:: MATRIX11 semantic presentation
begin
notation
let X be set ;
synonym 2Set X for TWOELEMENTSETS X;
end;
theorem Th1: :: MATRIX11:1
for X being set
for n being Nat holds
( X in 2Set (Seg n) iff ex i, j being Nat st
( i in Seg n & j in Seg n & i < j & X = {i,j} ) )
proof
let X be set ; ::_thesis: for n being Nat holds
( X in 2Set (Seg n) iff ex i, j being Nat st
( i in Seg n & j in Seg n & i < j & X = {i,j} ) )
let n be Nat; ::_thesis: ( X in 2Set (Seg n) iff ex i, j being Nat st
( i in Seg n & j in Seg n & i < j & X = {i,j} ) )
thus ( X in 2Set (Seg n) implies ex i, j being Nat st
( i in Seg n & j in Seg n & i < j & X = {i,j} ) ) ::_thesis: ( ex i, j being Nat st
( i in Seg n & j in Seg n & i < j & X = {i,j} ) implies X in 2Set (Seg n) )
proof
assume X in 2Set (Seg n) ; ::_thesis: ex i, j being Nat st
( i in Seg n & j in Seg n & i < j & X = {i,j} )
then consider x, y being set such that
A1: x in Seg n and
A2: y in Seg n and
A3: x <> y and
A4: X = {x,y} by SGRAPH1:8;
reconsider x = x, y = y as Element of NAT by A1, A2;
( x > y or y > x ) by A3, XXREAL_0:1;
hence ex i, j being Nat st
( i in Seg n & j in Seg n & i < j & X = {i,j} ) by A1, A2, A4; ::_thesis: verum
end;
assume ex i, j being Nat st
( i in Seg n & j in Seg n & i < j & X = {i,j} ) ; ::_thesis: X in 2Set (Seg n)
then consider i, j being Nat such that
A5: i in Seg n and
A6: j in Seg n and
A7: i < j and
A8: X = {i,j} ;
{i,j} c= Seg n by A5, A6, ZFMISC_1:32;
hence X in 2Set (Seg n) by A5, A6, A7, A8, SGRAPH1:8; ::_thesis: verum
end;
theorem Th2: :: MATRIX11:2
( 2Set (Seg 0) = {} & 2Set (Seg 1) = {} )
proof
thus 2Set (Seg 0) = {} ::_thesis: 2Set (Seg 1) = {}
proof
assume 2Set (Seg 0) <> {} ; ::_thesis: contradiction
then consider x being set such that
A1: x in 2Set (Seg 0) by XBOOLE_0:def_1;
ex i, j being Nat st
( i in Seg 0 & j in Seg 0 & i < j & x = {i,j} ) by A1, Th1;
hence contradiction ; ::_thesis: verum
end;
thus 2Set (Seg 1) = {} ::_thesis: verum
proof
assume 2Set (Seg 1) <> {} ; ::_thesis: contradiction
then consider x being set such that
A2: x in 2Set (Seg 1) by XBOOLE_0:def_1;
consider i, j being Nat such that
A3: i in Seg 1 and
A4: j in Seg 1 and
A5: i < j and
x = {i,j} by A2, Th1;
i = 1 by A3, FINSEQ_1:2, TARSKI:def_1;
hence contradiction by A4, A5, FINSEQ_1:2, TARSKI:def_1; ::_thesis: verum
end;
end;
theorem Th3: :: MATRIX11:3
for n being Nat st n >= 2 holds
{1,2} in 2Set (Seg n)
proof
let n be Nat; ::_thesis: ( n >= 2 implies {1,2} in 2Set (Seg n) )
assume A1: n >= 2 ; ::_thesis: {1,2} in 2Set (Seg n)
1 <= n by A1, XXREAL_0:2;
then A2: 1 in Seg n ;
2 in Seg n by A1;
hence {1,2} in 2Set (Seg n) by A2, Th1; ::_thesis: verum
end;
registration
let n be Nat;
cluster 2Set (Seg (n + 2)) -> non empty finite ;
coherence
( not 2Set (Seg (n + 2)) is empty & 2Set (Seg (n + 2)) is finite )
proof
n + 2 >= 0 + 2 by XREAL_1:6;
hence ( not 2Set (Seg (n + 2)) is empty & 2Set (Seg (n + 2)) is finite ) by Th3, SGRAPH1:13; ::_thesis: verum
end;
end;
registration
let n be Nat;
let x be set ;
let perm be Element of Permutations n;
clusterperm . x -> natural ;
coherence
perm . x is natural
proof
percases ( x in dom perm or not x in dom perm ) ;
supposeA1: x in dom perm ; ::_thesis: perm . x is natural
perm is Permutation of (Seg n) by MATRIX_2:def_9;
then A2: rng perm = Seg n by FUNCT_2:def_3;
perm . x in rng perm by A1, FUNCT_1:def_3;
hence perm . x is natural by A2; ::_thesis: verum
end;
suppose not x in dom perm ; ::_thesis: perm . x is natural
hence perm . x is natural by FUNCT_1:def_2; ::_thesis: verum
end;
end;
end;
end;
registration
let K be Field;
cluster the multF of K -> having_a_unity ;
coherence
the multF of K is having_a_unity ;
cluster the multF of K -> associative ;
coherence
the multF of K is associative ;
end;
definition
let n be Nat;
let K be Field;
let perm2 be Element of Permutations (n + 2);
func Part_sgn (perm2,K) -> Function of (2Set (Seg (n + 2))), the carrier of K means :Def1: :: MATRIX11:def 1
for i, j being Element of NAT st i in Seg (n + 2) & j in Seg (n + 2) & i < j holds
( ( perm2 . i < perm2 . j implies it . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies it . {i,j} = - (1_ K) ) );
existence
ex b1 being Function of (2Set (Seg (n + 2))), the carrier of K st
for i, j being Element of NAT st i in Seg (n + 2) & j in Seg (n + 2) & i < j holds
( ( perm2 . i < perm2 . j implies b1 . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies b1 . {i,j} = - (1_ K) ) )
proof
set n9 = n + 2;
defpred S1[ set , set ] means for i, j being Element of NAT st i in Seg (n + 2) & j in Seg (n + 2) & i < j & $1 = {i,j} holds
( ( perm2 . i < perm2 . j implies $2 = 1_ K ) & ( perm2 . i > perm2 . j implies $2 = - (1_ K) ) );
A1: for x being set st x in 2Set (Seg (n + 2)) holds
ex y being set st
( y in the carrier of K & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in 2Set (Seg (n + 2)) implies ex y being set st
( y in the carrier of K & S1[x,y] ) )
assume x in 2Set (Seg (n + 2)) ; ::_thesis: ex y being set st
( y in the carrier of K & S1[x,y] )
then consider i, j being Nat such that
A2: i in Seg (n + 2) and
A3: j in Seg (n + 2) and
A4: i < j and
A5: x = {i,j} by Th1;
perm2 is Permutation of (Seg (n + 2)) by MATRIX_2:def_9;
then A6: perm2 . i <> perm2 . j by A2, A3, A4, FUNCT_2:19;
now__::_thesis:_(_(_perm2_._i_<_perm2_._j_&_ex_y_being_set_st_
(_y_in_the_carrier_of_K_&_S1[x,y]_)_)_or_(_perm2_._i_>_perm2_._j_&_ex_y_being_set_st_
(_y_in_the_carrier_of_K_&_S1[x,y]_)_)_)
percases ( perm2 . i < perm2 . j or perm2 . i > perm2 . j ) by A6, XXREAL_0:1;
caseA7: perm2 . i < perm2 . j ; ::_thesis: ex y being set st
( y in the carrier of K & S1[x,y] )
S1[x, 1_ K]
proof
let i9, j9 be Element of NAT ; ::_thesis: ( i9 in Seg (n + 2) & j9 in Seg (n + 2) & i9 < j9 & x = {i9,j9} implies ( ( perm2 . i9 < perm2 . j9 implies 1_ K = 1_ K ) & ( perm2 . i9 > perm2 . j9 implies 1_ K = - (1_ K) ) ) )
assume that
i9 in Seg (n + 2) and
j9 in Seg (n + 2) and
A8: i9 < j9 and
A9: x = {i9,j9} ; ::_thesis: ( ( perm2 . i9 < perm2 . j9 implies 1_ K = 1_ K ) & ( perm2 . i9 > perm2 . j9 implies 1_ K = - (1_ K) ) )
( ( i = i9 & j = j9 ) or ( i = j9 & j = i9 ) ) by A5, A9, ZFMISC_1:22;
hence ( ( perm2 . i9 < perm2 . j9 implies 1_ K = 1_ K ) & ( perm2 . i9 > perm2 . j9 implies 1_ K = - (1_ K) ) ) by A4, A7, A8; ::_thesis: verum
end;
hence ex y being set st
( y in the carrier of K & S1[x,y] ) ; ::_thesis: verum
end;
caseA10: perm2 . i > perm2 . j ; ::_thesis: ex y being set st
( y in the carrier of K & S1[x,y] )
S1[x, - (1_ K)]
proof
let i9, j9 be Element of NAT ; ::_thesis: ( i9 in Seg (n + 2) & j9 in Seg (n + 2) & i9 < j9 & x = {i9,j9} implies ( ( perm2 . i9 < perm2 . j9 implies - (1_ K) = 1_ K ) & ( perm2 . i9 > perm2 . j9 implies - (1_ K) = - (1_ K) ) ) )
assume that
i9 in Seg (n + 2) and
j9 in Seg (n + 2) and
A11: i9 < j9 and
A12: x = {i9,j9} ; ::_thesis: ( ( perm2 . i9 < perm2 . j9 implies - (1_ K) = 1_ K ) & ( perm2 . i9 > perm2 . j9 implies - (1_ K) = - (1_ K) ) )
( ( i = i9 & j = j9 ) or ( i = j9 & j = i9 ) ) by A5, A12, ZFMISC_1:22;
hence ( ( perm2 . i9 < perm2 . j9 implies - (1_ K) = 1_ K ) & ( perm2 . i9 > perm2 . j9 implies - (1_ K) = - (1_ K) ) ) by A4, A10, A11; ::_thesis: verum
end;
hence ex y being set st
( y in the carrier of K & S1[x,y] ) ; ::_thesis: verum
end;
end;
end;
hence ex y being set st
( y in the carrier of K & S1[x,y] ) ; ::_thesis: verum
end;
consider Path being Function of (2Set (Seg (n + 2))), the carrier of K such that
A13: for x being set st x in 2Set (Seg (n + 2)) holds
S1[x,Path . x] from FUNCT_2:sch_1(A1);
take Path ; ::_thesis: for i, j being Element of NAT st i in Seg (n + 2) & j in Seg (n + 2) & i < j holds
( ( perm2 . i < perm2 . j implies Path . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies Path . {i,j} = - (1_ K) ) )
let i, j be Element of NAT ; ::_thesis: ( i in Seg (n + 2) & j in Seg (n + 2) & i < j implies ( ( perm2 . i < perm2 . j implies Path . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies Path . {i,j} = - (1_ K) ) ) )
assume that
A14: i in Seg (n + 2) and
A15: j in Seg (n + 2) and
A16: i < j ; ::_thesis: ( ( perm2 . i < perm2 . j implies Path . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies Path . {i,j} = - (1_ K) ) )
{i,j} in 2Set (Seg (n + 2)) by A14, A15, A16, Th1;
hence ( ( perm2 . i < perm2 . j implies Path . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies Path . {i,j} = - (1_ K) ) ) by A13, A14, A15, A16; ::_thesis: verum
end;
uniqueness
for b1, b2 being Function of (2Set (Seg (n + 2))), the carrier of K st ( for i, j being Element of NAT st i in Seg (n + 2) & j in Seg (n + 2) & i < j holds
( ( perm2 . i < perm2 . j implies b1 . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies b1 . {i,j} = - (1_ K) ) ) ) & ( for i, j being Element of NAT st i in Seg (n + 2) & j in Seg (n + 2) & i < j holds
( ( perm2 . i < perm2 . j implies b2 . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies b2 . {i,j} = - (1_ K) ) ) ) holds
b1 = b2
proof
set n9 = n + 2;
let P1, P2 be Function of (2Set (Seg (n + 2))), the carrier of K; ::_thesis: ( ( for i, j being Element of NAT st i in Seg (n + 2) & j in Seg (n + 2) & i < j holds
( ( perm2 . i < perm2 . j implies P1 . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies P1 . {i,j} = - (1_ K) ) ) ) & ( for i, j being Element of NAT st i in Seg (n + 2) & j in Seg (n + 2) & i < j holds
( ( perm2 . i < perm2 . j implies P2 . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies P2 . {i,j} = - (1_ K) ) ) ) implies P1 = P2 )
assume that
A17: for i, j being Element of NAT st i in Seg (n + 2) & j in Seg (n + 2) & i < j holds
( ( perm2 . i < perm2 . j implies P1 . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies P1 . {i,j} = - (1_ K) ) ) and
A18: for i, j being Element of NAT st i in Seg (n + 2) & j in Seg (n + 2) & i < j holds
( ( perm2 . i < perm2 . j implies P2 . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies P2 . {i,j} = - (1_ K) ) ) ; ::_thesis: P1 = P2
for x being set st x in 2Set (Seg (n + 2)) holds
P1 . x = P2 . x
proof
let x be set ; ::_thesis: ( x in 2Set (Seg (n + 2)) implies P1 . x = P2 . x )
assume x in 2Set (Seg (n + 2)) ; ::_thesis: P1 . x = P2 . x
then consider i, j being Nat such that
A19: i in Seg (n + 2) and
A20: j in Seg (n + 2) and
A21: i < j and
A22: x = {i,j} by Th1;
perm2 is Permutation of (Seg (n + 2)) by MATRIX_2:def_9;
then A23: perm2 . i <> perm2 . j by A19, A20, A21, FUNCT_2:19;
now__::_thesis:_(_(_perm2_._i_<_perm2_._j_&_P1_._x_=_P2_._x_)_or_(_perm2_._i_>_perm2_._j_&_P1_._x_=_P2_._x_)_)
percases ( perm2 . i < perm2 . j or perm2 . i > perm2 . j ) by A23, XXREAL_0:1;
caseA24: perm2 . i < perm2 . j ; ::_thesis: P1 . x = P2 . x
then P1 . {i,j} = 1_ K by A17, A19, A20, A21;
hence P1 . x = P2 . x by A18, A19, A20, A21, A22, A24; ::_thesis: verum
end;
caseA25: perm2 . i > perm2 . j ; ::_thesis: P1 . x = P2 . x
then P1 . {i,j} = - (1_ K) by A17, A19, A20, A21;
hence P1 . x = P2 . x by A18, A19, A20, A21, A22, A25; ::_thesis: verum
end;
end;
end;
hence P1 . x = P2 . x ; ::_thesis: verum
end;
hence P1 = P2 by FUNCT_2:12; ::_thesis: verum
end;
end;
:: deftheorem Def1 defines Part_sgn MATRIX11:def_1_:_
for n being Nat
for K being Field
for perm2 being Element of Permutations (n + 2)
for b4 being Function of (2Set (Seg (n + 2))), the carrier of K holds
( b4 = Part_sgn (perm2,K) iff for i, j being Element of NAT st i in Seg (n + 2) & j in Seg (n + 2) & i < j holds
( ( perm2 . i < perm2 . j implies b4 . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies b4 . {i,j} = - (1_ K) ) ) );
theorem Th4: :: MATRIX11:4
for n being Nat
for K being Field
for p2 being Element of Permutations (n + 2)
for X being Element of Fin (2Set (Seg (n + 2))) st ( for x being set st x in X holds
(Part_sgn (p2,K)) . x = 1_ K ) holds
the multF of K $$ (X,(Part_sgn (p2,K))) = 1_ K
proof
let n be Nat; ::_thesis: for K being Field
for p2 being Element of Permutations (n + 2)
for X being Element of Fin (2Set (Seg (n + 2))) st ( for x being set st x in X holds
(Part_sgn (p2,K)) . x = 1_ K ) holds
the multF of K $$ (X,(Part_sgn (p2,K))) = 1_ K
let K be Field; ::_thesis: for p2 being Element of Permutations (n + 2)
for X being Element of Fin (2Set (Seg (n + 2))) st ( for x being set st x in X holds
(Part_sgn (p2,K)) . x = 1_ K ) holds
the multF of K $$ (X,(Part_sgn (p2,K))) = 1_ K
let p2 be Element of Permutations (n + 2); ::_thesis: for X being Element of Fin (2Set (Seg (n + 2))) st ( for x being set st x in X holds
(Part_sgn (p2,K)) . x = 1_ K ) holds
the multF of K $$ (X,(Part_sgn (p2,K))) = 1_ K
let X be Element of Fin (2Set (Seg (n + 2))); ::_thesis: ( ( for x being set st x in X holds
(Part_sgn (p2,K)) . x = 1_ K ) implies the multF of K $$ (X,(Part_sgn (p2,K))) = 1_ K )
assume A1: for x being set st x in X holds
(Part_sgn (p2,K)) . x = 1_ K ; ::_thesis: the multF of K $$ (X,(Part_sgn (p2,K))) = 1_ K
set Path = Part_sgn (p2,K);
set 2S = 2Set (Seg (n + 2));
set KK = the carrier of K;
set mm = the multF of K;
consider G being Function of (Fin (2Set (Seg (n + 2)))), the carrier of K such that
A2: the multF of K $$ (X,(Part_sgn (p2,K))) = G . X and
A3: for e being Element of the carrier of K st e is_a_unity_wrt the multF of K holds
G . {} = e and
A4: for x being Element of 2Set (Seg (n + 2)) holds G . {x} = (Part_sgn (p2,K)) . x and
A5: for B being Element of Fin (2Set (Seg (n + 2))) st B c= X & B <> {} holds
for x being Element of 2Set (Seg (n + 2)) st x in X \ B holds
G . (B \/ {x}) = the multF of K . ((G . B),((Part_sgn (p2,K)) . x)) by SETWISEO:def_3;
defpred S1[ Nat] means for B being Element of Fin (2Set (Seg (n + 2))) st card B = $1 & B c= X holds
G . B = 1_ K;
A6: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A7: S1[k] ; ::_thesis: S1[k + 1]
set k1 = k + 1;
let B be Element of Fin (2Set (Seg (n + 2))); ::_thesis: ( card B = k + 1 & B c= X implies G . B = 1_ K )
assume that
A8: card B = k + 1 and
A9: B c= X ; ::_thesis: G . B = 1_ K
now__::_thesis:_(_(_k_=_0_&_G_._B_=_1__K_)_or_(_k_>_0_&_G_._B_=_1__K_)_)
percases ( k = 0 or k > 0 ) ;
case k = 0 ; ::_thesis: G . B = 1_ K
then consider x being set such that
A10: B = {x} by A8, CARD_2:42;
A11: x in B by A10, TARSKI:def_1;
B c= 2Set (Seg (n + 2)) by FINSUB_1:def_5;
then reconsider x = x as Element of 2Set (Seg (n + 2)) by A11;
G . B = (Part_sgn (p2,K)) . x by A4, A10;
hence G . B = 1_ K by A1, A9, A11; ::_thesis: verum
end;
caseA12: k > 0 ; ::_thesis: G . B = 1_ K
consider x being set such that
A13: x in B by A8, CARD_1:27, XBOOLE_0:def_1;
B c= 2Set (Seg (n + 2)) by FINSUB_1:def_5;
then reconsider x = x as Element of 2Set (Seg (n + 2)) by A13;
A14: (Part_sgn (p2,K)) . x = 1_ K by A1, A9, A13;
B c= 2Set (Seg (n + 2)) by FINSUB_1:def_5;
then B \ {x} c= 2Set (Seg (n + 2)) by XBOOLE_1:1;
then reconsider B9 = B \ {x} as Element of Fin (2Set (Seg (n + 2))) by FINSUB_1:def_5;
A15: not x in B9 by ZFMISC_1:56;
then A16: x in X \ B9 by A9, A13, XBOOLE_0:def_5;
A17: {x} \/ B9 = B by A13, ZFMISC_1:116;
then A18: k + 1 = (card B9) + 1 by A8, A15, CARD_2:41;
then G . B9 = 1_ K by A7, A9, XBOOLE_1:1;
then G . B = (1_ K) * (1_ K) by A5, A9, A12, A17, A18, A16, A14, CARD_1:27, XBOOLE_1:1;
hence G . B = 1_ K by VECTSP_1:def_4; ::_thesis: verum
end;
end;
end;
hence G . B = 1_ K ; ::_thesis: verum
end;
A19: S1[ 0 ]
proof
let B be Element of Fin (2Set (Seg (n + 2))); ::_thesis: ( card B = 0 & B c= X implies G . B = 1_ K )
assume that
A20: card B = 0 and
B c= X ; ::_thesis: G . B = 1_ K
B = {} by A20;
hence G . B = 1_ K by A3, FVSUM_1:4; ::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A19, A6);
then S1[ card X] ;
hence the multF of K $$ (X,(Part_sgn (p2,K))) = 1_ K by A2; ::_thesis: verum
end;
theorem Th5: :: MATRIX11:5
for n being Nat
for K being Field
for p2 being Element of Permutations (n + 2)
for s being Element of 2Set (Seg (n + 2)) holds
( (Part_sgn (p2,K)) . s = 1_ K or (Part_sgn (p2,K)) . s = - (1_ K) )
proof
let n be Nat; ::_thesis: for K being Field
for p2 being Element of Permutations (n + 2)
for s being Element of 2Set (Seg (n + 2)) holds
( (Part_sgn (p2,K)) . s = 1_ K or (Part_sgn (p2,K)) . s = - (1_ K) )
let K be Field; ::_thesis: for p2 being Element of Permutations (n + 2)
for s being Element of 2Set (Seg (n + 2)) holds
( (Part_sgn (p2,K)) . s = 1_ K or (Part_sgn (p2,K)) . s = - (1_ K) )
let p2 be Element of Permutations (n + 2); ::_thesis: for s being Element of 2Set (Seg (n + 2)) holds
( (Part_sgn (p2,K)) . s = 1_ K or (Part_sgn (p2,K)) . s = - (1_ K) )
let s be Element of 2Set (Seg (n + 2)); ::_thesis: ( (Part_sgn (p2,K)) . s = 1_ K or (Part_sgn (p2,K)) . s = - (1_ K) )
consider i, j being Nat such that
A1: i in Seg (n + 2) and
A2: j in Seg (n + 2) and
A3: i < j and
A4: s = {i,j} by Th1;
p2 is Permutation of (Seg (n + 2)) by MATRIX_2:def_9;
then p2 . i <> p2 . j by A1, A2, A3, FUNCT_2:19;
then ( p2 . i > p2 . j or p2 . i < p2 . j ) by XXREAL_0:1;
hence ( (Part_sgn (p2,K)) . s = 1_ K or (Part_sgn (p2,K)) . s = - (1_ K) ) by A1, A2, A3, A4, Def1; ::_thesis: verum
end;
theorem Th6: :: MATRIX11:6
for n being Nat
for K being Field
for p2, q2 being Element of Permutations (n + 2)
for i, j being Nat st i in Seg (n + 2) & j in Seg (n + 2) & i < j & p2 . i = q2 . i & p2 . j = q2 . j holds
(Part_sgn (p2,K)) . {i,j} = (Part_sgn (q2,K)) . {i,j}
proof
let n be Nat; ::_thesis: for K being Field
for p2, q2 being Element of Permutations (n + 2)
for i, j being Nat st i in Seg (n + 2) & j in Seg (n + 2) & i < j & p2 . i = q2 . i & p2 . j = q2 . j holds
(Part_sgn (p2,K)) . {i,j} = (Part_sgn (q2,K)) . {i,j}
let K be Field; ::_thesis: for p2, q2 being Element of Permutations (n + 2)
for i, j being Nat st i in Seg (n + 2) & j in Seg (n + 2) & i < j & p2 . i = q2 . i & p2 . j = q2 . j holds
(Part_sgn (p2,K)) . {i,j} = (Part_sgn (q2,K)) . {i,j}
let p2, q2 be Element of Permutations (n + 2); ::_thesis: for i, j being Nat st i in Seg (n + 2) & j in Seg (n + 2) & i < j & p2 . i = q2 . i & p2 . j = q2 . j holds
(Part_sgn (p2,K)) . {i,j} = (Part_sgn (q2,K)) . {i,j}
set n2 = n + 2;
let i, j be Nat; ::_thesis: ( i in Seg (n + 2) & j in Seg (n + 2) & i < j & p2 . i = q2 . i & p2 . j = q2 . j implies (Part_sgn (p2,K)) . {i,j} = (Part_sgn (q2,K)) . {i,j} )
assume that
A1: i in Seg (n + 2) and
A2: j in Seg (n + 2) and
A3: i < j and
A4: p2 . i = q2 . i and
A5: p2 . j = q2 . j ; ::_thesis: (Part_sgn (p2,K)) . {i,j} = (Part_sgn (q2,K)) . {i,j}
reconsider p29 = p2 as Permutation of (Seg (n + 2)) by MATRIX_2:def_9;
A6: p29 . i <> p29 . j by A1, A2, A3, FUNCT_2:19;
now__::_thesis:_(Part_sgn_(p2,K))_._{i,j}_=_(Part_sgn_(q2,K))_._{i,j}
percases ( p2 . i < p2 . j or p2 . i > p2 . j ) by A6, XXREAL_0:1;
supposeA7: p2 . i < p2 . j ; ::_thesis: (Part_sgn (p2,K)) . {i,j} = (Part_sgn (q2,K)) . {i,j}
then (Part_sgn (p2,K)) . {i,j} = 1_ K by A1, A2, A3, Def1;
hence (Part_sgn (p2,K)) . {i,j} = (Part_sgn (q2,K)) . {i,j} by A1, A2, A3, A4, A5, A7, Def1; ::_thesis: verum
end;
supposeA8: p2 . i > p2 . j ; ::_thesis: (Part_sgn (p2,K)) . {i,j} = (Part_sgn (q2,K)) . {i,j}
then (Part_sgn (p2,K)) . {i,j} = - (1_ K) by A1, A2, A3, Def1;
hence (Part_sgn (p2,K)) . {i,j} = (Part_sgn (q2,K)) . {i,j} by A1, A2, A3, A4, A5, A8, Def1; ::_thesis: verum
end;
end;
end;
hence (Part_sgn (p2,K)) . {i,j} = (Part_sgn (q2,K)) . {i,j} ; ::_thesis: verum
end;
theorem Th7: :: MATRIX11:7
for n being Nat
for K being Field
for X being Element of Fin (2Set (Seg (n + 2)))
for p2, q2 being Element of Permutations (n + 2)
for F being finite set st F = { s where s is Element of 2Set (Seg (n + 2)) : ( s in X & (Part_sgn (p2,K)) . s <> (Part_sgn (q2,K)) . s ) } holds
( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) )
proof
let n be Nat; ::_thesis: for K being Field
for X being Element of Fin (2Set (Seg (n + 2)))
for p2, q2 being Element of Permutations (n + 2)
for F being finite set st F = { s where s is Element of 2Set (Seg (n + 2)) : ( s in X & (Part_sgn (p2,K)) . s <> (Part_sgn (q2,K)) . s ) } holds
( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) )
let K be Field; ::_thesis: for X being Element of Fin (2Set (Seg (n + 2)))
for p2, q2 being Element of Permutations (n + 2)
for F being finite set st F = { s where s is Element of 2Set (Seg (n + 2)) : ( s in X & (Part_sgn (p2,K)) . s <> (Part_sgn (q2,K)) . s ) } holds
( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) )
let X be Element of Fin (2Set (Seg (n + 2))); ::_thesis: for p2, q2 being Element of Permutations (n + 2)
for F being finite set st F = { s where s is Element of 2Set (Seg (n + 2)) : ( s in X & (Part_sgn (p2,K)) . s <> (Part_sgn (q2,K)) . s ) } holds
( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) )
let p2, q2 be Element of Permutations (n + 2); ::_thesis: for F being finite set st F = { s where s is Element of 2Set (Seg (n + 2)) : ( s in X & (Part_sgn (p2,K)) . s <> (Part_sgn (q2,K)) . s ) } holds
( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) )
let F be finite set ; ::_thesis: ( F = { s where s is Element of 2Set (Seg (n + 2)) : ( s in X & (Part_sgn (p2,K)) . s <> (Part_sgn (q2,K)) . s ) } implies ( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) ) )
assume A1: F = { s where s is Element of 2Set (Seg (n + 2)) : ( s in X & (Part_sgn (p2,K)) . s <> (Part_sgn (q2,K)) . s ) } ; ::_thesis: ( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) )
set Pq = Part_sgn (q2,K);
set Pp = Part_sgn (p2,K);
set 2S = 2Set (Seg (n + 2));
X c= 2Set (Seg (n + 2)) by FINSUB_1:def_5;
then X \ F c= 2Set (Seg (n + 2)) by XBOOLE_1:1;
then reconsider Y = X \ F as Element of Fin (2Set (Seg (n + 2))) by FINSUB_1:def_5;
A2: F c= X
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F or x in X )
assume x in F ; ::_thesis: x in X
then ex s being Element of 2Set (Seg (n + 2)) st
( x = s & s in X & (Part_sgn (p2,K)) . s <> (Part_sgn (q2,K)) . s ) by A1;
hence x in X ; ::_thesis: verum
end;
then A3: F \/ Y = X by XBOOLE_1:45;
X c= 2Set (Seg (n + 2)) by FINSUB_1:def_5;
then F c= 2Set (Seg (n + 2)) by A2, XBOOLE_1:1;
then reconsider F9 = F as Element of Fin (2Set (Seg (n + 2))) by FINSUB_1:def_5;
set KK = the carrier of K;
set mm = the multF of K;
consider Gp being Function of (Fin (2Set (Seg (n + 2)))), the carrier of K such that
A4: the multF of K $$ (F9,(Part_sgn (p2,K))) = Gp . F and
A5: for e being Element of the carrier of K st e is_a_unity_wrt the multF of K holds
Gp . {} = e and
A6: for x being Element of 2Set (Seg (n + 2)) holds Gp . {x} = (Part_sgn (p2,K)) . x and
A7: for B being Element of Fin (2Set (Seg (n + 2))) st B c= F & B <> {} holds
for x being Element of 2Set (Seg (n + 2)) st x in F9 \ B holds
Gp . (B \/ {x}) = the multF of K . ((Gp . B),((Part_sgn (p2,K)) . x)) by SETWISEO:def_3;
A8: Y c= 2Set (Seg (n + 2)) by FINSUB_1:def_5;
consider Gq being Function of (Fin (2Set (Seg (n + 2)))), the carrier of K such that
A9: the multF of K $$ (F9,(Part_sgn (q2,K))) = Gq . F and
A10: for e being Element of the carrier of K st e is_a_unity_wrt the multF of K holds
Gq . {} = e and
A11: for x being Element of 2Set (Seg (n + 2)) holds Gq . {x} = (Part_sgn (q2,K)) . x and
A12: for B being Element of Fin (2Set (Seg (n + 2))) st B c= F & B <> {} holds
for x being Element of 2Set (Seg (n + 2)) st x in F \ B holds
Gq . (B \/ {x}) = the multF of K . ((Gq . B),((Part_sgn (q2,K)) . x)) by SETWISEO:def_3;
defpred S1[ Nat] means for B being Element of Fin (2Set (Seg (n + 2))) st card B = $1 & B c= F holds
( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) );
A13: for s being Element of 2Set (Seg (n + 2)) st s in F holds
(Part_sgn (p2,K)) . s = - ((Part_sgn (q2,K)) . s)
proof
let s be Element of 2Set (Seg (n + 2)); ::_thesis: ( s in F implies (Part_sgn (p2,K)) . s = - ((Part_sgn (q2,K)) . s) )
assume s in F ; ::_thesis: (Part_sgn (p2,K)) . s = - ((Part_sgn (q2,K)) . s)
then A14: ex s9 being Element of 2Set (Seg (n + 2)) st
( s9 = s & s9 in X & (Part_sgn (p2,K)) . s9 <> (Part_sgn (q2,K)) . s9 ) by A1;
A15: ( (Part_sgn (q2,K)) . s = 1_ K or (Part_sgn (q2,K)) . s = - (1_ K) ) by Th5;
( (Part_sgn (p2,K)) . s = 1_ K or (Part_sgn (p2,K)) . s = - (1_ K) ) by Th5;
then ((Part_sgn (p2,K)) . s) + ((Part_sgn (q2,K)) . s) = 0. K by A14, A15, RLVECT_1:def_10;
hence (Part_sgn (p2,K)) . s = - ((Part_sgn (q2,K)) . s) by VECTSP_1:16; ::_thesis: verum
end;
A16: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A17: S1[k] ; ::_thesis: S1[k + 1]
set k1 = k + 1;
let B be Element of Fin (2Set (Seg (n + 2))); ::_thesis: ( card B = k + 1 & B c= F implies ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ) )
assume that
A18: card B = k + 1 and
A19: B c= F ; ::_thesis: ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) )
now__::_thesis:_(_(_k_=_0_&_(_(card_B)_mod_2_=_0_implies_Gp_._B_=_Gq_._B_)_&_(_(card_B)_mod_2_=_1_implies_Gp_._B_=_-_(Gq_._B)_)_)_or_(_k_>_0_&_(_(card_B)_mod_2_=_0_implies_Gp_._B_=_Gq_._B_)_&_(_(card_B)_mod_2_=_1_implies_Gp_._B_=_-_(Gq_._B)_)_)_)
percases ( k = 0 or k > 0 ) ;
caseA20: k = 0 ; ::_thesis: ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) )
then consider x being set such that
A21: B = {x} by A18, CARD_2:42;
A22: x in B by A21, TARSKI:def_1;
B c= 2Set (Seg (n + 2)) by FINSUB_1:def_5;
then reconsider x = x as Element of 2Set (Seg (n + 2)) by A22;
A23: Gq . B = (Part_sgn (q2,K)) . x by A11, A21;
Gp . B = (Part_sgn (p2,K)) . x by A6, A21;
hence ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ) by A13, A18, A19, A20, A22, A23, NAT_D:14; ::_thesis: verum
end;
caseA24: k > 0 ; ::_thesis: ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) )
consider x being set such that
A25: x in B by A18, CARD_1:27, XBOOLE_0:def_1;
B c= 2Set (Seg (n + 2)) by FINSUB_1:def_5;
then reconsider x = x as Element of 2Set (Seg (n + 2)) by A25;
B c= 2Set (Seg (n + 2)) by FINSUB_1:def_5;
then B \ {x} c= 2Set (Seg (n + 2)) by XBOOLE_1:1;
then reconsider B9 = B \ {x} as Element of Fin (2Set (Seg (n + 2))) by FINSUB_1:def_5;
A26: not x in B9 by ZFMISC_1:56;
then A27: x in F \ B9 by A19, A25, XBOOLE_0:def_5;
A28: B9 c= F by A19, XBOOLE_1:1;
A29: {x} \/ B9 = B by A25, ZFMISC_1:116;
then A30: k + 1 = (card B9) + 1 by A18, A26, CARD_2:41;
then A31: Gq . B = the multF of K . ((Gq . B9),((Part_sgn (q2,K)) . x)) by A12, A19, A24, A29, A27, CARD_1:27, XBOOLE_1:1;
A32: Gp . B = the multF of K . ((Gp . B9),((Part_sgn (p2,K)) . x)) by A7, A19, A24, A29, A30, A27, CARD_1:27, XBOOLE_1:1;
now__::_thesis:_(_(_k_mod_2_=_0_&_(_(card_B)_mod_2_=_0_implies_Gp_._B_=_Gq_._B_)_&_(_(card_B)_mod_2_=_1_implies_Gp_._B_=_-_(Gq_._B)_)_)_or_(_k_mod_2_=_1_&_(_(card_B)_mod_2_=_0_implies_Gp_._B_=_Gq_._B_)_&_(_(card_B)_mod_2_=_1_implies_Gp_._B_=_-_(Gq_._B)_)_)_)
percases ( k mod 2 = 0 or k mod 2 = 1 ) by NAT_D:12;
caseA33: k mod 2 = 0 ; ::_thesis: ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) )
0 < 2 - 1 ;
then A34: (k + 1) mod 2 = 0 + 1 by A33, NAT_D:70;
A35: Gp . B = (Gp . B9) * (- ((Part_sgn (q2,K)) . x)) by A13, A19, A25, A32;
Gq . B = (Gp . B9) * ((Part_sgn (q2,K)) . x) by A17, A30, A28, A31, A33;
hence ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ) by A18, A35, A34, VECTSP_1:8; ::_thesis: verum
end;
caseA36: k mod 2 = 1 ; ::_thesis: ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) )
A37: (Part_sgn (p2,K)) . x = - ((Part_sgn (q2,K)) . x) by A13, A19, A25;
Gp . B9 = - (Gq . B9) by A17, A30, A28, A36;
then A38: Gp . B = (- (Gq . B9)) * (- ((Part_sgn (q2,K)) . x)) by A7, A19, A24, A29, A30, A27, A37, CARD_1:27, XBOOLE_1:1;
A39: 2 - 1 = 1 ;
Gq . B = (Gq . B9) * ((Part_sgn (q2,K)) . x) by A12, A19, A24, A29, A30, A27, CARD_1:27, XBOOLE_1:1;
hence ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ) by A18, A36, A38, A39, NAT_D:69, VECTSP_1:10; ::_thesis: verum
end;
end;
end;
hence ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ) ; ::_thesis: verum
end;
end;
end;
hence ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ) ; ::_thesis: verum
end;
A40: S1[ 0 ]
proof
let B be Element of Fin (2Set (Seg (n + 2))); ::_thesis: ( card B = 0 & B c= F implies ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ) )
assume that
A41: card B = 0 and
B c= F ; ::_thesis: ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) )
A42: 0 = 0 mod 2 by NAT_D:26;
A43: B = {} by A41;
then Gp . B = 1_ K by A5, FVSUM_1:4;
hence ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ) by A10, A43, A42, FVSUM_1:4; ::_thesis: verum
end;
A44: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A40, A16);
A45: Y misses F by XBOOLE_1:79;
then A46: the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K . (( the multF of K $$ (Y,(Part_sgn (p2,K)))),( the multF of K $$ (F9,(Part_sgn (p2,K))))) by A3, SETWOP_2:4;
A47: the multF of K $$ (X,(Part_sgn (q2,K))) = the multF of K . (( the multF of K $$ (Y,(Part_sgn (q2,K)))),( the multF of K $$ (F9,(Part_sgn (q2,K))))) by A45, A3, SETWOP_2:4;
A48: dom (Part_sgn (p2,K)) = 2Set (Seg (n + 2)) by FUNCT_2:def_1;
then A49: dom ((Part_sgn (p2,K)) | Y) = Y by A8, RELAT_1:62;
dom (Part_sgn (q2,K)) = 2Set (Seg (n + 2)) by FUNCT_2:def_1;
then A50: dom ((Part_sgn (q2,K)) | Y) = Y by A8, RELAT_1:62;
for x being set st x in dom ((Part_sgn (p2,K)) | Y) holds
((Part_sgn (p2,K)) | Y) . x = ((Part_sgn (q2,K)) | Y) . x
proof
let x be set ; ::_thesis: ( x in dom ((Part_sgn (p2,K)) | Y) implies ((Part_sgn (p2,K)) | Y) . x = ((Part_sgn (q2,K)) | Y) . x )
assume A51: x in dom ((Part_sgn (p2,K)) | Y) ; ::_thesis: ((Part_sgn (p2,K)) | Y) . x = ((Part_sgn (q2,K)) | Y) . x
Y c= 2Set (Seg (n + 2)) by FINSUB_1:def_5;
then reconsider x9 = x as Element of 2Set (Seg (n + 2)) by A49, A51;
A52: ((Part_sgn (p2,K)) | Y) . x9 = (Part_sgn (p2,K)) . x9 by A51, FUNCT_1:47;
A53: not x9 in F by A49, A51, XBOOLE_0:def_5;
assume A54: ((Part_sgn (p2,K)) | Y) . x <> ((Part_sgn (q2,K)) | Y) . x ; ::_thesis: contradiction
((Part_sgn (q2,K)) | Y) . x9 = (Part_sgn (q2,K)) . x9 by A49, A50, A51, FUNCT_1:47;
hence contradiction by A1, A49, A51, A54, A52, A53; ::_thesis: verum
end;
then A55: (Part_sgn (p2,K)) | Y = (Part_sgn (q2,K)) | Y by A48, A8, A50, FUNCT_1:2, RELAT_1:62;
then A56: the multF of K $$ (Y,(Part_sgn (p2,K))) = the multF of K $$ (Y,(Part_sgn (q2,K))) by SETWOP_2:7;
now__::_thesis:_(_(_(card_F)_mod_2_=_0_&_(_(card_F)_mod_2_=_0_implies_the_multF_of_K_$$_(X,(Part_sgn_(p2,K)))_=_the_multF_of_K_$$_(X,(Part_sgn_(q2,K)))_)_&_(_(card_F)_mod_2_=_1_implies_the_multF_of_K_$$_(X,(Part_sgn_(p2,K)))_=_-_(_the_multF_of_K_$$_(X,(Part_sgn_(q2,K))))_)_)_or_(_(card_F)_mod_2_=_1_&_(_(card_F)_mod_2_=_0_implies_the_multF_of_K_$$_(X,(Part_sgn_(p2,K)))_=_the_multF_of_K_$$_(X,(Part_sgn_(q2,K)))_)_&_(_(card_F)_mod_2_=_1_implies_the_multF_of_K_$$_(X,(Part_sgn_(p2,K)))_=_-_(_the_multF_of_K_$$_(X,(Part_sgn_(q2,K))))_)_)_)
percases ( (card F) mod 2 = 0 or (card F) mod 2 = 1 ) by NAT_D:12;
case (card F) mod 2 = 0 ; ::_thesis: ( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) )
hence ( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) ) by A4, A9, A44, A56, A46, A47; ::_thesis: verum
end;
caseA57: (card F) mod 2 = 1 ; ::_thesis: ( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) )
A58: the multF of K $$ (X,(Part_sgn (q2,K))) = ( the multF of K $$ (Y,(Part_sgn (p2,K)))) * ( the multF of K $$ (F9,(Part_sgn (q2,K)))) by A55, A47, SETWOP_2:7;
the multF of K $$ (X,(Part_sgn (p2,K))) = ( the multF of K $$ (Y,(Part_sgn (p2,K)))) * (- ( the multF of K $$ (F9,(Part_sgn (q2,K))))) by A4, A9, A44, A46, A57;
hence ( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) ) by A57, A58, VECTSP_1:8; ::_thesis: verum
end;
end;
end;
hence ( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) ) ; ::_thesis: verum
end;
theorem Th8: :: MATRIX11:8
for n being Nat
for P being Permutation of (Seg n) st P is being_transposition holds
for i, j being Nat st i < j holds
( P . i = j iff ( i in dom P & j in dom P & P . i = j & P . j = i & ( for k being Nat st k <> i & k <> j & k in dom P holds
P . k = k ) ) )
proof
let n be Nat; ::_thesis: for P being Permutation of (Seg n) st P is being_transposition holds
for i, j being Nat st i < j holds
( P . i = j iff ( i in dom P & j in dom P & P . i = j & P . j = i & ( for k being Nat st k <> i & k <> j & k in dom P holds
P . k = k ) ) )
let P be Permutation of (Seg n); ::_thesis: ( P is being_transposition implies for i, j being Nat st i < j holds
( P . i = j iff ( i in dom P & j in dom P & P . i = j & P . j = i & ( for k being Nat st k <> i & k <> j & k in dom P holds
P . k = k ) ) ) )
assume P is being_transposition ; ::_thesis: for i, j being Nat st i < j holds
( P . i = j iff ( i in dom P & j in dom P & P . i = j & P . j = i & ( for k being Nat st k <> i & k <> j & k in dom P holds
P . k = k ) ) )
then consider i9, j9 being Nat such that
i9 in dom P and
j9 in dom P and
i9 <> j9 and
A1: P . i9 = j9 and
A2: P . j9 = i9 and
A3: for k being Nat st k <> i9 & k <> j9 & k in dom P holds
P . k = k by MATRIX_2:def_11;
let i, j be Nat; ::_thesis: ( i < j implies ( P . i = j iff ( i in dom P & j in dom P & P . i = j & P . j = i & ( for k being Nat st k <> i & k <> j & k in dom P holds
P . k = k ) ) ) )
assume A4: i < j ; ::_thesis: ( P . i = j iff ( i in dom P & j in dom P & P . i = j & P . j = i & ( for k being Nat st k <> i & k <> j & k in dom P holds
