * v) ; ::_thesis: verum
end;
deffunc H1( set ) -> Element of the carrier of K = 0. K;
given a, b being Element of K such that A7: x = (a * u) + (b * v) ; ::_thesis: x in Lin {u,v}
consider L being Function of the carrier of V, the carrier of K such that
A8: ( L . u = a & L . v = b ) and
A9: for z being Element of V st z <> u & z <> v holds
L . z = H1(z) from FUNCT_2:sch_7(A5);
reconsider L = L as Element of Funcs ( the carrier of V, the carrier of K) by FUNCT_2:8;
now__::_thesis:_for_z_being_Vector_of_V_st_not_z_in_{u,v}_holds_
L_._z_=_0._K
let z be Vector of V; ::_thesis: ( not z in {u,v} implies L . z = 0. K )
assume A10: not z in {u,v} ; ::_thesis: L . z = 0. K
A11: z <> u by A10, TARSKI:def_2;
z <> v by A10, TARSKI:def_2;
hence L . z = 0. K by A9, A11; ::_thesis: verum
end;
then reconsider L = L as Linear_Combination of V by VECTSP_6:def_1;
Carrier L c= {u,v}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Carrier L or x in {u,v} )
assume A12: x in Carrier L ; ::_thesis: x in {u,v}
L . x <> 0. K by A12, VECTSP_6:2;
then ( x = v or x = u ) by A9, A12;
hence x in {u,v} by TARSKI:def_2; ::_thesis: verum
end;
then reconsider L = L as Linear_Combination of {u,v} by VECTSP_6:def_4;
Sum L = x by A5, A7, A8, VECTSP_6:18;
hence x in Lin {u,v} by VECTSP_7:7; ::_thesis: verum
end;
end;
end;
theorem Th117: :: MATRIX13:117
for n, m being Nat
for K being Field
for a being Element of K
for M being Matrix of m,n,K st lines M is linearly-independent & M is without_repeated_line holds
for i, j being Nat st j in Seg (len M) & i <> j holds
( RLine (M,i,((Line (M,i)) + (a * (Line (M,j))))) is without_repeated_line & lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) is linearly-independent )
proof
let n, m be Nat; ::_thesis: for K being Field
for a being Element of K
for M being Matrix of m,n,K st lines M is linearly-independent & M is without_repeated_line holds
for i, j being Nat st j in Seg (len M) & i <> j holds
( RLine (M,i,((Line (M,i)) + (a * (Line (M,j))))) is without_repeated_line & lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) is linearly-independent )
let K be Field; ::_thesis: for a being Element of K
for M being Matrix of m,n,K st lines M is linearly-independent & M is without_repeated_line holds
for i, j being Nat st j in Seg (len M) & i <> j holds
( RLine (M,i,((Line (M,i)) + (a * (Line (M,j))))) is without_repeated_line & lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) is linearly-independent )
let a be Element of K; ::_thesis: for M being Matrix of m,n,K st lines M is linearly-independent & M is without_repeated_line holds
for i, j being Nat st j in Seg (len M) & i <> j holds
( RLine (M,i,((Line (M,i)) + (a * (Line (M,j))))) is without_repeated_line & lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) is linearly-independent )
let M be Matrix of m,n,K; ::_thesis: ( lines M is linearly-independent & M is without_repeated_line implies for i, j being Nat st j in Seg (len M) & i <> j holds
( RLine (M,i,((Line (M,i)) + (a * (Line (M,j))))) is without_repeated_line & lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) is linearly-independent ) )
assume that
A1: lines M is linearly-independent and
A2: M is without_repeated_line ; ::_thesis: for i, j being Nat st j in Seg (len M) & i <> j holds
( RLine (M,i,((Line (M,i)) + (a * (Line (M,j))))) is without_repeated_line & lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) is linearly-independent )
set V = n -VectSp_over K;
let i, j be Nat; ::_thesis: ( j in Seg (len M) & i <> j implies ( RLine (M,i,((Line (M,i)) + (a * (Line (M,j))))) is without_repeated_line & lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) is linearly-independent ) )
assume that
A3: j in Seg (len M) and
A4: i <> j ; ::_thesis: ( RLine (M,i,((Line (M,i)) + (a * (Line (M,j))))) is without_repeated_line & lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) is linearly-independent )
set Lj = Line (M,j);
set Li = Line (M,i);
set R = RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))));
percases ( not i in Seg (len M) or i in Seg (len M) ) ;
suppose not i in Seg (len M) ; ::_thesis: ( RLine (M,i,((Line (M,i)) + (a * (Line (M,j))))) is without_repeated_line & lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) is linearly-independent )
hence ( RLine (M,i,((Line (M,i)) + (a * (Line (M,j))))) is without_repeated_line & lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) is linearly-independent ) by A1, A2, Th40; ::_thesis: verum
end;
supposeA5: i in Seg (len M) ; ::_thesis: ( RLine (M,i,((Line (M,i)) + (a * (Line (M,j))))) is without_repeated_line & lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) is linearly-independent )
reconsider N = n as Element of NAT by ORDINAL1:def_12;
A6: dom M = Seg (len M) by FINSEQ_1:def_3;
then A7: M . i <> M . j by A2, A3, A4, A5, FUNCT_1:def_4;
A8: len M = m by MATRIX_1:def_2;
then A9: Line (M,j) in lines M by A3, Th103;
A10: Line (M,i) in lines M by A5, A8, Th103;
then reconsider LI = Line (M,i), LJ = Line (M,j) as Vector of (n -VectSp_over K) by A9;
reconsider li = LI, lj = LJ as Element of N -tuples_on the carrier of K by Th102;
A11: M . i = Line (M,i) by A5, A8, MATRIX_2:8;
m <> 0 by A5, A8;
then A12: n = width M by Th1;
A13: M . j = Line (M,j) by A3, A8, MATRIX_2:8;
A14: for k being Nat st k in Seg m & k <> i holds
Line ((RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))),k) <> (Line (M,i)) + (a * (Line (M,j)))
proof
a * lj = a * LJ by Th102;
then li + (a * lj) = LI + (a * LJ) by Th102
.= ((1_ K) * LI) + (a * LJ) by VECTSP_1:def_17 ;
then A15: li + (a * lj) in Lin {LI,LJ} by Th116;
let k be Nat; ::_thesis: ( k in Seg m & k <> i implies Line ((RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))),k) <> (Line (M,i)) + (a * (Line (M,j))) )
assume that
A16: k in Seg m and
A17: k <> i ; ::_thesis: Line ((RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))),k) <> (Line (M,i)) + (a * (Line (M,j)))
set Lk = Line (M,k);
assume A18: Line ((RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))),k) = (Line (M,i)) + (a * (Line (M,j))) ; ::_thesis: contradiction
A19: Line ((RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))),k) = Line (M,k) by A16, A17, MATRIX11:28;
A20: Line (M,j) <> Line (M,k)
proof
{LI,LJ} c= lines M by A10, A9, ZFMISC_1:32;
then A21: {LI,LJ} is linearly-independent by A1, VECTSP_7:1;
assume A22: Line (M,j) = Line (M,k) ; ::_thesis: contradiction
A23: ((1_ K) + ((- (1_ K)) * a)) * LJ = ((1_ K) + ((- (1_ K)) * a)) * lj by Th102;
A24: (- (1_ K)) * LI = (- (1_ K)) * li by Th102;
0. (n -VectSp_over K) = n |-> (0. K) by Th102
.= lj + (- (li + (a * lj))) by A19, A18, A22, FVSUM_1:26
.= lj + ((- li) + (- (a * lj))) by FVSUM_1:31
.= lj + (((- (1_ K)) * li) + (- (a * lj))) by FVSUM_1:59
.= lj + (((- (1_ K)) * li) + ((- (1_ K)) * (a * lj))) by FVSUM_1:59
.= lj + (((- (1_ K)) * li) + (((- (1_ K)) * a) * lj)) by FVSUM_1:54
.= lj + ((((- (1_ K)) * a) * lj) + ((- (1_ K)) * li)) by FINSEQOP:33
.= (lj + (((- (1_ K)) * a) * lj)) + ((- (1_ K)) * li) by FINSEQOP:28
.= (((1_ K) * lj) + (((- (1_ K)) * a) * lj)) + ((- (1_ K)) * li) by FVSUM_1:57
.= (((1_ K) + ((- (1_ K)) * a)) * lj) + ((- (1_ K)) * li) by FVSUM_1:55
.= (((1_ K) + ((- (1_ K)) * a)) * LJ) + ((- (1_ K)) * LI) by A24, A23, Th102 ;
then - (1_ K) = 0. K by A7, A11, A13, A21, VECTSP_7:6;
hence contradiction by VECTSP_1:28; ::_thesis: verum
end;
A25: Line (M,k) in lines M by A16, Th103;
then reconsider LK = Line (M,k) as Vector of (n -VectSp_over K) ;
reconsider KIJ = {LK,LI,LJ} as Subset of (n -VectSp_over K) ;
A26: KIJ is linearly-independent by A1, A10, A9, A25, VECTSP_7:1, ZFMISC_1:133;
A27: Line (M,k) in KIJ by ENUMSET1:def_1;
A28: M . k = Line (M,k) by A16, MATRIX_2:8;
M . i <> M . k by A2, A5, A8, A6, A16, A17, FUNCT_1:def_4;
then KIJ \ {LK} = {LI,LJ} by A11, A20, A28, ENUMSET1:86;
hence contradiction by A19, A18, A15, A27, A26, VECTSP_9:14; ::_thesis: verum
end;
reconsider LiaLj = li + (a * lj) as Element of the carrier of K * by FINSEQ_1:def_11;
reconsider LL = LiaLj as set ;
set iLL = i .--> LL;
A29: len (li + (a * lj)) = n by CARD_1:def_7;
then RLine (M,i,(li + (a * lj))) = Replace (M,i,LiaLj) by A12, MATRIX11:29
.= M +* (i .--> LL) by A5, A6, FUNCT_7:def_3 ;
then A30: lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) = (M .: ((dom M) \ (dom (i .--> LL)))) \/ (rng (i .--> LL)) by FRECHET:12
.= (M .: ((dom M) \ {i})) \/ (rng (i .--> LL)) by FUNCOP_1:13
.= (M .: ((dom M) \ {i})) \/ {LL} by FUNCOP_1:8
.= ((M .: (dom M)) \ (M .: {i})) \/ {LL} by A2, FUNCT_1:64
.= ((lines M) \ (Im (M,i))) \/ {LL} by RELAT_1:113
.= ((lines M) \ {LI}) \/ {(li + (a * lj))} by A5, A6, A11, FUNCT_1:59 ;
A31: Line ((RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))),i) = (Line (M,i)) + (a * (Line (M,j))) by A5, A8, A29, A12, MATRIX11:28;
now__::_thesis:_for_x1,_x2_being_set_st_x1_in_dom_(RLine_(M,i,((Line_(M,i))_+_(a_*_(Line_(M,j))))))_&_x2_in_dom_(RLine_(M,i,((Line_(M,i))_+_(a_*_(Line_(M,j))))))_&_(RLine_(M,i,((Line_(M,i))_+_(a_*_(Line_(M,j))))))_._x1_=_(RLine_(M,i,((Line_(M,i))_+_(a_*_(Line_(M,j))))))_._x2_holds_
x1_=_x2
A32: len (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) = m by MATRIX_1:def_2;
let x1, x2 be set ; ::_thesis: ( x1 in dom (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) & x2 in dom (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) & (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) . x1 = (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) . x2 implies b1 = b2 )
assume that
A33: x1 in dom (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) and
A34: x2 in dom (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) and
A35: (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) . x1 = (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) . x2 ; ::_thesis: b1 = b2
reconsider i1 = x1, i2 = x2 as Element of NAT by A33, A34;
A36: dom (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) = Seg (len (RLine (M,i,((Line (M,i)) + (a * (Line (M,j))))))) by FINSEQ_1:def_3;
then A37: (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) . i1 = Line ((RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))),i1) by A33, A32, MATRIX_2:8;
A38: (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) . i2 = Line ((RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))),i2) by A34, A36, A32, MATRIX_2:8;
percases ( ( i1 = i & i2 = i ) or ( i1 = i & i2 <> i ) or ( i1 <> i & i2 = i ) or ( i1 <> i & i2 <> i ) ) ;
suppose ( i1 = i & i2 = i ) ; ::_thesis: b1 = b2
hence x1 = x2 ; ::_thesis: verum
end;
suppose ( ( i1 = i & i2 <> i ) or ( i1 <> i & i2 = i ) ) ; ::_thesis: b1 = b2
hence x1 = x2 by A14, A31, A33, A34, A35, A36, A32, A37, A38; ::_thesis: verum
end;
supposeA39: ( i1 <> i & i2 <> i ) ; ::_thesis: b1 = b2
then A40: (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) . i2 = Line (M,i2) by A34, A36, A32, A38, MATRIX11:28;
A41: Line (M,i1) = M . i1 by A33, A36, A32, MATRIX_2:8;
A42: Line (M,i2) = M . i2 by A34, A36, A32, MATRIX_2:8;
(RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) . i1 = Line (M,i1) by A33, A36, A32, A37, A39, MATRIX11:28;
hence x1 = x2 by A2, A8, A6, A33, A34, A35, A36, A32, A41, A40, A42, FUNCT_1:def_4; ::_thesis: verum
end;
end;
end;
hence RLine (M,i,((Line (M,i)) + (a * (Line (M,j))))) is without_repeated_line by FUNCT_1:def_4; ::_thesis: lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) is linearly-independent
A43: a * lj = a * LJ by Th102;
((lines M) \ {LI}) \/ {(LI + (a * LJ))} is linearly-independent by A1, A7, A11, A13, A10, A9, Th115;
hence lines (RLine (M,i,((Line (M,i)) + (a * (Line (M,j)))))) is linearly-independent by A43, A30, Th102; ::_thesis: verum
end;
end;
end;
theorem Th118: :: MATRIX13:118
for m, n being Nat
for K being Field
for P being finite without_zero Subset of NAT
for M being Matrix of m,n,K st P c= Seg m holds
lines (Segm (M,P,(Seg n))) c= lines M
proof
let m, n be Nat; ::_thesis: for K being Field
for P being finite without_zero Subset of NAT