P . k = k ) ) )
thus ( P . i = j implies ( i in dom P & j in dom P & P . i = j & P . j = i & ( for k being Nat st k <> i & k <> j & k in dom P holds
P . k = k ) ) ) ::_thesis: ( i in dom P & j in dom P & P . i = j & P . j = i & ( for k being Nat st k <> i & k <> j & k in dom P holds
P . k = k ) implies P . i = j )
proof
A5: dom P = Seg n by FUNCT_2:52;
A6: rng P = Seg n by FUNCT_2:def_3;
assume A7: P . i = j ; ::_thesis: ( i in dom P & j in dom P & P . i = j & P . j = i & ( for k being Nat st k <> i & k <> j & k in dom P holds
P . k = k ) )
then A8: i in dom P by A4, FUNCT_1:def_2;
then ( i = j9 or i = i9 ) by A4, A3, A7;
hence ( i in dom P & j in dom P & P . i = j & P . j = i & ( for k being Nat st k <> i & k <> j & k in dom P holds
P . k = k ) ) by A1, A2, A3, A7, A8, A6, A5, FUNCT_1:def_3; ::_thesis: verum
end;
thus ( i in dom P & j in dom P & P . i = j & P . j = i & ( for k being Nat st k <> i & k <> j & k in dom P holds
P . k = k ) implies P . i = j ) ; ::_thesis: verum
end;
theorem Th9: :: MATRIX11:9
for n being Nat
for K being Field
for p2, q2, pq2 being Element of Permutations (n + 2)
for i, j being Nat st pq2 = p2 * q2 & q2 is being_transposition & q2 . i = j & i < j holds
for s being Element of 2Set (Seg (n + 2)) holds
( not (Part_sgn (p2,K)) . s <> (Part_sgn (pq2,K)) . s or i in s or j in s )
proof
let n be Nat; ::_thesis: for K being Field
for p2, q2, pq2 being Element of Permutations (n + 2)
for i, j being Nat st pq2 = p2 * q2 & q2 is being_transposition & q2 . i = j & i < j holds
for s being Element of 2Set (Seg (n + 2)) holds
( not (Part_sgn (p2,K)) . s <> (Part_sgn (pq2,K)) . s or i in s or j in s )
let K be Field; ::_thesis: for p2, q2, pq2 being Element of Permutations (n + 2)
for i, j being Nat st pq2 = p2 * q2 & q2 is being_transposition & q2 . i = j & i < j holds
for s being Element of 2Set (Seg (n + 2)) holds
( not (Part_sgn (p2,K)) . s <> (Part_sgn (pq2,K)) . s or i in s or j in s )
set n2 = n + 2;
let p, q, pq be Element of Permutations (n + 2); ::_thesis: for i, j being Nat st pq = p * q & q is being_transposition & q . i = j & i < j holds
for s being Element of 2Set (Seg (n + 2)) holds
( not (Part_sgn (p,K)) . s <> (Part_sgn (pq,K)) . s or i in s or j in s )
let i, j be Nat; ::_thesis: ( pq = p * q & q is being_transposition & q . i = j & i < j implies for s being Element of 2Set (Seg (n + 2)) holds
( not (Part_sgn (p,K)) . s <> (Part_sgn (pq,K)) . s or i in s or j in s ) )
assume that
A1: pq = p * q and
A2: q is being_transposition and
A3: q . i = j and
A4: i < j ; ::_thesis: for s being Element of 2Set (Seg (n + 2)) holds
( not (Part_sgn (p,K)) . s <> (Part_sgn (pq,K)) . s or i in s or j in s )
reconsider q9 = q, pq9 = pq as Permutation of (Seg (n + 2)) by MATRIX_2:def_9;
let s be Element of 2Set (Seg (n + 2)); ::_thesis: ( not (Part_sgn (p,K)) . s <> (Part_sgn (pq,K)) . s or i in s or j in s )
assume A5: (Part_sgn (p,K)) . s <> (Part_sgn (pq,K)) . s ; ::_thesis: ( i in s or j in s )
A6: dom q9 = Seg (n + 2) by FUNCT_2:52;
A7: dom pq9 = Seg (n + 2) by FUNCT_2:52;
assume that
A8: not i in s and
A9: not j in s ; ::_thesis: contradiction
consider i9, j9 being Nat such that
A10: i9 in Seg (n + 2) and
A11: j9 in Seg (n + 2) and
A12: i9 < j9 and
A13: s = {i9,j9} by Th1;
A14: j <> j9 by A13, A9, TARSKI:def_2;
A15: j <> i9 by A13, A9, TARSKI:def_2;
i <> j9 by A13, A8, TARSKI:def_2;
then q . j9 = j9 by A2, A3, A4, A11, A14, A6, Th8;
then A16: pq . j9 = p . j9 by A1, A11, A7, FUNCT_1:12;
i <> i9 by A13, A8, TARSKI:def_2;
then q . i9 = i9 by A2, A3, A4, A10, A15, A6, Th8;
then pq . i9 = p . i9 by A1, A10, A7, FUNCT_1:12;
hence contradiction by A5, A10, A11, A12, A13, A16, Th6; ::_thesis: verum
end;
Lm1: for n being Nat
for K being Field
for p2 being Element of Permutations (n + 2)
for i, j being Nat st i in Seg (n + 2) & j in Seg (n + 2) & i < j & 1_ K <> - (1_ K) holds
( ( (Part_sgn (p2,K)) . {i,j} = 1_ K implies p2 . i < p2 . j ) & ( (Part_sgn (p2,K)) . {i,j} = - (1_ K) implies p2 . i > p2 . j ) )
proof
let n be Nat; ::_thesis: for K being Field
for p2 being Element of Permutations (n + 2)
for i, j being Nat st i in Seg (n + 2) & j in Seg (n + 2) & i < j & 1_ K <> - (1_ K) holds
( ( (Part_sgn (p2,K)) . {i,j} = 1_ K implies p2 . i < p2 . j ) & ( (Part_sgn (p2,K)) . {i,j} = - (1_ K) implies p2 . i > p2 . j ) )
let K be Field; ::_thesis: for p2 being Element of Permutations (n + 2)
for i, j being Nat st i in Seg (n + 2) & j in Seg (n + 2) & i < j & 1_ K <> - (1_ K) holds
( ( (Part_sgn (p2,K)) . {i,j} = 1_ K implies p2 . i < p2 . j ) & ( (Part_sgn (p2,K)) . {i,j} = - (1_ K) implies p2 . i > p2 . j ) )
let p2 be Element of Permutations (n + 2); ::_thesis: for i, j being Nat st i in Seg (n + 2) & j in Seg (n + 2) & i < j & 1_ K <> - (1_ K) holds
( ( (Part_sgn (p2,K)) . {i,j} = 1_ K implies p2 . i < p2 . j ) & ( (Part_sgn (p2,K)) . {i,j} = - (1_ K) implies p2 . i > p2 . j ) )
set n2 = n + 2;
let i, j be Nat; ::_thesis: ( i in Seg (n + 2) & j in Seg (n + 2) & i < j & 1_ K <> - (1_ K) implies ( ( (Part_sgn (p2,K)) . {i,j} = 1_ K implies p2 . i < p2 . j ) & ( (Part_sgn (p2,K)) . {i,j} = - (1_ K) implies p2 . i > p2 . j ) ) )
assume that
A1: i in Seg (n + 2) and
A2: j in Seg (n + 2) and
A3: i < j and
A4: 1_ K <> - (1_ K) ; ::_thesis: ( ( (Part_sgn (p2,K)) . {i,j} = 1_ K implies p2 . i < p2 . j ) & ( (Part_sgn (p2,K)) . {i,j} = - (1_ K) implies p2 . i > p2 . j ) )
reconsider p9 = p2 as Permutation of (Seg (n + 2)) by MATRIX_2:def_9;
p9 . i <> p9 . j by A1, A2, A3, FUNCT_2:19;
then A5: ( p2 . i < p2 . j or p2 . i > p2 . j ) by XXREAL_0:1;
thus ( (Part_sgn (p2,K)) . {i,j} = 1_ K implies p2 . i < p2 . j ) ::_thesis: ( (Part_sgn (p2,K)) . {i,j} = - (1_ K) implies p2 . i > p2 . j )
proof
p9 . i <> p9 . j by A1, A2, A3, FUNCT_2:19;
then A6: ( p2 . i < p2 . j or p2 . i > p2 . j ) by XXREAL_0:1;
assume (Part_sgn (p2,K)) . {i,j} = 1_ K ; ::_thesis: p2 . i < p2 . j
hence p2 . i < p2 . j by A1, A2, A3, A4, A6, Def1; ::_thesis: verum
end;
assume (Part_sgn (p2,K)) . {i,j} = - (1_ K) ; ::_thesis: p2 . i > p2 . j
hence p2 . i > p2 . j by A1, A2, A3, A4, A5, Def1; ::_thesis: verum
end;
theorem Th10: :: MATRIX11:10
for n being Nat
for p2, q2, pq2 being Element of Permutations (n + 2)
for i, j being Nat
for K being Field st pq2 = p2 * q2 & q2 is being_transposition & q2 . i = j & i < j & 1_ K <> - (1_ K) holds
( (Part_sgn (p2,K)) . {i,j} <> (Part_sgn (pq2,K)) . {i,j} & ( for k being Nat st k in Seg (n + 2) & i <> k & j <> k holds
( (Part_sgn (p2,K)) . {i,k} <> (Part_sgn (pq2,K)) . {i,k} iff (Part_sgn (p2,K)) . {j,k} <> (Part_sgn (pq2,K)) . {j,k} ) ) )
proof
let n be Nat; ::_thesis: for p2, q2, pq2 being Element of Permutations (n + 2)
for i, j being Nat
for K being Field st pq2 = p2 * q2 & q2 is being_transposition & q2 . i = j & i < j & 1_ K <> - (1_ K) holds
( (Part_sgn (p2,K)) . {i,j} <> (Part_sgn (pq2,K)) . {i,j} & ( for k being Nat st k in Seg (n + 2) & i <> k & j <> k holds
( (Part_sgn (p2,K)) . {i,k} <> (Part_sgn (pq2,K)) . {i,k} iff (Part_sgn (p2,K)) . {j,k} <> (Part_sgn (pq2,K)) . {j,k} ) ) )
set n2 = n + 2;
let p, q, pq be Element of Permutations (n + 2); ::_thesis: for i, j being Nat
for K being Field st pq = p * q & q is being_transposition & q . i = j & i < j & 1_ K <> - (1_ K) holds
( (Part_sgn (p,K)) . {i,j} <> (Part_sgn (pq,K)) . {i,j} & ( for k being Nat st k in Seg (n + 2) & i <> k & j <> k holds
( (Part_sgn (p,K)) . {i,k} <> (Part_sgn (pq,K)) . {i,k} iff (Part_sgn (p,K)) . {j,k} <> (Part_sgn (pq,K)) . {j,k} ) ) )
let i, j be Nat; ::_thesis: for K being Field st pq = p * q & q is being_transposition & q . i = j & i < j & 1_ K <> - (1_ K) holds
( (Part_sgn (p,K)) . {i,j} <> (Part_sgn (pq,K)) . {i,j} & ( for k being Nat st k in Seg (n + 2) & i <> k & j <> k holds
( (Part_sgn (p,K)) . {i,k} <> (Part_sgn (pq,K)) . {i,k} iff (Part_sgn (p,K)) . {j,k} <> (Part_sgn (pq,K)) . {j,k} ) ) )
let K be Field; ::_thesis: ( pq = p * q & q is being_transposition & q . i = j & i < j & 1_ K <> - (1_ K) implies ( (Part_sgn (p,K)) . {i,j} <> (Part_sgn (pq,K)) . {i,j} & ( for k being Nat st k in Seg (n + 2) & i <> k & j <> k holds
( (Part_sgn (p,K)) . {i,k} <> (Part_sgn (pq,K)) . {i,k} iff (Part_sgn (p,K)) . {j,k} <> (Part_sgn (pq,K)) . {j,k} ) ) ) )
assume that
A1: pq = p * q and
A2: q is being_transposition and
A3: q . i = j and
A4: i < j and
A5: 1_ K <> - (1_ K) ; ::_thesis: ( (Part_sgn (p,K)) . {i,j} <> (Part_sgn (pq,K)) . {i,j} & ( for k being Nat st k in Seg (n + 2) & i <> k & j <> k holds
( (Part_sgn (p,K)) . {i,k} <> (Part_sgn (pq,K)) . {i,k} iff (Part_sgn (p,K)) . {j,k} <> (Part_sgn (pq,K)) . {j,k} ) ) )
A6: i in dom q by A2, A3, A4, Th8;
set P2 = Part_sgn (pq,K);
set P1 = Part_sgn (p,K);
reconsider p9 = p, q9 = q, pq9 = pq as Permutation of (Seg (n + 2)) by MATRIX_2:def_9;
A7: dom q9 = Seg (n + 2) by FUNCT_2:52;
A8: j in dom q by A2, A3, A4, Th8;
A9: dom pq9 = Seg (n + 2) by FUNCT_2:52;
then A10: pq . i = p . j by A1, A3, A6, A7, FUNCT_1:12;
q . j = i by A2, A3, A4, Th8;
then A11: pq . j = p . i by A1, A8, A9, A7, FUNCT_1:12;
dom p9 = Seg (n + 2) by FUNCT_2:52;
then A12: p9 . i <> p9 . j by A4, A6, A8, A7, FUNCT_1:def_4;
now__::_thesis:_(Part_sgn_(p,K))_._{i,j}_<>_(Part_sgn_(pq,K))_._{i,j}
percases ( p . i < p . j or p . i > p . j ) by A12, XXREAL_0:1;
supposeA13: p . i < p . j ; ::_thesis: (Part_sgn (p,K)) . {i,j} <> (Part_sgn (pq,K)) . {i,j}
then (Part_sgn (p,K)) . {i,j} = 1_ K by A4, A6, A8, A7, Def1;
hence (Part_sgn (p,K)) . {i,j} <> (Part_sgn (pq,K)) . {i,j} by A4, A5, A6, A8, A7, A10, A11, A13, Def1; ::_thesis: verum
end;
supposeA14: p . i > p . j ; ::_thesis: (Part_sgn (p,K)) . {i,j} <> (Part_sgn (pq,K)) . {i,j}
then (Part_sgn (p,K)) . {i,j} = - (1_ K) by A4, A6, A8, A7, Def1;
hence (Part_sgn (p,K)) . {i,j} <> (Part_sgn (pq,K)) . {i,j} by A4, A5, A6, A8, A7, A10, A11, A14, Def1; ::_thesis: verum
end;
end;
end;
hence (Part_sgn (p,K)) . {i,j} <> (Part_sgn (pq,K)) . {i,j} ; ::_thesis: for k being Nat st k in Seg (n + 2) & i <> k & j <> k holds
( (Part_sgn (p,K)) . {i,k} <> (Part_sgn (pq,K)) . {i,k} iff (Part_sgn (p,K)) . {j,k} <> (Part_sgn (pq,K)) . {j,k} )
let k be Nat; ::_thesis: ( k in Seg (n + 2) & i <> k & j <> k implies ( (Part_sgn (p,K)) . {i,k} <> (Part_sgn (pq,K)) . {i,k} iff (Part_sgn (p,K)) . {j,k} <> (Part_sgn (pq,K)) . {j,k} ) )
assume that
A15: k in Seg (n + 2) and
A16: i <> k and
A17: j <> k ; ::_thesis: ( (Part_sgn (p,K)) . {i,k} <> (Part_sgn (pq,K)) . {i,k} iff (Part_sgn (p,K)) . {j,k} <> (Part_sgn (pq,K)) . {j,k} )
A18: q . k = k by A2, A3, A4, A7, A15, A16, A17, Th8;
A19: pq . k = p . (q . k) by A1, A9, A15, FUNCT_1:12;
( i < k or k < i ) by A16, XXREAL_0:1;
then A20: {i,k} in 2Set (Seg (n + 2)) by A6, A7, A15, Th1;
A21: ( (Part_sgn (p,K)) . {i,k} = (Part_sgn (pq,K)) . {i,k} implies (Part_sgn (p,K)) . {j,k} = (Part_sgn (pq,K)) . {j,k} )
proof
A22: ( j < k or k < j ) by A17, XXREAL_0:1;
A23: ( i < k or i > k ) by A16, XXREAL_0:1;
assume A24: (Part_sgn (p,K)) . {i,k} = (Part_sgn (pq,K)) . {i,k} ; ::_thesis: (Part_sgn (p,K)) . {j,k} = (Part_sgn (pq,K)) . {j,k}
( (Part_sgn (p,K)) . {k,i} = 1_ K or (Part_sgn (p,K)) . {k,i} = - (1_ K) ) by A20, Th5;
then ( ( pq . j < pq . k & p . j < p . k ) or ( pq . j > pq . k & p . j > p . k ) ) by A5, A6, A7, A10, A11, A15, A18, A19, A24, A23, Lm1;
then ( ( (Part_sgn (pq,K)) . {j,k} = 1_ K & (Part_sgn (p,K)) . {j,k} = 1_ K ) or ( (Part_sgn (pq,K)) . {j,k} = - (1_ K) & (Part_sgn (p,K)) . {j,k} = - (1_ K) ) ) by A8, A7, A15, A22, Def1;
hence (Part_sgn (p,K)) . {j,k} = (Part_sgn (pq,K)) . {j,k} ; ::_thesis: verum
end;
( j < k or k < j ) by A17, XXREAL_0:1;
then A25: {j,k} in 2Set (Seg (n + 2)) by A8, A7, A15, Th1;
( (Part_sgn (p,K)) . {j,k} = (Part_sgn (pq,K)) . {j,k} implies (Part_sgn (p,K)) . {i,k} = (Part_sgn (pq,K)) . {i,k} )
proof
A26: ( i < k or k < i ) by A16, XXREAL_0:1;
A27: ( j < k or j > k ) by A17, XXREAL_0:1;
assume A28: (Part_sgn (p,K)) . {j,k} = (Part_sgn (pq,K)) . {j,k} ; ::_thesis: (Part_sgn (p,K)) . {i,k} = (Part_sgn (pq,K)) . {i,k}
( (Part_sgn (p,K)) . {k,j} = 1_ K or (Part_sgn (p,K)) . {k,j} = - (1_ K) ) by A25, Th5;
then ( ( pq . i < pq . k & p . i < p . k ) or ( pq . i > pq . k & p . i > p . k ) ) by A5, A8, A7, A10, A11, A15, A18, A19, A28, A27, Lm1;
then ( ( (Part_sgn (pq,K)) . {i,k} = 1_ K & (Part_sgn (p,K)) . {i,k} = 1_ K ) or ( (Part_sgn (pq,K)) . {i,k} = - (1_ K) & (Part_sgn (p,K)) . {i,k} = - (1_ K) ) ) by A6, A7, A15, A26, Def1;
hence (Part_sgn (p,K)) . {i,k} = (Part_sgn (pq,K)) . {i,k} ; ::_thesis: verum
end;
hence ( (Part_sgn (p,K)) . {i,k} <> (Part_sgn (pq,K)) . {i,k} iff (Part_sgn (p,K)) . {j,k} <> (Part_sgn (pq,K)) . {j,k} ) by A21; ::_thesis: verum
end;
definition
let n be Nat;
let K be Field;
let perm2 be Element of Permutations (n + 2);
func sgn (perm2,K) -> Element of K equals :: MATRIX11:def 2
the multF of K $$ ((FinOmega (2Set (Seg (n + 2)))),(Part_sgn (perm2,K)));
coherence
the multF of K $$ ((FinOmega (2Set (Seg (n + 2)))),(Part_sgn (perm2,K))) is Element of K ;
end;
:: deftheorem defines sgn MATRIX11:def_2_:_
for n being Nat
for K being Field
for perm2 being Element of Permutations (n + 2) holds sgn (perm2,K) = the multF of K $$ ((FinOmega (2Set (Seg (n + 2)))),(Part_sgn (perm2,K)));
theorem Th11: :: MATRIX11:11
for n being Nat
for K being Field
for p2 being Element of Permutations (n + 2) holds
( sgn (p2,K) = 1_ K or sgn (p2,K) = - (1_ K) )
proof
let n be Nat; ::_thesis: for K being Field
for p2 being Element of Permutations (n + 2) holds
( sgn (p2,K) = 1_ K or sgn (p2,K) = - (1_ K) )
let K be Field; ::_thesis: for p2 being Element of Permutations (n + 2) holds
( sgn (p2,K) = 1_ K or sgn (p2,K) = - (1_ K) )
let p2 be Element of Permutations (n + 2); ::_thesis: ( sgn (p2,K) = 1_ K or sgn (p2,K) = - (1_ K) )
set KK = the carrier of K;
set n2 = n + 2;
set 2S = 2Set (Seg (n + 2));
set mm = the multF of K;
set Path = Part_sgn (p2,K);
A1: FinOmega (2Set (Seg (n + 2))) = 2Set (Seg (n + 2)) by MATRIX_2:def_14;
then reconsider 2S9 = 2Set (Seg (n + 2)) as Element of Fin (2Set (Seg (n + 2))) ;
consider G being Function of (Fin (2Set (Seg (n + 2)))), the carrier of K such that
A2: the multF of K $$ (2S9,(Part_sgn (p2,K))) = G . 2S9 and
A3: for e being Element of the carrier of K st e is_a_unity_wrt the multF of K holds
G . {} = e and
A4: for s being Element of 2Set (Seg (n + 2)) holds G . {s} = (Part_sgn (p2,K)) . s and
A5: for B being Element of Fin (2Set (Seg (n + 2))) st B c= 2S9 & B <> {} holds
for s being Element of 2Set (Seg (n + 2)) st s in 2S9 \ B holds
G . (B \/ {s}) = the multF of K . ((G . B),((Part_sgn (p2,K)) . s)) by SETWISEO:def_3;
defpred S1[ Nat] means for B being Element of Fin (2Set (Seg (n + 2))) st card B = $1 & B c= 2Set (Seg (n + 2)) & not G . B = 1_ K holds
G . B = - (1_ K);
A6: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A7: S1[k] ; ::_thesis: S1[k + 1]
set k1 = k + 1;
let B be Element of Fin (2Set (Seg (n + 2))); ::_thesis: ( card B = k + 1 & B c= 2Set (Seg (n + 2)) & not G . B = 1_ K implies G . B = - (1_ K) )
assume that
A8: card B = k + 1 and
A9: B c= 2Set (Seg (n + 2)) ; ::_thesis: ( G . B = 1_ K or G . B = - (1_ K) )
now__::_thesis:_(_(_k_=_0_&_(_G_._B_=_1__K_or_G_._B_=_-_(1__K)_)_)_or_(_k_>_0_&_(_G_._B_=_1__K_or_G_._B_=_-_(1__K)_)_)_)
percases ( k = 0 or k > 0 ) ;
case k = 0 ; ::_thesis: ( G . B = 1_ K or G . B = - (1_ K) )
then consider x being set such that
A10: B = {x} by A8, CARD_2:42;
x in B by A10, TARSKI:def_1;
then reconsider x = x as Element of 2Set (Seg (n + 2)) by A9;
G . B = (Part_sgn (p2,K)) . x by A4, A10;
hence ( G . B = 1_ K or G . B = - (1_ K) ) by Th5; ::_thesis: verum
end;
caseA11: k > 0 ; ::_thesis: ( G . B = 1_ K or G . B = - (1_ K) )
consider x being set such that
A12: x in B by A8, CARD_1:27, XBOOLE_0:def_1;
reconsider x = x as Element of 2Set (Seg (n + 2)) by A9, A12;
B \ {x} c= 2Set (Seg (n + 2)) by A9, XBOOLE_1:1;
then reconsider B9 = B \ {x} as Element of Fin (2Set (Seg (n + 2))) by FINSUB_1:def_5;
A13: not x in B9 by ZFMISC_1:56;
A14: {x} \/ B9 = B by A12, ZFMISC_1:116;
then A15: k + 1 = (card B9) + 1 by A8, A13, CARD_2:41;
then A16: ( G . B9 = 1_ K or G . B9 = - (1_ K) ) by A7, A9, XBOOLE_1:1;
x in (2Set (Seg (n + 2))) \ B9 by A13, XBOOLE_0:def_5;
then G . B = the multF of K . ((G . B9),((Part_sgn (p2,K)) . x)) by A5, A9, A11, A14, A15, CARD_1:27, XBOOLE_1:1;
then ( G . B = (1_ K) * (1_ K) or G . B = (1_ K) * (- (1_ K)) or G . B = (- (1_ K)) * (1_ K) or G . B = (- (1_ K)) * (- (1_ K)) ) by A16, Th5;
then ( G . B = (1_ K) * (1_ K) or G . B = (1_ K) * (- (1_ K)) ) by VECTSP_1:10;
hence ( G . B = 1_ K or G . B = - (1_ K) ) by VECTSP_1:def_4; ::_thesis: verum
end;
end;
end;
hence ( G . B = 1_ K or G . B = - (1_ K) ) ; ::_thesis: verum
end;
A17: S1[ 0 ]
proof
let B be Element of Fin (2Set (Seg (n + 2))); ::_thesis: ( card B = 0 & B c= 2Set (Seg (n + 2)) & not G . B = 1_ K implies G . B = - (1_ K) )
assume that
A18: card B = 0 and
B c= 2Set (Seg (n + 2)) ; ::_thesis: ( G . B = 1_ K or G . B = - (1_ K) )
B = {} by A18;
hence ( G . B = 1_ K or G . B = - (1_ K) ) by A3, FVSUM_1:4; ::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A17, A6);
then S1[ card 2S9] ;
hence ( sgn (p2,K) = 1_ K or sgn (p2,K) = - (1_ K) ) by A1, A2; ::_thesis: verum
end;
theorem Th12: :: MATRIX11:12
for n being Nat
for K being Field
for Id being Element of Permutations (n + 2) st Id = idseq (n + 2) holds
sgn (Id,K) = 1_ K
proof
let n be Nat; ::_thesis: for K being Field
for Id being Element of Permutations (n + 2) st Id = idseq (n + 2) holds
sgn (Id,K) = 1_ K
let K be Field; ::_thesis: for Id being Element of Permutations (n + 2) st Id = idseq (n + 2) holds
sgn (Id,K) = 1_ K
set n2 = n + 2;
let Id be Element of Permutations (n + 2); ::_thesis: ( Id = idseq (n + 2) implies sgn (Id,K) = 1_ K )
assume A1: Id = idseq (n + 2) ; ::_thesis: sgn (Id,K) = 1_ K
set Path = Part_sgn (Id,K);
set 2S = 2Set (Seg (n + 2));
A2: FinOmega (2Set (Seg (n + 2))) = 2Set (Seg (n + 2)) by MATRIX_2:def_14;
then reconsider 2S9 = 2Set (Seg (n + 2)) as Element of Fin (2Set (Seg (n + 2))) ;
now__::_thesis:_for_x_being_set_st_x_in_2S9_holds_
(Part_sgn_(Id,K))_._x_=_1__K
let x be set ; ::_thesis: ( x in 2S9 implies (Part_sgn (Id,K)) . x = 1_ K )
assume x in 2S9 ; ::_thesis: (Part_sgn (Id,K)) . x = 1_ K
then consider i, j being Nat such that
A3: i in Seg (n + 2) and
A4: j in Seg (n + 2) and
A5: i < j and
A6: x = {i,j} by Th1;
A7: Id . j = j by A1, A4, FUNCT_1:18;
Id . i = i by A1, A3, FUNCT_1:18;
hence (Part_sgn (Id,K)) . x = 1_ K by A3, A4, A5, A6, A7, Def1; ::_thesis: verum
end;
hence sgn (Id,K) = 1_ K by A2, Th4; ::_thesis: verum
end;
Lm2: for X being set
for n, i being Nat st X in 2Set (Seg n) & i in X holds
( i in Seg n & ex j being Nat st
( j in Seg n & i <> j & X = {i,j} ) )
proof
let X be set ; ::_thesis: for n, i being Nat st X in 2Set (Seg n) & i in X holds
( i in Seg n & ex j being Nat st
( j in Seg n & i <> j & X = {i,j} ) )
let n, i be Nat; ::_thesis: ( X in 2Set (Seg n) & i in X implies ( i in Seg n & ex j being Nat st
( j in Seg n & i <> j & X = {i,j} ) ) )
assume that
A1: X in 2Set (Seg n) and
A2: i in X ; ::_thesis: ( i in Seg n & ex j being Nat st
( j in Seg n & i <> j & X = {i,j} ) )
consider i9, j9 being Nat such that
A3: i9 in Seg n and
A4: j9 in Seg n and
A5: i9 < j9 and
A6: X = {i9,j9} by A1, Th1;
now__::_thesis:_(_(_i_=_i9_&_i_in_Seg_n_&_ex_j_being_Nat_st_
(_j_in_Seg_n_&_i_<>_j_&_X_=_{i,j}_)_)_or_(_i_=_j9_&_i_in_Seg_n_&_ex_j_being_Nat_st_
(_j_in_Seg_n_&_i_<>_j_&_X_=_{i,j}_)_)_)
percases ( i = i9 or i = j9 ) by A2, A6, TARSKI:def_2;
case i = i9 ; ::_thesis: ( i in Seg n & ex j being Nat st
( j in Seg n & i <> j & X = {i,j} ) )
hence ( i in Seg n & ex j being Nat st
( j in Seg n & i <> j & X = {i,j} ) ) by A3, A4, A5, A6; ::_thesis: verum
end;
case i = j9 ; ::_thesis: ( i in Seg n & ex j being Nat st
( j in Seg n & i <> j & X = {i,j} ) )
hence ( i in Seg n & ex j being Nat st
( j in Seg n & i <> j & X = {i,j} ) ) by A3, A4, A5, A6; ::_thesis: verum
end;
end;
end;
hence ( i in Seg n & ex j being Nat st
( j in Seg n & i <> j & X = {i,j} ) ) ; ::_thesis: verum
end;
theorem Th13: :: MATRIX11:13
for n being Nat
for K being Field
for p2, q2, pq2 being Element of Permutations (n + 2) st pq2 = p2 * q2 & q2 is being_transposition holds
sgn (pq2,K) = - (sgn (p2,K))
proof
let n be Nat; ::_thesis: for K being Field
for p2, q2, pq2 being Element of Permutations (n + 2) st pq2 = p2 * q2 & q2 is being_transposition holds
sgn (pq2,K) = - (sgn (p2,K))
let K be Field; ::_thesis: for p2, q2, pq2 being Element of Permutations (n + 2) st pq2 = p2 * q2 & q2 is being_transposition holds
sgn (pq2,K) = - (sgn (p2,K))
set n2 = n + 2;
set 2SS = 2Set (Seg (n + 2));
let p, q, pq be Element of Permutations (n + 2); ::_thesis: ( pq = p * q & q is being_transposition implies sgn (pq,K) = - (sgn (p,K)) )
assume that
A1: pq = p * q and
A2: q is being_transposition ; ::_thesis: sgn (pq,K) = - (sgn (p,K))
A3: FinOmega (2Set (Seg (n + 2))) = 2Set (Seg (n + 2)) by MATRIX_2:def_14;
then reconsider 2S = 2Set (Seg (n + 2)) as Element of Fin (2Set (Seg (n + 2))) ;
A4: for i, j being Nat st i < j & q . i = j holds
sgn (pq,K) = - (sgn (p,K))
proof
let i, j be Nat; ::_thesis: ( i < j & q . i = j implies sgn (pq,K) = - (sgn (p,K)) )
assume that
A5: i < j and
A6: q . i = j ; ::_thesis: sgn (pq,K) = - (sgn (p,K))
now__::_thesis:_sgn_(pq,K)_=_-_(sgn_(p,K))
percases ( 1_ K = - (1_ K) or 1_ K <> - (1_ K) ) ;
supposeA7: 1_ K = - (1_ K) ; ::_thesis: sgn (pq,K) = - (sgn (p,K))
then sgn (pq,K) = - (1_ K) by Th11;
hence sgn (pq,K) = - (sgn (p,K)) by A7, Th11; ::_thesis: verum
end;
supposeA8: 1_ K <> - (1_ K) ; ::_thesis: sgn (pq,K) = - (sgn (p,K))
set P2 = Part_sgn (p,K);
set P1 = Part_sgn (pq,K);
A9: (Part_sgn (pq,K)) . {i,j} <> (Part_sgn (p,K)) . {i,j} by A1, A2, A5, A6, A8, Th10;
defpred S1[ set , set ] means for k being Nat st k in $1 & k <> i holds
( ( k <> j implies $2 = {j,k} ) & ( k = j implies $2 = {i,j} ) );
set D = { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s ) } ;
{ s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s ) } c= 2S
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s ) } or x in 2S )
assume x in { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s ) } ; ::_thesis: x in 2S
then ex s being Element of 2Set (Seg (n + 2)) st
( x = s & s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s ) ;
hence x in 2S ; ::_thesis: verum
end;
then reconsider D = { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s ) } as finite set ;
set D1 = { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & i in s ) } ;
set D2 = { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & j in s ) } ;
A10: { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & i in s ) } c= D
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & i in s ) } or x in D )
assume x in { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & i in s ) } ; ::_thesis: x in D
then ex s being Element of 2Set (Seg (n + 2)) st
( x = s & s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & i in s ) ;
hence x in D ; ::_thesis: verum
end;
A11: { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & j in s ) } c= D
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & j in s ) } or x in D )
assume x in { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & j in s ) } ; ::_thesis: x in D
then ex s being Element of 2Set (Seg (n + 2)) st
( x = s & s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & j in s ) ;
hence x in D ; ::_thesis: verum
end;
then reconsider D1 = { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & i in s ) } , D2 = { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & j in s ) } as finite set by A10;
A12: j in dom q by A2, A5, A6, Th8;
A13: D c= D1 \/ D2
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in D or x in D1 \/ D2 )
assume x in D ; ::_thesis: x in D1 \/ D2
then consider s being Element of 2Set (Seg (n + 2)) such that
A14: x = s and
s in 2S and
A15: (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s ;
( i in s or j in s ) by A1, A2, A5, A6, A15, Th9;
then ( x in D1 or x in D2 ) by A14, A15;
hence x in D1 \/ D2 by XBOOLE_0:def_3; ::_thesis: verum
end;
D1 \/ D2 c= D by A10, A11, XBOOLE_1:8;
then A16: D1 \/ D2 = D by A13, XBOOLE_0:def_10;
A17: D1 /\ D2 c= {{i,j}}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in D1 /\ D2 or x in {{i,j}} )
assume A18: x in D1 /\ D2 ; ::_thesis: x in {{i,j}}
then x in D1 by XBOOLE_0:def_4;
then A19: ex s1 being Element of 2Set (Seg (n + 2)) st
( x = s1 & s1 in 2S & (Part_sgn (pq,K)) . s1 <> (Part_sgn (p,K)) . s1 & i in s1 ) ;
then consider i9, j9 being Nat such that
i9 in Seg (n + 2) and
j9 in Seg (n + 2) and
i9 < j9 and
A20: {i9,j9} = x by Th1;
x in D2 by A18, XBOOLE_0:def_4;
then ex s2 being Element of 2Set (Seg (n + 2)) st
( x = s2 & s2 in 2S & (Part_sgn (pq,K)) . s2 <> (Part_sgn (p,K)) . s2 & j in s2 ) ;
then A21: ( j = i9 or j = j9 ) by A20, TARSKI:def_2;
( i = i9 or i = j9 ) by A19, A20, TARSKI:def_2;
hence x in {{i,j}} by A5, A20, A21, TARSKI:def_1; ::_thesis: verum
end;
q is Permutation of (Seg (n + 2)) by MATRIX_2:def_9;
then A22: dom q = Seg (n + 2) by FUNCT_2:52;
A23: i in dom q by A2, A5, A6, Th8;
then A24: {i,j} in 2S by A5, A12, A22, Th1;
A25: i in {i,j} by TARSKI:def_2;
then {i,j} in D1 by A24, A9;
then card D1 > 0 ;
then reconsider c1 = (card D1) - 1 as Nat by NAT_1:20;
A26: j in {i,j} by TARSKI:def_2;
then A27: {i,j} in D2 by A24, A9;
A28: for x being set st x in D1 holds
ex y being set st
( y in D2 & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in D1 implies ex y being set st
( y in D2 & S1[x,y] ) )
assume x in D1 ; ::_thesis: ex y being set st
( y in D2 & S1[x,y] )
then consider s being Element of 2Set (Seg (n + 2)) such that
A29: x = s and
s in 2S and
A30: (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s and
A31: i in s ;
consider j9 being Nat such that
A32: j9 in Seg (n + 2) and
A33: j9 <> i and
A34: s = {i,j9} by A31, Lm2;
now__::_thesis:_ex_X_being_set_st_
(_X_in_D2_&_(_for_k_being_Nat_st_k_in_x_&_k_<>_i_holds_
(_(_k_<>_j_implies_X_=_{j,k}_)_&_(_k_=_j_implies_X_=_{i,j}_)_)_)_)
percases ( j9 = j or j9 <> j ) ;
supposeA35: j9 = j ; ::_thesis: ex X being set st
( X in D2 & ( for k being Nat st k in x & k <> i holds
( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) ) ) )
take X = {i,j}; ::_thesis: ( X in D2 & ( for k being Nat st k in x & k <> i holds
( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) ) ) )
thus X in D2 by A26, A24, A9; ::_thesis: for k being Nat st k in x & k <> i holds
( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) )
let k be Nat; ::_thesis: ( k in x & k <> i implies ( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) ) )
assume that
A36: k in x and
k <> i ; ::_thesis: ( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) )
thus ( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) ) by A29, A34, A35, A36, TARSKI:def_2; ::_thesis: verum
end;
supposeA37: j9 <> j ; ::_thesis: ex X being set st
( X in D2 & ( for k being Nat st k in x & k <> i holds
( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) ) ) )
take X = {j,j9}; ::_thesis: ( X in D2 & ( for k being Nat st k in x & k <> i holds
( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) ) ) )
( j < j9 or j > j9 ) by A37, XXREAL_0:1;
then A38: X in 2Set (Seg (n + 2)) by A12, A22, A32, Th1;
A39: j in X by TARSKI:def_2;
(Part_sgn (pq,K)) . X <> (Part_sgn (p,K)) . X by A1, A2, A5, A6, A8, A30, A32, A33, A34, A37, Th10;
hence X in D2 by A39, A38; ::_thesis: for k being Nat st k in x & k <> i holds
( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) )
let k be Nat; ::_thesis: ( k in x & k <> i implies ( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) ) )
assume that
A40: k in x and
A41: k <> i ; ::_thesis: ( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) )
thus ( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) ) by A29, A34, A37, A40, A41, TARSKI:def_2; ::_thesis: verum
end;
end;
end;
hence ex y being set st
( y in D2 & S1[x,y] ) ; ::_thesis: verum
end;
consider f being Function of D1,D2 such that
A42: for x being set st x in D1 holds
S1[x,f . x] from FUNCT_2:sch_1(A28);
A43: {i,j} in D2 by A26, A24, A9;
then A44: dom f = D1 by FUNCT_2:def_1;
for y being set st y in D2 holds
ex x being set st
( x in D1 & y = f . x )
proof
let y be set ; ::_thesis: ( y in D2 implies ex x being set st
( x in D1 & y = f . x ) )
assume y in D2 ; ::_thesis: ex x being set st
( x in D1 & y = f . x )
then consider s being Element of 2Set (Seg (n + 2)) such that
A45: s = y and
s in 2S and
A46: (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s and
A47: j in s ;
consider i1 being Nat such that
A48: i1 in Seg (n + 2) and
A49: i1 <> j and
A50: s = {j,i1} by A47, Lm2;
now__::_thesis:_ex_x_being_set_st_
(_x_in_D1_&_y_=_f_._x_)
percases ( i1 = i or i1 <> i ) ;
supposeA51: i1 = i ; ::_thesis: ex x being set st
( x in D1 & y = f . x )
A52: {i,j} in D1 by A25, A24, A9;
then f . s = y by A5, A26, A42, A45, A50, A51;
hence ex x being set st
( x in D1 & y = f . x ) by A50, A51, A52; ::_thesis: verum
end;
supposeA53: i1 <> i ; ::_thesis: ex x being set st
( x in D1 & y = f . x )
then ( i < i1 or i > i1 ) by XXREAL_0:1;
then A54: {i,i1} in 2Set (Seg (n + 2)) by A23, A22, A48, Th1;
A55: i in {i,i1} by TARSKI:def_2;
(Part_sgn (pq,K)) . {i,i1} <> (Part_sgn (p,K)) . {i,i1} by A1, A2, A5, A6, A8, A46, A48, A49, A50, A53, Th10;
then A56: {i,i1} in D1 by A54, A55;
i1 in {i,i1} by TARSKI:def_2;
then f . {i,i1} = {j,i1} by A42, A49, A53, A56;
hence ex x being set st
( x in D1 & y = f . x ) by A45, A50, A56; ::_thesis: verum
end;
end;
end;
hence ex x being set st
( x in D1 & y = f . x ) ; ::_thesis: verum
end;
then A57: rng f = D2 by FUNCT_2:10;
for x1, x2 being set st x1 in D1 & x2 in D1 & f . x1 = f . x2 holds
x1 = x2
proof
let x1, x2 be set ; ::_thesis: ( x1 in D1 & x2 in D1 & f . x1 = f . x2 implies x1 = x2 )
assume that
A58: x1 in D1 and
A59: x2 in D1 and
A60: f . x1 = f . x2 ; ::_thesis: x1 = x2
consider s1 being Element of 2Set (Seg (n + 2)) such that
A61: x1 = s1 and
s1 in 2S and
(Part_sgn (pq,K)) . s1 <> (Part_sgn (p,K)) . s1 and
A62: i in s1 by A58;
consider j1 being Nat such that
j1 in Seg (n + 2) and
A63: i <> j1 and
A64: {i,j1} = s1 by A62, Lm2;
consider s2 being Element of 2Set (Seg (n + 2)) such that
A65: x2 = s2 and
s2 in 2S and
(Part_sgn (pq,K)) . s2 <> (Part_sgn (p,K)) . s2 and
A66: i in s2 by A59;
consider j2 being Nat such that
j2 in Seg (n + 2) and
A67: i <> j2 and
A68: {i,j2} = s2 by A66, Lm2;
A69: j2 in s2 by A68, TARSKI:def_2;
A70: j1 in s1 by A64, TARSKI:def_2;
now__::_thesis:_(_(_j_=_j1_&_j_=_j2_&_x1_=_x2_)_or_(_j_<>_j1_&_j_=_j2_&_x1_=_x2_)_or_(_j_=_j1_&_j_<>_j2_&_x1_=_x2_)_or_(_j_<>_j1_&_j_<>_j2_&_x1_=_x2_)_)
percases ( ( j = j1 & j = j2 ) or ( j <> j1 & j = j2 ) or ( j = j1 & j <> j2 ) or ( j <> j1 & j <> j2 ) ) ;
case ( j = j1 & j = j2 ) ; ::_thesis: x1 = x2
hence x1 = x2 by A61, A64, A65, A68; ::_thesis: verum
end;
caseA71: ( j <> j1 & j = j2 ) ; ::_thesis: x1 = x2
then A72: f . x2 = {i,j} by A42, A59, A65, A67, A69;
f . x1 = {j,j1} by A42, A58, A61, A63, A70, A71;
hence x1 = x2 by A5, A60, A63, A66, A68, A71, A72, TARSKI:def_2; ::_thesis: verum
end;
caseA73: ( j = j1 & j <> j2 ) ; ::_thesis: x1 = x2
then A74: f . x2 = {j,j2} by A42, A59, A65, A67, A69;
f . x1 = {i,j} by A42, A58, A61, A63, A70, A73;
hence x1 = x2 by A5, A60, A62, A64, A67, A73, A74, TARSKI:def_2; ::_thesis: verum
end;
caseA75: ( j <> j1 & j <> j2 ) ; ::_thesis: x1 = x2
then A76: f . x2 = {j,j2} by A42, A59, A65, A67, A69;
A77: j1 in {j,j1} by TARSKI:def_2;
f . x1 = {j,j1} by A42, A58, A61, A63, A70, A75;
hence x1 = x2 by A60, A61, A64, A65, A68, A75, A76, A77, TARSKI:def_2; ::_thesis: verum
end;
end;
end;
hence x1 = x2 ; ::_thesis: verum
end;
then f is one-to-one by A43, FUNCT_2:19;
then D1,D2 are_equipotent by A57, A44, WELLORD2:def_4;
then A78: card D1 = card D2 by CARD_1:5;
{i,j} in D1 by A25, A24, A9;
then {i,j} in D1 /\ D2 by A27, XBOOLE_0:def_4;
then {{i,j}} c= D1 /\ D2 by ZFMISC_1:31;
then D1 /\ D2 = {{i,j}} by A17, XBOOLE_0:def_10;
then card D = ((card D1) + (card D1)) - (card {{i,j}}) by A78, A16, CARD_2:45
.= ((c1 + 1) + (c1 + 1)) - 1 by CARD_1:30
.= (2 * c1) + 1 ;
then (card D) mod 2 = 1 mod 2 by NAT_D:21;
hence sgn (pq,K) = - (sgn (p,K)) by A3, Th7, NAT_D:14; ::_thesis: verum
end;
end;
end;
hence sgn (pq,K) = - (sgn (p,K)) ; ::_thesis: verum
end;
consider i, j being Nat such that
i in dom q and
j in dom q and
A79: i <> j and
A80: q . i = j and
A81: q . j = i and
for k being Nat st k <> i & k <> j & k in dom q holds
q . k = k by A2, MATRIX_2:def_11;
( i < j or j < i ) by A79, XXREAL_0:1;
hence sgn (pq,K) = - (sgn (p,K)) by A4, A80, A81; ::_thesis: verum
end;
theorem Th14: :: MATRIX11:14
for n being Nat
for K being Field
for tr being Element of Permutations (n + 2) st tr is being_transposition holds
sgn (tr,K) = - (1_ K)
proof
let n be Nat; ::_thesis: for K being Field
for tr being Element of Permutations (n + 2) st tr is being_transposition holds
sgn (tr,K) = - (1_ K)