for M being Matrix of m,n,K st P c= Seg m holds
lines (Segm (M,P,(Seg n))) c= lines M
let K be Field; ::_thesis: for P being finite without_zero Subset of NAT
for M being Matrix of m,n,K st P c= Seg m holds
lines (Segm (M,P,(Seg n))) c= lines M
let P be finite without_zero Subset of NAT; ::_thesis: for M being Matrix of m,n,K st P c= Seg m holds
lines (Segm (M,P,(Seg n))) c= lines M
let M be Matrix of m,n,K; ::_thesis: ( P c= Seg m implies lines (Segm (M,P,(Seg n))) c= lines M )
set S = Segm (M,P,(Seg n));
assume A1: P c= Seg m ; ::_thesis: lines (Segm (M,P,(Seg n))) c= lines M
then A2: rng (Sgm P) = P by FINSEQ_1:def_13;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in lines (Segm (M,P,(Seg n))) or x in lines M )
assume x in lines (Segm (M,P,(Seg n))) ; ::_thesis: x in lines M
then consider i being Nat such that
A3: i in Seg (card P) and
A4: x = Line ((Segm (M,P,(Seg n))),i) by Th103;
Seg m <> {} by A1, A3;
then m <> 0 ;
then width M = n by Th1;
then A5: Line ((Segm (M,P,(Seg n))),i) = Line (M,((Sgm P) . i)) by A3, Th48;
dom (Sgm P) = Seg (card P) by A1, FINSEQ_3:40;
then (Sgm P) . i in rng (Sgm P) by A3, FUNCT_1:def_3;
hence x in lines M by A1, A4, A2, A5, Th103; ::_thesis: verum
end;
theorem Th119: :: MATRIX13:119
for m, n being Nat
for K being Field
for P being finite without_zero Subset of NAT
for M being Matrix of m,n,K st P c= Seg m & lines M is linearly-independent holds
lines (Segm (M,P,(Seg n))) is linearly-independent
proof
let m, n be Nat; ::_thesis: for K being Field
for P being finite without_zero Subset of NAT
for M being Matrix of m,n,K st P c= Seg m & lines M is linearly-independent holds
lines (Segm (M,P,(Seg n))) is linearly-independent
let K be Field; ::_thesis: for P being finite without_zero Subset of NAT
for M being Matrix of m,n,K st P c= Seg m & lines M is linearly-independent holds
lines (Segm (M,P,(Seg n))) is linearly-independent
let P be finite without_zero Subset of NAT; ::_thesis: for M being Matrix of m,n,K st P c= Seg m & lines M is linearly-independent holds
lines (Segm (M,P,(Seg n))) is linearly-independent
let M be Matrix of m,n,K; ::_thesis: ( P c= Seg m & lines M is linearly-independent implies lines (Segm (M,P,(Seg n))) is linearly-independent )
assume that
A1: P c= Seg m and
A2: lines M is linearly-independent ; ::_thesis: lines (Segm (M,P,(Seg n))) is linearly-independent
card (Seg n) = n by FINSEQ_1:57;
hence lines (Segm (M,P,(Seg n))) is linearly-independent by A1, A2, Th118, VECTSP_7:1; ::_thesis: verum
end;
theorem Th120: :: MATRIX13:120
for m, n being Nat
for K being Field
for P being finite without_zero Subset of NAT
for M being Matrix of m,n,K st P c= Seg m & M is without_repeated_line holds
Segm (M,P,(Seg n)) is without_repeated_line
proof
let m, n be Nat; ::_thesis: for K being Field
for P being finite without_zero Subset of NAT
for M being Matrix of m,n,K st P c= Seg m & M is without_repeated_line holds
Segm (M,P,(Seg n)) is without_repeated_line
let K be Field; ::_thesis: for P being finite without_zero Subset of NAT
for M being Matrix of m,n,K st P c= Seg m & M is without_repeated_line holds
Segm (M,P,(Seg n)) is without_repeated_line
let P be finite without_zero Subset of NAT; ::_thesis: for M being Matrix of m,n,K st P c= Seg m & M is without_repeated_line holds
Segm (M,P,(Seg n)) is without_repeated_line
let M be Matrix of m,n,K; ::_thesis: ( P c= Seg m & M is without_repeated_line implies Segm (M,P,(Seg n)) is without_repeated_line )
assume that
A1: P c= Seg m and
A2: M is without_repeated_line ; ::_thesis: Segm (M,P,(Seg n)) is without_repeated_line
set S = Segm (M,P,(Seg n));
let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in dom (Segm (M,P,(Seg n))) or not x2 in dom (Segm (M,P,(Seg n))) or not (Segm (M,P,(Seg n))) . x1 = (Segm (M,P,(Seg n))) . x2 or x1 = x2 )
assume that
A3: x1 in dom (Segm (M,P,(Seg n))) and
A4: x2 in dom (Segm (M,P,(Seg n))) and
A5: (Segm (M,P,(Seg n))) . x1 = (Segm (M,P,(Seg n))) . x2 ; ::_thesis: x1 = x2
reconsider i1 = x1, i2 = x2 as Element of NAT by A3, A4;
len (Segm (M,P,(Seg n))) = card P by MATRIX_1:def_2;
then A6: dom (Segm (M,P,(Seg n))) = Seg (card P) by FINSEQ_1:def_3;
then A7: Line ((Segm (M,P,(Seg n))),i1) = (Segm (M,P,(Seg n))) . i1 by A3, MATRIX_2:8;
A8: Line ((Segm (M,P,(Seg n))),i2) = (Segm (M,P,(Seg n))) . i2 by A4, A6, MATRIX_2:8;
A9: Sgm P is one-to-one by A1, FINSEQ_3:92;
A10: dom (Sgm P) = dom (Segm (M,P,(Seg n))) by A1, A6, FINSEQ_3:40;
Seg m <> {} by A1, A3, A6;
then m <> 0 ;
then A11: width M = n by Th1;
then A12: Line ((Segm (M,P,(Seg n))),i1) = Line (M,((Sgm P) . i1)) by A3, A6, Th48;
A13: Line ((Segm (M,P,(Seg n))),i2) = Line (M,((Sgm P) . i2)) by A4, A6, A11, Th48;
A14: len M = m by MATRIX_1:def_2;
A15: rng (Sgm P) = P by A1, FINSEQ_1:def_13;
then A16: (Sgm P) . i2 in P by A4, A10, FUNCT_1:def_3;
then A17: Line (M,((Sgm P) . i2)) = M . ((Sgm P) . i2) by A1, MATRIX_2:8;
A18: (Sgm P) . i1 in P by A3, A10, A15, FUNCT_1:def_3;
then (Sgm P) . i1 in Seg m by A1;
then A19: (Sgm P) . i1 in dom M by A14, FINSEQ_1:def_3;
(Sgm P) . i2 in Seg m by A1, A16;
then A20: (Sgm P) . i2 in dom M by A14, FINSEQ_1:def_3;
Line (M,((Sgm P) . i1)) = M . ((Sgm P) . i1) by A1, A18, MATRIX_2:8;
then (Sgm P) . i1 = (Sgm P) . i2 by A2, A5, A12, A13, A7, A8, A17, A19, A20, FUNCT_1:def_4;
hence x1 = x2 by A3, A4, A10, A9, FUNCT_1:def_4; ::_thesis: verum
end;
theorem Th121: :: MATRIX13:121
for m, n being Nat
for K being Field
for M being Matrix of m,n,K holds
( ( lines M is linearly-independent & M is without_repeated_line ) iff the_rank_of M = m )
proof
let m, n be Nat; ::_thesis: for K being Field
for M being Matrix of m,n,K holds
( ( lines M is linearly-independent & M is without_repeated_line ) iff the_rank_of M = m )
let K be Field; ::_thesis: for M being Matrix of m,n,K holds
( ( lines M is linearly-independent & M is without_repeated_line ) iff the_rank_of M = m )
defpred S1[ Nat] means for m, n being Nat st m = $1 holds
for M being Matrix of m,n,K holds