let K be Field; ::_thesis: for tr being Element of Permutations (n + 2) st tr is being_transposition holds
sgn (tr,K) = - (1_ K)
set n2 = n + 2;
set S = Seg (n + 2);
let tr be Element of Permutations (n + 2); ::_thesis: ( tr is being_transposition implies sgn (tr,K) = - (1_ K) )
assume A1: tr is being_transposition ; ::_thesis: sgn (tr,K) = - (1_ K)
reconsider Tr = tr as Permutation of (Seg (n + 2)) by MATRIX_2:def_9;
reconsider Id = idseq (n + 2), IdTr = (id (Seg (n + 2))) * Tr as Element of Permutations (n + 2) by MATRIX_2:def_9;
rng Tr = Seg (n + 2) by FUNCT_2:def_3;
then IdTr = Tr by RELAT_1:54;
then sgn (tr,K) = - (sgn (Id,K)) by A1, Th13;
hence sgn (tr,K) = - (1_ K) by Th12; ::_thesis: verum
end;
theorem Th15: :: MATRIX11:15
for n being Nat
for K being Field
for P being FinSequence of (Group_of_Perm (n + 2))
for p2 being Element of Permutations (n + 2) st p2 = Product P & ( for i being Nat st i in dom P holds
ex trans being Element of Permutations (n + 2) st
( P . i = trans & trans is being_transposition ) ) holds
( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) )
proof
let n be Nat; ::_thesis: for K being Field
for P being FinSequence of (Group_of_Perm (n + 2))
for p2 being Element of Permutations (n + 2) st p2 = Product P & ( for i being Nat st i in dom P holds
ex trans being Element of Permutations (n + 2) st
( P . i = trans & trans is being_transposition ) ) holds
( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) )
let K be Field; ::_thesis: for P being FinSequence of (Group_of_Perm (n + 2))
for p2 being Element of Permutations (n + 2) st p2 = Product P & ( for i being Nat st i in dom P holds
ex trans being Element of Permutations (n + 2) st
( P . i = trans & trans is being_transposition ) ) holds
( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) )
set n2 = n + 2;
set G = Group_of_Perm (n + 2);
defpred S1[ Nat] means for P being FinSequence of (Group_of_Perm (n + 2))
for p2 being Element of Permutations (n + 2) st p2 = Product P & len P = $1 & ( for i being Nat st i in dom P holds
ex trans being Element of Permutations (n + 2) st
( P . i = trans & trans is being_transposition ) ) holds
( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) );
A1: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A2: S1[k] ; ::_thesis: S1[k + 1]
set k1 = k + 1;
let P be FinSequence of (Group_of_Perm (n + 2)); ::_thesis: for p2 being Element of Permutations (n + 2) st p2 = Product P & len P = k + 1 & ( for i being Nat st i in dom P holds
ex trans being Element of Permutations (n + 2) st
( P . i = trans & trans is being_transposition ) ) holds
( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) )
let p2 be Element of Permutations (n + 2); ::_thesis: ( p2 = Product P & len P = k + 1 & ( for i being Nat st i in dom P holds
ex trans being Element of Permutations (n + 2) st
( P . i = trans & trans is being_transposition ) ) implies ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) )
assume that
A3: p2 = Product P and
A4: len P = k + 1 and
A5: for i being Nat st i in dom P holds
ex trans being Element of Permutations (n + 2) st
( P . i = trans & trans is being_transposition ) ; ::_thesis: ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) )
Seg (len P) = dom P by FINSEQ_1:def_3;
then consider x being set , Q being FinSequence such that
A6: P = <*x*> ^ Q and
A7: len P = (len Q) + 1 by A4, RELAT_1:38, REWRITE1:5;
reconsider X = <*x*>, Q = Q as FinSequence of (Group_of_Perm (n + 2)) by A6, FINSEQ_1:36;
A8: for i being Nat st i in dom Q holds
ex trans being Element of Permutations (n + 2) st
( Q . i = trans & trans is being_transposition )
proof
let i be Nat; ::_thesis: ( i in dom Q implies ex trans being Element of Permutations (n + 2) st
( Q . i = trans & trans is being_transposition ) )
assume A9: i in dom Q ; ::_thesis: ex trans being Element of Permutations (n + 2) st
( Q . i = trans & trans is being_transposition )
Q . i = P . ((len X) + i) by A6, A9, FINSEQ_1:def_7;
hence ex trans being Element of Permutations (n + 2) st
( Q . i = trans & trans is being_transposition ) by A5, A6, A9, FINSEQ_1:28; ::_thesis: verum
end;
1 + 0 <= k + 1 by XREAL_1:7;
then 1 in Seg (k + 1) ;
then A10: 1 in dom P by A4, FINSEQ_1:def_3;
P . 1 = x by A6, FINSEQ_1:41;
then consider tr being Element of Permutations (n + 2) such that
A11: x = tr and
A12: tr is being_transposition by A5, A10;
reconsider PQ = Product Q as Element of Permutations (n + 2) by MATRIX_2:def_10;
reconsider Tr = tr as Element of (Group_of_Perm (n + 2)) by MATRIX_2:def_10;
A13: p2 = Tr * (Product Q) by A3, A6, A11, GROUP_4:7
.= PQ * tr by MATRIX_2:def_10 ;
then A14: sgn (p2,K) = - (sgn (PQ,K)) by A12, Th13;
now__::_thesis:_(_(_(len_P)_mod_2_=_0_implies_sgn_(p2,K)_=_1__K_)_&_(_(len_P)_mod_2_=_1_implies_sgn_(p2,K)_=_-_(1__K)_)_)
percases ( (len Q) mod 2 = 0 or (len Q) mod 2 = 1 ) by NAT_D:12;
supposeA15: (len Q) mod 2 = 0 ; ::_thesis: ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) )
0 < 2 - 1 ;
then A16: (len P) mod 2 = 0 + 1 by A7, A15, NAT_D:70;
sgn (PQ,K) = 1_ K by A2, A4, A7, A8, A15;
hence ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) by A12, A13, A16, Th13; ::_thesis: verum
end;
supposeA17: (len Q) mod 2 = 1 ; ::_thesis: ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) )
A18: 2 - 1 = 1 ;
sgn (PQ,K) = - (1_ K) by A2, A4, A7, A8, A17;
hence ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) by A7, A14, A17, A18, NAT_D:69, RLVECT_1:17; ::_thesis: verum
end;
end;
end;
hence ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) ; ::_thesis: verum
end;
A19: S1[ 0 ]
proof
let P be FinSequence of (Group_of_Perm (n + 2)); ::_thesis: for p2 being Element of Permutations (n + 2) st p2 = Product P & len P = 0 & ( for i being Nat st i in dom P holds
ex trans being Element of Permutations (n + 2) st
( P . i = trans & trans is being_transposition ) ) holds
( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) )
let p2 be Element of Permutations (n + 2); ::_thesis: ( p2 = Product P & len P = 0 & ( for i being Nat st i in dom P holds
ex trans being Element of Permutations (n + 2) st
( P . i = trans & trans is being_transposition ) ) implies ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) )
assume that
A20: p2 = Product P and
A21: len P = 0 and
for i being Nat st i in dom P holds
ex trans being Element of Permutations (n + 2) st
( P . i = trans & trans is being_transposition ) ; ::_thesis: ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) )
P = <*> the carrier of (Group_of_Perm (n + 2)) by A21;
then Product P = 1_ (Group_of_Perm (n + 2)) by GROUP_4:8;
then Product P = idseq (n + 2) by MATRIX_2:24;
hence ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) by A20, A21, Th12, NAT_D:26; ::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A19, A1);
hence for P being FinSequence of (Group_of_Perm (n + 2))
for p2 being Element of Permutations (n + 2) st p2 = Product P & ( for i being Nat st i in dom P holds
ex trans being Element of Permutations (n + 2) st
( P . i = trans & trans is being_transposition ) ) holds
( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) ; ::_thesis: verum
end;
theorem Th16: :: MATRIX11:16
for i, j, n being Nat st i < j & i in Seg n & j in Seg n holds
ex tr being Element of Permutations n st
( tr is being_transposition & tr . i = j )
proof
let i, j, n be Nat; ::_thesis: ( i < j & i in Seg n & j in Seg n implies ex tr being Element of Permutations n st
( tr is being_transposition & tr . i = j ) )
assume that
A1: i < j and
A2: i in Seg n and
A3: j in Seg n ; ::_thesis: ex tr being Element of Permutations n st
( tr is being_transposition & tr . i = j )
defpred S1[ set , set ] means for k being Nat st k in Seg n & k = $1 holds
( ( k = i implies $2 = j ) & ( k = j implies $2 = i ) & ( k <> i & k <> j implies $2 = k ) );
A4: for x being set st x in Seg n holds
ex y being set st
( y in Seg n & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in Seg n implies ex y being set st
( y in Seg n & S1[x,y] ) )
assume A5: x in Seg n ; ::_thesis: ex y being set st
( y in Seg n & S1[x,y] )
reconsider m = x as Nat by A5;
now__::_thesis:_ex_y_being_set_st_
(_y_in_Seg_n_&_S1[x,y]_)
percases ( m = i or m = j or ( m <> i & m <> j ) ) ;
suppose m = i ; ::_thesis: ex y being set st
( y in Seg n & S1[x,y] )
then S1[x,j] ;
hence ex y being set st
( y in Seg n & S1[x,y] ) by A3; ::_thesis: verum
end;
suppose m = j ; ::_thesis: ex y being set st
( y in Seg n & S1[x,y] )
then S1[x,i] ;
hence ex y being set st
( y in Seg n & S1[x,y] ) by A2; ::_thesis: verum
end;
suppose ( m <> i & m <> j ) ; ::_thesis: ex y being set st
( y in Seg n & S1[x,y] )
then S1[x,x] ;
hence ex y being set st
( y in Seg n & S1[x,y] ) by A5; ::_thesis: verum
end;
end;
end;
hence ex y being set st
( y in Seg n & S1[x,y] ) ; ::_thesis: verum
end;
consider f being Function of (Seg n),(Seg n) such that
A6: for x being set st x in Seg n holds
S1[x,f . x] from FUNCT_2:sch_1(A4);
for x1, x2 being set st x1 in Seg n & x2 in Seg n & f . x1 = f . x2 holds
x1 = x2
proof
let x1, x2 be set ; ::_thesis: ( x1 in Seg n & x2 in Seg n & f . x1 = f . x2 implies x1 = x2 )
assume that
A7: x1 in Seg n and
A8: x2 in Seg n and
A9: f . x1 = f . x2 ; ::_thesis: x1 = x2
reconsider k1 = x1 as Nat by A7;
( x1 = i or x1 = j or ( x1 <> i & x1 <> j ) ) ;
then A10: ( ( x1 = i & f . x1 = j ) or ( x1 = j & f . x1 = i ) or ( x1 <> i & x1 <> j & f . x1 = k1 ) ) by A6, A7;
( x2 = i or x2 = j or ( x2 <> i & x2 <> j ) ) ;
hence x1 = x2 by A6, A8, A9, A10; ::_thesis: verum
end;
then A11: f is one-to-one by A2, FUNCT_2:19;
for y being set st y in Seg n holds
ex x being set st
( x in Seg n & y = f . x )
proof
let y be set ; ::_thesis: ( y in Seg n implies ex x being set st
( x in Seg n & y = f . x ) )
assume A12: y in Seg n ; ::_thesis: ex x being set st
( x in Seg n & y = f . x )
reconsider k = y as Nat by A12;
( ( k = i & f . j = i ) or ( k = j & f . i = j ) or ( k <> i & k <> j & f . k = k ) ) by A2, A3, A6, A12;
hence ex x being set st
( x in Seg n & y = f . x ) by A2, A3, A12; ::_thesis: verum
end;
then rng f = Seg n by FUNCT_2:10;
then f is onto by FUNCT_2:def_3;
then reconsider P = f as Element of Permutations n by A11, MATRIX_2:def_9;
A13: P . j = i by A3, A6;
dom P = Seg n by A2, FUNCT_2:def_1;
then A14: for k being Nat st k <> i & k <> j & k in dom P holds
P . k = k by A6;
take P ; ::_thesis: ( P is being_transposition & P . i = j )
A15: i in dom P by A2, FUNCT_2:def_1;
A16: j in dom P by A3, FUNCT_2:def_1;
P . i = j by A2, A6;
hence ( P is being_transposition & P . i = j ) by A1, A15, A16, A13, A14, MATRIX_2:def_11; ::_thesis: verum
end;
theorem Th17: :: MATRIX11:17
for k being Nat
for p being Element of Permutations (k + 1) st p . (k + 1) <> k + 1 holds
ex tr being Element of Permutations (k + 1) st
( tr is being_transposition & tr . (p . (k + 1)) = k + 1 & (tr * p) . (k + 1) = k + 1 )
proof
let k be Nat; ::_thesis: for p being Element of Permutations (k + 1) st p . (k + 1) <> k + 1 holds
ex tr being Element of Permutations (k + 1) st
( tr is being_transposition & tr . (p . (k + 1)) = k + 1 & (tr * p) . (k + 1) = k + 1 )
set k1 = k + 1;
let p be Element of Permutations (k + 1); ::_thesis: ( p . (k + 1) <> k + 1 implies ex tr being Element of Permutations (k + 1) st
( tr is being_transposition & tr . (p . (k + 1)) = k + 1 & (tr * p) . (k + 1) = k + 1 ) )
assume A1: p . (k + 1) <> k + 1 ; ::_thesis: ex tr being Element of Permutations (k + 1) st
( tr is being_transposition & tr . (p . (k + 1)) = k + 1 & (tr * p) . (k + 1) = k + 1 )
reconsider p9 = p as Permutation of (Seg (k + 1)) by MATRIX_2:def_9;
A2: dom p9 = Seg (k + 1) by FUNCT_2:52;
A3: rng p9 = Seg (k + 1) by FUNCT_2:def_3;
A4: k + 1 in Seg (k + 1) by FINSEQ_1:3;
then A5: p . (k + 1) in Seg (k + 1) by A2, A3, FUNCT_1:def_3;
then p . (k + 1) <= k + 1 by FINSEQ_1:1;
then p . (k + 1) < k + 1 by A1, XXREAL_0:1;
then consider tr being Element of Permutations (k + 1) such that
A6: tr is being_transposition and
A7: tr . (p . (k + 1)) = k + 1 by A4, A5, Th16;
reconsider tr9 = tr as Permutation of (Seg (k + 1)) by MATRIX_2:def_9;
dom tr9 = Seg (k + 1) by FUNCT_2:52;
then dom (tr * p) = Seg (k + 1) by A2, A3, RELAT_1:27;
then (tr * p) . (k + 1) = tr . (p . (k + 1)) by FINSEQ_1:3, FUNCT_1:12;
hence ex tr being Element of Permutations (k + 1) st
( tr is being_transposition & tr . (p . (k + 1)) = k + 1 & (tr * p) . (k + 1) = k + 1 ) by A6, A7; ::_thesis: verum
end;
theorem Th18: :: MATRIX11:18
for X, x being set st not x in X holds
for p1 being Permutation of (X \/ {x}) st p1 . x = x holds
ex p being Permutation of X st p1 | X = p
proof
let X, x be set ; ::_thesis: ( not x in X implies for p1 being Permutation of (X \/ {x}) st p1 . x = x holds
ex p being Permutation of X st p1 | X = p )
assume A1: not x in X ; ::_thesis: for p1 being Permutation of (X \/ {x}) st p1 . x = x holds
ex p being Permutation of X st p1 | X = p
let p1 be Permutation of (X \/ {x}); ::_thesis: ( p1 . x = x implies ex p being Permutation of X st p1 | X = p )
assume A2: p1 . x = x ; ::_thesis: ex p being Permutation of X st p1 | X = p
A3: X c= X \/ {x} by XBOOLE_1:7;
set pX = p1 | X;
A4: dom p1 = X \/ {x} by FUNCT_2:52;
then A5: dom (p1 | X) = X by RELAT_1:62, XBOOLE_1:7;
A6: rng p1 = X \/ {x} by FUNCT_2:def_3;
then A7: rng (p1 | X) c= X \/ {x} by RELAT_1:70;
A8: rng (p1 | X) c= X
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (p1 | X) or y in X )
assume A9: y in rng (p1 | X) ; ::_thesis: y in X
consider x9 being set such that
A10: x9 in dom (p1 | X) and
A11: (p1 | X) . x9 = y by A9, FUNCT_1:def_3;
assume A12: not y in X ; ::_thesis: contradiction
y in rng (p1 | X) by A10, A11, FUNCT_1:def_3;
then y in {x} by A7, A12, XBOOLE_0:def_3;
then A13: y = x by TARSKI:def_1;
(p1 | X) . x9 = p1 . x9 by A10, FUNCT_1:47;
hence contradiction by A1, A2, A3, A4, A5, A7, A9, A10, A11, A13, FUNCT_1:def_4; ::_thesis: verum
end;
X c= rng (p1 | X)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in X or y in rng (p1 | X) )
assume A14: y in X ; ::_thesis: y in rng (p1 | X)
consider x9 being set such that
A15: x9 in dom p1 and
A16: p1 . x9 = y by A3, A6, A14, FUNCT_1:def_3;
A17: x9 in X
proof
assume not x9 in X ; ::_thesis: contradiction
then x9 in {x} by A4, A15, XBOOLE_0:def_3;
hence contradiction by A1, A2, A14, A16, TARSKI:def_1; ::_thesis: verum
end;
then (p1 | X) . x9 = p1 . x9 by A5, FUNCT_1:47;
hence y in rng (p1 | X) by A5, A16, A17, FUNCT_1:def_3; ::_thesis: verum
end;
then A18: rng (p1 | X) = X by A8, XBOOLE_0:def_10;
A19: p1 | X is one-to-one by FUNCT_1:52;
reconsider pX = p1 | X as Function of X,X by A5, A18, FUNCT_2:1;
pX is onto by A18, FUNCT_2:def_3;
hence ex p being Permutation of X st p1 | X = p by A19; ::_thesis: verum
end;
theorem Th19: :: MATRIX11:19
for X, x being set
for p, q being Permutation of X
for p1, q1 being Permutation of (X \/ {x}) st p1 | X = p & q1 | X = q & p1 . x = x & q1 . x = x holds
( (p1 * q1) | X = p * q & (p1 * q1) . x = x )
proof
let X, x be set ; ::_thesis: for p, q being Permutation of X
for p1, q1 being Permutation of (X \/ {x}) st p1 | X = p & q1 | X = q & p1 . x = x & q1 . x = x holds
( (p1 * q1) | X = p * q & (p1 * q1) . x = x )
let p, q be Permutation of X; ::_thesis: for p1, q1 being Permutation of (X \/ {x}) st p1 | X = p & q1 | X = q & p1 . x = x & q1 . x = x holds
( (p1 * q1) | X = p * q & (p1 * q1) . x = x )
let p1, q1 be Permutation of (X \/ {x}); ::_thesis: ( p1 | X = p & q1 | X = q & p1 . x = x & q1 . x = x implies ( (p1 * q1) | X = p * q & (p1 * q1) . x = x ) )
assume that
A1: p1 | X = p and
A2: q1 | X = q and
A3: p1 . x = x and
A4: q1 . x = x ; ::_thesis: ( (p1 * q1) | X = p * q & (p1 * q1) . x = x )
set pq = p * q;
set pq1 = p1 * q1;
set X1 = X \/ {x};
A5: X c= X \/ {x} by XBOOLE_1:7;
A6: rng q = X by FUNCT_2:def_3;
A7: dom q = X by FUNCT_2:52;
dom (p1 * q1) = X \/ {x} by FUNCT_2:52;
then A8: dom ((p1 * q1) | X) = X by RELAT_1:62, XBOOLE_1:7;
A9: dom (p * q) = X by FUNCT_2:52;
A10: dom p = X by FUNCT_2:52;
for y being set st y in dom (p * q) holds
(p * q) . y = ((p1 * q1) | X) . y
proof
let y be set ; ::_thesis: ( y in dom (p * q) implies (p * q) . y = ((p1 * q1) | X) . y )
assume A11: y in dom (p * q) ; ::_thesis: (p * q) . y = ((p1 * q1) | X) . y
A12: (p * q) . y = p . (q . y) by A9, A11, FUNCT_2:15;
A13: (p1 * q1) . y = ((p1 * q1) | X) . y by A9, A8, A11, FUNCT_1:47;
A14: q . y in rng q by A7, A9, A11, FUNCT_1:def_3;
A15: (p1 * q1) . y = p1 . (q1 . y) by A5, A9, A11, FUNCT_2:15;
q1 . y = q . y by A2, A7, A9, A11, FUNCT_1:47;
hence (p * q) . y = ((p1 * q1) | X) . y by A1, A10, A6, A14, A13, A12, A15, FUNCT_1:47; ::_thesis: verum
end;
hence (p1 * q1) | X = p * q by A8, FUNCT_1:2, FUNCT_2:52; ::_thesis: (p1 * q1) . x = x
x in {x} by TARSKI:def_1;
then x in X \/ {x} by XBOOLE_0:def_3;
hence (p1 * q1) . x = x by A3, A4, FUNCT_2:15; ::_thesis: verum
end;
theorem Th20: :: MATRIX11:20
for k being Nat
for tr being Element of Permutations k st tr is being_transposition holds
( tr * tr = idseq k & tr = tr " )
proof
let k be Nat; ::_thesis: for tr being Element of Permutations k st tr is being_transposition holds
( tr * tr = idseq k & tr = tr " )
set I = idseq k;
let tr be Element of Permutations k; ::_thesis: ( tr is being_transposition implies ( tr * tr = idseq k & tr = tr " ) )
assume tr is being_transposition ; ::_thesis: ( tr * tr = idseq k & tr = tr " )
then consider i, j being Nat such that
i in dom tr and
j in dom tr and
i <> j and
A1: tr . i = j and
A2: tr . j = i and
A3: for m being Nat st m <> i & m <> j & m in dom tr holds
tr . m = m by MATRIX_2:def_11;
reconsider TR = tr as Permutation of (Seg k) by MATRIX_2:def_9;
set TT = TR * TR;
A4: dom (TR * TR) = Seg k by FUNCT_2:52;
A5: dom TR = Seg k by FUNCT_2:52;
A6: for x being set st x in dom (TR * TR) holds
(TR * TR) . x = (idseq k) . x
proof
let x be set ; ::_thesis: ( x in dom (TR * TR) implies (TR * TR) . x = (idseq k) . x )
assume A7: x in dom (TR * TR) ; ::_thesis: (TR * TR) . x = (idseq k) . x
reconsider m = x as Nat by A4, A7;
now__::_thesis:_(TR_*_TR)_._m_=_m
percases ( m = i or m = j or ( m <> i & m <> j ) ) ;
suppose ( m = i or m = j ) ; ::_thesis: (TR * TR) . m = m
hence (TR * TR) . m = m by A1, A2, A7, FUNCT_1:12; ::_thesis: verum
end;
suppose ( m <> i & m <> j ) ; ::_thesis: (TR * TR) . m = m
then tr . m = m by A3, A4, A5, A7;
hence (TR * TR) . m = m by A7, FUNCT_1:12; ::_thesis: verum
end;
end;
end;
hence (TR * TR) . x = (idseq k) . x by A4, A7, FUNCT_1:18; ::_thesis: verum
end;
A8: dom (idseq k) = Seg k by FUNCT_2:52;
hence tr * tr = idseq k by A6, FUNCT_1:2, FUNCT_2:52; ::_thesis: tr = tr "
rng TR = Seg k by FUNCT_2:def_3;
hence tr = tr " by A4, A8, A5, A6, FUNCT_1:2, FUNCT_1:42; ::_thesis: verum
end;
theorem Th21: :: MATRIX11:21
for n being Nat
for perm being Element of Permutations n ex P being FinSequence of (Group_of_Perm n) st
( perm = Product P & ( for i being Nat st i in dom P holds
ex trans being Element of Permutations n st
( P . i = trans & trans is being_transposition ) ) )
proof
let n be Nat; ::_thesis: for perm being Element of Permutations n ex P being FinSequence of (Group_of_Perm n) st
( perm = Product P & ( for i being Nat st i in dom P holds
ex trans being Element of Permutations n st
( P . i = trans & trans is being_transposition ) ) )
defpred S1[ Nat] means for perm being Element of Permutations $1 ex P being FinSequence of (Group_of_Perm $1) st
( perm = Product P & ( for i being Nat st i in dom P holds
ex trans being Element of Permutations $1 st
( P . i = trans & trans is being_transposition ) ) );
let perm be Element of Permutations n; ::_thesis: ex P being FinSequence of (Group_of_Perm n) st
( perm = Product P & ( for i being Nat st i in dom P holds
ex trans being Element of Permutations n st
( P . i = trans & trans is being_transposition ) ) )
A1: n is Element of NAT by ORDINAL1:def_12;
A2: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A3: S1[k] ; ::_thesis: S1[k + 1]
set k1 = k + 1;
let p be Element of Permutations (k + 1); ::_thesis: ex P being FinSequence of (Group_of_Perm (k + 1)) st
( p = Product P & ( for i being Nat st i in dom P holds
ex trans being Element of Permutations (k + 1) st
( P . i = trans & trans is being_transposition ) ) )
reconsider p9 = p as Permutation of (Seg (k + 1)) by MATRIX_2:def_9;
set Gk1 = Group_of_Perm (k + 1);
A4: for p being Element of Permutations (k + 1) st p . (k + 1) = k + 1 holds
ex P being FinSequence of (Group_of_Perm (k + 1)) st
( p = Product P & ( for i being Nat st i in dom P holds
ex trans being Element of Permutations (k + 1) st
( P . i = trans & trans is being_transposition ) ) )
proof
set Ik = idseq k;
set Ik1 = idseq (k + 1);
set Gk1 = Group_of_Perm (k + 1);
set Gk = Group_of_Perm k;
let p be Element of Permutations (k + 1); ::_thesis: ( p . (k + 1) = k + 1 implies ex P being FinSequence of (Group_of_Perm (k + 1)) st
( p = Product P & ( for i being Nat st i in dom P holds
ex trans being Element of Permutations (k + 1) st
( P . i = trans & trans is being_transposition ) ) ) )
assume A5: p . (k + 1) = k + 1 ; ::_thesis: ex P being FinSequence of (Group_of_Perm (k + 1)) st
( p = Product P & ( for i being Nat st i in dom P holds
ex trans being Element of Permutations (k + 1) st
( P . i = trans & trans is being_transposition ) ) )
set mG1 = the multF of (Group_of_Perm (k + 1));
set mG = the multF of (Group_of_Perm k);
reconsider p9 = p as Permutation of (Seg (k + 1)) by MATRIX_2:def_9;
A6: Seg (k + 1) = (Seg k) \/ {(k + 1)} by FINSEQ_1:9;
then consider pk being Permutation of (Seg k) such that
A7: p9 | (Seg k) = pk by A5, Th18, FINSEQ_3:8;
reconsider pk9 = pk as Element of Permutations k by MATRIX_2:def_9;
consider P being FinSequence of (Group_of_Perm k) such that
A8: pk9 = Product P and
A9: for i being Nat st i in dom P holds
ex trans being Element of Permutations k st
( P . i = trans & trans is being_transposition ) by A3;
A10: pk9 = the multF of (Group_of_Perm k) "**" P by A8, GROUP_4:def_2;
defpred S2[ set , set ] means for i being Nat
for tr being Element of Permutations k st i in dom P & P . i = tr & i = $1 holds
ex newtr being Element of Permutations (k + 1) st
( newtr = $2 & newtr is being_transposition & newtr . (k + 1) = k + 1 & tr = newtr | (Seg k) );
A11: not k + 1 in Seg k by FINSEQ_3:8;
A12: for m being Nat st m in Seg (len P) holds
ex x being Element of (Group_of_Perm (k + 1)) st S2[m,x]
proof
let m be Nat; ::_thesis: ( m in Seg (len P) implies ex x being Element of (Group_of_Perm (k + 1)) st S2[m,x] )
assume m in Seg (len P) ; ::_thesis: ex x being Element of (Group_of_Perm (k + 1)) st S2[m,x]
then m in dom P by FINSEQ_1:def_3;
then consider tr being Element of Permutations k such that
A13: P . m = tr and
A14: tr is being_transposition by A9;
consider i9, j9 being Nat such that
A15: i9 in dom tr and
A16: j9 in dom tr and
A17: i9 <> j9 and
A18: tr . i9 = j9 and
A19: tr . j9 = i9 and
A20: for k being Nat st k <> i9 & k <> j9 & k in dom tr holds
tr . k = k by A14, MATRIX_2:def_11;
reconsider tr9 = tr as Permutation of (Seg k) by MATRIX_2:def_9;
consider newt being Function of (Seg (k + 1)),(Seg (k + 1)) such that
A21: newt | (Seg k) = tr9 and
A22: newt . (k + 1) = k + 1 by A6, A11, STIRL2_1:57;
A23: newt . j9 = tr . j9 by A21, A16, FUNCT_1:47;
A24: ( Seg k is empty implies Seg k is empty ) ;
then A25: newt is onto by A6, A21, A22, STIRL2_1:58;
newt is one-to-one by A6, A11, A24, A21, A22, STIRL2_1:58;
then reconsider NT = newt as Element of Permutations (k + 1) by A25, MATRIX_2:def_9;
reconsider NT9 = NT as Element of (Group_of_Perm (k + 1)) by MATRIX_2:def_10;
take NT9 ; ::_thesis: S2[m,NT9]
let I be Nat; ::_thesis: for tr being Element of Permutations k st I in dom P & P . I = tr & I = m holds
ex newtr being Element of Permutations (k + 1) st
( newtr = NT9 & newtr is being_transposition & newtr . (k + 1) = k + 1 & tr = newtr | (Seg k) )
let TR be Element of Permutations k; ::_thesis: ( I in dom P & P . I = TR & I = m implies ex newtr being Element of Permutations (k + 1) st
( newtr = NT9 & newtr is being_transposition & newtr . (k + 1) = k + 1 & TR = newtr | (Seg k) ) )
assume that
I in dom P and
A26: P . I = TR and
A27: I = m ; ::_thesis: ex newtr being Element of Permutations (k + 1) st
( newtr = NT9 & newtr is being_transposition & newtr . (k + 1) = k + 1 & TR = newtr | (Seg k) )
take NT ; ::_thesis: ( NT = NT9 & NT is being_transposition & NT . (k + 1) = k + 1 & TR = NT | (Seg k) )
A28: dom tr c= dom newt by A21, RELAT_1:60;
A29: for m being Nat st m <> i9 & m <> j9 & m in dom newt holds
newt . m = m
proof
A30: dom tr9 = Seg k by FUNCT_2:52;
let m be Nat; ::_thesis: ( m <> i9 & m <> j9 & m in dom newt implies newt . m = m )
assume that
A31: m <> i9 and
A32: m <> j9 and
A33: m in dom newt ; ::_thesis: newt . m = m
dom newt = Seg (k + 1) by FUNCT_2:52;
then ( m in Seg k or m in {(k + 1)} ) by A6, A33, XBOOLE_0:def_3;
then ( m in dom tr or m = k + 1 ) by A30, TARSKI:def_1;
then ( ( tr . m = newt . m & tr . m = m ) or newt . m = m ) by A21, A22, A20, A31, A32, FUNCT_1:47;
hence newt . m = m ; ::_thesis: verum
end;
newt . i9 = tr . i9 by A21, A15, FUNCT_1:47;
hence ( NT = NT9 & NT is being_transposition & NT . (k + 1) = k + 1 & TR = NT | (Seg k) ) by A13, A21, A22, A26, A27, A15, A16, A17, A18, A19, A28, A23, A29, MATRIX_2:def_11; ::_thesis: verum
end;
consider Pr being FinSequence of (Group_of_Perm (k + 1)) such that
A34: dom Pr = Seg (len P) and
A35: for m being Nat st m in Seg (len P) holds
S2[m,Pr . m] from FINSEQ_1:sch_5(A12);
take Pr ; ::_thesis: ( p = Product Pr & ( for i being Nat st i in dom Pr holds
ex trans being Element of Permutations (k + 1) st
( Pr . i = trans & trans is being_transposition ) ) )
A36: Product Pr = the multF of (Group_of_Perm (k + 1)) "**" Pr by GROUP_4:def_2;
now__::_thesis:_(_p_=_Product_Pr_&_(_for_i_being_Nat_st_i_in_dom_Pr_holds_
ex_trans_being_Element_of_Permutations_(k_+_1)_st_
(_Pr_._i_=_trans_&_trans_is_being_transposition_)_)_)
percases ( len Pr = 0 or len Pr > 0 ) ;
supposeA37: len Pr = 0 ; ::_thesis: ( p = Product Pr & ( for i being Nat st i in dom Pr holds
ex trans being Element of Permutations (k + 1) st
( Pr . i = trans & trans is being_transposition ) ) )
then A38: Seg (len Pr) = 0 ;
A39: Product Pr = the_unity_wrt the multF of (Group_of_Perm (k + 1)) by A36, A37, FINSOP_1:def_1;
the_unity_wrt the multF of (Group_of_Perm (k + 1)) = 1_ (Group_of_Perm (k + 1)) by GROUP_1:22;
then A40: Product Pr = idseq (k + 1) by A39, MATRIX_2:24;
len P = 0 by A34, A37, FINSEQ_1:def_3;
then A41: pk9 = the_unity_wrt the multF of (Group_of_Perm k) by A10, FINSOP_1:def_1;
A42: dom p9 = Seg (k + 1) by FUNCT_2:52;
A43: the_unity_wrt the multF of (Group_of_Perm k) = 1_ (Group_of_Perm k) by GROUP_1:22;
A44: for y being set st y in dom p holds
p . y = (idseq (k + 1)) . y
proof
let y be set ; ::_thesis: ( y in dom p implies p . y = (idseq (k + 1)) . y )
assume A45: y in dom p ; ::_thesis: p . y = (idseq (k + 1)) . y
reconsider y9 = y as Nat by A42, A45;
A46: (idseq (k + 1)) . y9 = y9 by A42, A45, FUNCT_1:18;
A47: dom pk = Seg k by FUNCT_2:52;
( y in Seg k or y in {(k + 1)} ) by A6, A42, A45, XBOOLE_0:def_3;
then ( ( pk . y = p . y & (idseq k) . y9 = y9 ) or ( p . (k + 1) = k + 1 & y = k + 1 ) ) by A5, A7, A47, FUNCT_1:18, FUNCT_1:47, TARSKI:def_1;
hence p . y = (idseq (k + 1)) . y by A41, A43, A46, MATRIX_2:24; ::_thesis: verum
end;
dom (idseq (k + 1)) = Seg (k + 1) by FUNCT_2:52;
hence ( p = Product Pr & ( for i being Nat st i in dom Pr holds
ex trans being Element of Permutations (k + 1) st
( Pr . i = trans & trans is being_transposition ) ) ) by A40, A42, A44, A38, FINSEQ_1:def_3, FUNCT_1:2; ::_thesis: verum
end;
supposeA48: len Pr > 0 ; ::_thesis: ( p = Product Pr & ( for i being Nat st i in dom Pr holds
ex trans being Element of Permutations (k + 1) st
( Pr . i = trans & trans is being_transposition ) ) )
consider fPr being Function of NAT,(Group_of_Perm (k + 1)) such that
A49: fPr . 1 = Pr . 1 and
A50: for n being Element of NAT st 0 <> n & n < len Pr holds
fPr . (n + 1) = the multF of (Group_of_Perm (k + 1)) . ((fPr . n),(Pr . (n + 1))) and
A51: Product Pr = fPr . (len Pr) by A36, A48, FINSOP_1:def_1;
len P > 0 by A34, A48, FINSEQ_1:def_3;
then consider fP being Function of NAT,(Group_of_Perm k) such that
A52: fP . 1 = P . 1 and
A53: for n being Element of NAT st 0 <> n & n < len P holds
fP . (n + 1) = the multF of (Group_of_Perm k) . ((fP . n),(P . (n + 1))) and
A54: pk = fP . (len P) by A10, FINSOP_1:def_1;
A55: len P = len Pr by A34, FINSEQ_1:def_3;
defpred S3[ Nat] means ( $1 > 0 & $1 <= len P implies ex Prod1 being Element of Permutations (k + 1) ex Prod being Element of Permutations k st
( Prod1 = fPr . $1 & fP . $1 = Prod & Prod1 | (Seg k) = Prod & Prod1 . (k + 1) = k + 1 ) );
A56: for m being Element of NAT st S3[m] holds
S3[m + 1]
proof
let m be Element of NAT ; ::_thesis: ( S3[m] implies S3[m + 1] )
assume A57: S3[m] ; ::_thesis: S3[m + 1]
set m1 = m + 1;
assume that
m + 1 > 0 and
A58: m + 1 <= len P ; ::_thesis: ex Prod1 being Element of Permutations (k + 1) ex Prod being Element of Permutations k st
( Prod1 = fPr . (m + 1) & fP . (m + 1) = Prod & Prod1 | (Seg k) = Prod & Prod1 . (k + 1) = k + 1 )
(m + 1) + 0 > 0 ;
then m + 1 >= 1 by NAT_1:19;
then A59: m + 1 in Seg (len P) by A58;
A60: dom P = Seg (len P) by FINSEQ_1:def_3;
then consider tr being Element of Permutations k such that
A61: P . (m + 1) = tr and
tr is being_transposition by A9, A59;
consider tr1 being Element of Permutations (k + 1) such that
A62: tr1 = Pr . (m + 1) and
tr1 is being_transposition and
A63: tr1 . (k + 1) = k + 1 and
A64: tr = tr1 | (Seg k) by A35, A59, A60, A61;
now__::_thesis:_ex_Prod1_being_Element_of_Permutations_(k_+_1)_ex_Prod_being_Element_of_Permutations_k_st_
(_Prod1_=_fPr_._(m_+_1)_&_fP_._(m_+_1)_=_Prod_&_Prod1_|_(Seg_k)_=_Prod_&_Prod1_._(k_+_1)_=_k_+_1_)
percases ( m = 0 or m > 0 ) ;
suppose m = 0 ; ::_thesis: ex Prod1 being Element of Permutations (k + 1) ex Prod being Element of Permutations k st
( Prod1 = fPr . (m + 1) & fP . (m + 1) = Prod & Prod1 | (Seg k) = Prod & Prod1 . (k + 1) = k + 1 )
hence ex Prod1 being Element of Permutations (k + 1) ex Prod being Element of Permutations k st
( Prod1 = fPr . (m + 1) & fP . (m + 1) = Prod & Prod1 | (Seg k) = Prod & Prod1 . (k + 1) = k + 1 ) by A52, A49, A61, A62, A63, A64; ::_thesis: verum
end;
supposeA65: m > 0 ; ::_thesis: ex Prod1 being Element of Permutations (k + 1) ex Prod being Element of Permutations k st
( Prod1 = fPr . (m + 1) & fP . (m + 1) = Prod & Prod1 | (Seg k) = Prod & Prod1 . (k + 1) = k + 1 )
A66: m + 0 < m + 1 by XREAL_1:6;
then consider Q1 being Element of Permutations (k + 1), Q being Element of Permutations k such that
A67: Q1 = fPr . m and
A68: fP . m = Q and
A69: Q1 | (Seg k) = Q and
A70: Q1 . (k + 1) = k + 1 by A57, A58, A65, XXREAL_0:2;
reconsider Q = Q, tr = tr as Permutation of (Seg k) by MATRIX_2:def_9;
reconsider trQ = tr * Q as Element of Permutations k by MATRIX_2:def_9;
A71: m < len P by A58, A66, XXREAL_0:2;
then A72: fP . (m + 1) = the multF of (Group_of_Perm k) . (Q,tr) by A53, A61, A65, A68;
then A73: fP . (m + 1) = trQ by MATRIX_2:def_10;
reconsider Q1 = Q1, tr1 = tr1 as Permutation of (Seg (k + 1)) by MATRIX_2:def_9;
reconsider trQ1 = tr1 * Q1 as Element of Permutations (k + 1) by MATRIX_2:def_9;
A74: trQ1 | (Seg k) = trQ by A6, A63, A64, A69, A70, Th19;
len P = len Pr by A34, FINSEQ_1:def_3;
then fPr . (m + 1) = the multF of (Group_of_Perm (k + 1)) . (Q1,tr1) by A50, A62, A65, A67, A71;
then A75: fPr . (m + 1) = trQ1 by MATRIX_2:def_10;
trQ1 . (k + 1) = k + 1 by A6, A63, A64, A69, A70, A72, Th19;
hence ex Prod1 being Element of Permutations (k + 1) ex Prod being Element of Permutations k st
( Prod1 = fPr . (m + 1) & fP . (m + 1) = Prod & Prod1 | (Seg k) = Prod & Prod1 . (k + 1) = k + 1 ) by A73, A75, A74; ::_thesis: verum
end;
end;
end;
hence ex Prod1 being Element of Permutations (k + 1) ex Prod being Element of Permutations k st
( Prod1 = fPr . (m + 1) & fP . (m + 1) = Prod & Prod1 | (Seg k) = Prod & Prod1 . (k + 1) = k + 1 ) ; ::_thesis: verum
end;
A76: S3[ 0 ] ;
for m being Element of NAT holds S3[m] from NAT_1:sch_1(A76, A56);
then consider Prod1 being Element of Permutations (k + 1), Prod being Element of Permutations k such that
A77: Prod1 = fPr . (len P) and
A78: fP . (len P) = Prod and
A79: Prod1 | (Seg k) = Prod and
A80: Prod1 . (k + 1) = k + 1 by A48, A55;
reconsider Prod1 = Prod1 as Permutation of (Seg (k + 1)) by MATRIX_2:def_9;
A81: dom p9 = Seg (k + 1) by FUNCT_2:52;
A82: for y being set st y in dom p holds
p . y = Prod1 . y
proof
let y be set ; ::_thesis: ( y in dom p implies p . y = Prod1 . y )
assume y in dom p ; ::_thesis: p . y = Prod1 . y
then A83: ( y in Seg k or y in {(k + 1)} ) by A6, A81, XBOOLE_0:def_3;
dom pk = Seg k by FUNCT_2:52;
then ( ( Prod . y = p . y & Prod . y = Prod1 . y ) or ( y = k + 1 & p . (k + 1) = Prod1 . (k + 1) ) ) by A5, A7, A54, A78, A79, A80, A83, FUNCT_1:47, TARSKI:def_1;
hence p . y = Prod1 . y ; ::_thesis: verum
end;
dom Prod1 = Seg (k + 1) by FUNCT_2:52;
hence p = Product Pr by A51, A55, A77, A81, A82, FUNCT_1:2; ::_thesis: for i being Nat st i in dom Pr holds
ex trans being Element of Permutations (k + 1) st
( Pr . i = trans & trans is being_transposition )
thus for i being Nat st i in dom Pr holds
ex trans being Element of Permutations (k + 1) st
( Pr . i = trans & trans is being_transposition ) ::_thesis: verum
proof
A84: Seg (len P) = dom P by FINSEQ_1:def_3;
let m be Nat; ::_thesis: ( m in dom Pr implies ex trans being Element of Permutations (k + 1) st
( Pr . m = trans & trans is being_transposition ) )
assume A85: m in dom Pr ; ::_thesis: ex trans being Element of Permutations (k + 1) st
( Pr . m = trans & trans is being_transposition )
consider t being Element of Permutations k such that
A86: P . m = t and
t is being_transposition by A9, A34, A85, A84;
reconsider m9 = m as Element of NAT by ORDINAL1:def_12;
ex T being Element of Permutations (k + 1) st
( T = Pr . m9 & T is being_transposition & T . (k + 1) = k + 1 & t = T | (Seg k) ) by A34, A35, A85, A84, A86;
hence ex trans being Element of Permutations (k + 1) st
( Pr . m = trans & trans is being_transposition ) ; ::_thesis: verum
end;
end;
end;
end;
hence ( p = Product Pr & ( for i being Nat st i in dom Pr holds
ex trans being Element of Permutations (k + 1) st
( Pr . i = trans & trans is being_transposition ) ) ) ; ::_thesis: verum
end;
now__::_thesis:_ex_P_being_FinSequence_of_(Group_of_Perm_(k_+_1))_st_
(_p_=_Product_P_&_(_for_i_being_Nat_st_i_in_dom_P_holds_
ex_trans_being_Element_of_Permutations_(k_+_1)_st_
(_P_._i_=_trans_&_trans_is_being_transposition_)_)_)
percases ( p . (k + 1) = k + 1 or p . (k + 1) <> k + 1 ) ;
suppose p . (k + 1) = k + 1 ; ::_thesis: ex P being FinSequence of (Group_of_Perm (k + 1)) st
( p = Product P & ( for i being Nat st i in dom P holds
ex trans being Element of Permutations (k + 1) st
( P . i = trans & trans is being_transposition ) ) )
hence ex P being FinSequence of (Group_of_Perm (k + 1)) st
( p = Product P & ( for i being Nat st i in dom P holds
ex trans being Element of Permutations (k + 1) st
( P . i = trans & trans is being_transposition ) ) ) by A4; ::_thesis: verum
end;
supposeA87: p . (k + 1) <> k + 1 ; ::_thesis: ex PT being FinSequence of the carrier of (Group_of_Perm (k + 1)) st
( Product PT = p & ( for m being Nat st m in dom PT holds
ex trans being Element of Permutations (k + 1) st
( PT . m = trans & trans is being_transposition ) ) )
A88: rng p9 = Seg (k + 1) by FUNCT_2:def_3;
consider tr being Element of Permutations (k + 1) such that
A89: tr is being_transposition and
tr . (p . (k + 1)) = k + 1 and
A90: (tr * p) . (k + 1) = k + 1 by A87, Th17;
reconsider tr9 = tr as Permutation of (Seg (k + 1)) by MATRIX_2:def_9;
reconsider trp = tr9 * p9 as Element of Permutations (k + 1) by MATRIX_2:def_9;
consider P being FinSequence of (Group_of_Perm (k + 1)) such that
A91: trp = Product P and
A92: for i being Nat st i in dom P holds
ex trans being Element of Permutations (k + 1) st
( P . i = trans & trans is being_transposition ) by A4, A90;
reconsider TRP = trp as Element of (Group_of_Perm (k + 1)) by MATRIX_2:def_10;
reconsider T = tr as Element of (Group_of_Perm (k + 1)) by MATRIX_2:def_10;
take PT = P ^ <*T*>; ::_thesis: ( Product PT = p & ( for m being Nat st m in dom PT holds
ex trans being Element of Permutations (k + 1) st
( PT . m = trans & trans is being_transposition ) ) )
Product PT = TRP * T by A91, GROUP_4:6;
hence Product PT = tr * (tr * p) by MATRIX_2:def_10
.= (tr * tr) * p by RELAT_1:36
.= (idseq (k + 1)) * p by A89, Th20
.= p by A88, RELAT_1:54 ;
::_thesis: for m being Nat st m in dom PT holds
ex trans being Element of Permutations (k + 1) st
( PT . m = trans & trans is being_transposition )
thus for m being Nat st m in dom PT holds
ex trans being Element of Permutations (k + 1) st
( PT . m = trans & trans is being_transposition ) ::_thesis: verum
proof
set L = len P;
set L1 = (len P) + 1;
A93: Seg ((len P) + 1) = (Seg (len P)) \/ {((len P) + 1)} by FINSEQ_1:9;
len PT = (len P) + 1 by FINSEQ_2:16;
then A94: dom PT = Seg ((len P) + 1) by FINSEQ_1:def_3;
let m be Nat; ::_thesis: ( m in dom PT implies ex trans being Element of Permutations (k + 1) st
( PT . m = trans & trans is being_transposition ) )
assume A95: m in dom PT ; ::_thesis: ex trans being Element of Permutations (k + 1) st
( PT . m = trans & trans is being_transposition )
now__::_thesis:_ex_trans_being_Element_of_Permutations_(k_+_1)_st_
(_PT_._m_=_trans_&_trans_is_being_transposition_)
percases ( m in Seg (len P) or m in {((len P) + 1)} ) by A95, A94, A93, XBOOLE_0:def_3;
suppose m in Seg (len P) ; ::_thesis: ex trans being Element of Permutations (k + 1) st
( PT . m = trans & trans is being_transposition )
then A96: m in dom P by FINSEQ_1:def_3;
then PT . m = P . m by FINSEQ_1:def_7;
hence ex trans being Element of Permutations (k + 1) st
( PT . m = trans & trans is being_transposition ) by A92, A96; ::_thesis: verum
end;
suppose m in {((len P) + 1)} ; ::_thesis: ex trans being Element of Permutations (k + 1) st
( PT . m = trans & trans is being_transposition )
then m = (len P) + 1 by TARSKI:def_1;
hence ex trans being Element of Permutations (k + 1) st
( PT . m = trans & trans is being_transposition ) by A89, FINSEQ_1:42; ::_thesis: verum
end;
end;
end;
hence ex trans being Element of Permutations (k + 1) st
( PT . m = trans & trans is being_transposition ) ; ::_thesis: verum
end;
end;
end;
end;
hence ex P being FinSequence of (Group_of_Perm (k + 1)) st
( p = Product P & ( for i being Nat st i in dom P holds
ex trans being Element of Permutations (k + 1) st
( P . i = trans & trans is being_transposition ) ) ) ; ::_thesis: verum
end;
A97: S1[ 0 ]
proof
let perm be Element of Permutations 0; ::_thesis: ex P being FinSequence of (Group_of_Perm 0) st
( perm = Product P & ( for i being Nat st i in dom P holds
ex trans being Element of Permutations 0 st
( P . i = trans & trans is being_transposition ) ) )
take <*> the carrier of (Group_of_Perm 0) ; ::_thesis: ( perm = Product (<*> the carrier of (Group_of_Perm 0)) & ( for i being Nat st i in dom (<*> the carrier of (Group_of_Perm 0)) holds
ex trans being Element of Permutations 0 st
( (<*> the carrier of (Group_of_Perm 0)) . i = trans & trans is being_transposition ) ) )
perm is Permutation of (Seg 0) by MATRIX_2:def_9;
then perm = idseq 0 ;
then perm = 1_ (Group_of_Perm 0) by MATRIX_2:24;
hence ( perm = Product (<*> the carrier of (Group_of_Perm 0)) & ( for i being Nat st i in dom (<*> the carrier of (Group_of_Perm 0)) holds
ex trans being Element of Permutations 0 st
( (<*> the carrier of (Group_of_Perm 0)) . i = trans & trans is being_transposition ) ) ) by GROUP_4:8; ::_thesis: verum
end;
for n being Element of NAT holds S1[n] from NAT_1:sch_1(A97, A2);
hence ex P being FinSequence of (Group_of_Perm n) st
( perm = Product P & ( for i being Nat st i in dom P holds
ex trans being Element of Permutations n st
( P . i = trans & trans is being_transposition ) ) ) by A1; ::_thesis: verum
end;
theorem Th22: :: MATRIX11:22
for K being Field holds
( K is Fanoian iff 1_ K <> - (1_ K) )
proof
let K be Field; ::_thesis: ( K is Fanoian iff 1_ K <> - (1_ K) )
thus ( K is Fanoian implies 1_ K <> - (1_ K) ) ::_thesis: ( 1_ K <> - (1_ K) implies K is Fanoian )
proof
assume A1: K is Fanoian ; ::_thesis: 1_ K <> - (1_ K)
assume 1_ K = - (1_ K) ; ::_thesis: contradiction
then (1_ K) + (1_ K) = 0. K by RLVECT_1:def_10;
hence contradiction by A1, VECTSP_1:def_18; ::_thesis: verum
end;
assume A2: 1_ K <> - (1_ K) ; ::_thesis: K is Fanoian
assume not K is Fanoian ; ::_thesis: contradiction
then consider a being Element of K such that
A3: a + a = 0. K and
A4: a <> 0. K by VECTSP_1:def_18;
a = a * (1_ K) by VECTSP_1:def_4;
then 0. K = a * ((1_ K) + (1_ K)) by A3, VECTSP_1:def_7;
then 0. K = (1_ K) + (1_ K) by A4, VECTSP_1:12;
hence contradiction by A2, VECTSP_1:16; ::_thesis: verum
end;
theorem Th23: :: MATRIX11:23
for n being Nat
for perm2 being Element of Permutations (n + 2)
for K being Fanoian Field holds
( ( perm2 is even implies sgn (perm2,K) = 1_ K ) & ( sgn (perm2,K) = 1_ K implies perm2 is even ) & ( perm2 is odd implies sgn (perm2,K) = - (1_ K) ) & ( sgn (perm2,K) = - (1_ K) implies perm2 is odd ) )
proof
let n be Nat; ::_thesis: for perm2 being Element of Permutations (n + 2)
for K being Fanoian Field holds
( ( perm2 is even implies sgn (perm2,K) = 1_ K ) & ( sgn (perm2,K) = 1_ K implies perm2 is even ) & ( perm2 is odd implies sgn (perm2,K) = - (1_ K) ) & ( sgn (perm2,K) = - (1_ K) implies perm2 is odd ) )
let perm2 be Element of Permutations (n + 2); ::_thesis: for K being Fanoian Field holds
( ( perm2 is even implies sgn (perm2,K) = 1_ K ) & ( sgn (perm2,K) = 1_ K implies perm2 is even ) & ( perm2 is odd implies sgn (perm2,K) = - (1_ K) ) & ( sgn (perm2,K) = - (1_ K) implies perm2 is odd ) )
set n2 = n + 2;
let K be Fanoian Field; ::_thesis: ( ( perm2 is even implies sgn (perm2,K) = 1_ K ) & ( sgn (perm2,K) = 1_ K implies perm2 is even ) & ( perm2 is odd implies sgn (perm2,K) = - (1_ K) ) & ( sgn (perm2,K) = - (1_ K) implies perm2 is odd ) )
A1: len (Permutations (n + 2)) = n + 2 by MATRIX_2:18;
thus A2: ( perm2 is even implies sgn (perm2,K) = 1_ K ) ::_thesis: ( ( sgn (perm2,K) = 1_ K implies perm2 is even ) & ( perm2 is odd implies sgn (perm2,K) = - (1_ K) ) & ( sgn (perm2,K) = - (1_ K) implies perm2 is odd ) )
proof
assume perm2 is even ; ::_thesis: sgn (perm2,K) = 1_ K
then ex L being FinSequence of (Group_of_Perm (n + 2)) st
( (len L) mod 2 = 0 & perm2 = Product L & ( for i being Nat st i in dom L holds
ex q2 being Element of Permutations (n + 2) st
( L . i = q2 & q2 is being_transposition ) ) ) by A1, MATRIX_2:def_12;
hence sgn (perm2,K) = 1_ K by Th15; ::_thesis: verum
end;
thus ( sgn (perm2,K) = 1_ K implies perm2 is even ) ::_thesis: ( perm2 is odd iff sgn (perm2,K) = - (1_ K) )
proof
assume A3: sgn (perm2,K) = 1_ K ; ::_thesis: perm2 is even
consider P being FinSequence of (Group_of_Perm (n + 2)) such that
A4: perm2 = Product P and
A5: for i being Nat st i in dom P holds
ex trans being Element of Permutations (n + 2) st
( P . i = trans & trans is being_transposition ) by Th21;
assume perm2 is odd ; ::_thesis: contradiction
then (len P) mod 2 <> 0 by A1, A4, A5, MATRIX_2:def_12;
then (len P) mod 2 = 1 by NAT_D:12;
then sgn (perm2,K) = - (1_ K) by A4, A5, Th15;
hence contradiction by A3, Th22; ::_thesis: verum
end;
hence ( perm2 is odd iff sgn (perm2,K) = - (1_ K) ) by A2, Th11, Th22; ::_thesis: verum
end;
theorem Th24: :: MATRIX11:24
for n being Nat
for K being Field
for p2, q2, pq2 being Element of Permutations (n + 2) st pq2 = p2 * q2 holds
sgn (pq2,K) = (sgn (p2,K)) * (sgn (q2,K))
proof
let n be Nat; ::_thesis: for K being Field
for p2, q2, pq2 being Element of Permutations (n + 2) st pq2 = p2 * q2 holds
sgn (pq2,K) = (sgn (p2,K)) * (sgn (q2,K))
let K be Field; ::_thesis: for p2, q2, pq2 being Element of Permutations (n + 2) st pq2 = p2 * q2 holds
sgn (pq2,K) = (sgn (p2,K)) * (sgn (q2,K))
set n2 = n + 2;
let p, q, pq be Element of Permutations (n + 2); ::_thesis: ( pq = p * q implies sgn (pq,K) = (sgn (p,K)) * (sgn (q,K)) )
assume A1: pq = p * q ; ::_thesis: sgn (pq,K) = (sgn (p,K)) * (sgn (q,K))
consider P2 being FinSequence of (Group_of_Perm (n + 2)) such that
A2: q = Product P2 and
A3: for i being Nat st i in dom P2 holds
ex trans being Element of Permutations (n + 2) st
( P2 . i = trans & trans is being_transposition ) by Th21;
consider P1 being FinSequence of (Group_of_Perm (n + 2)) such that
A4: p = Product P1 and
A5: for i being Nat st i in dom P1 holds
ex trans being Element of Permutations (n + 2) st
( P1 . i = trans & trans is being_transposition ) by Th21;
set PP = P2 ^ P1;
A6: for i being Nat st i in dom (P2 ^ P1) holds
ex trans being Element of Permutations (n + 2) st
( (P2 ^ P1) . i = trans & trans is being_transposition )
proof
let i be Nat; ::_thesis: ( i in dom (P2 ^ P1) implies ex trans being Element of Permutations (n + 2) st
( (P2 ^ P1) . i = trans & trans is being_transposition ) )
assume A7: i in dom (P2 ^ P1) ; ::_thesis: ex trans being Element of Permutations (n + 2) st
( (P2 ^ P1) . i = trans & trans is being_transposition )
now__::_thesis:_ex_trans_being_Element_of_Permutations_(n_+_2)_st_
(_(P2_^_P1)_._i_=_trans_&_trans_is_being_transposition_)
percases ( i in dom P2 or ex k being Nat st
( k in dom P1 & i = (len P2) + k ) ) by A7, FINSEQ_1:25;
supposeA8: i in dom P2 ; ::_thesis: ex trans being Element of Permutations (n + 2) st
( (P2 ^ P1) . i = trans & trans is being_transposition )
then P2 . i = (P2 ^ P1) . i by FINSEQ_1:def_7;
hence ex trans being Element of Permutations (n + 2) st
( (P2 ^ P1) . i = trans & trans is being_transposition ) by A3, A8; ::_thesis: verum
end;
suppose ex k being Nat st
( k in dom P1 & i = (len P2) + k ) ; ::_thesis: ex trans being Element of Permutations (n + 2) st
( (P2 ^ P1) . i = trans & trans is being_transposition )
then consider k being Nat such that
A9: k in dom P1 and
A10: i = (len P2) + k ;
P1 . k = (P2 ^ P1) . i by A9, A10, FINSEQ_1:def_7;
hence ex trans being Element of Permutations (n + 2) st
( (P2 ^ P1) . i = trans & trans is being_transposition ) by A5, A9; ::_thesis: verum
end;
end;
end;
hence ex trans being Element of Permutations (n + 2) st
( (P2 ^ P1) . i = trans & trans is being_transposition ) ; ::_thesis: verum
end;
A11: Product (P2 ^ P1) = (Product P2) * (Product P1) by GROUP_4:5
.= pq by A1, A4, A2, MATRIX_2:def_10 ;
now__::_thesis:_sgn_(pq,K)_=_(sgn_(p,K))_*_(sgn_(q,K))
percases ( ( (len P1) mod 2 = 0 & (len P2) mod 2 = 0 ) or ( (len P1) mod 2 = 1 & (len P2) mod 2 = 0 ) or ( (len P1) mod 2 = 0 & (len P2) mod 2 = 1 ) or ( (len P1) mod 2 = 1 & (len P2) mod 2 = 1 ) ) by NAT_D:12;
supposeA12: ( (len P1) mod 2 = 0 & (len P2) mod 2 = 0 ) ; ::_thesis: sgn (pq,K) = (sgn (p,K)) * (sgn (q,K))
(len (P2 ^ P1)) mod 2 = ((len P2) + (len P1)) mod 2 by FINSEQ_1:22
.= ((0 + (len P1)) + 0) mod 2 by A12, NAT_D:22
.= 0 by A12 ;
then A13: sgn (pq,K) = 1_ K by A11, A6, Th15;
A14: sgn (q,K) = 1_ K by A2, A3, A12, Th15;
sgn (p,K) = 1_ K by A4, A5, A12, Th15;
hence sgn (pq,K) = (sgn (p,K)) * (sgn (q,K)) by A13, A14, VECTSP_1:def_4; ::_thesis: verum
end;
supposeA15: ( (len P1) mod 2 = 1 & (len P2) mod 2 = 0 ) ; ::_thesis: sgn (pq,K) = (sgn (p,K)) * (sgn (q,K))
(len (P2 ^ P1)) mod 2 = ((len P2) + (len P1)) mod 2 by FINSEQ_1:22
.= ((0 + (len P1)) + 0) mod 2 by A15, NAT_D:22
.= 1 by A15 ;
then A16: sgn (pq,K) = - (1_ K) by A11, A6, Th15;
A17: sgn (q,K) = 1_ K by A2, A3, A15, Th15;
sgn (p,K) = - (1_ K) by A4, A5, A15, Th15;
hence sgn (pq,K) = (sgn (p,K)) * (sgn (q,K)) by A16, A17, VECTSP_1:def_4; ::_thesis: verum
end;
supposeA18: ( (len P1) mod 2 = 0 & (len P2) mod 2 = 1 ) ; ::_thesis: sgn (pq,K) = (sgn (p,K)) * (sgn (q,K))
(len (P2 ^ P1)) mod 2 = ((len P2) + (len P1)) mod 2 by FINSEQ_1:22
.= (1 + (len P1)) mod 2 by A18, NAT_D:22
.= (1 + 0) mod 2 by A18, NAT_D:22
.= 1 by NAT_D:14 ;
then A19: sgn (pq,K) = - (1_ K) by A11, A6, Th15;
A20: sgn (q,K) = - (1_ K) by A2, A3, A18, Th15;
sgn (p,K) = 1_ K by A4, A5, A18, Th15;
hence sgn (pq,K) = (sgn (p,K)) * (sgn (q,K)) by A19, A20, VECTSP_1:def_4; ::_thesis: verum
end;
supposeA21: ( (len P1) mod 2 = 1 & (len P2) mod 2 = 1 ) ; ::_thesis: sgn (pq,K) = (sgn (p,K)) * (sgn (q,K))
(len (P2 ^ P1)) mod 2 = ((len P2) + (len P1)) mod 2 by FINSEQ_1:22
.= (1 + (len P1)) mod 2 by A21, NAT_D:22
.= (1 + 1) mod 2 by A21, NAT_D:22
.= 0 by NAT_D:25 ;
then A22: sgn (pq,K) = 1_ K by A11, A6, Th15;
A23: (1_ K) * (1_ K) = 1_ K by VECTSP_1:def_4;
A24: sgn (q,K) = - (1_ K) by A2, A3, A21, Th15;
sgn (p,K) = - (1_ K) by A4, A5, A21, Th15;
hence sgn (pq,K) = (sgn (p,K)) * (sgn (q,K)) by A22, A24, A23, VECTSP_1:10; ::_thesis: verum
end;
end;
end;
hence sgn (pq,K) = (sgn (p,K)) * (sgn (q,K)) ; ::_thesis: verum
end;
Lm3: for n being Nat
for p being Element of Permutations n st n < 2 holds
( p is even & p = idseq n )
proof
let n be Nat; ::_thesis: for p being Element of Permutations n st n < 2 holds
( p is even & p = idseq n )
let p be Element of Permutations n; ::_thesis: ( n < 2 implies ( p is even & p = idseq n ) )
reconsider P = p as Permutation of (Seg n) by MATRIX_2:def_9;
assume A1: n < 2 ; ::_thesis: ( p is even & p = idseq n )
now__::_thesis:_(_p_is_even_&_p_=_idseq_n_)
percases ( n = 0 or n = 1 ) by A1, NAT_1:23;
supposeA2: n = 0 ; ::_thesis: ( p is even & p = idseq n )
then A3: Seg n = {} ;
A4: len (Permutations n) = n by MATRIX_2:18;
P = {} by A2;
hence ( p is even & p = idseq n ) by A4, A3, MATRIX_2:25, RELAT_1:55; ::_thesis: verum
end;
supposeA5: n = 1 ; ::_thesis: ( p is even & p = idseq n )
A6: len (Permutations n) = n by MATRIX_2:18;
P = id (Seg n) by A5, MATRIX_2:19, TARSKI:def_1;
hence ( p is even & p = idseq n ) by A6, MATRIX_2:25; ::_thesis: verum
end;
end;
end;
hence ( p is even & p = idseq n ) ; ::_thesis: verum
end;
theorem Th25: :: MATRIX11:25
for n being Nat
for p, q being Element of Permutations n holds
( ( ( p is even & q is even ) or ( p is odd & q is odd ) ) iff p * q is even )
proof
let n be Nat; ::_thesis: for p, q being Element of Permutations n holds
( ( ( p is even & q is even ) or ( p is odd & q is odd ) ) iff p * q is even )
let p, q be Element of Permutations n; ::_thesis: ( ( ( p is even & q is even ) or ( p is odd & q is odd ) ) iff p * q is even )
reconsider pq = p * q as Element of Permutations n by MATRIX_9:39;
now__::_thesis:_(_(_(_p_is_even_&_q_is_even_)_or_(_p_is_odd_&_q_is_odd_)_)_iff_p_*_q_is_even_)
percases ( n < 2 or n >= 2 ) ;
supposeA1: n < 2 ; ::_thesis: ( ( ( p is even & q is even ) or ( p is odd & q is odd ) ) iff p * q is even )
then pq is even by Lm3;
hence ( ( ( p is even & q is even ) or ( p is odd & q is odd ) ) iff p * q is even ) by A1, Lm3; ::_thesis: verum
end;
suppose n >= 2 ; ::_thesis: ( ( ( ( p is even & q is even ) or ( p is odd & q is odd ) ) implies p * q is even ) & ( not p * q is even or ( p is even & q is even ) or ( p is odd & q is odd ) ) )
then reconsider n2 = n - 2 as Nat by NAT_1:21;
set K = the Fanoian Field;
reconsider p9 = p, q9 = q, pq = pq as Element of Permutations (n2 + 2) ;
thus ( ( ( p is even & q is even ) or ( p is odd & q is odd ) ) implies p * q is even ) ::_thesis: ( not p * q is even or ( p is even & q is even ) or ( p is odd & q is odd ) )
proof
assume ( ( p is even & q is even ) or ( p is odd & q is odd ) ) ; ::_thesis: p * q is even
then ( ( sgn (p9, the Fanoian Field) = 1_ the Fanoian Field & sgn (q9, the Fanoian Field) = 1_ the Fanoian Field ) or ( sgn (p9, the Fanoian Field) = - (1_ the Fanoian Field) & sgn (q9, the Fanoian Field) = - (1_ the Fanoian Field) ) ) by Th23;
then A2: (sgn (p9, the Fanoian Field)) * (sgn (q9, the Fanoian Field)) = (1_ the Fanoian Field) * (1_ the Fanoian Field) by VECTSP_1:10;
(1_ the Fanoian Field) * (1_ the Fanoian Field) = 1_ the Fanoian Field by VECTSP_1:def_4;
then sgn (pq, the Fanoian Field) = 1_ the Fanoian Field by A2, Th24;
hence p * q is even by Th23; ::_thesis: verum
end;
thus ( not p * q is even or ( p is even & q is even ) or ( p is odd & q is odd ) ) ::_thesis: verum
proof
assume p * q is even ; ::_thesis: ( ( p is even & q is even ) or ( p is odd & q is odd ) )
then sgn (pq, the Fanoian Field) = 1_ the Fanoian Field by Th23;
then A3: (sgn (p9, the Fanoian Field)) * (sgn (q9, the Fanoian Field)) = 1_ the Fanoian Field by Th24;
assume A4: ( not ( p is even & q is even ) & not ( p is odd & q is odd ) ) ; ::_thesis: contradiction
now__::_thesis:_contradiction
percases ( ( p is even & q is odd ) or ( p is odd & q is even ) ) by A4;
supposeA5: ( p is even & q is odd ) ; ::_thesis: contradiction
then A6: sgn (q9, the Fanoian Field) = - (1_ the Fanoian Field) by Th23;
sgn (p9, the Fanoian Field) = 1_ the Fanoian Field by A5, Th23;
then (sgn (p9, the Fanoian Field)) * (sgn (q9, the Fanoian Field)) = - (1_ the Fanoian Field) by A6, VECTSP_1:def_4;
hence contradiction by A3, Th22; ::_thesis: verum
end;
supposeA7: ( p is odd & q is even ) ; ::_thesis: contradiction
then A8: sgn (q9, the Fanoian Field) = 1_ the Fanoian Field by Th23;
sgn (p9, the Fanoian Field) = - (1_ the Fanoian Field) by A7, Th23;
then (sgn (p9, the Fanoian Field)) * (sgn (q9, the Fanoian Field)) = - (1_ the Fanoian Field) by A8, VECTSP_1:def_4;
hence contradiction by A3, Th22; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;
end;
hence ( ( ( p is even & q is even ) or ( p is odd & q is odd ) ) iff p * q is even ) ; ::_thesis: verum
end;
theorem Th26: :: MATRIX11:26
for n being Nat
for K being Field
for a being Element of K
for perm2 being Element of Permutations (n + 2) holds - (a,perm2) = (sgn (perm2,K)) * a
proof
let n be Nat; ::_thesis: for K being Field
for a being Element of K
for perm2 being Element of Permutations (n + 2) holds - (a,perm2) = (sgn (perm2,K)) * a
let K be Field; ::_thesis: for a being Element of K
for perm2 being Element of Permutations (n + 2) holds - (a,perm2) = (sgn (perm2,K)) * a
let a be Element of K; ::_thesis: for perm2 being Element of Permutations (n + 2) holds - (a,perm2) = (sgn (perm2,K)) * a
let perm2 be Element of Permutations (n + 2); ::_thesis: - (a,perm2) = (sgn (perm2,K)) * a
percases ( ( perm2 is even & K is Fanoian ) or ( perm2 is odd & K is Fanoian ) or ( perm2 is even & not K is Fanoian ) or ( perm2 is odd & not K is Fanoian ) ) ;
supposeA1: ( perm2 is even & K is Fanoian ) ; ::_thesis: - (a,perm2) = (sgn (perm2,K)) * a
then A2: - (a,perm2) = a by MATRIX_2:def_13;
sgn (perm2,K) = 1_ K by A1, Th23;
hence - (a,perm2) = (sgn (perm2,K)) * a by A2, VECTSP_1:def_4; ::_thesis: verum
end;
supposeA3: ( perm2 is odd & K is Fanoian ) ; ::_thesis: - (a,perm2) = (sgn (perm2,K)) * a
then A4: - (a,perm2) = - a by MATRIX_2:def_13;
A5: (- (1_ K)) * a = - ((1_ K) * a) by VECTSP_1:8;
sgn (perm2,K) = - (1_ K) by A3, Th23;
hence - (a,perm2) = (sgn (perm2,K)) * a by A4, A5, VECTSP_1:def_4; ::_thesis: verum
end;
supposeA6: ( perm2 is even & not K is Fanoian ) ; ::_thesis: - (a,perm2) = (sgn (perm2,K)) * a
then A7: - (a,perm2) = a by MATRIX_2:def_13;
A8: ( sgn (perm2,K) = 1_ K or sgn (perm2,K) = - (1_ K) ) by Th11;
1_ K = - (1_ K) by A6, Th22;
hence - (a,perm2) = (sgn (perm2,K)) * a by A7, A8, VECTSP_1:def_4; ::_thesis: verum
end;
supposeA9: ( perm2 is odd & not K is Fanoian ) ; ::_thesis: - (a,perm2) = (sgn (perm2,K)) * a
then A10: - (a,perm2) = - a by MATRIX_2:def_13;
A11: (- (1_ K)) * a = - ((1_ K) * a) by VECTSP_1:8;
A12: ( sgn (perm2,K) = 1_ K or sgn (perm2,K) = - (1_ K) ) by Th11;
1_ K = - (1_ K) by A9, Th22;
hence - (a,perm2) = (sgn (perm2,K)) * a by A10, A11, A12, VECTSP_1:def_4; ::_thesis: verum
end;
end;
end;
theorem Th27: :: MATRIX11:27
for n being Nat
for tr being Element of Permutations (n + 2) st tr is being_transposition holds
tr is odd
proof
let n be Nat; ::_thesis: for tr being Element of Permutations (n + 2) st tr is being_transposition holds
tr is odd
set K = the Fanoian Field;
let tr be Element of Permutations (n + 2); ::_thesis: ( tr is being_transposition implies tr is odd )
assume tr is being_transposition ; ::_thesis: tr is odd
then sgn (tr, the Fanoian Field) = - (1_ the Fanoian Field) by Th14;
hence tr is odd by Th23; ::_thesis: verum
end;
registration
let n be Nat;
cluster Relation-like Seg (n + 2) -defined Seg (n + 2) -valued Function-like one-to-one non empty total quasi_total onto bijective finite odd for Element of bool [:(Seg (n + 2)),(Seg (n + 2)):];
existence
ex b1 being Permutation of (Seg (n + 2)) st b1 is odd
proof
set n2 = n + 2;
A1: len (Permutations (n + 2)) = n + 2 by MATRIX_2:18;
n + 2 >= 2 + 0 by XREAL_1:6;
then {1,2} in 2Set (Seg (n + 2)) by Th3;
then consider i, j being Nat such that
A2: i in Seg (n + 2) and
A3: j in Seg (n + 2) and
A4: i < j and
{1,2} = {i,j} by Th1;
consider tr being Element of Permutations (n + 2) such that
A5: tr is being_transposition and
tr . i = j by A2, A3, A4, Th16;
tr is odd by A5, Th27;
hence ex b1 being Permutation of (Seg (n + 2)) st b1 is odd by A1; ::_thesis: verum
end;
end;
begin
definition
let l, n, m be Nat;
let D be non empty set ;
let M be Matrix of n,m,D;
let pD be FinSequence of D;
func ReplaceLine (M,l,pD) -> Matrix of n,m,D means :Def3: :: MATRIX11:def 3
( len it = len M & width it = width M & ( for i, j being Nat st [i,j] in Indices M holds
( ( i <> l implies it * (i,j) = M * (i,j) ) & ( i = l implies it * (l,j) = pD . j ) ) ) ) if len pD = width M
otherwise it = M;
consistency
for b1 being Matrix of n,m,D holds verum ;
existence
( ( len pD = width M implies ex b1 being Matrix of n,m,D st
( len b1 = len M & width b1 = width M & ( for i, j being Nat st [i,j] in Indices M holds
( ( i <> l implies b1 * (i,j) = M * (i,j) ) & ( i = l implies b1 * (l,j) = pD . j ) ) ) ) ) & ( not len pD = width M implies ex b1 being Matrix of n,m,D st b1 = M ) )
proof
thus ( len pD = width M implies ex M1 being Matrix of n,m,D st
( len M1 = len M & width M1 = width M & ( for i, j being Nat st [i,j] in Indices M holds
( ( i <> l implies M1 * (i,j) = M * (i,j) ) & ( i = l implies M1 * (l,j) = pD . j ) ) ) ) ) ::_thesis: ( not len pD = width M implies ex b1 being Matrix of n,m,D st b1 = M )
proof
reconsider M9 = M as Matrix of len M, width M,D by MATRIX_2:7;
reconsider n1 = n, m1 = m as Element of NAT by ORDINAL1:def_12;
defpred S1[ set , set , set ] means for i, j being Nat st i = $1 & j = $2 holds
( ( i <> l implies $3 = M * (i,j) ) & ( i = l implies $3 = pD . j ) );
assume A1: len pD = width M ; ::_thesis: ex M1 being Matrix of n,m,D st
( len M1 = len M & width M1 = width M & ( for i, j being Nat st [i,j] in Indices M holds
( ( i <> l implies M1 * (i,j) = M * (i,j) ) & ( i = l implies M1 * (l,j) = pD . j ) ) ) )
A2: for i, j being Nat st [i,j] in [:(Seg n1),(Seg m1):] holds
ex x being Element of D st S1[i,j,x]
proof
let i, j be Nat; ::_thesis: ( [i,j] in [:(Seg n1),(Seg m1):] implies ex x being Element of D st S1[i,j,x] )
assume A3: [i,j] in [:(Seg n1),(Seg m1):] ; ::_thesis: ex x being Element of D st S1[i,j,x]
now__::_thesis:_(_(_i_=_l_&_ex_x_being_Element_of_D_st_S1[i,j,x]_)_or_(_i_<>_l_&_ex_x_being_Element_of_D_st_S1[i,j,x]_)_)
percases ( i = l or i <> l ) ;
caseA4: i = l ; ::_thesis: ex x being Element of D st S1[i,j,x]
A5: rng pD c= D by FINSEQ_1:def_4;
n1 <> 0 by A3, ZFMISC_1:87;
then len pD = m by A1, MATRIX_1:23;
then j in Seg (len pD) by A3, ZFMISC_1:87;
then j in dom pD by FINSEQ_1:def_3;
then A6: pD . j in rng pD by FUNCT_1:def_3;
S1[i,j,pD . j] by A4;
hence ex x being Element of D st S1[i,j,x] by A6, A5; ::_thesis: verum
end;
case i <> l ; ::_thesis: ex x being Element of D st S1[i,j,x]
then S1[i,j,M * (i,j)] ;
hence ex x being Element of D st S1[i,j,x] ; ::_thesis: verum
end;
end;
end;
hence ex x being Element of D st S1[i,j,x] ; ::_thesis: verum
end;
consider M1 being Matrix of n1,m1,D such that
A7: for i, j being Nat st [i,j] in Indices M1 holds
S1[i,j,M1 * (i,j)] from MATRIX_1:sch_2(A2);
reconsider M1 = M1 as Matrix of n,m,D ;
take M1 ; ::_thesis: ( len M1 = len M & width M1 = width M & ( for i, j being Nat st [i,j] in Indices M holds
( ( i <> l implies M1 * (i,j) = M * (i,j) ) & ( i = l implies M1 * (l,j) = pD . j ) ) ) )
A8: now__::_thesis:_(_len_M_=_len_M1_&_width_M1_=_width_M_)
percases ( n = 0 or n > 0 ) ;
supposeA9: n = 0 ; ::_thesis: ( len M = len M1 & width M1 = width M )
then len M1 = 0 by MATRIX_1:def_2;
then A10: width M1 = 0 by MATRIX_1:def_3;
len M = 0 by A9, MATRIX_1:def_2;
hence ( len M = len M1 & width M1 = width M ) by A9, A10, MATRIX_1:def_2, MATRIX_1:def_3; ::_thesis: verum
end;
supposeA11: n > 0 ; ::_thesis: ( len M = len M1 & width M = width M1 )
then A12: width M = m by MATRIX_1:23;
len M = n by A11, MATRIX_1:23;
hence ( len M = len M1 & width M = width M1 ) by A11, A12, MATRIX_1:23; ::_thesis: verum
end;
end;
end;
Indices M9 = Indices M1 by MATRIX_1:26;
hence ( len M1 = len M & width M1 = width M & ( for i, j being Nat st [i,j] in Indices M holds
( ( i <> l implies M1 * (i,j) = M * (i,j) ) & ( i = l implies M1 * (l,j) = pD . j ) ) ) ) by A7, A8; ::_thesis: verum
end;
thus ( not len pD = width M implies ex b1 being Matrix of n,m,D st b1 = M ) ; ::_thesis: verum
end;
uniqueness
for b1, b2 being Matrix of n,m,D holds
( ( len pD = width M & len b1 = len M & width b1 = width M & ( for i, j being Nat st [i,j] in Indices M holds
( ( i <> l implies b1 * (i,j) = M * (i,j) ) & ( i = l implies b1 * (l,j) = pD . j ) ) ) & len b2 = len M & width b2 = width M & ( for i, j being Nat st [i,j] in Indices M holds
( ( i <> l implies b2 * (i,j) = M * (i,j) ) & ( i = l implies b2 * (l,j) = pD . j ) ) ) implies b1 = b2 ) & ( not len pD = width M & b1 = M & b2 = M implies b1 = b2 ) )
proof
let M1, M2 be Matrix of n,m,D; ::_thesis: ( ( len pD = width M & len M1 = len M & width M1 = width M & ( for i, j being Nat st [i,j] in Indices M holds
( ( i <> l implies M1 * (i,j) = M * (i,j) ) & ( i = l implies M1 * (l,j) = pD . j ) ) ) & len M2 = len M & width M2 = width M & ( for i, j being Nat st [i,j] in Indices M holds
( ( i <> l implies M2 * (i,j) = M * (i,j) ) & ( i = l implies M2 * (l,j) = pD . j ) ) ) implies M1 = M2 ) & ( not len pD = width M & M1 = M & M2 = M implies M1 = M2 ) )
thus ( len pD = width M & len M1 = len M & width M1 = width M & ( for i, j being Nat st [i,j] in Indices M holds
( ( i <> l implies M1 * (i,j) = M * (i,j) ) & ( i = l implies M1 * (l,j) = pD . j ) ) ) & len M2 = len M & width M2 = width M & ( for i, j being Nat st [i,j] in Indices M holds
( ( i <> l implies M2 * (i,j) = M * (i,j) ) & ( i = l implies M2 * (l,j) = pD . j ) ) ) implies M1 = M2 ) ::_thesis: ( not len pD = width M & M1 = M & M2 = M implies M1 = M2 )
proof
assume len pD = width M ; ::_thesis: ( not len M1 = len M or not width M1 = width M or ex i, j being Nat st
( [i,j] in Indices M & not ( ( i <> l implies M1 * (i,j) = M * (i,j) ) & ( i = l implies M1 * (l,j) = pD . j ) ) ) or not len M2 = len M or not width M2 = width M or ex i, j being Nat st
( [i,j] in Indices M & not ( ( i <> l implies M2 * (i,j) = M * (i,j) ) & ( i = l implies M2 * (l,j) = pD . j ) ) ) or M1 = M2 )
assume that
A13: len M1 = len M and
A14: width M1 = width M and
A15: for i, j being Nat st [i,j] in Indices M holds
( ( i <> l implies M1 * (i,j) = M * (i,j) ) & ( i = l implies M1 * (l,j) = pD . j ) ) ; ::_thesis: ( not len M2 = len M or not width M2 = width M or ex i, j being Nat st
( [i,j] in Indices M & not ( ( i <> l implies M2 * (i,j) = M * (i,j) ) & ( i = l implies M2 * (l,j) = pD . j ) ) ) or M1 = M2 )
assume that
len M2 = len M and
width M2 = width M and
A16: for i, j being Nat st [i,j] in Indices M holds
( ( i <> l implies M2 * (i,j) = M * (i,j) ) & ( i = l implies M2 * (l,j) = pD . j ) ) ; ::_thesis: M1 = M2
for i, j being Nat st [i,j] in Indices M1 holds
M1 * (i,j) = M2 * (i,j)
proof
let i, j be Nat; ::_thesis: ( [i,j] in Indices M1 implies M1 * (i,j) = M2 * (i,j) )
assume [i,j] in Indices M1 ; ::_thesis: M1 * (i,j) = M2 * (i,j)
then A17: [i,j] in Indices M by A13, A14, MATRIX_4:55;
then A18: ( i = l implies M1 * (l,j) = pD . j ) by A15;
A19: ( i <> l implies M2 * (i,j) = M * (i,j) ) by A16, A17;
( i <> l implies M1 * (i,j) = M * (i,j) ) by A15, A17;
hence M1 * (i,j) = M2 * (i,j) by A16, A17, A18, A19; ::_thesis: verum
end;
hence M1 = M2 by MATRIX_1:27; ::_thesis: verum
end;
thus ( not len pD = width M & M1 = M & M2 = M implies M1 = M2 ) ; ::_thesis: verum
end;
end;
:: deftheorem Def3 defines ReplaceLine MATRIX11:def_3_:_
for l, n, m being Nat
for D being non empty set
for M being Matrix of n,m,D
for pD being FinSequence of D
for b7 being Matrix of n,m,D holds
( ( len pD = width M implies ( b7 = ReplaceLine (M,l,pD) iff ( len b7 = len M & width b7 = width M & ( for i, j being Nat st [i,j] in Indices M holds
( ( i <> l implies b7 * (i,j) = M * (i,j) ) & ( i = l implies b7 * (l,j) = pD . j ) ) ) ) ) ) & ( not len pD = width M implies ( b7 = ReplaceLine (M,l,pD) iff b7 = M ) ) );
notation
let l, n, m be Nat;
let D be non empty set ;
let M be Matrix of n,m,D;
let pD be FinSequence of D;
synonym RLine (M,l,pD) for ReplaceLine (M,l,pD);
end;
Lm4: for m, n being Nat
for D being non empty set
for l being Nat
for M being Matrix of n,m,D
for pD being FinSequence of D holds
( Indices M = Indices (RLine (M,l,pD)) & len M = len (RLine (M,l,pD)) & width M = width (RLine (M,l,pD)) )
proof
let m, n be Nat; ::_thesis: for D being non empty set
for l being Nat
for M being Matrix of n,m,D
for pD being FinSequence of D holds
( Indices M = Indices (RLine (M,l,pD)) & len M = len (RLine (M,l,pD)) & width M = width (RLine (M,l,pD)) )
let D be non empty set ; ::_thesis: for l being Nat
for M being Matrix of n,m,D
for pD being FinSequence of D holds
( Indices M = Indices (RLine (M,l,pD)) & len M = len (RLine (M,l,pD)) & width M = width (RLine (M,l,pD)) )
let l be Nat; ::_thesis: for M being Matrix of n,m,D
for pD being FinSequence of D holds
( Indices M = Indices (RLine (M,l,pD)) & len M = len (RLine (M,l,pD)) & width M = width (RLine (M,l,pD)) )
let M be Matrix of n,m,D; ::_thesis: for pD being FinSequence of D holds
( Indices M = Indices (RLine (M,l,pD)) & len M = len (RLine (M,l,pD)) & width M = width (RLine (M,l,pD)) )