( ( lines M is linearly-independent & M is without_repeated_line ) iff the_rank_of M = m );
A1: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A2: S1[k] ; ::_thesis: S1[k + 1]
let m, n be Nat; ::_thesis: ( m = k + 1 implies for M being Matrix of m,n,K holds
( ( lines M is linearly-independent & M is without_repeated_line ) iff the_rank_of M = m ) )
assume A3: m = k + 1 ; ::_thesis: for M being Matrix of m,n,K holds
( ( lines M is linearly-independent & M is without_repeated_line ) iff the_rank_of M = m )
let M be Matrix of m,n,K; ::_thesis: ( ( lines M is linearly-independent & M is without_repeated_line ) iff the_rank_of M = m )
thus ( lines M is linearly-independent & M is without_repeated_line implies the_rank_of M = m ) ::_thesis: ( the_rank_of M = m implies ( lines M is linearly-independent & M is without_repeated_line ) )
proof
A4: k < m by A3, NAT_1:13;
A5: len (n |-> (0. K)) = n by CARD_1:def_7;
k < m by A3, NAT_1:13;
then A6: Seg k c= Seg m by FINSEQ_1:5;
A7: m in Seg m by A3, FINSEQ_1:4;
then Line (M,m) in lines M by Th103;
then reconsider LM = Line (M,m) as Element of n -tuples_on the carrier of K by Th102;
A8: len LM = n by CARD_1:def_7;
assume that
A9: lines M is linearly-independent and
A10: M is without_repeated_line ; ::_thesis: the_rank_of M = m
LM <> n |-> (0. K) by A9, A10, A7, Th109;
then consider i being Nat such that
A11: 1 <= i and
A12: i <= n and
A13: LM . i <> (n |-> (0. K)) . i by A8, A5, FINSEQ_1:14;
defpred S2[ Nat] means ( $1 < m implies ex M1 being Matrix of m,n,K st
( Line (M1,m) = Line (M,m) & lines M1 is linearly-independent & M1 is without_repeated_line & the_rank_of M1 = the_rank_of M & ( for j being Nat st j <= $1 & j in Seg m holds
M1 * (j,i) = 0. K ) ) );
i in NAT by ORDINAL1:def_12;
then A14: i in Seg n by A11, A12;
then A15: LM . i <> 0. K by A13, FINSEQ_2:57;
len (Line (M,m)) = width M by MATRIX_1:def_7;
then A16: LM . i = M * (m,i) by A8, A14, MATRIX_1:def_7;
A17: for l being Nat st S2[l] holds
S2[l + 1]
proof
set Mmi = M * (m,i);
let L be Nat; ::_thesis: ( S2[L] implies S2[L + 1] )
assume A18: S2[L] ; ::_thesis: S2[L + 1]
set L1 = L + 1;
assume A19: L + 1 < m ; ::_thesis: ex M1 being Matrix of m,n,K st
( Line (M1,m) = Line (M,m) & lines M1 is linearly-independent & M1 is without_repeated_line & the_rank_of M1 = the_rank_of M & ( for j being Nat st j <= L + 1 & j in Seg m holds
M1 * (j,i) = 0. K ) )
then consider M1 being Matrix of m,n,K such that
A20: Line (M1,m) = Line (M,m) and
A21: lines M1 is linearly-independent and
A22: M1 is without_repeated_line and
A23: the_rank_of M1 = the_rank_of M and
A24: for j being Nat st j <= L & j in Seg m holds
M1 * (j,i) = 0. K by A18, NAT_1:13;
set MLi = M1 * ((L + 1),i);
take R = RLine (M1,(L + 1),((Line (M1,(L + 1))) + ((- (((M * (m,i)) ") * (M1 * ((L + 1),i)))) * (Line (M1,m))))); ::_thesis: ( Line (R,m) = Line (M,m) & lines R is linearly-independent & R is without_repeated_line & the_rank_of R = the_rank_of M & ( for j being Nat st j <= L + 1 & j in Seg m holds
R * (j,i) = 0. K ) )
len M1 = m by MATRIX_1:def_2;
hence ( Line (R,m) = Line (M,m) & lines R is linearly-independent & R is without_repeated_line & the_rank_of R = the_rank_of M ) by A7, A19, A20, A21, A22, A23, Th92, Th117, MATRIX11:28; ::_thesis: for j being Nat st j <= L + 1 & j in Seg m holds
R * (j,i) = 0. K
set LMm = Line (M1,m);
set LML = Line (M1,(L + 1));
let j be Nat; ::_thesis: ( j <= L + 1 & j in Seg m implies R * (j,i) = 0. K )
assume that
A25: j <= L + 1 and
A26: j in Seg m ; ::_thesis: R * (j,i) = 0. K
m <> 0 by A26;
then A27: width M1 = n by Th1;
then A28: (Line (M1,(L + 1))) . i = M1 * ((L + 1),i) by A14, MATRIX_1:def_7;
0 + 1 <= L + 1 by NAT_1:13;
then A29: L + 1 in Seg m by A19;
len ((Line (M1,(L + 1))) + ((- (((M * (m,i)) ") * (M1 * ((L + 1),i)))) * (Line (M1,m)))) = width M1 by CARD_1:def_7;
then A30: Line (R,(L + 1)) = (Line (M1,(L + 1))) + ((- (((M * (m,i)) ") * (M1 * ((L + 1),i)))) * (Line (M1,m))) by A29, MATRIX11:28;
m <> 0 by A26;
then width M = n by Th1;
then (Line (M1,m)) . i = M * (m,i) by A14, A20, MATRIX_1:def_7;
then ((- (((M * (m,i)) ") * (M1 * ((L + 1),i)))) * (Line (M1,m))) . i = (- (((M * (m,i)) ") * (M1 * ((L + 1),i)))) * (M * (m,i)) by A14, A27, FVSUM_1:51;
then A31: (Line (R,(L + 1))) . i = (M1 * ((L + 1),i)) + ((- (((M * (m,i)) ") * (M1 * ((L + 1),i)))) * (M * (m,i))) by A14, A27, A28, A30, FVSUM_1:18
.= (M1 * ((L + 1),i)) + ((- ((1_ K) * (((M * (m,i)) ") * (M1 * ((L + 1),i))))) * (M * (m,i))) by VECTSP_1:def_4
.= (M1 * ((L + 1),i)) + (((- (1_ K)) * (((M * (m,i)) ") * (M1 * ((L + 1),i)))) * (M * (m,i))) by VECTSP_1:9
.= (M1 * ((L + 1),i)) + ((- (1_ K)) * ((((M * (m,i)) ") * (M1 * ((L + 1),i))) * (M * (m,i)))) by GROUP_1:def_3
.= (M1 * ((L + 1),i)) + ((- (1_ K)) * ((((M * (m,i)) ") * (M * (m,i))) * (M1 * ((L + 1),i)))) by GROUP_1:def_3
.= (M1 * ((L + 1),i)) + ((- (1_ K)) * ((1_ K) * (M1 * ((L + 1),i)))) by A15, A16, VECTSP_1:def_10
.= (M1 * ((L + 1),i)) + ((- (1_ K)) * (M1 * ((L + 1),i))) by VECTSP_1:def_4
.= (M1 * ((L + 1),i)) + (- ((1_ K) * (M1 * ((L + 1),i)))) by VECTSP_1:9
.= (M1 * ((L + 1),i)) + (- (M1 * ((L + 1),i))) by VECTSP_1:def_4
.= 0. K by RLVECT_1:5 ;
m <> 0 by A26;
then A32: width R = n by Th1;
percases ( j = L + 1 or j <= L ) by A25, NAT_1:9;
suppose j = L + 1 ; ::_thesis: R * (j,i) = 0. K
hence R * (j,i) = 0. K by A14, A32, A31, MATRIX_1:def_7; ::_thesis: verum
end;
supposeA33: j <= L ; ::_thesis: R * (j,i) = 0. K
then A34: j < L + 1 by NAT_1:13;
thus 0. K = M1 * (j,i) by A24, A26, A33
.= (Line (M1,j)) . i by A14, A27, MATRIX_1:def_7
.= (Line (R,j)) . i by A26, A34, MATRIX11:28
.= R * (j,i) by A14, A32, MATRIX_1:def_7 ; ::_thesis: verum
end;
end;
end;
A35: S2[ 0 ]
proof
assume 0 < m ; ::_thesis: ex M1 being Matrix of m,n,K st
( Line (M1,m) = Line (M,m) & lines M1 is linearly-independent & M1 is without_repeated_line & the_rank_of M1 = the_rank_of M & ( for j being Nat st j <= 0 & j in Seg m holds
M1 * (j,i) = 0. K ) )