let pD be FinSequence of D; ::_thesis: ( Indices M = Indices (RLine (M,l,pD)) & len M = len (RLine (M,l,pD)) & width M = width (RLine (M,l,pD)) )
now__::_thesis:_(_(_len_pD_=_width_M_&_Indices_M_=_Indices_(RLine_(M,l,pD))_&_len_M_=_len_(RLine_(M,l,pD))_&_width_M_=_width_(RLine_(M,l,pD))_)_or_(_len_pD_<>_width_M_&_Indices_M_=_Indices_(RLine_(M,l,pD))_&_len_M_=_len_(RLine_(M,l,pD))_&_width_M_=_width_(RLine_(M,l,pD))_)_)
percases ( len pD = width M or len pD <> width M ) ;
caseA1: len pD = width M ; ::_thesis: ( Indices M = Indices (RLine (M,l,pD)) & len M = len (RLine (M,l,pD)) & width M = width (RLine (M,l,pD)) )
then A2: width M = width (RLine (M,l,pD)) by Def3;
len M = len (ReplaceLine (M,l,pD)) by A1, Def3;
hence ( Indices M = Indices (RLine (M,l,pD)) & len M = len (RLine (M,l,pD)) & width M = width (RLine (M,l,pD)) ) by A2, MATRIX_4:55; ::_thesis: verum
end;
case len pD <> width M ; ::_thesis: ( Indices M = Indices (RLine (M,l,pD)) & len M = len (RLine (M,l,pD)) & width M = width (RLine (M,l,pD)) )
hence ( Indices M = Indices (RLine (M,l,pD)) & len M = len (RLine (M,l,pD)) & width M = width (RLine (M,l,pD)) ) by Def3; ::_thesis: verum
end;
end;
end;
hence ( Indices M = Indices (RLine (M,l,pD)) & len M = len (RLine (M,l,pD)) & width M = width (RLine (M,l,pD)) ) ; ::_thesis: verum
end;
theorem Th28: :: MATRIX11:28
for m, n being Nat
for D being non empty set
for l being Nat
for M being Matrix of n,m,D
for pD being FinSequence of D
for i being Nat st i in Seg n holds
( ( i = l & len pD = width M implies Line ((RLine (M,l,pD)),i) = pD ) & ( i <> l implies Line ((RLine (M,l,pD)),i) = Line (M,i) ) )
proof
let m, n be Nat; ::_thesis: for D being non empty set
for l being Nat
for M being Matrix of n,m,D
for pD being FinSequence of D
for i being Nat st i in Seg n holds
( ( i = l & len pD = width M implies Line ((RLine (M,l,pD)),i) = pD ) & ( i <> l implies Line ((RLine (M,l,pD)),i) = Line (M,i) ) )
let D be non empty set ; ::_thesis: for l being Nat
for M being Matrix of n,m,D
for pD being FinSequence of D
for i being Nat st i in Seg n holds
( ( i = l & len pD = width M implies Line ((RLine (M,l,pD)),i) = pD ) & ( i <> l implies Line ((RLine (M,l,pD)),i) = Line (M,i) ) )
let l be Nat; ::_thesis: for M being Matrix of n,m,D
for pD being FinSequence of D
for i being Nat st i in Seg n holds
( ( i = l & len pD = width M implies Line ((RLine (M,l,pD)),i) = pD ) & ( i <> l implies Line ((RLine (M,l,pD)),i) = Line (M,i) ) )
let M be Matrix of n,m,D; ::_thesis: for pD being FinSequence of D
for i being Nat st i in Seg n holds
( ( i = l & len pD = width M implies Line ((RLine (M,l,pD)),i) = pD ) & ( i <> l implies Line ((RLine (M,l,pD)),i) = Line (M,i) ) )
let pD be FinSequence of D; ::_thesis: for i being Nat st i in Seg n holds
( ( i = l & len pD = width M implies Line ((RLine (M,l,pD)),i) = pD ) & ( i <> l implies Line ((RLine (M,l,pD)),i) = Line (M,i) ) )
let i be Nat; ::_thesis: ( i in Seg n implies ( ( i = l & len pD = width M implies Line ((RLine (M,l,pD)),i) = pD ) & ( i <> l implies Line ((RLine (M,l,pD)),i) = Line (M,i) ) ) )
assume A1: i in Seg n ; ::_thesis: ( ( i = l & len pD = width M implies Line ((RLine (M,l,pD)),i) = pD ) & ( i <> l implies Line ((RLine (M,l,pD)),i) = Line (M,i) ) )
set R = RLine (M,l,pD);
set LR = Line ((RLine (M,l,pD)),i);
thus ( i = l & len pD = width M implies Line ((RLine (M,l,pD)),i) = pD ) ::_thesis: ( i <> l implies Line ((RLine (M,l,pD)),i) = Line (M,i) )
proof
assume that
A2: i = l and
A3: len pD = width M ; ::_thesis: Line ((RLine (M,l,pD)),i) = pD
A4: width (RLine (M,l,pD)) = len pD by A3, Def3;
A5: now__::_thesis:_for_j_being_Nat_st_1_<=_j_&_j_<=_len_pD_holds_
(Line_((RLine_(M,l,pD)),i))_._j_=_pD_._j
let j be Nat; ::_thesis: ( 1 <= j & j <= len pD implies (Line ((RLine (M,l,pD)),i)) . j = pD . j )
assume that
A6: 1 <= j and
A7: j <= len pD ; ::_thesis: (Line ((RLine (M,l,pD)),i)) . j = pD . j
j in NAT by ORDINAL1:def_12;
then A8: j in Seg (width (RLine (M,l,pD))) by A4, A6, A7;
n = len (RLine (M,l,pD)) by MATRIX_1:def_2;
then i in dom (RLine (M,l,pD)) by A1, FINSEQ_1:def_3;
then A9: [i,j] in Indices (RLine (M,l,pD)) by A8, ZFMISC_1:87;
A10: Indices (RLine (M,l,pD)) = Indices M by Lm4;
(Line ((RLine (M,l,pD)),i)) . j = (RLine (M,l,pD)) * (i,j) by A8, MATRIX_1:def_7;
hence (Line ((RLine (M,l,pD)),i)) . j = pD . j by A2, A3, A9, A10, Def3; ::_thesis: verum
end;
len (Line ((RLine (M,l,pD)),i)) = len pD by A4, MATRIX_1:def_7;
hence Line ((RLine (M,l,pD)),i) = pD by A5, FINSEQ_1:14; ::_thesis: verum
end;
set LM = Line (M,i);
A11: width M = len (Line (M,i)) by MATRIX_1:def_7;
A12: width M = width (RLine (M,l,pD)) by Lm4;
assume A13: i <> l ; ::_thesis: Line ((RLine (M,l,pD)),i) = Line (M,i)
A14: now__::_thesis:_for_j_being_Nat_st_1_<=_j_&_j_<=_len_(Line_(M,i))_holds_
(Line_(M,i))_._j_=_(Line_((RLine_(M,l,pD)),i))_._j
let j be Nat; ::_thesis: ( 1 <= j & j <= len (Line (M,i)) implies (Line (M,i)) . j = (Line ((RLine (M,l,pD)),i)) . j )
assume that
A15: 1 <= j and
A16: j <= len (Line (M,i)) ; ::_thesis: (Line (M,i)) . j = (Line ((RLine (M,l,pD)),i)) . j
j in NAT by ORDINAL1:def_12;
then A17: j in Seg (len (Line (M,i))) by A15, A16;
then A18: (Line (M,i)) . j = M * (i,j) by A11, MATRIX_1:def_7;
i in Seg (len M) by A1, MATRIX_1:def_2;
then i in dom M by FINSEQ_1:def_3;
then A19: [i,j] in Indices M by A11, A17, ZFMISC_1:87;
A20: (Line ((RLine (M,l,pD)),i)) . j = (RLine (M,l,pD)) * (i,j) by A12, A11, A17, MATRIX_1:def_7;
now__::_thesis:_(_(_len_pD_=_width_M_&_(Line_(M,i))_._j_=_(Line_((RLine_(M,l,pD)),i))_._j_)_or_(_len_pD_<>_width_M_&_(Line_(M,i))_._j_=_(Line_((RLine_(M,l,pD)),i))_._j_)_)
percases ( len pD = width M or len pD <> width M ) ;
case len pD = width M ; ::_thesis: (Line (M,i)) . j = (Line ((RLine (M,l,pD)),i)) . j
hence (Line (M,i)) . j = (Line ((RLine (M,l,pD)),i)) . j by A13, A18, A20, A19, Def3; ::_thesis: verum
end;
case len pD <> width M ; ::_thesis: (Line (M,i)) . j = (Line ((RLine (M,l,pD)),i)) . j
hence (Line (M,i)) . j = (Line ((RLine (M,l,pD)),i)) . j by Def3; ::_thesis: verum
end;
end;
end;
hence (Line (M,i)) . j = (Line ((RLine (M,l,pD)),i)) . j ; ::_thesis: verum
end;
len (Line ((RLine (M,l,pD)),i)) = width (RLine (M,l,pD)) by MATRIX_1:def_7;
hence Line ((RLine (M,l,pD)),i) = Line (M,i) by A12, A11, A14, FINSEQ_1:14; ::_thesis: verum
end;
theorem :: MATRIX11:29
for m, n, l being Nat
for D being non empty set
for M being Matrix of n,m,D
for pD being FinSequence of D st len pD = width M holds
for p9 being Element of D * st pD = p9 holds
RLine (M,l,pD) = Replace (M,l,p9)
proof
let m, n, l be Nat; ::_thesis: for D being non empty set
for M being Matrix of n,m,D
for pD being FinSequence of D st len pD = width M holds
for p9 being Element of D * st pD = p9 holds
RLine (M,l,pD) = Replace (M,l,p9)
let D be non empty set ; ::_thesis: for M being Matrix of n,m,D
for pD being FinSequence of D st len pD = width M holds
for p9 being Element of D * st pD = p9 holds
RLine (M,l,pD) = Replace (M,l,p9)
let M be Matrix of n,m,D; ::_thesis: for pD being FinSequence of D st len pD = width M holds
for p9 being Element of D * st pD = p9 holds
RLine (M,l,pD) = Replace (M,l,p9)
let pD be FinSequence of D; ::_thesis: ( len pD = width M implies for p9 being Element of D * st pD = p9 holds
RLine (M,l,pD) = Replace (M,l,p9) )
assume A1: len pD = width M ; ::_thesis: for p9 being Element of D * st pD = p9 holds
RLine (M,l,pD) = Replace (M,l,p9)
set RL = RLine (M,l,pD);
let p9 be Element of D * ; ::_thesis: ( pD = p9 implies RLine (M,l,pD) = Replace (M,l,p9) )
assume A2: pD = p9 ; ::_thesis: RLine (M,l,pD) = Replace (M,l,p9)
set R = Replace (M,l,p9);
A3: len (Replace (M,l,p9)) = len M by FINSEQ_7:5;
A4: now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_len_(Replace_(M,l,p9))_holds_
(Replace_(M,l,p9))_._i_=_(RLine_(M,l,pD))_._i
let i be Nat; ::_thesis: ( 1 <= i & i <= len (Replace (M,l,p9)) implies (Replace (M,l,p9)) . i = (RLine (M,l,pD)) . i )
assume that
A5: 1 <= i and
A6: i <= len (Replace (M,l,p9)) ; ::_thesis: (Replace (M,l,p9)) . i = (RLine (M,l,pD)) . i
i in NAT by ORDINAL1:def_12;
then A7: i in Seg (len (Replace (M,l,p9))) by A5, A6;
then A8: i in dom (Replace (M,l,p9)) by FINSEQ_1:def_3;
A9: i in Seg n by A3, A7, MATRIX_1:def_2;
A10: i in dom M by A3, A7, FINSEQ_1:def_3;
now__::_thesis:_(_(_i_=_l_&_(Replace_(M,l,p9))_._i_=_(RLine_(M,l,pD))_._i_)_or_(_i_<>_l_&_(Replace_(M,l,p9))_._i_=_(RLine_(M,l,pD))_._i_)_)
percases ( i = l or i <> l ) ;
caseA11: i = l ; ::_thesis: (Replace (M,l,p9)) . i = (RLine (M,l,pD)) . i
then A12: Line ((RLine (M,l,pD)),i) = pD by A1, A9, Th28;
A13: (Replace (M,l,p9)) /. i = (Replace (M,l,p9)) . i by A8, PARTFUN1:def_6;
(Replace (M,l,p9)) /. i = p9 by A3, A5, A6, A11, FINSEQ_7:8;
hence (Replace (M,l,p9)) . i = (RLine (M,l,pD)) . i by A2, A9, A13, A12, MATRIX_2:8; ::_thesis: verum
end;
caseA14: i <> l ; ::_thesis: (Replace (M,l,p9)) . i = (RLine (M,l,pD)) . i
then A15: Line (M,i) = Line ((RLine (M,l,pD)),i) by A9, Th28;
A16: (Replace (M,l,p9)) . i = (Replace (M,l,p9)) /. i by A8, PARTFUN1:def_6;
A17: M . i = Line (M,i) by A9, MATRIX_2:8;
A18: M /. i = M . i by A10, PARTFUN1:def_6;
(Replace (M,l,p9)) /. i = M /. i by A3, A5, A6, A14, FINSEQ_7:10;
hence (Replace (M,l,p9)) . i = (RLine (M,l,pD)) . i by A9, A16, A18, A17, A15, MATRIX_2:8; ::_thesis: verum
end;
end;
end;
hence (Replace (M,l,p9)) . i = (RLine (M,l,pD)) . i ; ::_thesis: verum
end;
len M = len (RLine (M,l,pD)) by Lm4;
hence RLine (M,l,pD) = Replace (M,l,p9) by A4, FINSEQ_1:14, FINSEQ_7:5; ::_thesis: verum
end;
theorem Th30: :: MATRIX11:30
for n, m, l being Nat
for D being non empty set
for M being Matrix of n,m,D holds M = RLine (M,l,(Line (M,l)))
proof
let n, m, l be Nat; ::_thesis: for D being non empty set
for M being Matrix of n,m,D holds M = RLine (M,l,(Line (M,l)))
let D be non empty set ; ::_thesis: for M being Matrix of n,m,D holds M = RLine (M,l,(Line (M,l)))
let M be Matrix of n,m,D; ::_thesis: M = RLine (M,l,(Line (M,l)))
set L = Line (M,l);
set RL = RLine (M,l,(Line (M,l)));
A1: width M = len (Line (M,l)) by MATRIX_1:def_7;
A2: now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_len_M_holds_
(RLine_(M,l,(Line_(M,l))))_._i_=_M_._i
let i be Nat; ::_thesis: ( 1 <= i & i <= len M implies (RLine (M,l,(Line (M,l)))) . i = M . i )
assume that
A3: 1 <= i and
A4: i <= len M ; ::_thesis: (RLine (M,l,(Line (M,l)))) . i = M . i
i in NAT by ORDINAL1:def_12;
then A5: i in Seg (len M) by A3, A4;
A6: n = len M by MATRIX_1:def_2;
then A7: (RLine (M,l,(Line (M,l)))) . i = Line ((RLine (M,l,(Line (M,l)))),i) by A5, MATRIX_2:8;
A8: Line (M,i) = M . i by A5, A6, MATRIX_2:8;
now__::_thesis:_(_(_i_=_l_&_(RLine_(M,l,(Line_(M,l))))_._i_=_M_._i_)_or_(_i_<>_l_&_(RLine_(M,l,(Line_(M,l))))_._i_=_M_._i_)_)
percases ( i = l or i <> l ) ;
case i = l ; ::_thesis: (RLine (M,l,(Line (M,l)))) . i = M . i
hence (RLine (M,l,(Line (M,l)))) . i = M . i by A1, A5, A6, A8, A7, Th28; ::_thesis: verum
end;
case i <> l ; ::_thesis: (RLine (M,l,(Line (M,l)))) . i = M . i
hence (RLine (M,l,(Line (M,l)))) . i = M . i by A5, A6, A8, A7, Th28; ::_thesis: verum
end;
end;
end;
hence (RLine (M,l,(Line (M,l)))) . i = M . i ; ::_thesis: verum
end;
len M = len (RLine (M,l,(Line (M,l)))) by Lm4;
hence M = RLine (M,l,(Line (M,l))) by A2, FINSEQ_1:14; ::_thesis: verum
end;
Lm5: for K being Field
for pK being FinSequence of K
for a being Element of K holds len pK = len (a * pK)
proof
let K be Field; ::_thesis: for pK being FinSequence of K
for a being Element of K holds len pK = len (a * pK)
let pK be FinSequence of K; ::_thesis: for a being Element of K holds len pK = len (a * pK)
let a be Element of K; ::_thesis: len pK = len (a * pK)
pK is Element of (len pK) -tuples_on the carrier of K by FINSEQ_2:92;
then a * pK is Element of (len pK) -tuples_on the carrier of K by FINSEQ_2:113;
hence len pK = len (a * pK) by CARD_1:def_7; ::_thesis: verum
end;
Lm6: for K being Field
for pK, qK being FinSequence of K st len pK = len qK holds
len pK = len (pK + qK)
proof
let K be Field; ::_thesis: for pK, qK being FinSequence of K st len pK = len qK holds
len pK = len (pK + qK)
let pK, qK be FinSequence of K; ::_thesis: ( len pK = len qK implies len pK = len (pK + qK) )
assume len pK = len qK ; ::_thesis: len pK = len (pK + qK)
then A1: qK is Element of (len pK) -tuples_on the carrier of K by FINSEQ_2:92;
pK is Element of (len pK) -tuples_on the carrier of K by FINSEQ_2:92;
then pK + qK is Element of (len pK) -tuples_on the carrier of K by A1, FINSEQ_2:120;
hence len pK = len (pK + qK) by CARD_1:def_7; ::_thesis: verum
end;
theorem Th31: :: MATRIX11:31
for n being Nat
for K being Field
for a, b being Element of K
for l being Nat
for pK, qK being FinSequence of K
for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds
for M being Matrix of n,K holds the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK))))))
proof
let n be Nat; ::_thesis: for K being Field
for a, b being Element of K
for l being Nat
for pK, qK being FinSequence of K
for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds
for M being Matrix of n,K holds the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK))))))
let K be Field; ::_thesis: for a, b being Element of K
for l being Nat
for pK, qK being FinSequence of K
for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds
for M being Matrix of n,K holds the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK))))))
let a, b be Element of K; ::_thesis: for l being Nat
for pK, qK being FinSequence of K
for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds
for M being Matrix of n,K holds the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK))))))
let l be Nat; ::_thesis: for pK, qK being FinSequence of K
for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds
for M being Matrix of n,K holds the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK))))))
let pK, qK be FinSequence of K; ::_thesis: for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds
for M being Matrix of n,K holds the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK))))))
let perm be Element of Permutations n; ::_thesis: ( l in Seg n & len pK = n & len qK = n implies for M being Matrix of n,K holds the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK)))))) )
assume that
A1: l in Seg n and
A2: len pK = n and
A3: len qK = n ; ::_thesis: for M being Matrix of n,K holds the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK))))))
Seg n <> {} by A1;
then A4: n <> 0 ;
reconsider L = l as Element of NAT by ORDINAL1:def_12;
set mm = the multF of K;
let M be Matrix of n,K; ::_thesis: the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK))))))
set Rpq = RLine (M,l,((a * pK) + (b * qK)));
set Ppq = Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))));
A5: len (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = n by MATRIX_3:def_7;
then consider fpq being Function of NAT, the carrier of K such that
A6: fpq . 1 = (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . 1 and
A7: for k being Element of NAT st 0 <> k & k < len (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) holds
fpq . (k + 1) = the multF of K . ((fpq . k),((Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . (k + 1))) and
A8: the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = fpq . (len (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK))))))) by A4, FINSOP_1:def_1;
set Rq = RLine (M,l,qK);
set Pq = Path_matrix (perm,(RLine (M,l,qK)));
A9: len (Path_matrix (perm,(RLine (M,l,qK)))) = n by MATRIX_3:def_7;
then consider fq being Function of NAT, the carrier of K such that
A10: fq . 1 = (Path_matrix (perm,(RLine (M,l,qK)))) . 1 and
A11: for k being Element of NAT st 0 <> k & k < len (Path_matrix (perm,(RLine (M,l,qK)))) holds
fq . (k + 1) = the multF of K . ((fq . k),((Path_matrix (perm,(RLine (M,l,qK)))) . (k + 1))) and
A12: the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK)))) = fq . (len (Path_matrix (perm,(RLine (M,l,qK))))) by A4, FINSOP_1:def_1;
set Rp = RLine (M,l,pK);
set Pp = Path_matrix (perm,(RLine (M,l,pK)));
A13: len (Path_matrix (perm,(RLine (M,l,pK)))) = n by MATRIX_3:def_7;
then consider fp being Function of NAT, the carrier of K such that
A14: fp . 1 = (Path_matrix (perm,(RLine (M,l,pK)))) . 1 and
A15: for k being Element of NAT st 0 <> k & k < len (Path_matrix (perm,(RLine (M,l,pK)))) holds
fp . (k + 1) = the multF of K . ((fp . k),((Path_matrix (perm,(RLine (M,l,pK)))) . (k + 1))) and
A16: the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))) = fp . (len (Path_matrix (perm,(RLine (M,l,pK))))) by A4, FINSOP_1:def_1;
A17: n >= 1 by A4, NAT_1:14;
defpred S1[ Nat] means ( 1 <= $1 & $1 < L implies ( fp . $1 = fq . $1 & fpq . $1 = fp . $1 ) );
A18: for k being Element of NAT st k in Seg n holds
( ( k <> L implies ( (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (Path_matrix (perm,(RLine (M,l,pK)))) . k & (Path_matrix (perm,(RLine (M,l,pK)))) . k = (Path_matrix (perm,(RLine (M,l,qK)))) . k ) ) & ( k = L implies ex Ppk, Pqk being Element of K st
( Ppk = (Path_matrix (perm,(RLine (M,l,pK)))) . k & Pqk = (Path_matrix (perm,(RLine (M,l,qK)))) . k & (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (a * Ppk) + (b * Pqk) ) ) )
proof
let k be Element of NAT ; ::_thesis: ( k in Seg n implies ( ( k <> L implies ( (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (Path_matrix (perm,(RLine (M,l,pK)))) . k & (Path_matrix (perm,(RLine (M,l,pK)))) . k = (Path_matrix (perm,(RLine (M,l,qK)))) . k ) ) & ( k = L implies ex Ppk, Pqk being Element of K st
( Ppk = (Path_matrix (perm,(RLine (M,l,pK)))) . k & Pqk = (Path_matrix (perm,(RLine (M,l,qK)))) . k & (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (a * Ppk) + (b * Pqk) ) ) ) )
assume A19: k in Seg n ; ::_thesis: ( ( k <> L implies ( (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (Path_matrix (perm,(RLine (M,l,pK)))) . k & (Path_matrix (perm,(RLine (M,l,pK)))) . k = (Path_matrix (perm,(RLine (M,l,qK)))) . k ) ) & ( k = L implies ex Ppk, Pqk being Element of K st
( Ppk = (Path_matrix (perm,(RLine (M,l,pK)))) . k & Pqk = (Path_matrix (perm,(RLine (M,l,qK)))) . k & (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (a * Ppk) + (b * Pqk) ) ) )
A20: perm . k in Seg n by A5, A19, MATRIX_7:14;
then reconsider pk = perm . k as Element of NAT ;
A21: k in dom (Path_matrix (perm,(RLine (M,l,pK)))) by A13, A19, FINSEQ_1:def_3;
A22: k in dom (Path_matrix (perm,(RLine (M,l,qK)))) by A9, A19, FINSEQ_1:def_3;
[k,pk] in [:(Seg n),(Seg n):] by A19, A20, ZFMISC_1:87;
then A23: [k,pk] in Indices M by MATRIX_1:24;
dom qK = Seg n by A3, FINSEQ_1:def_3;
then A24: qK /. pk = qK . pk by A5, A19, MATRIX_7:14, PARTFUN1:def_6;
dom pK = Seg n by A2, FINSEQ_1:def_3;
then pK /. pk = pK . pk by A5, A19, MATRIX_7:14, PARTFUN1:def_6;
then reconsider ppk = pK . pk, qpk = qK . pk as Element of K by A24;
A25: len (b * qK) = n by A3, Lm5;
then dom (b * qK) = Seg n by FINSEQ_1:def_3;
then A26: (b * qK) . pk = b * qpk by A5, A19, FVSUM_1:50, MATRIX_7:14;
A27: len (a * pK) = n by A2, Lm5;
then A28: len ((a * pK) + (b * qK)) = n by A25, Lm6;
then A29: dom ((a * pK) + (b * qK)) = Seg n by FINSEQ_1:def_3;
dom (a * pK) = Seg n by A27, FINSEQ_1:def_3;
then A30: (a * pK) . pk = a * ppk by A5, A19, FVSUM_1:50, MATRIX_7:14;
A31: width M = n by MATRIX_1:24;
A32: k in dom (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) by A5, A19, FINSEQ_1:def_3;
thus ( k <> L implies ( (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (Path_matrix (perm,(RLine (M,l,pK)))) . k & (Path_matrix (perm,(RLine (M,l,pK)))) . k = (Path_matrix (perm,(RLine (M,l,qK)))) . k ) ) ::_thesis: ( k = L implies ex Ppk, Pqk being Element of K st
( Ppk = (Path_matrix (perm,(RLine (M,l,pK)))) . k & Pqk = (Path_matrix (perm,(RLine (M,l,qK)))) . k & (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (a * Ppk) + (b * Pqk) ) )
proof
assume A33: k <> L ; ::_thesis: ( (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (Path_matrix (perm,(RLine (M,l,pK)))) . k & (Path_matrix (perm,(RLine (M,l,pK)))) . k = (Path_matrix (perm,(RLine (M,l,qK)))) . k )
then A34: (RLine (M,l,qK)) * (k,pk) = M * (k,pk) by A3, A23, A31, Def3;
(RLine (M,l,pK)) * (k,pk) = M * (k,pk) by A2, A23, A31, A33, Def3;
then A35: (Path_matrix (perm,(RLine (M,l,pK)))) . k = M * (k,pk) by A21, MATRIX_3:def_7;
(RLine (M,l,((a * pK) + (b * qK)))) * (k,pk) = M * (k,pk) by A28, A23, A31, A33, Def3;
hence ( (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (Path_matrix (perm,(RLine (M,l,pK)))) . k & (Path_matrix (perm,(RLine (M,l,pK)))) . k = (Path_matrix (perm,(RLine (M,l,qK)))) . k ) by A32, A22, A34, A35, MATRIX_3:def_7; ::_thesis: verum
end;
assume A36: k = L ; ::_thesis: ex Ppk, Pqk being Element of K st
( Ppk = (Path_matrix (perm,(RLine (M,l,pK)))) . k & Pqk = (Path_matrix (perm,(RLine (M,l,qK)))) . k & (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (a * Ppk) + (b * Pqk) )
then A37: (RLine (M,l,pK)) * (k,pk) = pK . pk by A2, A23, A31, Def3;
A38: (RLine (M,l,qK)) * (k,pk) = qK . pk by A3, A23, A31, A36, Def3;
take ppk ; ::_thesis: ex Pqk being Element of K st
( ppk = (Path_matrix (perm,(RLine (M,l,pK)))) . k & Pqk = (Path_matrix (perm,(RLine (M,l,qK)))) . k & (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (a * ppk) + (b * Pqk) )
take qpk ; ::_thesis: ( ppk = (Path_matrix (perm,(RLine (M,l,pK)))) . k & qpk = (Path_matrix (perm,(RLine (M,l,qK)))) . k & (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (a * ppk) + (b * qpk) )
(RLine (M,l,((a * pK) + (b * qK)))) * (k,pk) = ((a * pK) + (b * qK)) . pk by A28, A23, A31, A36, Def3;
then (RLine (M,l,((a * pK) + (b * qK)))) * (k,pk) = (a * ppk) + (b * qpk) by A5, A19, A29, A30, A26, FVSUM_1:17, MATRIX_7:14;
hence ( ppk = (Path_matrix (perm,(RLine (M,l,pK)))) . k & qpk = (Path_matrix (perm,(RLine (M,l,qK)))) . k & (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (a * ppk) + (b * qpk) ) by A32, A21, A22, A37, A38, MATRIX_3:def_7; ::_thesis: verum
end;
A39: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A40: S1[k] ; ::_thesis: S1[k + 1]
set k1 = k + 1;
assume that
A41: 1 <= k + 1 and
A42: k + 1 < L ; ::_thesis: ( fp . (k + 1) = fq . (k + 1) & fpq . (k + 1) = fp . (k + 1) )
L <= n by A1, FINSEQ_1:1;
then A43: k + 1 <= n by A42, XXREAL_0:2;
then A44: k < n by NAT_1:13;
A45: k + 1 in Seg n by A41, A43;
now__::_thesis:_(_(_k_=_0_&_fp_._(k_+_1)_=_fq_._(k_+_1)_&_fpq_._(k_+_1)_=_fp_._(k_+_1)_)_or_(_k_>_0_&_fp_._(k_+_1)_=_fq_._(k_+_1)_&_fpq_._(k_+_1)_=_fp_._(k_+_1)_)_)
percases ( k = 0 or k > 0 ) ;
case k = 0 ; ::_thesis: ( fp . (k + 1) = fq . (k + 1) & fpq . (k + 1) = fp . (k + 1) )
hence ( fp . (k + 1) = fq . (k + 1) & fpq . (k + 1) = fp . (k + 1) ) by A6, A14, A10, A18, A42, A45; ::_thesis: verum
end;
caseA46: k > 0 ; ::_thesis: ( fp . (k + 1) = fq . (k + 1) & fpq . (k + 1) = fp . (k + 1) )
then A47: fp . (k + 1) = the multF of K . ((fp . k),((Path_matrix (perm,(RLine (M,l,pK)))) . (k + 1))) by A13, A15, A44;
A48: 0 < k + 0 by A46;
A49: fq . (k + 1) = the multF of K . ((fq . k),((Path_matrix (perm,(RLine (M,l,qK)))) . (k + 1))) by A9, A11, A44, A46;
fpq . (k + 1) = the multF of K . ((fpq . k),((Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . (k + 1))) by A5, A7, A44, A46;
hence ( fp . (k + 1) = fq . (k + 1) & fpq . (k + 1) = fp . (k + 1) ) by A18, A40, A42, A45, A48, A47, A49, NAT_1:13, NAT_1:19; ::_thesis: verum
end;
end;
end;
hence ( fp . (k + 1) = fq . (k + 1) & fpq . (k + 1) = fp . (k + 1) ) ; ::_thesis: verum
end;
defpred S2[ Nat] means ( 1 <= $1 & L <= $1 & $1 <= n implies for k being Element of NAT st $1 = k holds
fpq . k = (a * (fp . k)) + (b * (fq . k)) );
A50: S1[ 0 ] ;
A51: fpq . L = (a * (fp . L)) + (b * (fq . L))
proof
consider PpL, PqL being Element of K such that
A52: PpL = (Path_matrix (perm,(RLine (M,l,pK)))) . L and
A53: PqL = (Path_matrix (perm,(RLine (M,l,qK)))) . L and
A54: (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . L = (a * PpL) + (b * PqL) by A1, A18;
A55: L >= 1 by A1, FINSEQ_1:1;
now__::_thesis:_(_(_L_>_1_&_fpq_._L_=_(a_*_(fp_._L))_+_(b_*_(fq_._L))_)_or_(_L_=_1_&_fpq_._L_=_(a_*_(fp_._L))_+_(b_*_(fq_._L))_)_)
percases ( L > 1 or L = 1 ) by A55, XXREAL_0:1;
caseA56: L > 1 ; ::_thesis: fpq . L = (a * (fp . L)) + (b * (fq . L))
then reconsider L1 = L - 1 as Element of NAT by NAT_1:20;
A57: L1 + 1 > 1 + 0 by A56;
A58: L1 < L1 + 1 by NAT_1:19;
L <= n by A1, FINSEQ_1:1;
then A59: L1 < n by A58, XXREAL_0:2;
then fp . L = (fp . L1) * PpL by A13, A15, A52, A57;
then A60: (fp . L1) * (a * PpL) = a * (fp . L) by GROUP_1:def_3;
A61: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A50, A39);
A62: 1 <= L1 by A57, NAT_1:19;
then fp . L1 = fq . L1 by A61, A58;
then fq . L = (fp . L1) * PqL by A9, A11, A53, A57, A59;
then A63: (fp . L1) * (b * PqL) = b * (fq . L) by GROUP_1:def_3;
fpq . L1 = fp . L1 by A61, A58, A62;
then fpq . L = (fp . L1) * ((a * PpL) + (b * PqL)) by A5, A7, A54, A57, A59;
hence fpq . L = (a * (fp . L)) + (b * (fq . L)) by A60, A63, VECTSP_1:def_7; ::_thesis: verum
end;
case L = 1 ; ::_thesis: fpq . L = (a * (fp . L)) + (b * (fq . L))
hence fpq . L = (a * (fp . L)) + (b * (fq . L)) by A6, A14, A10, A52, A53, A54; ::_thesis: verum
end;
end;
end;
hence fpq . L = (a * (fp . L)) + (b * (fq . L)) ; ::_thesis: verum
end;
A64: for k being Element of NAT st S2[k] holds
S2[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S2[k] implies S2[k + 1] )
assume A65: S2[k] ; ::_thesis: S2[k + 1]
set k1 = k + 1;
assume that
A66: 1 <= k + 1 and
A67: L <= k + 1 and
A68: k + 1 <= n ; ::_thesis: for k being Element of NAT st k + 1 = k holds
fpq . k = (a * (fp . k)) + (b * (fq . k))
let k9 be Element of NAT ; ::_thesis: ( k + 1 = k9 implies fpq . k9 = (a * (fp . k9)) + (b * (fq . k9)) )
assume A69: k9 = k + 1 ; ::_thesis: fpq . k9 = (a * (fp . k9)) + (b * (fq . k9))
now__::_thesis:_(_(_k_+_1_=_L_&_fpq_._k9_=_(a_*_(fp_._k9))_+_(b_*_(fq_._k9))_)_or_(_k_+_1_>_L_&_fpq_._k9_=_(a_*_(fp_._k9))_+_(b_*_(fq_._k9))_)_)
percases ( k + 1 = L or k + 1 > L ) by A67, XXREAL_0:1;
case k + 1 = L ; ::_thesis: fpq . k9 = (a * (fp . k9)) + (b * (fq . k9))
hence fpq . k9 = (a * (fp . k9)) + (b * (fq . k9)) by A51, A69; ::_thesis: verum
end;
caseA70: k + 1 > L ; ::_thesis: fpq . k9 = (a * (fp . k9)) + (b * (fq . k9))
A71: k + 1 in Seg n by A66, A68;
then A72: (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . (k + 1) = (Path_matrix (perm,(RLine (M,l,pK)))) . (k + 1) by A18, A70;
k + 1 in dom (Path_matrix (perm,(RLine (M,l,pK)))) by A13, A71, FINSEQ_1:def_3;
then (Path_matrix (perm,(RLine (M,l,pK)))) /. (k + 1) = (Path_matrix (perm,(RLine (M,l,pK)))) . (k + 1) by PARTFUN1:def_6;
then reconsider Ppk1 = (Path_matrix (perm,(RLine (M,l,pK)))) . (k + 1) as Element of K ;
A73: k < n by A68, NAT_1:13;
A74: (b * (fq . k)) * Ppk1 = b * ((fq . k) * Ppk1) by GROUP_1:def_3;
A75: (a * (fp . k)) * Ppk1 = a * ((fp . k) * Ppk1) by GROUP_1:def_3;
A76: 1 <= L by A1, FINSEQ_1:1;
A77: k >= L by A70, NAT_1:13;
then fpq . k = (a * (fp . k)) + (b * (fq . k)) by A65, A68, A76, NAT_1:13, XXREAL_0:2;
then A78: fpq . (k + 1) = ((a * (fp . k)) + (b * (fq . k))) * Ppk1 by A5, A7, A77, A73, A76, A72;
(Path_matrix (perm,(RLine (M,l,pK)))) . (k + 1) = (Path_matrix (perm,(RLine (M,l,qK)))) . (k + 1) by A18, A70, A71;
then A79: fq . (k + 1) = (fq . k) * Ppk1 by A9, A11, A77, A73, A76;
fp . (k + 1) = (fp . k) * Ppk1 by A13, A15, A77, A73, A76;
hence fpq . k9 = (a * (fp . k9)) + (b * (fq . k9)) by A69, A78, A79, A75, A74, VECTSP_1:def_7; ::_thesis: verum
end;
end;
end;
hence fpq . k9 = (a * (fp . k9)) + (b * (fq . k9)) ; ::_thesis: verum
end;
A80: L <= n by A1, FINSEQ_1:1;
A81: S2[ 0 ] ;
for k being Element of NAT holds S2[k] from NAT_1:sch_1(A81, A64);
hence the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK)))))) by A17, A5, A13, A9, A8, A16, A12, A80; ::_thesis: verum
end;
theorem Th32: :: MATRIX11:32
for n being Nat
for K being Field
for a, b being Element of K
for l being Nat
for pK, qK being FinSequence of K
for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds
for M being Matrix of n,K holds (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm))
proof
let n be Nat; ::_thesis: for K being Field
for a, b being Element of K
for l being Nat
for pK, qK being FinSequence of K
for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds
for M being Matrix of n,K holds (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm))
let K be Field; ::_thesis: for a, b being Element of K
for l being Nat
for pK, qK being FinSequence of K
for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds
for M being Matrix of n,K holds (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm))
let a, b be Element of K; ::_thesis: for l being Nat
for pK, qK being FinSequence of K
for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds
for M being Matrix of n,K holds (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm))
let l be Nat; ::_thesis: for pK, qK being FinSequence of K
for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds
for M being Matrix of n,K holds (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm))
let pK, qK be FinSequence of K; ::_thesis: for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds
for M being Matrix of n,K holds (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm))
let perm be Element of Permutations n; ::_thesis: ( l in Seg n & len pK = n & len qK = n implies for M being Matrix of n,K holds (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm)) )
assume that
A1: l in Seg n and
A2: len pK = n and
A3: len qK = n ; ::_thesis: for M being Matrix of n,K holds (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm))
set mm = the multF of K;
let M be Matrix of n,K; ::_thesis: (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm))
set Rpq = RLine (M,l,((a * pK) + (b * qK)));
set Rp = RLine (M,l,pK);
set Rq = RLine (M,l,qK);
set Ppq = Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))));
set Pathpq = Path_product (RLine (M,l,((a * pK) + (b * qK))));
set Pp = Path_matrix (perm,(RLine (M,l,pK)));
set Pathp = Path_product (RLine (M,l,pK));
set Pq = Path_matrix (perm,(RLine (M,l,qK)));
set Pathq = Path_product (RLine (M,l,qK));
now__::_thesis:_(_(_perm_is_even_&_(Path_product_(RLine_(M,l,((a_*_pK)_+_(b_*_qK)))))_._perm_=_(a_*_((Path_product_(RLine_(M,l,pK)))_._perm))_+_(b_*_((Path_product_(RLine_(M,l,qK)))_._perm))_)_or_(_perm_is_odd_&_(Path_product_(RLine_(M,l,((a_*_pK)_+_(b_*_qK)))))_._perm_=_(a_*_((Path_product_(RLine_(M,l,pK)))_._perm))_+_(b_*_((Path_product_(RLine_(M,l,qK)))_._perm))_)_)
percases ( perm is even or perm is odd ) ;
caseA4: perm is even ; ::_thesis: (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm))
then the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = - (( the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK))))))),perm) by MATRIX_2:def_13;
then A5: (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) by MATRIX_3:def_8;
the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK)))) = - (( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK))))),perm) by A4, MATRIX_2:def_13;
then A6: (Path_product (RLine (M,l,qK))) . perm = the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK)))) by MATRIX_3:def_8;
the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))) = - (( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK))))),perm) by A4, MATRIX_2:def_13;
then (Path_product (RLine (M,l,pK))) . perm = the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))) by MATRIX_3:def_8;
hence (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm)) by A1, A2, A3, A6, A5, Th31; ::_thesis: verum
end;
caseA7: perm is odd ; ::_thesis: (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm))
then - ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK))))))) = - (( the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK))))))),perm) by MATRIX_2:def_13;
then A8: (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = - ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK))))))) by MATRIX_3:def_8;
- ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK))))) = - (( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK))))),perm) by A7, MATRIX_2:def_13;
then A9: (Path_product (RLine (M,l,pK))) . perm = - ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK))))) by MATRIX_3:def_8;
A10: - (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) = a * (- ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) by VECTSP_1:8;
- ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK))))) = - (( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK))))),perm) by A7, MATRIX_2:def_13;
then A11: (Path_product (RLine (M,l,qK))) . perm = - ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK))))) by MATRIX_3:def_8;
A12: - ((a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK))))))) = (- (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK))))))) - (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK)))))) by VECTSP_1:17;