take M ; ::_thesis: ( Line (M,m) = Line (M,m) & lines M is linearly-independent & M is without_repeated_line & the_rank_of M = the_rank_of M & ( for j being Nat st j <= 0 & j in Seg m holds
M * (j,i) = 0. K ) )
thus ( Line (M,m) = Line (M,m) & lines M is linearly-independent & M is without_repeated_line & the_rank_of M = the_rank_of M & ( for j being Nat st j <= 0 & j in Seg m holds
M * (j,i) = 0. K ) ) by A9, A10; ::_thesis: verum
end;
for l being Nat holds S2[l] from NAT_1:sch_2(A35, A17);
then consider M1 being Matrix of m,n,K such that
A36: Line (M1,m) = Line (M,m) and
A37: lines M1 is linearly-independent and
A38: M1 is without_repeated_line and
A39: the_rank_of M1 = the_rank_of M and
A40: for j being Nat st j <= k & j in Seg m holds
M1 * (j,i) = 0. K by A4;
set S = Segm (M1,(Seg k),(Seg n));
A41: card (Seg k) = k by FINSEQ_1:57;
then A42: len (Segm (M1,(Seg k),(Seg n))) = k by MATRIX_1:def_2;
A43: card (Seg n) = n by FINSEQ_1:57;
A44: now__::_thesis:_for_j_being_Nat_st_j_in_Seg_k_holds_
(Segm_(M1,(Seg_k),(Seg_n)))_*_(j,i)_=_0._K
A45: (Sgm (Seg n)) . i = (idseq n) . i by FINSEQ_3:48
.= i by A14, FINSEQ_2:49 ;
let j be Nat; ::_thesis: ( j in Seg k implies (Segm (M1,(Seg k),(Seg n))) * (j,i) = 0. K )
assume A46: j in Seg k ; ::_thesis: (Segm (M1,(Seg k),(Seg n))) * (j,i) = 0. K
A47: j <= k by A46, FINSEQ_1:1;
A48: (Sgm (Seg k)) . j = (idseq k) . j by FINSEQ_3:48
.= j by A46, FINSEQ_2:49 ;
width (Segm (M1,(Seg k),(Seg n))) = n by A43, A46, Th1;
then [j,i] in [:(Seg k),(Seg (width (Segm (M1,(Seg k),(Seg n))))):] by A14, A46, ZFMISC_1:87;
then [j,i] in Indices (Segm (M1,(Seg k),(Seg n))) by A42, FINSEQ_1:def_3;
hence (Segm (M1,(Seg k),(Seg n))) * (j,i) = M1 * (((Sgm (Seg k)) . j),((Sgm (Seg n)) . i)) by Def1
.= 0. K by A40, A6, A46, A48, A45, A47 ;
::_thesis: verum
end;
set SwS = Seg (width (Segm (M1,(Seg k),(Seg n))));
set SSS = Segm ((Segm (M1,(Seg k),(Seg n))),(Seg k),((Seg (width (Segm (M1,(Seg k),(Seg n))))) \ {i}));
A49: width M1 = n by A3, Th1;
Segm (M1,(Seg k),(Seg n)) is without_repeated_line by A38, A6, Th120;
then A50: Segm ((Segm (M1,(Seg k),(Seg n))),(Seg k),((Seg (width (Segm (M1,(Seg k),(Seg n))))) \ {i})) is without_repeated_line by A41, A42, A44, Th113;
lines (Segm (M1,(Seg k),(Seg n))) is linearly-independent by A37, A6, Th119;
then lines (Segm ((Segm (M1,(Seg k),(Seg n))),(Seg k),((Seg (width (Segm (M1,(Seg k),(Seg n))))) \ {i}))) is linearly-independent by A38, A6, A41, A42, A44, Th114, Th120;
then the_rank_of (Segm ((Segm (M1,(Seg k),(Seg n))),(Seg k),((Seg (width (Segm (M1,(Seg k),(Seg n))))) \ {i}))) = k by A2, A41, A50;
then consider P, Q being finite without_zero Subset of NAT such that
A51: [:P,Q:] c= Indices (Segm ((Segm (M1,(Seg k),(Seg n))),(Seg k),((Seg (width (Segm (M1,(Seg k),(Seg n))))) \ {i}))) and
A52: card P = card Q and
A53: card P = k and
A54: Det (EqSegm ((Segm ((Segm (M1,(Seg k),(Seg n))),(Seg k),((Seg (width (Segm (M1,(Seg k),(Seg n))))) \ {i}))),P,Q)) <> 0. K by Def4;
( P = {} iff Q = {} ) by A52;
then consider P1, Q1 being finite without_zero Subset of NAT such that
A55: P1 c= Seg k and
A56: Q1 c= (Seg (width (Segm (M1,(Seg k),(Seg n))))) \ {i} and
P1 = (Sgm (Seg k)) .: P and
Q1 = (Sgm ((Seg (width (Segm (M1,(Seg k),(Seg n))))) \ {i})) .: Q and
A57: card P1 = card P and
A58: card Q1 = card Q and
A59: Segm ((Segm ((Segm (M1,(Seg k),(Seg n))),(Seg k),((Seg (width (Segm (M1,(Seg k),(Seg n))))) \ {i}))),P,Q) = Segm ((Segm (M1,(Seg k),(Seg n))),P1,Q1) by A51, Th57;
(Seg (width (Segm (M1,(Seg k),(Seg n))))) \ {i} c= Seg (width (Segm (M1,(Seg k),(Seg n)))) by XBOOLE_1:36;
then A60: Q1 c= Seg (width (Segm (M1,(Seg k),(Seg n)))) by A56, XBOOLE_1:1;
then [:P1,Q1:] c= [:(Seg k),(Seg (width (Segm (M1,(Seg k),(Seg n))))):] by A55, ZFMISC_1:96;
then A61: [:P1,Q1:] c= Indices (Segm (M1,(Seg k),(Seg n))) by A42, FINSEQ_1:def_3;
A62: now__::_thesis:_Q1_c=_Seg_n
percases ( k = 0 or k > 0 ) ;
suppose k = 0 ; ::_thesis: Q1 c= Seg n
then width (Segm (M1,(Seg k),(Seg n))) = 0 by A42, MATRIX_1:def_3;
then Seg (width (Segm (M1,(Seg k),(Seg n)))) c= Seg n by FINSEQ_1:5;
hence Q1 c= Seg n by A60, XBOOLE_1:1; ::_thesis: verum
end;
suppose k > 0 ; ::_thesis: Q1 c= Seg n
hence Q1 c= Seg n by A43, A60, Th1; ::_thesis: verum
end;
end;
end;
( P1 = {} iff Q1 = {} ) by A52, A57, A58;
then consider P2, Q2 being finite without_zero Subset of NAT such that
A63: P2 c= Seg k and
A64: Q2 c= Seg n and
A65: P2 = (Sgm (Seg k)) .: P1 and
A66: Q2 = (Sgm (Seg n)) .: Q1 and
A67: card P2 = card P1 and
A68: card Q2 = card Q1 and
A69: Segm ((Segm (M1,(Seg k),(Seg n))),P1,Q1) = Segm (M1,P2,Q2) by A61, Th57;
A70: Q2 = (idseq n) .: Q1 by A66, FINSEQ_3:48
.= Q1 by A62, FRECHET:13 ;
reconsider i = i, m = m as non zero Element of NAT by A3, A11, ORDINAL1:def_12;
set Q2i = Q2 \/ {i};
set SQ2i = Sgm (Q2 \/ {i});
A71: {i} c= Seg n by A14, ZFMISC_1:31;
then A72: Q2 \/ {i} c= Seg n by A64, XBOOLE_1:8;
then A73: rng (Sgm (Q2 \/ {i})) = Q2 \/ {i} by FINSEQ_1:def_13;
A74: P2 = (idseq k) .: P1 by A65, FINSEQ_3:48
.= P1 by A55, FRECHET:13 ;
A75: EqSegm ((Segm ((Segm (M1,(Seg k),(Seg n))),(Seg k),((Seg (width (Segm (M1,(Seg k),(Seg n))))) \ {i}))),P,Q) = Segm ((Segm (M1,(Seg k),(Seg n))),P1,Q1) by A52, A59, Def3
.= EqSegm (M1,P1,Q1) by A52, A57, A58, A69, A74, A70, Def3 ;
A76: len (EqSegm (M1,P2,Q2)) = k by A53, A57, A67, MATRIX_1:def_2;
A77: len M1 = m by Th1;
then A78: the_rank_of M1 <= m by Th74;
set P2m = P2 \/ {m};
set ES = EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}));
A79: P2 c= Seg m by A6, A63, XBOOLE_1:1;
i in {i} by TARSKI:def_1;
then A80: i in Q2 \/ {i} by XBOOLE_0:def_3;
i in {i} by TARSKI:def_1;
then A81: not i in Q2 by A56, A70, XBOOLE_0:def_5;
then A82: card (Q2 \/ {i}) = m by A3, A52, A53, A58, A68, CARD_2:41;
then dom (Sgm (Q2 \/ {i})) = Seg m by A64, A71, FINSEQ_3:40, XBOOLE_1:8;
then consider Si being set such that
A83: Si in Seg m and
A84: (Sgm (Q2 \/ {i})) . Si = i by A80, A73, FUNCT_1:def_3;
reconsider Si = Si as Element of NAT by A83;
k < m by A3, NAT_1:13;
then A85: not m in P2 by A55, A74, FINSEQ_1:1;
then A86: card (P2 \/ {m}) = m by A3, A53, A57, A67, CARD_2:41;
then A87: len (Delete ((EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))),m,Si)) = m -' 1 by MATRIX_1:def_2;
A88: {m} c= Seg m by A7, ZFMISC_1:31;
then P2 \/ {m} c= Seg m by A79, XBOOLE_1:8;
then [:(P2 \/ {m}),(Q2 \/ {i}):] c= [:(Seg m),(Seg n):] by A72, ZFMISC_1:96;
then A89: [:(P2 \/ {m}),(Q2 \/ {i}):] c= Indices M1 by A77, A49, FINSEQ_1:def_3;
card (Seg m) = m by FINSEQ_1:57;
then A90: P2 \/ {m} = Seg m by A88, A79, A86, CARD_FIN:1, XBOOLE_1:8;
A91: now__::_thesis:_for_j_being_Nat_st_j_in_Seg_m_holds_
(EqSegm_(M1,(P2_\/_{m}),(Q2_\/_{i})))_*_(j,Si)_=_M1_*_(j,i)
A92: dom (EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))) = Seg (len (EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i})))) by FINSEQ_1:def_3;
A93: dom M1 = Seg (len M1) by FINSEQ_1:def_3;
A94: m = len (EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))) by A86, MATRIX_1:def_2;
let j be Nat; ::_thesis: ( j in Seg m implies (EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))) * (j,Si) = M1 * (j,i) )
assume A95: j in Seg m ; ::_thesis: (EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))) * (j,Si) = M1 * (j,i)
Col (M1,i) = Col ((Segm (M1,(P2 \/ {m}),(Q2 \/ {i}))),Si) by A77, A82, A83, A84, A90, Th50
.= Col ((EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))),Si) by A3, A53, A57, A67, A85, A82, Def3, CARD_2:41 ;
hence (EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))) * (j,Si) = (Col (M1,i)) . j by A95, A94, A92, MATRIX_1:def_8
.= M1 * (j,i) by A77, A95, A93, MATRIX_1:def_8 ;
::_thesis: verum
end;
then A96: (EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))) * (m,Si) = M1 * (m,i) by A3, FINSEQ_1:4
.= (Line (M,m)) . i by A14, A36, A49, MATRIX_1:def_7 ;
set LC = LaplaceExpC ((EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))),Si);
len (LaplaceExpC ((EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))),Si)) = m by A86, LAPLACE:def_8;
then A97: dom (LaplaceExpC ((EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))),Si)) = Seg m by FINSEQ_1:def_3;
now__::_thesis:_for_j_being_Nat_st_j_in_Seg_m_&_j_<>_m_holds_
0._K_=_(LaplaceExpC_((EqSegm_(M1,(P2_\/_{m}),(Q2_\/_{i}))),Si))_._j
let j be Nat; ::_thesis: ( j in Seg m & j <> m implies 0. K = (LaplaceExpC ((EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))),Si)) . j )
assume that
A98: j in Seg m and
A99: j <> m ; ::_thesis: 0. K = (LaplaceExpC ((EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))),Si)) . j
reconsider J = j as Element of NAT by ORDINAL1:def_12;
j <= m by A98, FINSEQ_1:1;
then j <= k by A3, A99, NAT_1:9;
then 0. K = M1 * (j,i) by A40, A98
.= (EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))) * (j,Si) by A91, A98 ;
hence 0. K = ((EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))) * (j,Si)) * (Cofactor ((EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))),J,Si)) by VECTSP_1:7
.= (LaplaceExpC ((EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))),Si)) . j by A97, A98, LAPLACE:def_8 ;
::_thesis: verum
end;
then A100: (LaplaceExpC ((EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))),Si)) . m = Sum (LaplaceExpC ((EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))),Si)) by A7, A97, MATRIX_3:12
.= Det (EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))) by A86, A83, LAPLACE:27 ;
reconsider mSi = m + Si as Element of NAT ;
- (1_ K) <> 0. K by VECTSP_1:28;
then A101: (power K) . ((- (1_ K)),mSi) <> 0. K by Lm6;
(Sgm (P2 \/ {m})) . m = (idseq m) . m by A90, FINSEQ_3:48
.= m by A7, FINSEQ_2:49 ;
then Delete ((EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))),m,Si) = EqSegm (M1,((P2 \/ {m}) \ {m}),((Q2 \/ {i}) \ {i})) by A7, A86, A82, A83, A84, Th64
.= EqSegm (M1,P2,((Q2 \/ {i}) \ {i})) by A85, ZFMISC_1:117
.= EqSegm (M1,P2,Q2) by A81, ZFMISC_1:117 ;
then ((power K) . ((- (1_ K)),mSi)) * (Minor ((EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))),m,Si)) <> 0. K by A53, A54, A74, A70, A75, A86, A87, A76, A101, VECTSP_1:12;
then ((EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))) * (m,Si)) * (Cofactor ((EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))),m,Si)) <> 0. K by A15, A96, VECTSP_1:12;
then Det (EqSegm (M1,(P2 \/ {m}),(Q2 \/ {i}))) <> 0. K by A7, A97, A100, LAPLACE:def_8;
then the_rank_of M1 >= m by A89, A86, A82, Def4;
hence the_rank_of M = m by A39, A78, XXREAL_0:1; ::_thesis: verum
end;
thus ( the_rank_of M = m implies ( lines M is linearly-independent & M is without_repeated_line ) ) by Th105, Th110; ::_thesis: verum
end;
A102: S1[ 0 ]
proof
let m, n be Nat; ::_thesis: ( m = 0 implies for M being Matrix of m,n,K holds
( ( lines M is linearly-independent & M is without_repeated_line ) iff the_rank_of M = m ) )
assume A103: m = 0 ; ::_thesis: for M being Matrix of m,n,K holds
( ( lines M is linearly-independent & M is without_repeated_line ) iff the_rank_of M = m )
let M be Matrix of m,n,K; ::_thesis: ( ( lines M is linearly-independent & M is without_repeated_line ) iff the_rank_of M = m )
A104: len M = 0 by A103, MATRIX_1:def_2;
then M = {} ;
hence ( ( lines M is linearly-independent & M is without_repeated_line ) iff the_rank_of M = m ) by A103, A104, Th74; ::_thesis: verum
end;
for k being Nat holds S1[k] from NAT_1:sch_2(A102, A1);
hence for M being Matrix of m,n,K holds
( ( lines M is linearly-independent & M is without_repeated_line ) iff the_rank_of M = m ) ; ::_thesis: verum
end;
theorem Th122: :: MATRIX13:122
for n, m being Nat
for K being Field
for M being Matrix of m,n,K
for U being Subset of (n -VectSp_over K) st U c= lines M holds
ex P being finite without_zero Subset of NAT st
( P c= Seg m & lines (Segm (M,P,(Seg n))) = U & Segm (M,P,(Seg n)) is without_repeated_line )
proof
let n, m be Nat; ::_thesis: for K being Field
for M being Matrix of m,n,K
for U being Subset of (n -VectSp_over K) st U c= lines M holds
ex P being finite without_zero Subset of NAT st