the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK)))))) by A1, A2, A3, Th31;
hence (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm)) by A9, A11, A8, A10, A12, VECTSP_1:8; ::_thesis: verum
end;
end;
end;
hence (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm)) ; ::_thesis: verum
end;
theorem Th33: :: MATRIX11:33
for n being Nat
for K being Field
for a, b being Element of K
for l being Nat
for pK, qK being FinSequence of K st l in Seg n & len pK = n & len qK = n holds
for M being Matrix of n,K holds Det (RLine (M,l,((a * pK) + (b * qK)))) = (a * (Det (RLine (M,l,pK)))) + (b * (Det (RLine (M,l,qK))))
proof
let n be Nat; ::_thesis: for K being Field
for a, b being Element of K
for l being Nat
for pK, qK being FinSequence of K st l in Seg n & len pK = n & len qK = n holds
for M being Matrix of n,K holds Det (RLine (M,l,((a * pK) + (b * qK)))) = (a * (Det (RLine (M,l,pK)))) + (b * (Det (RLine (M,l,qK))))
let K be Field; ::_thesis: for a, b being Element of K
for l being Nat
for pK, qK being FinSequence of K st l in Seg n & len pK = n & len qK = n holds
for M being Matrix of n,K holds Det (RLine (M,l,((a * pK) + (b * qK)))) = (a * (Det (RLine (M,l,pK)))) + (b * (Det (RLine (M,l,qK))))
let a, b be Element of K; ::_thesis: for l being Nat
for pK, qK being FinSequence of K st l in Seg n & len pK = n & len qK = n holds
for M being Matrix of n,K holds Det (RLine (M,l,((a * pK) + (b * qK)))) = (a * (Det (RLine (M,l,pK)))) + (b * (Det (RLine (M,l,qK))))
let l be Nat; ::_thesis: for pK, qK being FinSequence of K st l in Seg n & len pK = n & len qK = n holds
for M being Matrix of n,K holds Det (RLine (M,l,((a * pK) + (b * qK)))) = (a * (Det (RLine (M,l,pK)))) + (b * (Det (RLine (M,l,qK))))
let pK, qK be FinSequence of K; ::_thesis: ( l in Seg n & len pK = n & len qK = n implies for M being Matrix of n,K holds Det (RLine (M,l,((a * pK) + (b * qK)))) = (a * (Det (RLine (M,l,pK)))) + (b * (Det (RLine (M,l,qK)))) )
assume that
A1: l in Seg n and
A2: len pK = n and
A3: len qK = n ; ::_thesis: for M being Matrix of n,K holds Det (RLine (M,l,((a * pK) + (b * qK)))) = (a * (Det (RLine (M,l,pK)))) + (b * (Det (RLine (M,l,qK))))
set P = Permutations n;
set KK = the carrier of K;
set aa = the addF of K;
let M be Matrix of n,K; ::_thesis: Det (RLine (M,l,((a * pK) + (b * qK)))) = (a * (Det (RLine (M,l,pK)))) + (b * (Det (RLine (M,l,qK))))
set Rpq = RLine (M,l,((a * pK) + (b * qK)));
set Rp = RLine (M,l,pK);
set Rq = RLine (M,l,qK);
set Pathpq = Path_product (RLine (M,l,((a * pK) + (b * qK))));
set Pathp = Path_product (RLine (M,l,pK));
set Pathq = Path_product (RLine (M,l,qK));
set F = FinOmega (Permutations n);
A4: FinOmega (Permutations n) = Permutations n by MATRIX_2:26, MATRIX_2:def_14;
then consider Gpq being Function of (Fin (Permutations n)), the carrier of K such that
A5: Det (RLine (M,l,((a * pK) + (b * qK)))) = Gpq . (FinOmega (Permutations n)) and
A6: for e being Element of the carrier of K st e is_a_unity_wrt the addF of K holds
Gpq . {} = e and
A7: for x being Element of Permutations n holds Gpq . {x} = (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . x and
A8: for B9 being Element of Fin (Permutations n) st B9 c= FinOmega (Permutations n) & B9 <> {} holds
for x being Element of Permutations n st x in (FinOmega (Permutations n)) \ B9 holds
Gpq . (B9 \/ {x}) = the addF of K . ((Gpq . B9),((Path_product (RLine (M,l,((a * pK) + (b * qK))))) . x)) by SETWISEO:def_3;
consider Gq being Function of (Fin (Permutations n)), the carrier of K such that
A9: Det (RLine (M,l,qK)) = Gq . (FinOmega (Permutations n)) and
A10: for e being Element of the carrier of K st e is_a_unity_wrt the addF of K holds
Gq . {} = e and
A11: for x being Element of Permutations n holds Gq . {x} = (Path_product (RLine (M,l,qK))) . x and
A12: for B9 being Element of Fin (Permutations n) st B9 c= FinOmega (Permutations n) & B9 <> {} holds
for x being Element of Permutations n st x in (FinOmega (Permutations n)) \ B9 holds
Gq . (B9 \/ {x}) = the addF of K . ((Gq . B9),((Path_product (RLine (M,l,qK))) . x)) by A4, SETWISEO:def_3;
consider Gp being Function of (Fin (Permutations n)), the carrier of K such that
A13: Det (RLine (M,l,pK)) = Gp . (FinOmega (Permutations n)) and
A14: for e being Element of the carrier of K st e is_a_unity_wrt the addF of K holds
Gp . {} = e and
A15: for x being Element of Permutations n holds Gp . {x} = (Path_product (RLine (M,l,pK))) . x and
A16: for B9 being Element of Fin (Permutations n) st B9 c= FinOmega (Permutations n) & B9 <> {} holds
for x being Element of Permutations n st x in (FinOmega (Permutations n)) \ B9 holds
Gp . (B9 \/ {x}) = the addF of K . ((Gp . B9),((Path_product (RLine (M,l,pK))) . x)) by A4, SETWISEO:def_3;
defpred S1[ Nat] means for B being Element of Fin (Permutations n) st card B = $1 holds
Gpq . B = (a * (Gp . B)) + (b * (Gq . B));
A17: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A18: S1[k] ; ::_thesis: S1[k + 1]
let B be Element of Fin (Permutations n); ::_thesis: ( card B = k + 1 implies Gpq . B = (a * (Gp . B)) + (b * (Gq . B)) )
assume A19: card B = k + 1 ; ::_thesis: Gpq . B = (a * (Gp . B)) + (b * (Gq . B))
now__::_thesis:_(_(_k_=_0_&_Gpq_._B_=_(a_*_(Gp_._B))_+_(b_*_(Gq_._B))_)_or_(_k_>_0_&_Gpq_._B_=_(a_*_(Gp_._B))_+_(b_*_(Gq_._B))_)_)
percases ( k = 0 or k > 0 ) ;
case k = 0 ; ::_thesis: Gpq . B = (a * (Gp . B)) + (b * (Gq . B))
then consider x being set such that
A20: B = {x} by A19, CARD_2:42;
A21: x in B by A20, TARSKI:def_1;
B c= Permutations n by FINSUB_1:def_5;
then reconsider x = x as Element of Permutations n by A21;
A22: Gp . B = (Path_product (RLine (M,l,pK))) . x by A15, A20;
A23: Gq . B = (Path_product (RLine (M,l,qK))) . x by A11, A20;
Gpq . B = (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . x by A7, A20;
hence Gpq . B = (a * (Gp . B)) + (b * (Gq . B)) by A1, A2, A3, A22, A23, Th32; ::_thesis: verum
end;
caseA24: k > 0 ; ::_thesis: Gpq . B = (a * (Gp . B)) + (b * (Gq . B))
consider x being set such that
A25: x in B by A19, CARD_1:27, XBOOLE_0:def_1;
B c= Permutations n by FINSUB_1:def_5;
then reconsider x = x as Element of Permutations n by A25;
B c= Permutations n by FINSUB_1:def_5;
then B \ {x} c= Permutations n by XBOOLE_1:1;
then reconsider B9 = B \ {x} as Element of Fin (Permutations n) by FINSUB_1:def_5;
A26: not x in B9 by ZFMISC_1:56;
then A27: x in (FinOmega (Permutations n)) \ B9 by A4, XBOOLE_0:def_5;
A28: {x} \/ B9 = B by A25, ZFMISC_1:116;
then A29: k + 1 = (card B9) + 1 by A19, A26, CARD_2:41;
then A30: Gpq . B9 = (a * (Gp . B9)) + (b * (Gq . B9)) by A18;
A31: B9 c= FinOmega (Permutations n) by A4, FINSUB_1:def_5;
then Gpq . B = the addF of K . ((Gpq . B9),((Path_product (RLine (M,l,((a * pK) + (b * qK))))) . x)) by A8, A24, A28, A29, A27, CARD_1:27;
then A32: Gpq . B = ((a * (Gp . B9)) + (b * (Gq . B9))) + ((a * ((Path_product (RLine (M,l,pK))) . x)) + (b * ((Path_product (RLine (M,l,qK))) . x))) by A1, A2, A3, A30, Th32
.= (a * (Gp . B9)) + ((b * (Gq . B9)) + ((a * ((Path_product (RLine (M,l,pK))) . x)) + (b * ((Path_product (RLine (M,l,qK))) . x)))) by RLVECT_1:def_3
.= (a * (Gp . B9)) + ((a * ((Path_product (RLine (M,l,pK))) . x)) + ((b * (Gq . B9)) + (b * ((Path_product (RLine (M,l,qK))) . x)))) by RLVECT_1:def_3
.= ((a * (Gp . B9)) + (a * ((Path_product (RLine (M,l,pK))) . x))) + ((b * (Gq . B9)) + (b * ((Path_product (RLine (M,l,qK))) . x))) by RLVECT_1:def_3
.= (a * ((Gp . B9) + ((Path_product (RLine (M,l,pK))) . x))) + ((b * (Gq . B9)) + (b * ((Path_product (RLine (M,l,qK))) . x))) by VECTSP_1:def_7
.= (a * ( the addF of K . ((Gp . B9),((Path_product (RLine (M,l,pK))) . x)))) + (b * ((Gq . B9) + ((Path_product (RLine (M,l,qK))) . x))) by VECTSP_1:def_7
.= (a * ( the addF of K . ((Gp . B9),((Path_product (RLine (M,l,pK))) . x)))) + (b * ( the addF of K . ((Gq . B9),((Path_product (RLine (M,l,qK))) . x)))) ;
Gp . B = the addF of K . ((Gp . B9),((Path_product (RLine (M,l,pK))) . x)) by A16, A24, A28, A29, A27, A31, CARD_1:27;
hence Gpq . B = (a * (Gp . B)) + (b * (Gq . B)) by A12, A24, A28, A29, A27, A31, A32, CARD_1:27; ::_thesis: verum
end;
end;
end;
hence Gpq . B = (a * (Gp . B)) + (b * (Gq . B)) ; ::_thesis: verum
end;
A33: S1[ 0 ]
proof
A34: b * (0. K) = 0. K by VECTSP_1:6;
A35: a * (0. K) = 0. K by VECTSP_1:6;
let B be Element of Fin (Permutations n); ::_thesis: ( card B = 0 implies Gpq . B = (a * (Gp . B)) + (b * (Gq . B)) )
assume card B = 0 ; ::_thesis: Gpq . B = (a * (Gp . B)) + (b * (Gq . B))
then A36: B = {} ;
then A37: Gp . B = 0. K by A14, FVSUM_1:6;
A38: Gq . B = 0. K by A10, A36, FVSUM_1:6;
Gpq . B = 0. K by A6, A36, FVSUM_1:6;
hence Gpq . B = (a * (Gp . B)) + (b * (Gq . B)) by A37, A38, A35, A34, RLVECT_1:4; ::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A33, A17);
then S1[ card (FinOmega (Permutations n))] ;
hence Det (RLine (M,l,((a * pK) + (b * qK)))) = (a * (Det (RLine (M,l,pK)))) + (b * (Det (RLine (M,l,qK)))) by A5, A13, A9; ::_thesis: verum
end;
theorem Th34: :: MATRIX11:34
for l, n being Nat
for K being Field
for a being Element of K
for pK being FinSequence of K
for A being Matrix of n,K st l in Seg n & len pK = n holds
Det (RLine (A,l,(a * pK))) = a * (Det (RLine (A,l,pK)))
proof
let l, n be Nat; ::_thesis: for K being Field
for a being Element of K
for pK being FinSequence of K
for A being Matrix of n,K st l in Seg n & len pK = n holds
Det (RLine (A,l,(a * pK))) = a * (Det (RLine (A,l,pK)))
let K be Field; ::_thesis: for a being Element of K
for pK being FinSequence of K
for A being Matrix of n,K st l in Seg n & len pK = n holds
Det (RLine (A,l,(a * pK))) = a * (Det (RLine (A,l,pK)))
let a be Element of K; ::_thesis: for pK being FinSequence of K
for A being Matrix of n,K st l in Seg n & len pK = n holds
Det (RLine (A,l,(a * pK))) = a * (Det (RLine (A,l,pK)))
let pK be FinSequence of K; ::_thesis: for A being Matrix of n,K st l in Seg n & len pK = n holds
Det (RLine (A,l,(a * pK))) = a * (Det (RLine (A,l,pK)))
let A be Matrix of n,K; ::_thesis: ( l in Seg n & len pK = n implies Det (RLine (A,l,(a * pK))) = a * (Det (RLine (A,l,pK))) )
assume that
A1: l in Seg n and
A2: len pK = n ; ::_thesis: Det (RLine (A,l,(a * pK))) = a * (Det (RLine (A,l,pK)))
pK is Element of (len pK) -tuples_on the carrier of K by FINSEQ_2:92;
then A3: (a * pK) + ((0. K) * pK) = (a + (0. K)) * pK by FVSUM_1:55;
a + (0. K) = a by RLVECT_1:4;
hence Det (RLine (A,l,(a * pK))) = (a * (Det (RLine (A,l,pK)))) + ((0. K) * (Det (RLine (A,l,pK)))) by A1, A2, A3, Th33
.= (a * (Det (RLine (A,l,pK)))) + (0. K) by VECTSP_1:6
.= a * (Det (RLine (A,l,pK))) by RLVECT_1:4 ;
::_thesis: verum
end;
theorem :: MATRIX11:35
for l, n being Nat
for K being Field
for a being Element of K
for A being Matrix of n,K st l in Seg n holds
Det (RLine (A,l,(a * (Line (A,l))))) = a * (Det A)
proof
let l, n be Nat; ::_thesis: for K being Field
for a being Element of K
for A being Matrix of n,K st l in Seg n holds
Det (RLine (A,l,(a * (Line (A,l))))) = a * (Det A)
let K be Field; ::_thesis: for a being Element of K
for A being Matrix of n,K st l in Seg n holds
Det (RLine (A,l,(a * (Line (A,l))))) = a * (Det A)
let a be Element of K; ::_thesis: for A being Matrix of n,K st l in Seg n holds
Det (RLine (A,l,(a * (Line (A,l))))) = a * (Det A)
let A be Matrix of n,K; ::_thesis: ( l in Seg n implies Det (RLine (A,l,(a * (Line (A,l))))) = a * (Det A) )
A1: len (Line (A,l)) = width A by MATRIX_1:def_7;
assume l in Seg n ; ::_thesis: Det (RLine (A,l,(a * (Line (A,l))))) = a * (Det A)
then Det (RLine (A,l,(a * (Line (A,l))))) = a * (Det (RLine (A,l,(Line (A,l))))) by A1, Th34, MATRIX_1:24;
hence Det (RLine (A,l,(a * (Line (A,l))))) = a * (Det A) by Th30; ::_thesis: verum
end;
theorem Th36: :: MATRIX11:36
for l, n being Nat
for K being Field
for pK, qK being FinSequence of K
for A being Matrix of n,K st l in Seg n & len pK = n & len qK = n holds
Det (RLine (A,l,(pK + qK))) = (Det (RLine (A,l,pK))) + (Det (RLine (A,l,qK)))
proof
let l, n be Nat; ::_thesis: for K being Field
for pK, qK being FinSequence of K
for A being Matrix of n,K st l in Seg n & len pK = n & len qK = n holds
Det (RLine (A,l,(pK + qK))) = (Det (RLine (A,l,pK))) + (Det (RLine (A,l,qK)))
let K be Field; ::_thesis: for pK, qK being FinSequence of K
for A being Matrix of n,K st l in Seg n & len pK = n & len qK = n holds
Det (RLine (A,l,(pK + qK))) = (Det (RLine (A,l,pK))) + (Det (RLine (A,l,qK)))
let pK, qK be FinSequence of K; ::_thesis: for A being Matrix of n,K st l in Seg n & len pK = n & len qK = n holds
Det (RLine (A,l,(pK + qK))) = (Det (RLine (A,l,pK))) + (Det (RLine (A,l,qK)))
let A be Matrix of n,K; ::_thesis: ( l in Seg n & len pK = n & len qK = n implies Det (RLine (A,l,(pK + qK))) = (Det (RLine (A,l,pK))) + (Det (RLine (A,l,qK))) )
assume that
A1: l in Seg n and
A2: len pK = n and
A3: len qK = n ; ::_thesis: Det (RLine (A,l,(pK + qK))) = (Det (RLine (A,l,pK))) + (Det (RLine (A,l,qK)))
pK is Element of (len pK) -tuples_on the carrier of K by FINSEQ_2:92;
then A4: (1_ K) * pK = pK by FVSUM_1:57;
qK is Element of (len pK) -tuples_on the carrier of K by A2, A3, FINSEQ_2:92;
then (1_ K) * qK = qK by FVSUM_1:57;
hence Det (RLine (A,l,(pK + qK))) = ((1_ K) * (Det (RLine (A,l,pK)))) + ((1_ K) * (Det (RLine (A,l,qK)))) by A1, A2, A3, A4, Th33
.= (Det (RLine (A,l,pK))) + ((1_ K) * (Det (RLine (A,l,qK)))) by VECTSP_1:def_4
.= (Det (RLine (A,l,pK))) + (Det (RLine (A,l,qK))) by VECTSP_1:def_4 ;
::_thesis: verum
end;
Lm7: for n, m being Nat
for D being non empty set
for F being Function of (Seg n),(Seg n)
for M being Matrix of n,m,D holds M * F is Matrix of n,m,D
proof
let n, m be Nat; ::_thesis: for D being non empty set
for F being Function of (Seg n),(Seg n)
for M being Matrix of n,m,D holds M * F is Matrix of n,m,D
let D be non empty set ; ::_thesis: for F being Function of (Seg n),(Seg n)
for M being Matrix of n,m,D holds M * F is Matrix of n,m,D
let F be Function of (Seg n),(Seg n); ::_thesis: for M being Matrix of n,m,D holds M * F is Matrix of n,m,D
let M be Matrix of n,m,D; ::_thesis: M * F is Matrix of n,m,D
A1: rng F c= Seg n by RELAT_1:def_19;
len M = n by MATRIX_1:def_2;
then A2: dom M = Seg n by FINSEQ_1:def_3;
dom F = Seg n by FUNCT_2:52;
then A3: dom (M * F) = Seg n by A1, A2, RELAT_1:27;
then reconsider Mp = M * F as FinSequence by FINSEQ_1:def_2;
A4: for x being set st x in rng Mp holds
ex p being FinSequence of D st
( x = p & len p = m )
proof
A5: rng M c= D * by FINSEQ_1:def_4;
let x be set ; ::_thesis: ( x in rng Mp implies ex p being FinSequence of D st
( x = p & len p = m ) )
assume A6: x in rng Mp ; ::_thesis: ex p being FinSequence of D st
( x = p & len p = m )
rng Mp c= rng M by RELAT_1:26;
then x in D * by A6, A5, TARSKI:def_3;
then reconsider p = x as FinSequence of D by FINSEQ_1:def_11;
take p ; ::_thesis: ( x = p & len p = m )
p in rng M by A6, FUNCT_1:14;
hence ( x = p & len p = m ) by MATRIX_1:def_2; ::_thesis: verum
end;
then reconsider Mp = Mp as Matrix of D by MATRIX_1:9;
A7: n is Element of NAT by ORDINAL1:def_12;
( len Mp = n & ( for p being FinSequence of D st p in rng Mp holds
len p = m ) )
proof
thus len Mp = n by A3, A7, FINSEQ_1:def_3; ::_thesis: for p being FinSequence of D st p in rng Mp holds
len p = m
let p be FinSequence of D; ::_thesis: ( p in rng Mp implies len p = m )
assume p in rng Mp ; ::_thesis: len p = m
then ex q being FinSequence of D st
( p = q & len q = m ) by A4;
hence len p = m ; ::_thesis: verum
end;
hence M * F is Matrix of n,m,D by MATRIX_1:def_2; ::_thesis: verum
end;
begin
definition
let n, m be Nat;
let D be non empty set ;
let F be Function of (Seg n),(Seg n);
let M be Matrix of n,m,D;
:: original: *
redefine funcM * F -> Matrix of n,m,D means :Def4: :: MATRIX11:def 4
( len it = len M & width it = width M & ( for i, j, k being Nat st [i,j] in Indices M & F . i = k holds
it * (i,j) = M * (k,j) ) );
compatibility
for b1 being Matrix of n,m,D holds
( b1 = F * M iff ( len b1 = len M & width b1 = width M & ( for i, j, k being Nat st [i,j] in Indices M & F . i = k holds
b1 * (i,j) = M * (k,j) ) ) )
proof
reconsider Mf = M * F as Matrix of n,m,D by Lm7;
let Mp be Matrix of n,m,D; ::_thesis: ( Mp = F * M iff ( len Mp = len M & width Mp = width M & ( for i, j, k being Nat st [i,j] in Indices M & F . i = k holds
Mp * (i,j) = M * (k,j) ) ) )
thus ( Mp = M * F implies ( len Mp = len M & width Mp = width M & ( for i, j, k being Nat st [i,j] in Indices M & F . i = k holds
Mp * (i,j) = M * (k,j) ) ) ) ::_thesis: ( len Mp = len M & width Mp = width M & ( for i, j, k being Nat st [i,j] in Indices M & F . i = k holds
Mp * (i,j) = M * (k,j) ) implies Mp = F * M )
proof
A1: rng F c= Seg n by RELAT_1:def_19;
assume A2: Mp = M * F ; ::_thesis: ( len Mp = len M & width Mp = width M & ( for i, j, k being Nat st [i,j] in Indices M & F . i = k holds
Mp * (i,j) = M * (k,j) ) )
A3: len M = n by MATRIX_1:def_2;
A4: len Mp = n by MATRIX_1:def_2;
A5: now__::_thesis:_(_(_n_=_0_&_width_M_=_width_Mp_)_or_(_n_>_0_&_width_M_=_width_Mp_)_)
percases ( n = 0 or n > 0 ) ;
caseA6: n = 0 ; ::_thesis: width M = width Mp
then width M = 0 by A3, MATRIX_1:def_3;
hence width M = width Mp by A4, A6, MATRIX_1:def_3; ::_thesis: verum
end;
caseA7: n > 0 ; ::_thesis: width M = width Mp
then width M = m by A3, MATRIX_1:20;
hence width M = width Mp by A4, A7, MATRIX_1:20; ::_thesis: verum
end;
end;
end;
hence ( len Mp = len M & width Mp = width M ) by A3, MATRIX_1:def_2; ::_thesis: for i, j, k being Nat st [i,j] in Indices M & F . i = k holds
Mp * (i,j) = M * (k,j)
let i, j, k be Nat; ::_thesis: ( [i,j] in Indices M & F . i = k implies Mp * (i,j) = M * (k,j) )
assume that
A8: [i,j] in Indices M and
A9: F . i = k ; ::_thesis: Mp * (i,j) = M * (k,j)
Indices M = [:(Seg n),(Seg (width M)):] by MATRIX_1:25;
then A10: i in Seg n by A8, ZFMISC_1:87;
then A11: Line (Mp,i) = Mp . i by MATRIX_2:8;
dom F = Seg n by FUNCT_2:52;
then A12: F . i in rng F by A10, FUNCT_1:def_3;
len Mp = n by MATRIX_1:25;
then dom Mp = Seg n by FINSEQ_1:def_3;
then Mp . i = M . k by A2, A9, A10, FUNCT_1:12;
then A13: Line (Mp,i) = Line (M,k) by A9, A12, A1, A11, MATRIX_2:8;
A14: j in Seg (width M) by A8, ZFMISC_1:87;
then (Line (M,k)) . j = M * (k,j) by MATRIX_1:def_7;
hence Mp * (i,j) = M * (k,j) by A5, A14, A13, MATRIX_1:def_7; ::_thesis: verum
end;
assume that
A15: len Mp = len M and
A16: width Mp = width M ; ::_thesis: ( ex i, j, k being Nat st
( [i,j] in Indices M & F . i = k & not Mp * (i,j) = M * (k,j) ) or Mp = F * M )
assume A17: for i, j, k being Nat st [i,j] in Indices M & F . i = k holds
Mp * (i,j) = M * (k,j) ; ::_thesis: Mp = F * M
for i, j being Nat st [i,j] in Indices Mp holds
Mp * (i,j) = Mf * (i,j)
proof
A18: Indices Mp = Indices M by A15, A16, MATRIX_4:55;
let i, j be Nat; ::_thesis: ( [i,j] in Indices Mp implies Mp * (i,j) = Mf * (i,j) )
assume A19: [i,j] in Indices Mp ; ::_thesis: Mp * (i,j) = Mf * (i,j)
Indices Mp = [:(Seg n),(Seg (width M)):] by A16, MATRIX_1:25;
then A20: i in Seg n by A19, ZFMISC_1:87;
then A21: Line (Mf,i) = Mf . i by MATRIX_2:8;
A22: rng F c= Seg n by RELAT_1:def_19;
dom F = Seg n by FUNCT_2:52;
then A23: F . i in rng F by A20, FUNCT_1:def_3;
then F . i in Seg n by A22;
then reconsider k = F . i as Element of NAT ;
len Mf = n by MATRIX_1:25;
then dom Mf = Seg n by FINSEQ_1:def_3;
then Mf . i = M . k by A20, FUNCT_1:12;
then A24: Line (Mf,i) = Line (M,k) by A23, A22, A21, MATRIX_2:8;
A25: width M = len (Line (M,k)) by MATRIX_1:def_7;
A26: width Mf = len (Line (Mf,i)) by MATRIX_1:def_7;
A27: j in Seg (width M) by A16, A19, ZFMISC_1:87;
then (Line (M,k)) . j = M * (k,j) by MATRIX_1:def_7;
then Mf * (i,j) = M * (k,j) by A27, A24, A25, A26, MATRIX_1:def_7;
hence Mp * (i,j) = Mf * (i,j) by A17, A19, A18; ::_thesis: verum
end;
hence Mp = F * M by MATRIX_1:27; ::_thesis: verum
end;
correctness
coherence
F * M is Matrix of n,m,D;
by Lm7;
end;
:: deftheorem Def4 defines * MATRIX11:def_4_:_
for n, m being Nat
for D being non empty set
for F being Function of (Seg n),(Seg n)
for M, b6 being Matrix of n,m,D holds
( b6 = M * F iff ( len b6 = len M & width b6 = width M & ( for i, j, k being Nat st [i,j] in Indices M & F . i = k holds
b6 * (i,j) = M * (k,j) ) ) );
theorem Th37: :: MATRIX11:37
for n, m being Nat
for D being non empty set
for F being Function of (Seg n),(Seg n)
for M being Matrix of n,m,D holds
( Indices M = Indices (M * F) & ( for i, j being Nat st [i,j] in Indices M holds
ex k being Nat st
( F . i = k & [k,j] in Indices M & (M * F) * (i,j) = M * (k,j) ) ) )
proof
let n, m be Nat; ::_thesis: for D being non empty set
for F being Function of (Seg n),(Seg n)
for M being Matrix of n,m,D holds
( Indices M = Indices (M * F) & ( for i, j being Nat st [i,j] in Indices M holds
ex k being Nat st
( F . i = k & [k,j] in Indices M & (M * F) * (i,j) = M * (k,j) ) ) )
let D be non empty set ; ::_thesis: for F being Function of (Seg n),(Seg n)
for M being Matrix of n,m,D holds
( Indices M = Indices (M * F) & ( for i, j being Nat st [i,j] in Indices M holds
ex k being Nat st
( F . i = k & [k,j] in Indices M & (M * F) * (i,j) = M * (k,j) ) ) )
let F be Function of (Seg n),(Seg n); ::_thesis: for M being Matrix of n,m,D holds
( Indices M = Indices (M * F) & ( for i, j being Nat st [i,j] in Indices M holds
ex k being Nat st
( F . i = k & [k,j] in Indices M & (M * F) * (i,j) = M * (k,j) ) ) )
let M be Matrix of n,m,D; ::_thesis: ( Indices M = Indices (M * F) & ( for i, j being Nat st [i,j] in Indices M holds
ex k being Nat st
( F . i = k & [k,j] in Indices M & (M * F) * (i,j) = M * (k,j) ) ) )
set Mp = M * F;
A1: dom F = Seg n by FUNCT_2:52;
A2: width M = width (M * F) by Def4;
len M = len (M * F) by Def4;
hence Indices M = Indices (M * F) by A2, MATRIX_4:55; ::_thesis: for i, j being Nat st [i,j] in Indices M holds
ex k being Nat st
( F . i = k & [k,j] in Indices M & (M * F) * (i,j) = M * (k,j) )
let i, j be Nat; ::_thesis: ( [i,j] in Indices M implies ex k being Nat st
( F . i = k & [k,j] in Indices M & (M * F) * (i,j) = M * (k,j) ) )
assume A3: [i,j] in Indices M ; ::_thesis: ex k being Nat st
( F . i = k & [k,j] in Indices M & (M * F) * (i,j) = M * (k,j) )
Indices M = [:(Seg n),(Seg (width M)):] by MATRIX_1:25;
then i in Seg n by A3, ZFMISC_1:87;
then A4: F . i in rng F by A1, FUNCT_1:def_3;
A5: rng F c= Seg n by RELAT_1:def_19;
then F . i in Seg n by A4;
then reconsider k = F . i as Element of NAT ;
j in Seg (width M) by A3, ZFMISC_1:87;
then [k,j] in [:(Seg n),(Seg (width M)):] by A4, A5, ZFMISC_1:87;
then A6: [k,j] in Indices M by MATRIX_1:25;
(M * F) * (i,j) = M * (k,j) by A3, Def4;
hence ex k being Nat st
( F . i = k & [k,j] in Indices M & (M * F) * (i,j) = M * (k,j) ) by A6; ::_thesis: verum
end;
theorem Th38: :: MATRIX11:38
for n, m being Nat
for D being non empty set
for M being Matrix of n,m,D
for F being Function of (Seg n),(Seg n)
for k being Nat st k in Seg n holds
Line ((M * F),k) = M . (F . k)
proof
let n, m be Nat; ::_thesis: for D being non empty set
for M being Matrix of n,m,D
for F being Function of (Seg n),(Seg n)
for k being Nat st k in Seg n holds
Line ((M * F),k) = M . (F . k)
let D be non empty set ; ::_thesis: for M being Matrix of n,m,D
for F being Function of (Seg n),(Seg n)
for k being Nat st k in Seg n holds
Line ((M * F),k) = M . (F . k)
let M be Matrix of n,m,D; ::_thesis: for F being Function of (Seg n),(Seg n)
for k being Nat st k in Seg n holds
Line ((M * F),k) = M . (F . k)
let F be Function of (Seg n),(Seg n); ::_thesis: for k being Nat st k in Seg n holds
Line ((M * F),k) = M . (F . k)
let k be Nat; ::_thesis: ( k in Seg n implies Line ((M * F),k) = M . (F . k) )
assume A1: k in Seg n ; ::_thesis: Line ((M * F),k) = M . (F . k)
len (M * F) = n by MATRIX_1:def_2;
then k in dom (M * F) by A1, FINSEQ_1:def_3;
then (M * F) . k = M . (F . k) by FUNCT_1:12;
hence Line ((M * F),k) = M . (F . k) by A1, MATRIX_2:8; ::_thesis: verum
end;
theorem Th39: :: MATRIX11:39
for m, n being Nat
for D being non empty set
for M being Matrix of n,m,D holds M * (idseq n) = M
proof
let m, n be Nat; ::_thesis: for D being non empty set
for M being Matrix of n,m,D holds M * (idseq n) = M
let D be non empty set ; ::_thesis: for M being Matrix of n,m,D holds M * (idseq n) = M
let M be Matrix of n,m,D; ::_thesis: M * (idseq n) = M
reconsider I = idseq n as Permutation of (Seg n) ;
A1: width (M * I) = width M by Def4;
A2: for i, j being Nat st [i,j] in Indices M holds
M * (i,j) = (M * I) * (i,j)
proof
let i, j be Nat; ::_thesis: ( [i,j] in Indices M implies M * (i,j) = (M * I) * (i,j) )
assume A3: [i,j] in Indices M ; ::_thesis: M * (i,j) = (M * I) * (i,j)
[i,j] in [:(Seg n),(Seg (width M)):] by A3, MATRIX_1:25;
then A4: i in Seg n by ZFMISC_1:87;
ex k being Nat st
( I . i = k & [k,j] in Indices M & (M * I) * (i,j) = M * (k,j) ) by A3, Th37;
hence M * (i,j) = (M * I) * (i,j) by A4, FUNCT_1:17; ::_thesis: verum
end;
len (M * I) = len M by Def4;
hence M * (idseq n) = M by A1, A2, MATRIX_1:21; ::_thesis: verum
end;
theorem Th40: :: MATRIX11:40
for n being Nat
for K being Field
for A being Matrix of n,K
for p being Element of Permutations n
for Perm being Permutation of (Seg n)
for q being Element of Permutations n st q = p * (Perm ") holds
Path_matrix (p,(A * Perm)) = (Path_matrix (q,A)) * Perm
proof
let n be Nat; ::_thesis: for K being Field
for A being Matrix of n,K
for p being Element of Permutations n
for Perm being Permutation of (Seg n)
for q being Element of Permutations n st q = p * (Perm ") holds
Path_matrix (p,(A * Perm)) = (Path_matrix (q,A)) * Perm
let K be Field; ::_thesis: for A being Matrix of n,K
for p being Element of Permutations n
for Perm being Permutation of (Seg n)
for q being Element of Permutations n st q = p * (Perm ") holds
Path_matrix (p,(A * Perm)) = (Path_matrix (q,A)) * Perm
let A be Matrix of n,K; ::_thesis: for p being Element of Permutations n
for Perm being Permutation of (Seg n)
for q being Element of Permutations n st q = p * (Perm ") holds
Path_matrix (p,(A * Perm)) = (Path_matrix (q,A)) * Perm
let p be Element of Permutations n; ::_thesis: for Perm being Permutation of (Seg n)
for q being Element of Permutations n st q = p * (Perm ") holds
Path_matrix (p,(A * Perm)) = (Path_matrix (q,A)) * Perm
let Perm be Permutation of (Seg n); ::_thesis: for q being Element of Permutations n st q = p * (Perm ") holds
Path_matrix (p,(A * Perm)) = (Path_matrix (q,A)) * Perm
let q be Element of Permutations n; ::_thesis: ( q = p * (Perm ") implies Path_matrix (p,(A * Perm)) = (Path_matrix (q,A)) * Perm )
assume A1: q = p * (Perm ") ; ::_thesis: Path_matrix (p,(A * Perm)) = (Path_matrix (q,A)) * Perm
reconsider perm = Perm as Element of Permutations n by MATRIX_2:def_9;
set Ap = A * Perm;
set P2 = Path_matrix (q,A);
set P1 = Path_matrix (p,(A * Perm));
A2: dom perm = Seg n by FUNCT_2:52;
A3: p is Permutation of (Seg n) by MATRIX_2:def_9;
then A4: dom p = Seg n by FUNCT_2:52;
A5: rng p = Seg n by A3, FUNCT_2:def_3;
A6: q is Permutation of (Seg n) by MATRIX_2:def_9;
then A7: dom q = Seg n by FUNCT_2:52;
len (Path_matrix (q,A)) = n by MATRIX_3:def_7;
then A8: dom (Path_matrix (q,A)) = Seg n by FINSEQ_1:def_3;
A9: rng perm = Seg n by FUNCT_2:def_3;
then A10: dom ((Path_matrix (q,A)) * perm) = Seg n by A2, A8, RELAT_1:27;
then reconsider P2p = (Path_matrix (q,A)) * perm as FinSequence by FINSEQ_1:def_2;
A11: len (Path_matrix (p,(A * Perm))) = n by MATRIX_3:def_7;
A12: rng q = Seg n by A6, FUNCT_2:def_3;
A13: now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_k_<=_len_(Path_matrix_(p,(A_*_Perm)))_holds_
P2p_._k_=_(Path_matrix_(p,(A_*_Perm)))_._k
let k be Nat; ::_thesis: ( 1 <= k & k <= len (Path_matrix (p,(A * Perm))) implies P2p . k = (Path_matrix (p,(A * Perm))) . k )
assume that
A14: 1 <= k and
A15: k <= len (Path_matrix (p,(A * Perm))) ; ::_thesis: P2p . k = (Path_matrix (p,(A * Perm))) . k
k in NAT by ORDINAL1:def_12;
then A16: k in Seg n by A11, A14, A15;
then A17: p . k in Seg n by A4, A5, FUNCT_1:3;
then reconsider pk = p . k as Element of NAT ;
A18: k = (perm ") . (perm . k) by A2, A16, FUNCT_1:34;
[k,pk] in [:(Seg n),(Seg n):] by A16, A17, ZFMISC_1:87;
then [k,pk] in Indices A by MATRIX_1:24;
then consider permk being Nat such that
A19: perm . k = permk and
A20: [permk,pk] in Indices A and
A21: (A * Perm) * (k,pk) = A * (permk,pk) by Th37;
dom P2p = Seg n by A2, A9, A8, RELAT_1:27;
then A22: P2p . k = (Path_matrix (q,A)) . permk by A16, A19, FUNCT_1:12;
Indices A = [:(Seg n),(Seg n):] by MATRIX_1:24;
then A23: permk in Seg n by A20, ZFMISC_1:87;
then q . permk in Seg n by A7, A12, FUNCT_1:3;
then reconsider qpermk = q . permk as Element of NAT ;
A24: (Path_matrix (q,A)) . permk = A * (permk,qpermk) by A8, A23, MATRIX_3:def_7;
A25: dom (Path_matrix (p,(A * Perm))) = Seg n by A11, FINSEQ_1:def_3;
q . permk = p . ((perm ") . (perm . k)) by A1, A7, A19, A23, FUNCT_1:12;
hence P2p . k = (Path_matrix (p,(A * Perm))) . k by A16, A21, A24, A22, A18, A25, MATRIX_3:def_7; ::_thesis: verum
end;
n is Element of NAT by ORDINAL1:def_12;
then len P2p = n by A10, FINSEQ_1:def_3;
hence Path_matrix (p,(A * Perm)) = (Path_matrix (q,A)) * Perm by A11, A13, FINSEQ_1:14; ::_thesis: verum
end;
theorem Th41: :: MATRIX11:41
for n being Nat
for K being Field
for A being Matrix of n,K
for p being Element of Permutations n
for Perm being Permutation of (Seg n)
for q being Element of Permutations n st q = p * (Perm ") holds
the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A))
proof
let n be Nat; ::_thesis: for K being Field
for A being Matrix of n,K
for p being Element of Permutations n
for Perm being Permutation of (Seg n)
for q being Element of Permutations n st q = p * (Perm ") holds
the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A))
let K be Field; ::_thesis: for A being Matrix of n,K
for p being Element of Permutations n
for Perm being Permutation of (Seg n)
for q being Element of Permutations n st q = p * (Perm ") holds
the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A))
let A be Matrix of n,K; ::_thesis: for p being Element of Permutations n
for Perm being Permutation of (Seg n)
for q being Element of Permutations n st q = p * (Perm ") holds
the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A))
let p be Element of Permutations n; ::_thesis: for Perm being Permutation of (Seg n)
for q being Element of Permutations n st q = p * (Perm ") holds
the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A))
let Perm be Permutation of (Seg n); ::_thesis: for q being Element of Permutations n st q = p * (Perm ") holds
the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A))
let q be Element of Permutations n; ::_thesis: ( q = p * (Perm ") implies the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A)) )
assume A1: q = p * (Perm ") ; ::_thesis: the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A))
set mm = the multF of K;
set P2 = Path_matrix (q,A);
set P1 = Path_matrix (p,(A * Perm));
now__::_thesis:_(_(_n_=_0_&_the_multF_of_K_$$_(Path_matrix_(p,(A_*_Perm)))_=_the_multF_of_K_$$_(Path_matrix_(q,A))_)_or_(_n_+_0_>_0_&_the_multF_of_K_$$_(Path_matrix_(p,(A_*_Perm)))_=_the_multF_of_K_$$_(Path_matrix_(q,A))_)_)
percases ( n = 0 or n + 0 > 0 ) ;
caseA2: n = 0 ; ::_thesis: the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A))
then len (Path_matrix (p,(A * Perm))) = 0 by MATRIX_3:def_7;
then A3: the multF of K $$ (Path_matrix (p,(A * Perm))) = the_unity_wrt the multF of K by FINSOP_1:def_1;
len (Path_matrix (q,A)) = 0 by A2, MATRIX_3:def_7;
hence the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A)) by A3, FINSOP_1:def_1; ::_thesis: verum
end;
case n + 0 > 0 ; ::_thesis: the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A))
then A4: n >= 1 by NAT_1:19;
A5: len (Path_matrix (q,A)) = n by MATRIX_3:def_7;
A6: Perm is Element of Permutations n by MATRIX_2:def_9;
Path_matrix (p,(A * Perm)) = (Path_matrix (q,A)) * Perm by A1, Th40;
hence the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A)) by A4, A5, A6, MATRIX_7:33; ::_thesis: verum
end;
end;
end;
hence the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A)) ; ::_thesis: verum
end;
theorem Th42: :: MATRIX11:42
for n being Nat
for K being Field
for p2, q2 being Element of Permutations (n + 2) st q2 = p2 " holds
sgn (p2,K) = sgn (q2,K)
proof
let n be Nat; ::_thesis: for K being Field
for p2, q2 being Element of Permutations (n + 2) st q2 = p2 " holds
sgn (p2,K) = sgn (q2,K)
let K be Field; ::_thesis: for p2, q2 being Element of Permutations (n + 2) st q2 = p2 " holds
sgn (p2,K) = sgn (q2,K)
A1: (n + 1) + 1 >= 0 + 1 by XREAL_1:6;
let p2, q2 be Element of Permutations (n + 2); ::_thesis: ( q2 = p2 " implies sgn (p2,K) = sgn (q2,K) )
assume q2 = p2 " ; ::_thesis: sgn (p2,K) = sgn (q2,K)
then A2: - ((1_ K),p2) = - ((1_ K),q2) by A1, MATRIX_7:29;
A3: - ((1_ K),q2) = (sgn (q2,K)) * (1_ K) by Th26;
- ((1_ K),p2) = (sgn (p2,K)) * (1_ K) by Th26;
then (sgn (p2,K)) * (1_ K) = sgn (q2,K) by A2, A3, VECTSP_1:def_4;
hence sgn (p2,K) = sgn (q2,K) by VECTSP_1:def_4; ::_thesis: verum
end;
theorem Th43: :: MATRIX11:43
for n being Nat
for K being Field
for M being Matrix of n + 2,K
for perm2 being Element of Permutations (n + 2)
for Perm2 being Permutation of (Seg (n + 2)) st perm2 = Perm2 holds
for p2, q2 being Element of Permutations (n + 2) st q2 = p2 * (Perm2 ") holds
(Path_product M) . q2 = (sgn (perm2,K)) * ((Path_product (M * Perm2)) . p2)
proof
let n be Nat; ::_thesis: for K being Field
for M being Matrix of n + 2,K
for perm2 being Element of Permutations (n + 2)
for Perm2 being Permutation of (Seg (n + 2)) st perm2 = Perm2 holds
for p2, q2 being Element of Permutations (n + 2) st q2 = p2 * (Perm2 ") holds
(Path_product M) . q2 = (sgn (perm2,K)) * ((Path_product (M * Perm2)) . p2)
let K be Field; ::_thesis: for M being Matrix of n + 2,K
for perm2 being Element of Permutations (n + 2)
for Perm2 being Permutation of (Seg (n + 2)) st perm2 = Perm2 holds
for p2, q2 being Element of Permutations (n + 2) st q2 = p2 * (Perm2 ") holds
(Path_product M) . q2 = (sgn (perm2,K)) * ((Path_product (M * Perm2)) . p2)
let M be Matrix of n + 2,K; ::_thesis: for perm2 being Element of Permutations (n + 2)
for Perm2 being Permutation of (Seg (n + 2)) st perm2 = Perm2 holds
for p2, q2 being Element of Permutations (n + 2) st q2 = p2 * (Perm2 ") holds
(Path_product M) . q2 = (sgn (perm2,K)) * ((Path_product (M * Perm2)) . p2)
let perm2 be Element of Permutations (n + 2); ::_thesis: for Perm2 being Permutation of (Seg (n + 2)) st perm2 = Perm2 holds
for p2, q2 being Element of Permutations (n + 2) st q2 = p2 * (Perm2 ") holds
(Path_product M) . q2 = (sgn (perm2,K)) * ((Path_product (M * Perm2)) . p2)
let Perm2 be Permutation of (Seg (n + 2)); ::_thesis: ( perm2 = Perm2 implies for p2, q2 being Element of Permutations (n + 2) st q2 = p2 * (Perm2 ") holds
(Path_product M) . q2 = (sgn (perm2,K)) * ((Path_product (M * Perm2)) . p2) )
assume A1: perm2 = Perm2 ; ::_thesis: for p2, q2 being Element of Permutations (n + 2) st q2 = p2 * (Perm2 ") holds
(Path_product M) . q2 = (sgn (perm2,K)) * ((Path_product (M * Perm2)) . p2)
set P = Permutations (n + 2);
set mm = the multF of K;
let p2, q2 be Element of Permutations (n + 2); ::_thesis: ( q2 = p2 * (Perm2 ") implies (Path_product M) . q2 = (sgn (perm2,K)) * ((Path_product (M * Perm2)) . p2) )
assume A2: q2 = p2 * (Perm2 ") ; ::_thesis: (Path_product M) . q2 = (sgn (perm2,K)) * ((Path_product (M * Perm2)) . p2)
reconsider perm29 = perm2 " as Element of Permutations (n + 2) by MATRIX_7:18;
set PM = the multF of K $$ (Path_matrix (q2,M));
set PMp = the multF of K $$ (Path_matrix (p2,(M * Perm2)));
sgn (q2,K) = (sgn (p2,K)) * (sgn (perm29,K)) by A1, A2, Th24
.= (sgn (p2,K)) * (sgn (perm2,K)) by Th42 ;
then - (( the multF of K $$ (Path_matrix (q2,M))),q2) = ((sgn (perm2,K)) * (sgn (p2,K))) * ( the multF of K $$ (Path_matrix (q2,M))) by Th26
.= (sgn (perm2,K)) * ((sgn (p2,K)) * ( the multF of K $$ (Path_matrix (q2,M)))) by GROUP_1:def_3
.= (sgn (perm2,K)) * ((sgn (p2,K)) * ( the multF of K $$ (Path_matrix (p2,(M * Perm2))))) by A2, Th41
.= (sgn (perm2,K)) * (- (( the multF of K $$ (Path_matrix (p2,(M * Perm2)))),p2)) by Th26
.= (sgn (perm2,K)) * ((Path_product (M * Perm2)) . p2) by MATRIX_3:def_8 ;
hence (Path_product M) . q2 = (sgn (perm2,K)) * ((Path_product (M * Perm2)) . p2) by MATRIX_3:def_8; ::_thesis: verum
end;
theorem Th44: :: MATRIX11:44
for n being Nat
for perm being Element of Permutations n ex P being Permutation of (Permutations n) st
for p being Element of Permutations n holds P . p = p * perm
proof
let n be Nat; ::_thesis: for perm being Element of Permutations n ex P being Permutation of (Permutations n) st
for p being Element of Permutations n holds P . p = p * perm
let perm be Element of Permutations n; ::_thesis: ex P being Permutation of (Permutations n) st
for p being Element of Permutations n holds P . p = p * perm
set P = Permutations n;
defpred S1[ set , set ] means for p being Element of Permutations n st $1 = p holds
$2 = p * perm;
A1: card (Permutations n) = card (Permutations n) ;
A2: for x being set st x in Permutations n holds
ex y being set st
( y in Permutations n & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in Permutations n implies ex y being set st
( y in Permutations n & S1[x,y] ) )
assume x in Permutations n ; ::_thesis: ex y being set st
( y in Permutations n & S1[x,y] )
then reconsider p = x as Element of Permutations n ;
reconsider pp = p * perm as Element of Permutations n by MATRIX_9:39;
take pp ; ::_thesis: ( pp in Permutations n & S1[x,pp] )
thus ( pp in Permutations n & S1[x,pp] ) ; ::_thesis: verum
end;
consider G being Function of (Permutations n),(Permutations n) such that
A3: for x being set st x in Permutations n holds
S1[x,G . x] from FUNCT_2:sch_1(A2);
for x1, x2 being set st x1 in Permutations n & x2 in Permutations n & G . x1 = G . x2 holds
x1 = x2
proof
let x1, x2 be set ; ::_thesis: ( x1 in Permutations n & x2 in Permutations n & G . x1 = G . x2 implies x1 = x2 )
assume that
A4: x1 in Permutations n and
A5: x2 in Permutations n and
A6: G . x1 = G . x2 ; ::_thesis: x1 = x2
reconsider p1 = x1, p2 = x2 as Element of Permutations n by A4, A5;
p2 is Permutation of (Seg n) by MATRIX_2:def_9;
then A7: dom p2 = Seg n by FUNCT_2:52;
A8: G . p2 = p2 * perm by A3;
A9: G . p1 = p1 * perm by A3;
perm is Permutation of (Seg n) by MATRIX_2:def_9;
then A10: rng perm = Seg n by FUNCT_2:def_3;
p1 is Permutation of (Seg n) by MATRIX_2:def_9;
then dom p1 = Seg n by FUNCT_2:52;
then p1 = p1 * (id (rng perm)) by A10, RELAT_1:52
.= p1 * (perm * (perm ")) by FUNCT_1:39
.= (p2 * perm) * (perm ") by A6, A9, A8, RELAT_1:36
.= p2 * (perm * (perm ")) by RELAT_1:36
.= p2 * (id (rng perm)) by FUNCT_1:39
.= p2 by A10, A7, RELAT_1:52 ;
hence x1 = x2 ; ::_thesis: verum
end;
then A11: G is one-to-one by FUNCT_2:19;
Permutations n is finite set by MATRIX_2:26;
then G is onto by A11, A1, STIRL2_1:60;
then reconsider G = G as Permutation of (Permutations n) by A11;
take G ; ::_thesis: for p being Element of Permutations n holds G . p = p * perm
thus for p being Element of Permutations n holds G . p = p * perm by A3; ::_thesis: verum
end;
theorem Th45: :: MATRIX11:45
for n being Nat
for K being Field
for M being Matrix of n + 2,n + 2,K
for perm2 being Element of Permutations (n + 2)
for Perm2 being Permutation of (Seg (n + 2)) st perm2 = Perm2 holds
Det (M * Perm2) = (sgn (perm2,K)) * (Det M)
proof
let n be Nat; ::_thesis: for K being Field
for M being Matrix of n + 2,n + 2,K
for perm2 being Element of Permutations (n + 2)
for Perm2 being Permutation of (Seg (n + 2)) st perm2 = Perm2 holds
Det (M * Perm2) = (sgn (perm2,K)) * (Det M)
let K be Field; ::_thesis: for M being Matrix of n + 2,n + 2,K
for perm2 being Element of Permutations (n + 2)
for Perm2 being Permutation of (Seg (n + 2)) st perm2 = Perm2 holds
Det (M * Perm2) = (sgn (perm2,K)) * (Det M)
set n2 = n + 2;
let M be Matrix of n + 2,n + 2,K; ::_thesis: for perm2 being Element of Permutations (n + 2)
for Perm2 being Permutation of (Seg (n + 2)) st perm2 = Perm2 holds
Det (M * Perm2) = (sgn (perm2,K)) * (Det M)
let perm2 be Element of Permutations (n + 2); ::_thesis: for Perm2 being Permutation of (Seg (n + 2)) st perm2 = Perm2 holds
Det (M * Perm2) = (sgn (perm2,K)) * (Det M)
let Perm2 be Permutation of (Seg (n + 2)); ::_thesis: ( perm2 = Perm2 implies Det (M * Perm2) = (sgn (perm2,K)) * (Det M) )
assume A1: perm2 = Perm2 ; ::_thesis: Det (M * Perm2) = (sgn (perm2,K)) * (Det M)
set PathM = Path_product M;
set Mperm = M * Perm2;
set P = Permutations (n + 2);
set KK = the carrier of K;
set aa = the addF of K;
set PathMp = Path_product (M * Perm2);
set F = FinOmega (Permutations (n + 2));
reconsider perm29 = perm2 " as Element of Permutations (n + 2) by MATRIX_7:18;
A2: FinOmega (Permutations (n + 2)) = Permutations (n + 2) by MATRIX_2:26, MATRIX_2:def_14;
then consider GM being Function of (Fin (Permutations (n + 2))), the carrier of K such that
A3: Det M = GM . (FinOmega (Permutations (n + 2))) and
for e being Element of the carrier of K st e is_a_unity_wrt the addF of K holds
GM . {} = e and
A4: for x being Element of Permutations (n + 2) holds GM . {x} = (Path_product M) . x and
A5: for B9 being Element of Fin (Permutations (n + 2)) st B9 c= FinOmega (Permutations (n + 2)) & B9 <> {} holds
for x being Element of Permutations (n + 2) st x in (FinOmega (Permutations (n + 2))) \ B9 holds
GM . (B9 \/ {x}) = the addF of K . ((GM . B9),((Path_product M) . x)) by SETWISEO:def_3;
consider PERM being Permutation of (Permutations (n + 2)) such that
A6: for p being Element of Permutations (n + 2) holds PERM . p = p * perm29 by Th44;
consider GMp being Function of (Fin (Permutations (n + 2))), the carrier of K such that
A7: Det (M * Perm2) = GMp . (FinOmega (Permutations (n + 2))) and
for e being Element of the carrier of K st e is_a_unity_wrt the addF of K holds
GMp . {} = e and
A8: for x being Element of Permutations (n + 2) holds GMp . {x} = (Path_product (M * Perm2)) . x and
A9: for B9 being Element of Fin (Permutations (n + 2)) st B9 c= FinOmega (Permutations (n + 2)) & B9 <> {} holds
for x being Element of Permutations (n + 2) st x in (FinOmega (Permutations (n + 2))) \ B9 holds
GMp . (B9 \/ {x}) = the addF of K . ((GMp . B9),((Path_product (M * Perm2)) . x)) by A2, SETWISEO:def_3;
defpred S1[ Nat] means ( $1 <> 0 implies for B being Element of Fin (Permutations (n + 2)) st card B = $1 holds
(sgn (perm2,K)) * (GMp . B) = GM . (PERM .: B) );
A10: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A11: S1[k] ; ::_thesis: S1[k + 1]
set k1 = k + 1;
assume k + 1 <> 0 ; ::_thesis: for B being Element of Fin (Permutations (n + 2)) st card B = k + 1 holds
(sgn (perm2,K)) * (GMp . B) = GM . (PERM .: B)
let B be Element of Fin (Permutations (n + 2)); ::_thesis: ( card B = k + 1 implies (sgn (perm2,K)) * (GMp . B) = GM . (PERM .: B) )
assume A12: card B = k + 1 ; ::_thesis: (sgn (perm2,K)) * (GMp . B) = GM . (PERM .: B)
percases ( k = 0 or k > 0 ) ;
suppose k = 0 ; ::_thesis: (sgn (perm2,K)) * (GMp . B) = GM . (PERM .: B)
then consider x being set such that
A13: B = {x} by A12, CARD_2:42;
A14: x in B by A13, TARSKI:def_1;
B c= Permutations (n + 2) by FINSUB_1:def_5;
then reconsider x = x as Element of Permutations (n + 2) by A14;
A15: GM . {(PERM . x)} = (Path_product M) . (PERM . x) by A4;
A16: PERM . x = x * perm29 by A6;
A17: Permutations (n + 2) = dom PERM by FUNCT_2:52;
GMp . {x} = (Path_product (M * Perm2)) . x by A8;
then (sgn (perm2,K)) * (GMp . B) = GM . {(PERM . x)} by A1, A13, A15, A16, Th43;
then (sgn (perm2,K)) * (GMp . B) = GM . (Im (PERM,x)) by A17, FUNCT_1:59;
hence (sgn (perm2,K)) * (GMp . B) = GM . (PERM .: B) by A13; ::_thesis: verum
end;
supposeA18: k > 0 ; ::_thesis: (sgn (perm2,K)) * (GMp . B) = GM . (PERM .: B)
consider x being set such that
A19: x in B by A12, CARD_1:27, XBOOLE_0:def_1;
B c= Permutations (n + 2) by FINSUB_1:def_5;
then reconsider x = x as Element of Permutations (n + 2) by A19;
PERM .: (B \ {x}) c= rng PERM by RELAT_1:111;
then A20: PERM .: (B \ {x}) c= Permutations (n + 2) by FUNCT_2:def_3;
reconsider Px = PERM . x as Element of Permutations (n + 2) ;
A21: Px in {Px} by TARSKI:def_1;
dom PERM = Permutations (n + 2) by FUNCT_2:52;
then A22: Im (PERM,x) = {Px} by FUNCT_1:59;
A23: B c= Permutations (n + 2) by FINSUB_1:def_5;
then B \ {x} c= Permutations (n + 2) by XBOOLE_1:1;
then reconsider B9 = B \ {x}, PeBx = PERM .: (B \ {x}), PeB = PERM .: B as Element of Fin (Permutations (n + 2)) by A20, FINSUB_1:def_5;
A24: {x} \/ B9 = B by A19, ZFMISC_1:116;
then A25: PERM .: B = (Im (PERM,x)) \/ PeBx by RELAT_1:120;
PERM . x = x * perm29 by A6;
then A26: (sgn (perm2,K)) * ((Path_product (M * Perm2)) . x) = (Path_product M) . Px by A1, Th43;
A27: dom PERM = Permutations (n + 2) by FUNCT_2:52;
B9 misses {x} by XBOOLE_1:79;
then B9 /\ {x} = {} by XBOOLE_0:def_7;
then PERM .: {} = {Px} /\ PeBx by A22, FUNCT_1:62;
then not Px in PeBx by A21, XBOOLE_0:def_4;
then A28: Px in (FinOmega (Permutations (n + 2))) \ PeBx by A2, XBOOLE_0:def_5;
A29: B9 c= Permutations (n + 2) by FINSUB_1:def_5;
A30: not x in B9 by ZFMISC_1:56;
then A31: x in (FinOmega (Permutations (n + 2))) \ B9 by A2, XBOOLE_0:def_5;
A32: k + 1 = (card B9) + 1 by A12, A24, A30, CARD_2:41;
then ex y being set st y in B9 by A18, CARD_1:27, XBOOLE_0:def_1;
then GM . PeB = the addF of K . ((GM . PeBx),((Path_product M) . Px)) by A2, A5, A20, A25, A22, A28, A29, A27;
then GM . PeB = ((sgn (perm2,K)) * (GMp . B9)) + ((sgn (perm2,K)) * ((Path_product (M * Perm2)) . x)) by A11, A18, A32, A26
.= (sgn (perm2,K)) * ((GMp . B9) + ((Path_product (M * Perm2)) . x)) by VECTSP_1:def_7
.= (sgn (perm2,K)) * (GMp . B) by A2, A9, A18, A23, A24, A32, A31, CARD_1:27, XBOOLE_1:1 ;
hence (sgn (perm2,K)) * (GMp . B) = GM . (PERM .: B) ; ::_thesis: verum
end;
end;
end;
A33: S1[ 0 ] ;
A34: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A33, A10);
A35: rng PERM = Permutations (n + 2) by FUNCT_2:def_3;
A36: dom PERM = Permutations (n + 2) by FUNCT_2:52;
A37: PERM .: (dom PERM) = rng PERM by RELAT_1:113;
A38: (1_ K) * (1_ K) = (- (1_ K)) * (- (1_ K)) by VECTSP_1:10;
A39: ( sgn (perm2,K) = 1_ K or sgn (perm2,K) = - (1_ K) ) by Th11;
A40: (1_ K) * (1_ K) = 1_ K by VECTSP_1:def_4;
card (FinOmega (Permutations (n + 2))) <> 0 by A2;
then (sgn (perm2,K)) * (Det (M * Perm2)) = Det M by A2, A3, A7, A34, A37, A36, A35;
hence (sgn (perm2,K)) * (Det M) = (1_ K) * (Det (M * Perm2)) by A39, A40, A38, GROUP_1:def_3
.= Det (M * Perm2) by VECTSP_1:def_4 ;
::_thesis: verum
end;
theorem Th46: :: MATRIX11:46
for n being Nat
for K being Field
for M being Matrix of n,K
for perm being Element of Permutations n
for Perm being Permutation of (Seg n) st perm = Perm holds
Det (M * Perm) = - ((Det M),perm)
proof
let n be Nat; ::_thesis: for K being Field
for M being Matrix of n,K
for perm being Element of Permutations n
for Perm being Permutation of (Seg n) st perm = Perm holds
Det (M * Perm) = - ((Det M),perm)
let K be Field; ::_thesis: for M being Matrix of n,K
for perm being Element of Permutations n
for Perm being Permutation of (Seg n) st perm = Perm holds
Det (M * Perm) = - ((Det M),perm)
let M be Matrix of n,K; ::_thesis: for perm being Element of Permutations n
for Perm being Permutation of (Seg n) st perm = Perm holds
Det (M * Perm) = - ((Det M),perm)
let perm be Element of Permutations n; ::_thesis: for Perm being Permutation of (Seg n) st perm = Perm holds
Det (M * Perm) = - ((Det M),perm)
let Perm be Permutation of (Seg n); ::_thesis: ( perm = Perm implies Det (M * Perm) = - ((Det M),perm) )
assume A1: Perm = perm ; ::_thesis: Det (M * Perm) = - ((Det M),perm)
percases ( n < 2 or n >= 2 ) ;
supposeA2: n < 2 ; ::_thesis: Det (M * Perm) = - ((Det M),perm)
then perm = idseq n by Lm3;
then A3: M * perm = M by Th39;
perm is even by A2, Lm3;
hence Det (M * Perm) = - ((Det M),perm) by A1, A3, MATRIX_2:def_13; ::_thesis: verum
end;
suppose n >= 2 ; ::_thesis: Det (M * Perm) = - ((Det M),perm)
then reconsider n2 = n - 2 as Nat by NAT_1:21;
reconsider M9 = M as Matrix of n2 + 2,K ;
reconsider Perm2 = Perm as Permutation of (Seg (n2 + 2)) ;
reconsider perm2 = perm as Element of Permutations (n2 + 2) ;
Det (M9 * Perm2) = (sgn (perm2,K)) * (Det M9) by A1, Th45;
hence Det (M * Perm) = - ((Det M),perm) by Th26; ::_thesis: verum
end;
end;
end;
theorem Th47: :: MATRIX11:47
for n being Nat
for PERM being Permutation of (Permutations n)
for perm being Element of Permutations n st perm is odd & ( for p being Element of Permutations n holds PERM . p = p * perm ) holds
PERM .: { p where p is Element of Permutations n : p is even } = { q where q is Element of Permutations n : q is odd }
proof
let n be Nat; ::_thesis: for PERM being Permutation of (Permutations n)
for perm being Element of Permutations n st perm is odd & ( for p being Element of Permutations n holds PERM . p = p * perm ) holds
PERM .: { p where p is Element of Permutations n : p is even } = { q where q is Element of Permutations n : q is odd }
set P = Permutations n;
let PERM be Permutation of (Permutations n); ::_thesis: for perm being Element of Permutations n st perm is odd & ( for p being Element of Permutations n holds PERM . p = p * perm ) holds
PERM .: { p where p is Element of Permutations n : p is even } = { q where q is Element of Permutations n : q is odd }
let perm be Element of Permutations n; ::_thesis: ( perm is odd & ( for p being Element of Permutations n holds PERM . p = p * perm ) implies PERM .: { p where p is Element of Permutations n : p is even } = { q where q is Element of Permutations n : q is odd } )
assume that
A1: perm is odd and
A2: for p being Element of Permutations n holds PERM . p = p * perm ; ::_thesis: PERM .: { p where p is Element of Permutations n : p is even } = { q where q is Element of Permutations n : q is odd }
set E = { p where p is Element of Permutations n : p is even } ;
set OD = { q where q is Element of Permutations n : q is odd } ;
for y being set holds
( y in { q where q is Element of Permutations n : q is odd } iff ex x being set st
( x in dom PERM & x in { p where p is Element of Permutations n : p is even } & y = PERM . x ) )
proof
let y be set ; ::_thesis: ( y in { q where q is Element of Permutations n : q is odd } iff ex x being set st
( x in dom PERM & x in { p where p is Element of Permutations n : p is even } & y = PERM . x ) )
thus ( y in { q where q is Element of Permutations n : q is odd } implies ex x being set st
( x in dom PERM & x in { p where p is Element of Permutations n : p is even } & y = PERM . x ) ) ::_thesis: ( ex x being set st
( x in dom PERM & x in { p where p is Element of Permutations n : p is even } & y = PERM . x ) implies y in { q where q is Element of Permutations n : q is odd } )
proof
reconsider perm9 = perm " as Element of Permutations n by MATRIX_7:18;
A3: dom PERM = Permutations n by FUNCT_2:52;
n >= 2 by A1, Lm3;
then A4: n >= 1 by XXREAL_0:2;
assume y in { q where q is Element of Permutations n : q is odd } ; ::_thesis: ex x being set st
( x in dom PERM & x in { p where p is Element of Permutations n : p is even } & y = PERM . x )
then consider q being Element of Permutations n such that
A5: y = q and
A6: q is odd ;
A7: q * (idseq n) = q by MATRIX_2:21;
n is Element of NAT by ORDINAL1:def_12;
then perm9 is odd by A1, A4, MATRIX_7:28;
then A8: q * perm9 is even by A6, Th25;
reconsider qp9 = q * perm9 as Element of Permutations n by MATRIX_9:39;
take qp9 ; ::_thesis: ( qp9 in dom PERM & qp9 in { p where p is Element of Permutations n : p is even } & y = PERM . qp9 )
A9: perm9 * perm = idseq n by MATRIX_2:22;
PERM . qp9 = qp9 * perm by A2;
hence ( qp9 in dom PERM & qp9 in { p where p is Element of Permutations n : p is even } & y = PERM . qp9 ) by A5, A3, A9, A7, A8, RELAT_1:36; ::_thesis: verum
end;
assume ex x being set st
( x in dom PERM & x in { p where p is Element of Permutations n : p is even } & y = PERM . x ) ; ::_thesis: y in { q where q is Element of Permutations n : q is odd }
then consider x being set such that
x in dom PERM and
A10: x in { p where p is Element of Permutations n : p is even } and
A11: y = PERM . x ;
consider p being Element of Permutations n such that
A12: p = x and
A13: p is even by A10;
reconsider pp = p * perm as Element of Permutations n by MATRIX_9:39;
A14: PERM . x = p * perm by A2, A12;
pp is odd by A1, A13, Th25;
hence y in { q where q is Element of Permutations n : q is odd } by A11, A14; ::_thesis: verum
end;
hence PERM .: { p where p is Element of Permutations n : p is even } = { q where q is Element of Permutations n : q is odd } by FUNCT_1:def_6; ::_thesis: verum
end;
Lm8: for n, i, j being Nat st i in Seg n & j in Seg n & i < j holds
ex ODD, EVEN being finite set st
( EVEN = { p where p is Element of Permutations n : p is even } & ODD = { q where q is Element of Permutations n : q is odd } & EVEN /\ ODD = {} & EVEN \/ ODD = Permutations n & ex PERM being Function of EVEN,ODD ex perm being Element of Permutations n st
( perm is being_transposition & perm . i = j & dom PERM = EVEN & PERM is bijective & ( for p being Element of Permutations n st p in EVEN holds
PERM . p = p * perm ) ) )
proof
let n, i, j be Nat; ::_thesis: ( i in Seg n & j in Seg n & i < j implies ex ODD, EVEN being finite set st
( EVEN = { p where p is Element of Permutations n : p is even } & ODD = { q where q is Element of Permutations n : q is odd } & EVEN /\ ODD = {} & EVEN \/ ODD = Permutations n & ex PERM being Function of EVEN,ODD ex perm being Element of Permutations n st
( perm is being_transposition & perm . i = j & dom PERM = EVEN & PERM is bijective & ( for p being Element of Permutations n st p in EVEN holds
PERM . p = p * perm ) ) ) )
assume that
A1: i in Seg n and
A2: j in Seg n and
A3: i < j ; ::_thesis: ex ODD, EVEN being finite set st
( EVEN = { p where p is Element of Permutations n : p is even } & ODD = { q where q is Element of Permutations n : q is odd } & EVEN /\ ODD = {} & EVEN \/ ODD = Permutations n & ex PERM being Function of EVEN,ODD ex perm being Element of Permutations n st
( perm is being_transposition & perm . i = j & dom PERM = EVEN & PERM is bijective & ( for p being Element of Permutations n st p in EVEN holds
PERM . p = p * perm ) ) )
set P = Permutations n;
consider tr being Element of Permutations n such that
A4: tr is being_transposition and
A5: tr . i = j by A1, A2, A3, Th16;
{i,j} in 2Set (Seg n) by A1, A2, A3, Th1;
then reconsider n2 = n - 2 as Nat by Th2, NAT_1:21, NAT_1:23;
set ODD = { q where q is Element of Permutations n : q is odd } ;
set EVEN = { p where p is Element of Permutations n : p is even } ;
A6: { p where p is Element of Permutations n : p is even } c= Permutations n
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Element of Permutations n : p is even } or x in Permutations n )
assume x in { p where p is Element of Permutations n : p is even } ; ::_thesis: x in Permutations n
then ex p being Element of Permutations n st
( p = x & p is even ) ;
hence x in Permutations n ; ::_thesis: verum
end;
A7: { q where q is Element of Permutations n : q is odd } c= Permutations n
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { q where q is Element of Permutations n : q is odd } or x in Permutations n )
assume x in { q where q is Element of Permutations n : q is odd } ; ::_thesis: x in Permutations n
then ex q being Element of Permutations n st
( q = x & q is odd ) ;
hence x in Permutations n ; ::_thesis: verum
end;
then reconsider O = { q where q is Element of Permutations n : q is odd } , E = { p where p is Element of Permutations n : p is even } as finite set by A6, FINSET_1:1, MATRIX_2:26;
take O ; ::_thesis: ex EVEN being finite set st
( EVEN = { p where p is Element of Permutations n : p is even } & O = { q where q is Element of Permutations n : q is odd } & EVEN /\ O = {} & EVEN \/ O = Permutations n & ex PERM being Function of EVEN,O ex perm being Element of Permutations n st
( perm is being_transposition & perm . i = j & dom PERM = EVEN & PERM is bijective & ( for p being Element of Permutations n st p in EVEN holds
PERM . p = p * perm ) ) )
take E ; ::_thesis: ( E = { p where p is Element of Permutations n : p is even } & O = { q where q is Element of Permutations n : q is odd } & E /\ O = {} & E \/ O = Permutations n & ex PERM being Function of E,O ex perm being Element of Permutations n st
( perm is being_transposition & perm . i = j & dom PERM = E & PERM is bijective & ( for p being Element of Permutations n st p in E holds
PERM . p = p * perm ) ) )
thus ( E = { p where p is Element of Permutations n : p is even } & O = { q where q is Element of Permutations n : q is odd } ) ; ::_thesis: ( E /\ O = {} & E \/ O = Permutations n & ex PERM being Function of E,O ex perm being Element of Permutations n st
( perm is being_transposition & perm . i = j & dom PERM = E & PERM is bijective & ( for p being Element of Permutations n st p in E holds
PERM . p = p * perm ) ) )
thus E /\ O = {} ::_thesis: ( E \/ O = Permutations n & ex PERM being Function of E,O ex perm being Element of Permutations n st
( perm is being_transposition & perm . i = j & dom PERM = E & PERM is bijective & ( for p being Element of Permutations n st p in E holds
PERM . p = p * perm ) ) )
proof
assume E /\ O <> {} ; ::_thesis: contradiction
then consider x being set such that
A8: x in E /\ O by XBOOLE_0:def_1;
x in O by A8, XBOOLE_0:def_4;
then A9: ex q being Element of Permutations n st
( q = x & q is odd ) ;
x in E by A8, XBOOLE_0:def_4;
then ex p being Element of Permutations n st
( p = x & p is even ) ;
hence contradiction by A9; ::_thesis: verum
end;
thus E \/ O = Permutations n ::_thesis: ex PERM being Function of E,O ex perm being Element of Permutations n st
( perm is being_transposition & perm . i = j & dom PERM = E & PERM is bijective & ( for p being Element of Permutations n st p in E holds
PERM . p = p * perm ) )
proof
thus E \/ O c= Permutations n by A6, A7, XBOOLE_1:8; :: according to XBOOLE_0:def_10 ::_thesis: Permutations n c= E \/ O
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Permutations n or x in E \/ O )
assume x in Permutations n ; ::_thesis: x in E \/ O
then reconsider p = x as Element of Permutations n ;
( p is even or p is odd ) ;
then ( p in E or p in O ) ;
hence x in E \/ O by XBOOLE_0:def_3; ::_thesis: verum
end;
consider PE being Permutation of (Permutations n) such that
A10: for p being Element of Permutations n holds PE . p = p * tr by Th44;
set PERM = PE | E;
tr is Element of Permutations (n2 + 2) ;
then PE .: E = O by A4, A10, Th27, Th47;
then A11: rng (PE | E) = O by RELAT_1:115;
A12: dom PE = Permutations n by FUNCT_2:52;
then A13: dom (PE | E) = E by A6, RELAT_1:62;
then reconsider PERM = PE | E as Function of E,O by A11, FUNCT_2:1;
take PERM ; ::_thesis: ex perm being Element of Permutations n st
( perm is being_transposition & perm . i = j & dom PERM = E & PERM is bijective & ( for p being Element of Permutations n st p in E holds
PERM . p = p * perm ) )
take tr ; ::_thesis: ( tr is being_transposition & tr . i = j & dom PERM = E & PERM is bijective & ( for p being Element of Permutations n st p in E holds
PERM . p = p * tr ) )
( PERM is one-to-one & PERM is onto ) by A11, FUNCT_1:52, FUNCT_2:def_3;
hence ( tr is being_transposition & tr . i = j & dom PERM = E & PERM is bijective ) by A4, A5, A6, A12, RELAT_1:62; ::_thesis: for p being Element of Permutations n st p in E holds
PERM . p = p * tr
let p be Element of Permutations n; ::_thesis: ( p in E implies PERM . p = p * tr )
assume p in E ; ::_thesis: PERM . p = p * tr
then PERM . p = PE . p by A13, FUNCT_1:47;
hence PERM . p = p * tr by A10; ::_thesis: verum
end;
theorem :: MATRIX11:48
for n being Nat st n >= 2 holds
ex ODD, EVEN being finite set st
( EVEN = { p where p is Element of Permutations n : p is even } & ODD = { q where q is Element of Permutations n : q is odd } & EVEN /\ ODD = {} & EVEN \/ ODD = Permutations n & card EVEN = card ODD )
proof
let n be Nat; ::_thesis: ( n >= 2 implies ex ODD, EVEN being finite set st
( EVEN = { p where p is Element of Permutations n : p is even } & ODD = { q where q is Element of Permutations n : q is odd } & EVEN /\ ODD = {} & EVEN \/ ODD = Permutations n & card EVEN = card ODD ) )
assume A1: n >= 2 ; ::_thesis: ex ODD, EVEN being finite set st
( EVEN = { p where p is Element of Permutations n : p is even } & ODD = { q where q is Element of Permutations n : q is odd } & EVEN /\ ODD = {} & EVEN \/ ODD = Permutations n & card EVEN = card ODD )
1 <= n by A1, XXREAL_0:2;
then A2: 1 in Seg n ;
2 in Seg n by A1;
then consider O, E being finite set such that
A3: ( E = { p where p is Element of Permutations n : p is even } & O = { q where q is Element of Permutations n : q is odd } ) and
A4: ( E /\ O = {} & E \/ O = Permutations n ) and
A5: ex P being Function of E,O ex perm being Element of Permutations n st
( perm is being_transposition & perm . 1 = 2 & dom P = E & P is bijective & ( for p being Element of Permutations n st p in E holds
P . p = p * perm ) ) by A2, Lm8;
consider P being Function of E,O, perm being Element of Permutations n such that
perm is being_transposition and
perm . 1 = 2 and
A6: dom P = E and
A7: P is bijective and
for p being Element of Permutations n st p in E holds
P . p = p * perm by A5;
rng P = O by A7, FUNCT_2:def_3;
then E,O are_equipotent by A6, A7, WELLORD2:def_4;
then card E = card O by CARD_1:5;
hence ex ODD, EVEN being finite set st
( EVEN = { p where p is Element of Permutations n : p is even } & ODD = { q where q is Element of Permutations n : q is odd } & EVEN /\ ODD = {} & EVEN \/ ODD = Permutations n & card EVEN = card ODD ) by A3, A4; ::_thesis: verum
end;
theorem Th49: :: MATRIX11:49
for n being Nat
for K being Field
for i, j being Nat st i in Seg n & j in Seg n & i < j holds
for M being Matrix of n,K st Line (M,i) = Line (M,j) holds
for p, q, tr being Element of Permutations n st q = p * tr & tr is being_transposition & tr . i = j holds
(Path_product M) . q = - ((Path_product M) . p)
proof
let n be Nat; ::_thesis: for K being Field
for i, j being Nat st i in Seg n & j in Seg n & i < j holds
for M being Matrix of n,K st Line (M,i) = Line (M,j) holds
for p, q, tr being Element of Permutations n st q = p * tr & tr is being_transposition & tr . i = j holds
(Path_product M) . q = - ((Path_product M) . p)
let K be Field; ::_thesis: for i, j being Nat st i in Seg n & j in Seg n & i < j holds
for M being Matrix of n,K st Line (M,i) = Line (M,j) holds
for p, q, tr being Element of Permutations n st q = p * tr & tr is being_transposition & tr . i = j holds
(Path_product M) . q = - ((Path_product M) . p)
let i, j be Nat; ::_thesis: ( i in Seg n & j in Seg n & i < j implies for M being Matrix of n,K st Line (M,i) = Line (M,j) holds
for p, q, tr being Element of Permutations n st q = p * tr & tr is being_transposition & tr . i = j holds
(Path_product M) . q = - ((Path_product M) . p) )
assume that
A1: i in Seg n and
A2: j in Seg n and
A3: i < j ; ::_thesis: for M being Matrix of n,K st Line (M,i) = Line (M,j) holds
for p, q, tr being Element of Permutations n st q = p * tr & tr is being_transposition & tr . i = j holds
(Path_product M) . q = - ((Path_product M) . p)
{i,j} in 2Set (Seg n) by A1, A2, A3, Th1;
then reconsider n2 = n - 2 as Nat by Th2, NAT_1:21, NAT_1:23;
let M be Matrix of n,K; ::_thesis: ( Line (M,i) = Line (M,j) implies for p, q, tr being Element of Permutations n st q = p * tr & tr is being_transposition & tr . i = j holds
(Path_product M) . q = - ((Path_product M) . p) )
assume A4: Line (M,i) = Line (M,j) ; ::_thesis: for p, q, tr being Element of Permutations n st q = p * tr & tr is being_transposition & tr . i = j holds
(Path_product M) . q = - ((Path_product M) . p)
reconsider M9 = M as Matrix of n2 + 2,K ;
let p, q, tr be Element of Permutations n; ::_thesis: ( q = p * tr & tr is being_transposition & tr . i = j implies (Path_product M) . q = - ((Path_product M) . p) )
assume that
A5: q = p * tr and
A6: tr is being_transposition and
A7: tr . i = j ; ::_thesis: (Path_product M) . q = - ((Path_product M) . p)
reconsider TR = tr as Permutation of (Seg (n2 + 2)) by MATRIX_2:def_9;
set Mt = M9 * TR;
A8: for k being Nat st 1 <= k & k <= len M9 holds
M9 . k = (M9 * TR) . k
proof
let k be Nat; ::_thesis: ( 1 <= k & k <= len M9 implies M9 . k = (M9 * TR) . k )
assume that
A9: 1 <= k and
A10: k <= len M9 ; ::_thesis: M9 . k = (M9 * TR) . k
k in NAT by ORDINAL1:def_12;
then A11: k in Seg (len M9) by A9, A10;
A12: Line (M,j) = M . j by A2, MATRIX_2:8;
A13: dom TR = Seg n by FUNCT_2:52;
A14: Line (M,i) = M . i by A1, MATRIX_2:8;
A15: len M9 = n by MATRIX_1:def_2;
then A16: Line ((M9 * TR),k) = M . (tr . k) by A11, Th38;
percases ( k = i or k = j or ( k <> i & k <> j ) ) ;
suppose k = i ; ::_thesis: M9 . k = (M9 * TR) . k
hence M9 . k = (M9 * TR) . k by A1, A4, A7, A16, A14, A12, MATRIX_2:8; ::_thesis: verum
end;
supposeA17: k = j ; ::_thesis: M9 . k = (M9 * TR) . k
then A18: M . k = M . i by A2, A4, A14, MATRIX_2:8;
Line ((M9 * TR),k) = M . i by A3, A6, A7, A16, A17, Th8;
hence M9 . k = (M9 * TR) . k by A2, A17, A18, MATRIX_2:8; ::_thesis: verum
end;
suppose ( k <> i & k <> j ) ; ::_thesis: M9 . k = (M9 * TR) . k
then Line ((M9 * TR),k) = M . k by A3, A6, A7, A11, A15, A13, A16, Th8;
hence M9 . k = (M9 * TR) . k by A11, A15, MATRIX_2:8; ::_thesis: verum
end;
end;
end;
len (M9 * TR) = len M9 by Def4;
then A19: M9 * TR = M by A8, FINSEQ_1:14;
reconsider Tr = tr, p2 = p as Element of Permutations (n2 + 2) ;
A20: sgn (Tr,K) = - (1_ K) by A6, Th14;
tr = tr " by A6, Th20;
hence (Path_product M) . q = (- (1_ K)) * ((Path_product M9) . p2) by A5, A19, A20, Th43
.= - ((1_ K) * ((Path_product M9) . p2)) by VECTSP_1:9
.= - ((Path_product M) . p) by VECTSP_1:def_4 ;
::_thesis: verum
end;
theorem Th50: :: MATRIX11:50
for n being Nat
for K being Field
for i, j being Nat st i in Seg n & j in Seg n & i < j holds
for M being Matrix of n,K st Line (M,i) = Line (M,j) holds
Det M = 0. K
proof
let n be Nat; ::_thesis: for K being Field
for i, j being Nat st i in Seg n & j in Seg n & i < j holds