( P c= Seg m & lines (Segm (M,P,(Seg n))) = U & Segm (M,P,(Seg n)) is without_repeated_line )
let K be Field; ::_thesis: for M being Matrix of m,n,K
for U being Subset of (n -VectSp_over K) st U c= lines M holds
ex P being finite without_zero Subset of NAT st
( P c= Seg m & lines (Segm (M,P,(Seg n))) = U & Segm (M,P,(Seg n)) is without_repeated_line )
let M be Matrix of m,n,K; ::_thesis: for U being Subset of (n -VectSp_over K) st U c= lines M holds
ex P being finite without_zero Subset of NAT st
( P c= Seg m & lines (Segm (M,P,(Seg n))) = U & Segm (M,P,(Seg n)) is without_repeated_line )
defpred S1[ set , set ] means ex i being Nat st
( i in Seg m & Line (M,i) = $1 & $2 = i );
let U be Subset of (n -VectSp_over K); ::_thesis: ( U c= lines M implies ex P being finite without_zero Subset of NAT st
( P c= Seg m & lines (Segm (M,P,(Seg n))) = U & Segm (M,P,(Seg n)) is without_repeated_line ) )
assume A1: U c= lines M ; ::_thesis: ex P being finite without_zero Subset of NAT st
( P c= Seg m & lines (Segm (M,P,(Seg n))) = U & Segm (M,P,(Seg n)) is without_repeated_line )
A2: for x being set st x in U holds
ex y being set st
( y in Seg m & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in U implies ex y being set st
( y in Seg m & S1[x,y] ) )
assume x in U ; ::_thesis: ex y being set st
( y in Seg m & S1[x,y] )
then consider i being Nat such that
A3: i in Seg m and
A4: x = Line (M,i) by A1, Th103;
take i ; ::_thesis: ( i in Seg m & S1[x,i] )
thus ( i in Seg m & S1[x,i] ) by A3, A4; ::_thesis: verum
end;
consider f being Function of U,(Seg m) such that
A5: for x being set st x in U holds
S1[x,f . x] from FUNCT_2:sch_1(A2);
A6: rng f c= Seg m by RELAT_1:def_19;
then not 0 in rng f ;
then reconsider P = rng f as finite without_zero Subset of NAT by A6, MEASURE6:def_2, XBOOLE_1:1;
set S = Segm (M,P,(Seg n));
A7: rng (Sgm P) = P by A6, FINSEQ_1:def_13;
A8: lines (Segm (M,P,(Seg n))) c= U
proof
A9: rng (Sgm P) = P by A6, FINSEQ_1:def_13;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in lines (Segm (M,P,(Seg n))) or x in U )
A10: dom (Segm (M,P,(Seg n))) = Seg (len (Segm (M,P,(Seg n)))) by FINSEQ_1:def_3
.= Seg (card P) by MATRIX_1:def_2 ;
assume A11: x in lines (Segm (M,P,(Seg n))) ; ::_thesis: x in U
then consider y being set such that
A12: y in dom (Segm (M,P,(Seg n))) and
A13: (Segm (M,P,(Seg n))) . y = x by FUNCT_1:def_3;
lines (Segm (M,P,(Seg n))) c= lines M by A6, Th118;
then A14: M <> {} by A11;
len M = m by MATRIX_1:def_2;
then A15: width M = n by A14, Th1;
reconsider y = y as Element of NAT by A12;
dom (Sgm P) = Seg (card P) by A6, FINSEQ_3:40;
then (Sgm P) . y in rng (Sgm P) by A12, A10, FUNCT_1:def_3;
then consider z being set such that
A16: z in dom f and
A17: f . z = (Sgm P) . y by A9, FUNCT_1:def_3;
ex i being Nat st
( i in Seg m & Line (M,i) = z & f . z = i ) by A5, A16;
then z = Line ((Segm (M,P,(Seg n))),y) by A12, A10, A17, A15, Th48
.= x by A12, A13, A10, MATRIX_2:8 ;
hence x in U by A16; ::_thesis: verum
end;
take P ; ::_thesis: ( P c= Seg m & lines (Segm (M,P,(Seg n))) = U & Segm (M,P,(Seg n)) is without_repeated_line )
thus P c= Seg m by RELAT_1:def_19; ::_thesis: ( lines (Segm (M,P,(Seg n))) = U & Segm (M,P,(Seg n)) is without_repeated_line )
U c= lines (Segm (M,P,(Seg n)))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in U or x in lines (Segm (M,P,(Seg n))) )
A18: dom (Sgm P) = Seg (card P) by A6, FINSEQ_3:40;
assume A19: x in U ; ::_thesis: x in lines (Segm (M,P,(Seg n)))
then consider i being Nat such that
A20: i in Seg m and
A21: Line (M,i) = x and
A22: f . x = i by A5;
dom f = U by A20, FUNCT_2:def_1;
then i in P by A19, A22, FUNCT_1:def_3;
then i in rng (Sgm P) by A6, FINSEQ_1:def_13;
then consider y being set such that
A23: y in dom (Sgm P) and
A24: (Sgm P) . y = i by FUNCT_1:def_3;
reconsider y = y as Element of NAT by A23;
m <> 0 by A20;
then width M = n by Th1;
then Line ((Segm (M,P,(Seg n))),y) = x by A21, A23, A24, A18, Th48;
hence x in lines (Segm (M,P,(Seg n))) by A23, A18, Th103; ::_thesis: verum
end;
hence U = lines (Segm (M,P,(Seg n))) by A8, XBOOLE_0:def_10; ::_thesis: Segm (M,P,(Seg n)) is without_repeated_line
let x1, x2 be set ; :: according to FUNCT_1:def_4 ::_thesis: ( not x1 in dom (Segm (M,P,(Seg n))) or not x2 in dom (Segm (M,P,(Seg n))) or not (Segm (M,P,(Seg n))) . x1 = (Segm (M,P,(Seg n))) . x2 or x1 = x2 )
assume that
A25: x1 in dom (Segm (M,P,(Seg n))) and
A26: x2 in dom (Segm (M,P,(Seg n))) and
A27: (Segm (M,P,(Seg n))) . x1 = (Segm (M,P,(Seg n))) . x2 ; ::_thesis: x1 = x2
A28: dom (Segm (M,P,(Seg n))) = Seg (len (Segm (M,P,(Seg n)))) by FINSEQ_1:def_3
.= Seg (card P) by MATRIX_1:def_2 ;
then A29: dom (Sgm P) = dom (Segm (M,P,(Seg n))) by A6, FINSEQ_3:40;
reconsider i1 = x1, i2 = x2 as Element of NAT by A25, A26;
A30: dom (Sgm P) = Seg (card P) by A6, FINSEQ_3:40;
then (Sgm P) . i1 in rng (Sgm P) by A25, A28, FUNCT_1:def_3;
then consider y1 being set such that
A31: y1 in dom f and
A32: f . y1 = (Sgm P) . i1 by A7, FUNCT_1:def_3;
A33: ex j1 being Nat st
( j1 in Seg m & Line (M,j1) = y1 & f . y1 = j1 ) by A5, A31;
then m <> 0 ;
then A34: width M = n by Th1;
(Sgm P) . i2 in rng (Sgm P) by A26, A28, A30, FUNCT_1:def_3;
then consider y2 being set such that
A35: y2 in dom f and
A36: f . y2 = (Sgm P) . i2 by A7, FUNCT_1:def_3;
ex j2 being Nat st
( j2 in Seg m & Line (M,j2) = y2 & f . y2 = j2 ) by A5, A35;
then A37: Line ((Segm (M,P,(Seg n))),i2) = y2 by A26, A28, A36, A34, Th48;
A38: Sgm P is one-to-one by A6, FINSEQ_3:92;
A39: Line ((Segm (M,P,(Seg n))),i1) = (Segm (M,P,(Seg n))) . i1 by A25, A28, MATRIX_2:8;
Line ((Segm (M,P,(Seg n))),i1) = y1 by A25, A28, A32, A33, A34, Th48;
then (Sgm P) . i1 = (Sgm P) . i2 by A26, A27, A28, A32, A36, A37, A39, MATRIX_2:8;
hence x1 = x2 by A25, A26, A29, A38, FUNCT_1:def_4; ::_thesis: verum
end;
theorem :: MATRIX13:123
for m, n being Nat
for K being Field
for M being Matrix of m,n,K
for RANK being Element of NAT holds
( the_rank_of M = RANK iff ( ex U being finite Subset of (n -VectSp_over K) st
( U is linearly-independent & U c= lines M & card U = RANK ) & ( for W being finite Subset of (n -VectSp_over K) st W is linearly-independent & W c= lines M holds
card W <= RANK ) ) )
proof
let m, n be Nat; ::_thesis: for K being Field
for M being Matrix of m,n,K
for RANK being Element of NAT holds
( the_rank_of M = RANK iff ( ex U being finite Subset of (n -VectSp_over K) st
( U is linearly-independent & U c= lines M & card U = RANK ) & ( for W being finite Subset of (n -VectSp_over K) st W is linearly-independent & W c= lines M holds
card W <= RANK ) ) )
let K be Field; ::_thesis: for M being Matrix of m,n,K
for RANK being Element of NAT holds
( the_rank_of M = RANK iff ( ex U being finite Subset of (n -VectSp_over K) st
( U is linearly-independent & U c= lines M & card U = RANK ) & ( for W being finite Subset of (n -VectSp_over K) st W is linearly-independent & W c= lines M holds
card W <= RANK ) ) )
let M be Matrix of m,n,K; ::_thesis: for RANK being Element of NAT holds
( the_rank_of M = RANK iff ( ex U being finite Subset of (n -VectSp_over K) st
( U is linearly-independent & U c= lines M & card U = RANK ) & ( for W being finite Subset of (n -VectSp_over K) st W is linearly-independent & W c= lines M holds
card W <= RANK ) ) )
let R be Element of NAT ; ::_thesis: ( the_rank_of M = R iff ( ex U being finite Subset of (n -VectSp_over K) st
( U is linearly-independent & U c= lines M & card U = R ) & ( for W being finite Subset of (n -VectSp_over K) st W is linearly-independent & W c= lines M holds
card W <= R ) ) )
A1: len M = m by MATRIX_1:def_2;
A2: card (Seg n) = n by FINSEQ_1:57;
A3: for W being finite Subset of (n -VectSp_over K) st W is linearly-independent & W c= lines M holds
card W <= the_rank_of M
proof
let W be finite Subset of (n -VectSp_over K); ::_thesis: ( W is linearly-independent & W c= lines M implies card W <= the_rank_of M )
assume that
A4: W is linearly-independent and
A5: W c= lines M ; ::_thesis: card W <= the_rank_of M
consider P1 being finite without_zero Subset of NAT such that
A6: P1 c= Seg m and
A7: lines (Segm (M,P1,(Seg n))) = W and
A8: Segm (M,P1,(Seg n)) is without_repeated_line by A5, Th122;
set S1 = Segm (M,P1,(Seg n));
A9: (Segm (M,P1,(Seg n))) .: (dom (Segm (M,P1,(Seg n)))) = lines (Segm (M,P1,(Seg n))) by RELAT_1:113;
dom (Segm (M,P1,(Seg n))),(Segm (M,P1,(Seg n))) .: (dom (Segm (M,P1,(Seg n)))) are_equipotent by A8, CARD_1:33;
then A10: card W = card (dom (Segm (M,P1,(Seg n)))) by A7, A9, CARD_1:5
.= card (Seg (len (Segm (M,P1,(Seg n))))) by FINSEQ_1:def_3
.= card (Seg (card P1)) by MATRIX_1:def_2
.= card P1 by FINSEQ_1:57 ;
percases ( card P1 = 0 or card P1 > 0 ) ;
suppose card P1 = 0 ; ::_thesis: card W <= the_rank_of M
hence card W <= the_rank_of M by A10; ::_thesis: verum
end;
suppose card P1 > 0 ; ::_thesis: card W <= the_rank_of M
then Seg m <> {} by A6;
then A11: m <> 0 ;
then A12: len M = m by Th1;
width M = n by Th1, A11;
then [:P1,(Seg n):] c= [:(Seg (len M)),(Seg (width M)):] by A6, A12, ZFMISC_1:96;
then [:P1,(Seg n):] c= Indices M by FINSEQ_1:def_3;
then the_rank_of (Segm (M,P1,(Seg n))) <= the_rank_of M by Th79;
hence card W <= the_rank_of M by A2, A4, A7, A8, A10, Th121; ::_thesis: verum
end;
end;
end;
A13: now__::_thesis:_Seg_(width_M)_c=_Seg_n
percases ( len M = 0 or len M > 0 ) ;
suppose len M = 0 ; ::_thesis: Seg (width M) c= Seg n
then width M = 0 by MATRIX_1:def_3;
hence Seg (width M) c= Seg n by FINSEQ_1:5; ::_thesis: verum
end;
suppose len M > 0 ; ::_thesis: Seg (width M) c= Seg n
then m > 0 by MATRIX_1:def_2;
hence Seg (width M) c= Seg n by Th1; ::_thesis: verum
end;
end;
end;
consider P, Q being finite without_zero Subset of NAT such that
A14: [:P,Q:] c= Indices M and
A15: card P = card Q and
A16: card P = the_rank_of M and
A17: Det (EqSegm (M,P,Q)) <> 0. K by Def4;
Q c= Seg (width M) by A14, A15, Th67;
then A18: Q c= Seg n by A13, XBOOLE_1:1;
set S = Segm (M,P,(Seg n));
A19: len (Segm (M,P,(Seg n))) = card P by MATRIX_1:def_2;
Segm (M,P,Q) = EqSegm (M,P,Q) by A15, Def3;
then A20: the_rank_of (EqSegm (M,P,Q)) <= the_rank_of (Segm (M,P,(Seg n))) by A18, Th80;
A21: the_rank_of (Segm (M,P,(Seg n))) <= len (Segm (M,P,(Seg n))) by Th74;
the_rank_of (EqSegm (M,P,Q)) = card P by A17, Th83;
then A22: the_rank_of (Segm (M,P,(Seg n))) = card P by A20, A19, A21, XXREAL_0:1;
then A23: lines (Segm (M,P,(Seg n))) is linearly-independent by Th121;
Segm (M,P,(Seg n)) is without_repeated_line by A22, Th121;
then A24: dom (Segm (M,P,(Seg n))),(Segm (M,P,(Seg n))) .: (dom (Segm (M,P,(Seg n)))) are_equipotent by CARD_1:33;
(Segm (M,P,(Seg n))) .: (dom (Segm (M,P,(Seg n)))) = lines (Segm (M,P,(Seg n))) by RELAT_1:113;
then A25: card (lines (Segm (M,P,(Seg n)))) = card (dom (Segm (M,P,(Seg n)))) by A24, CARD_1:5
.= card (Seg (len (Segm (M,P,(Seg n))))) by FINSEQ_1:def_3
.= card (Seg (card P)) by MATRIX_1:def_2
.= card P by FINSEQ_1:57 ;
A26: P c= Seg (len M) by A14, A15, Th67;
then lines (Segm (M,P,(Seg n))) c= lines M by A1, Th118;
hence ( the_rank_of M = R implies ( ex U being finite Subset of (n -VectSp_over K) st
( U is linearly-independent & U c= lines M & card U = R ) & ( for W being finite Subset of (n -VectSp_over K) st W is linearly-independent & W c= lines M holds
card W <= R ) ) ) by A16, A23, A25, A2, A3; ::_thesis: ( ex U being finite Subset of (n -VectSp_over K) st
( U is linearly-independent & U c= lines M & card U = R ) & ( for W being finite Subset of (n -VectSp_over K) st W is linearly-independent & W c= lines M holds
card W <= R ) implies the_rank_of M = R )
assume that
A27: ex U being finite Subset of (n -VectSp_over K) st
( U is linearly-independent & U c= lines M & card U = R ) and
A28: for W being finite Subset of (n -VectSp_over K) st W is linearly-independent & W c= lines M holds
card W <= R ; ::_thesis: the_rank_of M = R
A29: R <= the_rank_of M by A3, A27;
the_rank_of M <= R by A16, A23, A25, A26, A1, A2, A28, Th118;
hence the_rank_of M = R by A29, XXREAL_0:1; ::_thesis: verum
end;