for M being Matrix of n,K st Line (M,i) = Line (M,j) holds
Det M = 0. K
let K be Field; ::_thesis: for i, j being Nat st i in Seg n & j in Seg n & i < j holds
for M being Matrix of n,K st Line (M,i) = Line (M,j) holds
Det M = 0. K
let i, j be Nat; ::_thesis: ( i in Seg n & j in Seg n & i < j implies for M being Matrix of n,K st Line (M,i) = Line (M,j) holds
Det M = 0. K )
assume that
A1: i in Seg n and
A2: j in Seg n and
A3: i < j ; ::_thesis: for M being Matrix of n,K st Line (M,i) = Line (M,j) holds
Det M = 0. K
set P = Permutations n;
consider Q, E being finite set such that
( E = { p where p is Element of Permutations n : p is even } & Q = { q where q is Element of Permutations n : q is odd } ) and
A4: ( E /\ Q = {} & E \/ Q = Permutations n ) and
A5: ex P being Function of E,Q ex tr being Element of Permutations n st
( tr is being_transposition & tr . i = j & dom P = E & P is bijective & ( for p being Element of Permutations n st p in E holds
P . p = p * tr ) ) by A1, A2, A3, Lm8;
A6: E c= Permutations n by A4, XBOOLE_1:7;
set KK = the carrier of K;
set aa = the addF of K;
let M be Matrix of n,K; ::_thesis: ( Line (M,i) = Line (M,j) implies Det M = 0. K )
assume A7: Line (M,i) = Line (M,j) ; ::_thesis: Det M = 0. K
A8: Q c= Permutations n by A4, XBOOLE_1:7;
set PathM = Path_product M;
consider PERM being Function of E,Q, tr being Element of Permutations n such that
A9: tr is being_transposition and
A10: tr . i = j and
A11: dom PERM = E and
A12: PERM is bijective and
A13: for p being Element of Permutations n st p in E holds
PERM . p = p * tr by A5;
reconsider E = E, Q = Q as Element of Fin (Permutations n) by A6, A8, FINSUB_1:def_5;
the addF of K is having_a_unity by FVSUM_1:8;
then consider GE being Function of (Fin (Permutations n)), the carrier of K such that
A14: the addF of K $$ (E,(Path_product M)) = GE . E and
A15: for e being Element of the carrier of K st e is_a_unity_wrt the addF of K holds
GE . {} = e and
A16: for x being Element of Permutations n holds GE . {x} = (Path_product M) . x and
A17: for B9 being Element of Fin (Permutations n) st B9 c= E & B9 <> {} holds
for x being Element of Permutations n st x in E \ B9 holds
GE . (B9 \/ {x}) = the addF of K . ((GE . B9),((Path_product M) . x)) by SETWISEO:def_3;
A18: E misses Q by A4, XBOOLE_0:def_7;
the addF of K is having_a_unity by FVSUM_1:8;
then consider GQ being Function of (Fin (Permutations n)), the carrier of K such that
A19: the addF of K $$ (Q,(Path_product M)) = GQ . Q and
A20: for e being Element of the carrier of K st e is_a_unity_wrt the addF of K holds
GQ . {} = e and
A21: for x being Element of Permutations n holds GQ . {x} = (Path_product M) . x and
A22: for B9 being Element of Fin (Permutations n) st B9 c= Q & B9 <> {} holds
for x being Element of Permutations n st x in Q \ B9 holds
GQ . (B9 \/ {x}) = the addF of K . ((GQ . B9),((Path_product M) . x)) by SETWISEO:def_3;
defpred S1[ Nat] means for B, PB being Element of Fin (Permutations n) st card B = $1 & B c= E & PERM .: B = PB holds
(GE . B) + (GQ . PB) = 0. K;
A23: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A24: S1[k] ; ::_thesis: S1[k + 1]
let B, PB be Element of Fin (Permutations n); ::_thesis: ( card B = k + 1 & B c= E & PERM .: B = PB implies (GE . B) + (GQ . PB) = 0. K )
assume that
A25: card B = k + 1 and
A26: B c= E and
A27: PERM .: B = PB ; ::_thesis: (GE . B) + (GQ . PB) = 0. K
now__::_thesis:_(_(_k_=_0_&_(GE_._B)_+_(GQ_._PB)_=_0._K_)_or_(_k_>_0_&_(GQ_._PB)_+_(GE_._B)_=_0._K_)_)
percases ( k = 0 or k > 0 ) ;
case k = 0 ; ::_thesis: (GE . B) + (GQ . PB) = 0. K
then consider x being set such that
A28: B = {x} by A25, CARD_2:42;
A29: x in B by A28, TARSKI:def_1;
B c= Permutations n by FINSUB_1:def_5;
then reconsider x = x as Element of Permutations n by A29;
x * tr is Element of Permutations n by MATRIX_9:39;
then reconsider Px = PERM . x as Element of Permutations n by A13, A26, A29;
A30: Im (PERM,x) = {Px} by A11, A26, A29, FUNCT_1:59;
A31: GE . {x} = (Path_product M) . x by A16;
A32: GQ . {(PERM . x)} = (Path_product M) . Px by A21;
Px = x * tr by A13, A26, A29;
then - (GE . B) = GQ . PB by A1, A2, A3, A7, A9, A10, A27, A28, A31, A32, A30, Th49;
hence (GE . B) + (GQ . PB) = 0. K by RLVECT_1:def_10; ::_thesis: verum
end;
caseA33: k > 0 ; ::_thesis: (GQ . PB) + (GE . B) = 0. K
consider x being set such that
A34: x in B by A25, CARD_1:27, XBOOLE_0:def_1;
B c= Permutations n by FINSUB_1:def_5;
then reconsider x = x as Element of Permutations n by A34;
x * tr is Element of Permutations n by MATRIX_9:39;
then reconsider Px = PERM . x as Element of Permutations n by A13, A26, A34;
A35: Im (PERM,x) = {Px} by A11, A26, A34, FUNCT_1:59;
Px = x * tr by A13, A26, A34;
then A36: - ((Path_product M) . x) = (Path_product M) . Px by A1, A2, A3, A7, A9, A10, Th49;
A37: Q c= Permutations n by FINSUB_1:def_5;
B c= Permutations n by FINSUB_1:def_5;
then A38: B \ {x} c= Permutations n by XBOOLE_1:1;
A39: rng PERM = Q by A12, FUNCT_2:def_3;
then A40: Px in Q by A11, A26, A34, FUNCT_1:def_3;
PERM .: (B \ {x}) c= rng PERM by RELAT_1:111;
then PERM .: (B \ {x}) c= Permutations n by A39, A37, XBOOLE_1:1;
then reconsider B9 = B \ {x}, PeBx = PERM .: (B \ {x}) as Element of Fin (Permutations n) by A38, FINSUB_1:def_5;
A41: Px in {Px} by TARSKI:def_1;
A42: {x} \/ B9 = B by A34, ZFMISC_1:116;
then A43: PERM .: B = (Im (PERM,x)) \/ PeBx by RELAT_1:120;
B9 misses {x} by XBOOLE_1:79;
then B9 /\ {x} = {} by XBOOLE_0:def_7;
then PERM .: {} = {Px} /\ PeBx by A12, A35, FUNCT_1:62;
then not Px in PeBx by A41, XBOOLE_0:def_4;
then A44: Px in Q \ PeBx by A40, XBOOLE_0:def_5;
A45: not x in B9 by ZFMISC_1:56;
then A46: x in E \ B9 by A26, A34, XBOOLE_0:def_5;
A47: k + 1 = (card B9) + 1 by A25, A42, A45, CARD_2:41;
then consider y being set such that
A48: y in B9 by A33, CARD_1:27, XBOOLE_0:def_1;
B \ {x} c= E by A26, XBOOLE_1:1;
then PERM . y in PeBx by A11, A48, FUNCT_1:def_6;
then GQ . PB = the addF of K . ((GQ . PeBx),((Path_product M) . Px)) by A22, A27, A39, A43, A35, A44, RELAT_1:111;
hence (GQ . PB) + (GE . B) = ((GQ . PeBx) - ((Path_product M) . x)) + ((GE . B9) + ((Path_product M) . x)) by A17, A26, A33, A42, A47, A46, A36, CARD_1:27, XBOOLE_1:1
.= (GQ . PeBx) + ((- ((Path_product M) . x)) + ((GE . B9) + ((Path_product M) . x))) by RLVECT_1:def_3
.= (GQ . PeBx) + ((GE . B9) + (((Path_product M) . x) - ((Path_product M) . x))) by RLVECT_1:def_3
.= (GQ . PeBx) + ((GE . B9) + (0. K)) by RLVECT_1:def_10
.= ((GQ . PeBx) + (GE . B9)) + (0. K) by RLVECT_1:def_3
.= (0. K) + (0. K) by A24, A26, A47, XBOOLE_1:1
.= 0. K by RLVECT_1:4 ;
::_thesis: verum
end;
end;
end;
hence (GE . B) + (GQ . PB) = 0. K ; ::_thesis: verum
end;
set F = FinOmega (Permutations n);
A49: Permutations n = FinOmega (Permutations n) by A4, MATRIX_2:def_14;
rng PERM = Q by A12, FUNCT_2:def_3;
then A50: PERM .: E = Q by A11, RELAT_1:113;
A51: S1[ 0 ]
proof
let B, PB be Element of Fin (Permutations n); ::_thesis: ( card B = 0 & B c= E & PERM .: B = PB implies (GE . B) + (GQ . PB) = 0. K )
assume that
A52: card B = 0 and
B c= E and
A53: PERM .: B = PB ; ::_thesis: (GE . B) + (GQ . PB) = 0. K
A54: B = {} by A52;
then A55: GE . B = 0. K by A15, FVSUM_1:6;
PERM .: {} = {} ;
then GQ . PB = 0. K by A20, A53, A54, FVSUM_1:6;
hence (GE . B) + (GQ . PB) = 0. K by A55, RLVECT_1:def_4; ::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A51, A23);
then S1[ card E] ;
then ( the addF of K $$ (E,(Path_product M))) + ( the addF of K $$ (Q,(Path_product M))) = 0. K by A14, A19, A50;
hence Det M = 0. K by A4, A18, A49, FVSUM_1:8, SETWOP_2:4; ::_thesis: verum
end;
theorem Th51: :: MATRIX11:51
for n being Nat
for K being Field
for A being Matrix of n,K
for i, j being Nat st i in Seg n & j in Seg n & i <> j holds
Det (RLine (A,i,(Line (A,j)))) = 0. K
proof
let n be Nat; ::_thesis: for K being Field
for A being Matrix of n,K
for i, j being Nat st i in Seg n & j in Seg n & i <> j holds
Det (RLine (A,i,(Line (A,j)))) = 0. K
let K be Field; ::_thesis: for A being Matrix of n,K
for i, j being Nat st i in Seg n & j in Seg n & i <> j holds
Det (RLine (A,i,(Line (A,j)))) = 0. K
let A be Matrix of n,K; ::_thesis: for i, j being Nat st i in Seg n & j in Seg n & i <> j holds
Det (RLine (A,i,(Line (A,j)))) = 0. K
let i, j be Nat; ::_thesis: ( i in Seg n & j in Seg n & i <> j implies Det (RLine (A,i,(Line (A,j)))) = 0. K )
assume that
A1: i in Seg n and
A2: j in Seg n and
A3: i <> j ; ::_thesis: Det (RLine (A,i,(Line (A,j)))) = 0. K
A4: ( i < j or j < i ) by A3, XXREAL_0:1;
len (Line (A,j)) = width A by MATRIX_1:def_7;
then A5: Line ((RLine (A,i,(Line (A,j)))),i) = Line (A,j) by A1, Th28;
Line ((RLine (A,i,(Line (A,j)))),j) = Line (A,j) by A2, A3, Th28;
hence Det (RLine (A,i,(Line (A,j)))) = 0. K by A1, A2, A5, A4, Th50; ::_thesis: verum
end;
theorem Th52: :: MATRIX11:52
for n being Nat
for K being Field
for a being Element of K
for A being Matrix of n,K
for i, j being Nat st i in Seg n & j in Seg n & i <> j holds
Det (RLine (A,i,(a * (Line (A,j))))) = 0. K
proof
let n be Nat; ::_thesis: for K being Field
for a being Element of K
for A being Matrix of n,K
for i, j being Nat st i in Seg n & j in Seg n & i <> j holds
Det (RLine (A,i,(a * (Line (A,j))))) = 0. K
let K be Field; ::_thesis: for a being Element of K
for A being Matrix of n,K
for i, j being Nat st i in Seg n & j in Seg n & i <> j holds
Det (RLine (A,i,(a * (Line (A,j))))) = 0. K
let a be Element of K; ::_thesis: for A being Matrix of n,K
for i, j being Nat st i in Seg n & j in Seg n & i <> j holds
Det (RLine (A,i,(a * (Line (A,j))))) = 0. K
let A be Matrix of n,K; ::_thesis: for i, j being Nat st i in Seg n & j in Seg n & i <> j holds
Det (RLine (A,i,(a * (Line (A,j))))) = 0. K
let i, j be Nat; ::_thesis: ( i in Seg n & j in Seg n & i <> j implies Det (RLine (A,i,(a * (Line (A,j))))) = 0. K )
assume that
A1: i in Seg n and
A2: j in Seg n and
A3: i <> j ; ::_thesis: Det (RLine (A,i,(a * (Line (A,j))))) = 0. K
width A = n by MATRIX_1:24;
then len (Line (A,j)) = n by MATRIX_1:def_7;
hence Det (RLine (A,i,(a * (Line (A,j))))) = a * (Det (RLine (A,i,(Line (A,j))))) by A1, Th34
.= a * (0. K) by A1, A2, A3, Th51
.= 0. K by VECTSP_1:6 ;
::_thesis: verum
end;
theorem :: MATRIX11:53
for n being Nat
for K being Field
for a being Element of K
for A being Matrix of n,K
for i, j being Nat st i in Seg n & j in Seg n & i <> j holds
Det A = Det (RLine (A,i,((Line (A,i)) + (a * (Line (A,j))))))
proof
let n be Nat; ::_thesis: for K being Field
for a being Element of K
for A being Matrix of n,K
for i, j being Nat st i in Seg n & j in Seg n & i <> j holds
Det A = Det (RLine (A,i,((Line (A,i)) + (a * (Line (A,j))))))
let K be Field; ::_thesis: for a being Element of K
for A being Matrix of n,K
for i, j being Nat st i in Seg n & j in Seg n & i <> j holds
Det A = Det (RLine (A,i,((Line (A,i)) + (a * (Line (A,j))))))
let a be Element of K; ::_thesis: for A being Matrix of n,K
for i, j being Nat st i in Seg n & j in Seg n & i <> j holds
Det A = Det (RLine (A,i,((Line (A,i)) + (a * (Line (A,j))))))
let A be Matrix of n,K; ::_thesis: for i, j being Nat st i in Seg n & j in Seg n & i <> j holds
Det A = Det (RLine (A,i,((Line (A,i)) + (a * (Line (A,j))))))
let i, j be Nat; ::_thesis: ( i in Seg n & j in Seg n & i <> j implies Det A = Det (RLine (A,i,((Line (A,i)) + (a * (Line (A,j)))))) )
assume that
A1: i in Seg n and
A2: j in Seg n and
A3: i <> j ; ::_thesis: Det A = Det (RLine (A,i,((Line (A,i)) + (a * (Line (A,j))))))
A4: width A = n by MATRIX_1:24;
then A5: len (Line (A,j)) = n by MATRIX_1:def_7;
A6: len (Line (A,j)) = len (a * (Line (A,j))) by Lm5;
len (Line (A,i)) = n by A4, MATRIX_1:def_7;
hence Det (RLine (A,i,((Line (A,i)) + (a * (Line (A,j)))))) = (Det (RLine (A,i,(Line (A,i))))) + (Det (RLine (A,i,(a * (Line (A,j)))))) by A1, A5, A6, Th36
.= (Det A) + (Det (RLine (A,i,(a * (Line (A,j)))))) by Th30
.= (Det A) + (0. K) by A1, A2, A3, Th52
.= Det A by RLVECT_1:4 ;
::_thesis: verum
end;
theorem Th54: :: MATRIX11:54
for n being Nat
for K being Field
for F being Function of (Seg n),(Seg n)
for A being Matrix of n,K st not F in Permutations n holds
Det (A * F) = 0. K
proof
let n be Nat; ::_thesis: for K being Field
for F being Function of (Seg n),(Seg n)
for A being Matrix of n,K st not F in Permutations n holds
Det (A * F) = 0. K
let K be Field; ::_thesis: for F being Function of (Seg n),(Seg n)
for A being Matrix of n,K st not F in Permutations n holds
Det (A * F) = 0. K
let F be Function of (Seg n),(Seg n); ::_thesis: for A being Matrix of n,K st not F in Permutations n holds
Det (A * F) = 0. K
let A be Matrix of n,K; ::_thesis: ( not F in Permutations n implies Det (A * F) = 0. K )
assume not F in Permutations n ; ::_thesis: Det (A * F) = 0. K
then A1: ( not F is onto or not F is one-to-one ) by MATRIX_2:def_9;
card (Seg n) = card (Seg n) ;
then not F is one-to-one by A1, STIRL2_1:60;
then consider x, y being set such that
A2: x in dom F and
A3: y in dom F and
A4: F . x = F . y and
A5: x <> y by FUNCT_1:def_4;
A6: dom F = Seg n by FUNCT_2:52;
then reconsider x = x, y = y as Nat by A2, A3;
Line ((A * F),x) = A . (F . x) by A2, A6, Th38;
then A7: Line ((A * F),x) = Line ((A * F),y) by A3, A4, A6, Th38;
( x > y or y > x ) by A5, XXREAL_0:1;
hence Det (A * F) = 0. K by A2, A3, A6, A7, Th50; ::_thesis: verum
end;
begin
definition
let K be non empty addLoopStr ;
func addFinS K -> BinOp of ( the carrier of K *) means :Def5: :: MATRIX11:def 5
for p1, p2 being Element of the carrier of K * holds it . (p1,p2) = p1 + p2;
existence
ex b1 being BinOp of ( the carrier of K *) st
for p1, p2 being Element of the carrier of K * holds b1 . (p1,p2) = p1 + p2
proof
set KK = the carrier of K;
defpred S1[ set , set , set ] means for p1, p2 being Element of the carrier of K * st $1 = p1 & $2 = p2 holds
$3 = p1 + p2;
A1: for x, y being Element of the carrier of K * ex z being Element of the carrier of K * st S1[x,y,z]
proof
let x, y be Element of the carrier of K * ; ::_thesis: ex z being Element of the carrier of K * st S1[x,y,z]
reconsider p1 = x, p2 = y as FinSequence of the carrier of K ;
reconsider pp = p1 + p2 as Element of the carrier of K * by FINSEQ_1:def_11;
take pp ; ::_thesis: S1[x,y,pp]
thus S1[x,y,pp] ; ::_thesis: verum
end;
consider A being Function of [:( the carrier of K *),( the carrier of K *):],( the carrier of K *) such that
A2: for x, y being Element of the carrier of K * holds S1[x,y,A . (x,y)] from BINOP_1:sch_3(A1);
take A ; ::_thesis: for p1, p2 being Element of the carrier of K * holds A . (p1,p2) = p1 + p2
thus for p1, p2 being Element of the carrier of K * holds A . (p1,p2) = p1 + p2 by A2; ::_thesis: verum
end;
uniqueness
for b1, b2 being BinOp of ( the carrier of K *) st ( for p1, p2 being Element of the carrier of K * holds b1 . (p1,p2) = p1 + p2 ) & ( for p1, p2 being Element of the carrier of K * holds b2 . (p1,p2) = p1 + p2 ) holds
b1 = b2
proof
set KK = the carrier of K;
let f1, f2 be Function of [:( the carrier of K *),( the carrier of K *):],( the carrier of K *); ::_thesis: ( ( for p1, p2 being Element of the carrier of K * holds f1 . (p1,p2) = p1 + p2 ) & ( for p1, p2 being Element of the carrier of K * holds f2 . (p1,p2) = p1 + p2 ) implies f1 = f2 )
assume that
A3: for p1, p2 being Element of the carrier of K * holds f1 . (p1,p2) = p1 + p2 and
A4: for p1, p2 being Element of the carrier of K * holds f2 . (p1,p2) = p1 + p2 ; ::_thesis: f1 = f2
now__::_thesis:_for_p1,_p2_being_Element_of_the_carrier_of_K_*_holds_f1_._(p1,p2)_=_f2_._(p1,p2)
let p1, p2 be Element of the carrier of K * ; ::_thesis: f1 . (p1,p2) = f2 . (p1,p2)
f1 . (p1,p2) = p1 + p2 by A3;
hence f1 . (p1,p2) = f2 . (p1,p2) by A4; ::_thesis: verum
end;
hence f1 = f2 by BINOP_1:2; ::_thesis: verum
end;
end;
:: deftheorem Def5 defines addFinS MATRIX11:def_5_:_
for K being non empty addLoopStr
for b2 being BinOp of ( the carrier of K *) holds
( b2 = addFinS K iff for p1, p2 being Element of the carrier of K * holds b2 . (p1,p2) = p1 + p2 );
Lm9: for K being non empty addLoopStr
for p1, p2 being Element of the carrier of K * holds dom (p1 + p2) = (dom p1) /\ (dom p2)
proof
let K be non empty addLoopStr ; ::_thesis: for p1, p2 being Element of the carrier of K * holds dom (p1 + p2) = (dom p1) /\ (dom p2)
let p1, p2 be Element of the carrier of K * ; ::_thesis: dom (p1 + p2) = (dom p1) /\ (dom p2)
A1: rng p2 c= the carrier of K by FINSEQ_1:def_4;
rng p1 c= the carrier of K by FINSEQ_1:def_4;
then [:(rng p1),(rng p2):] c= [: the carrier of K, the carrier of K:] by A1, ZFMISC_1:96;
then [:(rng p1),(rng p2):] c= dom the addF of K by FUNCT_2:def_1;
hence dom (p1 + p2) = (dom p1) /\ (dom p2) by FUNCOP_1:69; ::_thesis: verum
end;
registration
let K be non empty Abelian addLoopStr ;
cluster addFinS K -> commutative ;
coherence
addFinS K is commutative
proof
set KK = the carrier of K;
let p1, p2 be Element of the carrier of K * ; :: according to BINOP_1:def_2 ::_thesis: (addFinS K) . (p1,p2) = (addFinS K) . (p2,p1)
A1: rng p2 c= the carrier of K by FINSEQ_1:def_4;
A2: dom (p1 + p2) = (dom p1) /\ (dom p2) by Lm9;
A3: dom (p2 + p1) = (dom p2) /\ (dom p1) by Lm9;
A4: rng p1 c= the carrier of K by FINSEQ_1:def_4;
now__::_thesis:_for_k_being_Nat_st_k_in_dom_(p1_+_p2)_holds_
(p1_+_p2)_._k_=_(p2_+_p1)_._k
let k be Nat; ::_thesis: ( k in dom (p1 + p2) implies (p1 + p2) . k = (p2 + p1) . k )
assume A5: k in dom (p1 + p2) ; ::_thesis: (p1 + p2) . k = (p2 + p1) . k
k in dom p2 by A2, A5, XBOOLE_0:def_4;
then A6: p2 . k in rng p2 by FUNCT_1:def_3;
k in dom p1 by A2, A5, XBOOLE_0:def_4;
then p1 . k in rng p1 by FUNCT_1:def_3;
then reconsider p1k = p1 . k, p2k = p2 . k as Element of K by A4, A1, A6;
(p1 + p2) . k = p1k + p2k by A5, FVSUM_1:17;
hence (p1 + p2) . k = (p2 + p1) . k by A2, A3, A5, FVSUM_1:17; ::_thesis: verum
end;
then A7: p1 + p2 = p2 + p1 by A3, Lm9, FINSEQ_1:13;
(addFinS K) . (p1,p2) = p1 + p2 by Def5;
hence (addFinS K) . (p1,p2) = (addFinS K) . (p2,p1) by A7, Def5; ::_thesis: verum
end;
end;
registration
let K be non empty add-associative addLoopStr ;
cluster addFinS K -> associative ;
coherence
addFinS K is associative
proof
set aK = addFinS K;
set KK = the carrier of K;
let p1, p2, p3 be Element of the carrier of K * ; :: according to BINOP_1:def_3 ::_thesis: (addFinS K) . (p1,((addFinS K) . (p2,p3))) = (addFinS K) . (((addFinS K) . (p1,p2)),p3)
reconsider p12 = p1 + p2, p23 = p2 + p3 as Element of the carrier of K * by FINSEQ_1:def_11;
A1: rng p1 c= the carrier of K by FINSEQ_1:def_4;
A2: rng p2 c= the carrier of K by FINSEQ_1:def_4;
A3: rng p12 c= the carrier of K by FINSEQ_1:def_4;
A4: rng p3 c= the carrier of K by FINSEQ_1:def_4;
A5: rng p23 c= the carrier of K by FINSEQ_1:def_4;
A6: dom p12 = (dom p1) /\ (dom p2) by Lm9;
A7: dom p23 = (dom p2) /\ (dom p3) by Lm9;
A8: dom (p12 + p3) = (dom p12) /\ (dom p3) by Lm9;
A9: dom (p1 + p23) = (dom p1) /\ (dom p23) by Lm9;
then A10: dom (p12 + p3) = dom (p1 + p23) by A6, A8, A7, XBOOLE_1:16;
now__::_thesis:_for_k_being_Nat_st_k_in_dom_(p12_+_p3)_holds_
(p1_+_p23)_._k_=_(p12_+_p3)_._k
let k be Nat; ::_thesis: ( k in dom (p12 + p3) implies (p1 + p23) . k = (p12 + p3) . k )
assume A11: k in dom (p12 + p3) ; ::_thesis: (p1 + p23) . k = (p12 + p3) . k
A12: k in dom p12 by A8, A11, XBOOLE_0:def_4;
then A13: p12 . k in rng p12 by FUNCT_1:def_3;
k in dom p1 by A6, A12, XBOOLE_0:def_4;
then A14: p1 . k in rng p1 by FUNCT_1:def_3;
A15: k in dom p3 by A8, A11, XBOOLE_0:def_4;
then A16: p3 . k in rng p3 by FUNCT_1:def_3;
A17: k in dom p2 by A6, A12, XBOOLE_0:def_4;
then A18: p2 . k in rng p2 by FUNCT_1:def_3;
A19: k in dom p23 by A7, A15, A17, XBOOLE_0:def_4;
then p23 . k in rng p23 by FUNCT_1:def_3;
then reconsider p1k = p1 . k, p12k = p12 . k, p2k = p2 . k, p23k = p23 . k, p3k = p3 . k as Element of K by A1, A2, A4, A3, A5, A14, A13, A16, A18;
A20: p12 . k = p1k + p2k by A12, FVSUM_1:17;
A21: (p12 + p3) . k = p12k + p3k by A11, FVSUM_1:17;
A22: p23 . k = p2k + p3k by A19, FVSUM_1:17;
(p1 + p23) . k = p1k + p23k by A10, A11, FVSUM_1:17;
hence (p1 + p23) . k = (p12 + p3) . k by A20, A22, A21, RLVECT_1:def_3; ::_thesis: verum
end;
then A23: p1 + p23 = p12 + p3 by A6, A8, A7, A9, FINSEQ_1:13, XBOOLE_1:16;
thus (addFinS K) . (p1,((addFinS K) . (p2,p3))) = (addFinS K) . (p1,p23) by Def5
.= p1 + p23 by Def5
.= (addFinS K) . (p12,p3) by A23, Def5
.= (addFinS K) . (((addFinS K) . (p1,p2)),p3) by Def5 ; ::_thesis: verum
end;
end;
theorem Th55: :: MATRIX11:55
for K being Field
for A, B being Matrix of K st width A = len B & len B > 0 holds
for i being Nat st i in Seg (len A) holds
ex P being FinSequence of the carrier of K * st
( len P = len B & Line ((A * B),i) = (addFinS K) "**" P & ( for j being Nat st j in Seg (len B) holds
P . j = (A * (i,j)) * (Line (B,j)) ) )
proof
let K be Field; ::_thesis: for A, B being Matrix of K st width A = len B & len B > 0 holds
for i being Nat st i in Seg (len A) holds
ex P being FinSequence of the carrier of K * st
( len P = len B & Line ((A * B),i) = (addFinS K) "**" P & ( for j being Nat st j in Seg (len B) holds
P . j = (A * (i,j)) * (Line (B,j)) ) )
let A, B be Matrix of K; ::_thesis: ( width A = len B & len B > 0 implies for i being Nat st i in Seg (len A) holds
ex P being FinSequence of the carrier of K * st
( len P = len B & Line ((A * B),i) = (addFinS K) "**" P & ( for j being Nat st j in Seg (len B) holds
P . j = (A * (i,j)) * (Line (B,j)) ) ) )
assume that
A1: width A = len B and
A2: len B > 0 ; ::_thesis: for i being Nat st i in Seg (len A) holds
ex P being FinSequence of the carrier of K * st
( len P = len B & Line ((A * B),i) = (addFinS K) "**" P & ( for j being Nat st j in Seg (len B) holds
P . j = (A * (i,j)) * (Line (B,j)) ) )
set aa = the addF of K;
set mm = the multF of K;
set a = addFinS K;
set KK = the carrier of K;
reconsider m = len B, w = width B as Nat ;
let i be Nat; ::_thesis: ( i in Seg (len A) implies ex P being FinSequence of the carrier of K * st
( len P = len B & Line ((A * B),i) = (addFinS K) "**" P & ( for j being Nat st j in Seg (len B) holds
P . j = (A * (i,j)) * (Line (B,j)) ) ) )
assume A3: i in Seg (len A) ; ::_thesis: ex P being FinSequence of the carrier of K * st
( len P = len B & Line ((A * B),i) = (addFinS K) "**" P & ( for j being Nat st j in Seg (len B) holds
P . j = (A * (i,j)) * (Line (B,j)) ) )
deffunc H1( Nat) -> Element of (width B) -tuples_on the carrier of K = (A * (i,$1)) * (Line (B,$1));
consider P being FinSequence such that
A4: len P = len B and
A5: for k being Nat st k in dom P holds
P . k = H1(k) from FINSEQ_1:sch_2();
A6: dom P = dom B by A4, FINSEQ_3:29
.= Seg (len B) by FINSEQ_1:def_3 ;
rng P c= the carrier of K *
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng P or y in the carrier of K * )
assume y in rng P ; ::_thesis: y in the carrier of K *
then consider x being set such that
A7: x in dom P and
A8: y = P . x by FUNCT_1:def_3;
reconsider x = x as Element of NAT by A7;
P . x = (A * (i,x)) * (Line (B,x)) by A5, A7;
hence y in the carrier of K * by A8, FINSEQ_1:def_11; ::_thesis: verum
end;
then reconsider P = P as FinSequence of the carrier of K * by FINSEQ_1:def_4;
A9: m >= 1 by A2, NAT_1:14;
then consider F being Function of NAT,( the carrier of K *) such that
A10: F . 1 = P . 1 and
A11: for n being Element of NAT st 0 <> n & n < len P holds
F . (n + 1) = (addFinS K) . ((F . n),(P . (n + 1))) and
A12: (addFinS K) "**" P = F . (len P) by A4, FINSOP_1:def_1;
defpred S1[ Nat] means ( 1 <= $1 & $1 <= m implies for F1 being FinSequence of the carrier of K st F . $1 = F1 holds
( len F1 = w & ( for j being Element of NAT st j in Seg w holds
ex LC being FinSequence of the carrier of K st
( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg $1) & the addF of K "**" LC = F1 . j ) ) ) );
A13: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A14: S1[k] ; ::_thesis: S1[k + 1]
set k1 = k + 1;
assume that
A15: 1 <= k + 1 and
A16: k + 1 <= m ; ::_thesis: for F1 being FinSequence of the carrier of K st F . (k + 1) = F1 holds
( len F1 = w & ( for j being Element of NAT st j in Seg w holds
ex LC being FinSequence of the carrier of K st
( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC = F1 . j ) ) )
A17: k + 1 in Seg m by A15, A16;
let Fk1 be FinSequence of the carrier of K; ::_thesis: ( F . (k + 1) = Fk1 implies ( len Fk1 = w & ( for j being Element of NAT st j in Seg w holds
ex LC being FinSequence of the carrier of K st
( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC = Fk1 . j ) ) ) )
assume A18: F . (k + 1) = Fk1 ; ::_thesis: ( len Fk1 = w & ( for j being Element of NAT st j in Seg w holds
ex LC being FinSequence of the carrier of K st
( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC = Fk1 . j ) ) )
percases ( k = 0 or k > 0 ) ;
supposeA19: k = 0 ; ::_thesis: ( len Fk1 = w & ( for j being Element of NAT st j in Seg w holds
ex LC being FinSequence of the carrier of K st
( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC = Fk1 . j ) ) )
then A20: P . 1 = (A * (i,1)) * (Line (B,1)) by A5, A6, A17;
A21: len (Line (B,1)) = w by MATRIX_1:def_7;
hence len Fk1 = w by A10, A18, A19, A20, Lm5; ::_thesis: for j being Element of NAT st j in Seg w holds
ex LC being FinSequence of the carrier of K st
( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC = Fk1 . j )
let j be Element of NAT ; ::_thesis: ( j in Seg w implies ex LC being FinSequence of the carrier of K st
( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC = Fk1 . j ) )
assume A22: j in Seg w ; ::_thesis: ex LC being FinSequence of the carrier of K st
( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC = Fk1 . j )
len Fk1 = w by A10, A18, A19, A20, A21, Lm5;
then A23: j in dom Fk1 by A22, FINSEQ_1:def_3;
(Line (B,1)) . j = B * (1,j) by A22, MATRIX_1:def_7;
then A24: Fk1 . j = (A * (i,1)) * (B * (1,j)) by A10, A18, A19, A20, A23, FVSUM_1:50;
set C = Col (B,j);
set L = Line (A,i);
reconsider LC1 = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)), mLC = mlt ((Line (A,i)),(Col (B,j))) as FinSequence of the carrier of K by FINSEQ_1:18;
A25: the multF of K .: ((Line (A,i)),(Col (B,j))) is Element of m -tuples_on the carrier of K by A1, FINSEQ_2:120;
then len mLC = m by CARD_1:def_7;
then A26: dom mLC = Seg m by FINSEQ_1:def_3;
Seg 1 = (Seg m) /\ (Seg 1) by A2, FINSEQ_1:7, NAT_1:14;
then A27: dom LC1 = Seg 1 by A19, A26, RELAT_1:61;
then A28: len LC1 = 1 by FINSEQ_1:def_3;
1 in Seg 1 ;
then LC1 . 1 = mLC . 1 by A27, FUNCT_1:47;
then A29: LC1 = <*(mLC . 1)*> by A28, FINSEQ_1:40;
take LC1 ; ::_thesis: ( LC1 = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC1 = Fk1 . j )
A30: 1 in Seg m by A9;
len mLC = m by A25, CARD_1:def_7;
then A31: 1 in dom mLC by A30, FINSEQ_1:def_3;
Seg m = dom B by FINSEQ_1:def_3;
then A32: (Col (B,j)) . 1 = B * (1,j) by A30, MATRIX_1:def_8;
(Line (A,i)) . 1 = A * (i,1) by A1, A30, MATRIX_1:def_7;
then mLC . 1 = (A * (i,1)) * (B * (1,j)) by A32, A31, FVSUM_1:60;
hence ( LC1 = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC1 = Fk1 . j ) by A24, A29, FINSOP_1:11; ::_thesis: verum
end;
supposeA33: k > 0 ; ::_thesis: ( len Fk1 = w & ( for j being Element of NAT st j in Seg w holds
ex LC being FinSequence of the carrier of K st
( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC = Fk1 . j ) ) )
dom P = Seg m by A4, FINSEQ_1:def_3;
then A34: P . (k + 1) in rng P by A17, FUNCT_1:def_3;
rng P c= the carrier of K * by FINSEQ_1:def_4;
then reconsider Pk1 = P . (k + 1), Fk = F . k as FinSequence of the carrier of K by A34, FINSEQ_1:def_11;
A35: Pk1 is Element of the carrier of K * by FINSEQ_1:def_11;
then A36: (addFinS K) . (Fk,Pk1) = Fk + Pk1 by Def5;
A37: k + 0 < k + 1 by XREAL_1:8;
then k < m by A16, XXREAL_0:2;
then A38: Fk1 = Fk + Pk1 by A4, A11, A18, A33, A36;
A39: len (Line (B,(k + 1))) = w by MATRIX_1:def_7;
Pk1 = H1(k + 1) by A5, A6, A17;
then len Pk1 = w by A39, Lm5;
then A40: dom Pk1 = Seg w by FINSEQ_1:def_3;
A41: len Fk = w by A14, A16, A33, A37, NAT_1:14, XXREAL_0:2;
then dom Fk = Seg w by FINSEQ_1:def_3;
then A42: dom (Fk + Pk1) = (Seg w) /\ (Seg w) by A40, A35, Lm9;
hence len Fk1 = w by A38, FINSEQ_1:def_3; ::_thesis: for j being Element of NAT st j in Seg w holds
ex LC being FinSequence of the carrier of K st
( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC = Fk1 . j )
A43: rng Fk c= the carrier of K by FINSEQ_1:def_4;
set L = Line (A,i);
A44: Pk1 = H1(k + 1) by A5, A6, A17;
let j be Element of NAT ; ::_thesis: ( j in Seg w implies ex LC being FinSequence of the carrier of K st
( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC = Fk1 . j ) )
assume A45: j in Seg w ; ::_thesis: ex LC being FinSequence of the carrier of K st
( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC = Fk1 . j )
A46: (Line (B,(k + 1))) . j = B * ((k + 1),j) by A45, MATRIX_1:def_7;
then Pk1 . j = (A * (i,(k + 1))) * (B * ((k + 1),j)) by A40, A45, A44, FVSUM_1:50;
then reconsider Pk1j = Pk1 . j as Element of the carrier of K ;
set C = Col (B,j);
consider LC being FinSequence of the carrier of K such that
A47: LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg k) and
A48: the addF of K "**" LC = Fk . j by A14, A16, A33, A37, A45, NAT_1:14, XXREAL_0:2;
reconsider LC1 = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)), mLC = mlt ((Line (A,i)),(Col (B,j))) as FinSequence of the carrier of K by FINSEQ_1:18;
take LC1 ; ::_thesis: ( LC1 = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC1 = Fk1 . j )
the multF of K .: ((Line (A,i)),(Col (B,j))) is Element of m -tuples_on the carrier of K by A1, FINSEQ_2:120;
then len mLC = m by CARD_1:def_7;
then A49: k + 1 in dom mLC by A17, FINSEQ_1:def_3;
A50: k + 1 in Seg m by A15, A16;
then A51: (Line (A,i)) . (k + 1) = A * (i,(k + 1)) by A1, MATRIX_1:def_7;
Seg m = dom B by FINSEQ_1:def_3;
then (Col (B,j)) . (k + 1) = B * ((k + 1),j) by A50, MATRIX_1:def_8;
then A52: mLC . (k + 1) = (A * (i,(k + 1))) * (B * ((k + 1),j)) by A51, A49, FVSUM_1:60;
LC1 = LC ^ <*(mLC . (k + 1))*> by A47, A49, FINSEQ_5:10;
then A53: the addF of K "**" LC1 = the addF of K . ((Fk . j),((A * (i,(k + 1))) * (B * ((k + 1),j)))) by A48, A52, FINSOP_1:4, FVSUM_1:8;
j in dom Fk by A41, A45, FINSEQ_1:def_3;
then Fk . j in rng Fk by FUNCT_1:def_3;
then reconsider Fkj = Fk . j as Element of the carrier of K by A43;
Fk1 . j = Fkj + Pk1j by A42, A38, A45, FVSUM_1:17;
hence ( LC1 = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC1 = Fk1 . j ) by A40, A45, A46, A44, A53, FVSUM_1:50; ::_thesis: verum
end;
end;
end;
set L = Line ((A * B),i);
width (A * B) = w by A1, MATRIX_3:def_4;
then A54: len (Line ((A * B),i)) = w by MATRIX_1:def_7;
(addFinS K) "**" P is Element of the carrier of K * ;
then reconsider Fm = F . m as FinSequence of the carrier of K by A4, A12;
A55: S1[ 0 ] ;
A56: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A55, A13);
A57: for j being Nat st 1 <= j & j <= len (Line ((A * B),i)) holds
(Line ((A * B),i)) . j = Fm . j
proof
set AB = A * B;
set LA = Line (A,i);
let j be Nat; ::_thesis: ( 1 <= j & j <= len (Line ((A * B),i)) implies (Line ((A * B),i)) . j = Fm . j )
assume that
A58: 1 <= j and
A59: j <= len (Line ((A * B),i)) ; ::_thesis: (Line ((A * B),i)) . j = Fm . j
set CB = Col (B,j);
the multF of K .: ((Line (A,i)),(Col (B,j))) is Element of m -tuples_on the carrier of K by A1, FINSEQ_2:120;
then len (mlt ((Line (A,i)),(Col (B,j)))) = m by CARD_1:def_7;
then A60: dom (mlt ((Line (A,i)),(Col (B,j)))) = Seg m by FINSEQ_1:def_3;
j in NAT by ORDINAL1:def_12;
then A61: j in Seg w by A54, A58, A59;
then ex LC being FinSequence of the carrier of K st
( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg m) & the addF of K "**" LC = Fm . j ) by A9, A56;
then A62: Fm . j = (Line (A,i)) "*" (Col (B,j)) by A60, RELAT_1:69;
A63: len (A * B) = len A by A1, MATRIX_3:def_4;
A64: width (A * B) = w by A1, MATRIX_3:def_4;
len A <> 0 by A3;
then A * B is Matrix of len A,w,K by A63, A64, MATRIX_1:20;
then Indices (A * B) = [:(Seg (len A)),(Seg w):] by A64, MATRIX_1:25;
then [i,j] in Indices (A * B) by A3, A61, ZFMISC_1:87;
then (A * B) * (i,j) = (Line (A,i)) "*" (Col (B,j)) by A1, MATRIX_3:def_4;
hence (Line ((A * B),i)) . j = Fm . j by A61, A62, A64, MATRIX_1:def_7; ::_thesis: verum
end;
take P ; ::_thesis: ( len P = len B & Line ((A * B),i) = (addFinS K) "**" P & ( for j being Nat st j in Seg (len B) holds
P . j = (A * (i,j)) * (Line (B,j)) ) )
thus len P = len B by A4; ::_thesis: ( Line ((A * B),i) = (addFinS K) "**" P & ( for j being Nat st j in Seg (len B) holds
P . j = (A * (i,j)) * (Line (B,j)) ) )
len Fm = w by A9, A56;
hence Line ((A * B),i) = (addFinS K) "**" P by A4, A12, A54, A57, FINSEQ_1:14; ::_thesis: for j being Nat st j in Seg (len B) holds
P . j = (A * (i,j)) * (Line (B,j))
let j be Nat; ::_thesis: ( j in Seg (len B) implies P . j = (A * (i,j)) * (Line (B,j)) )
assume j in Seg (len B) ; ::_thesis: P . j = (A * (i,j)) * (Line (B,j))
hence P . j = (A * (i,j)) * (Line (B,j)) by A5, A6; ::_thesis: verum
end;
theorem Th56: :: MATRIX11:56
for n being Nat
for K being Field
for A, B, C being Matrix of n,K
for i being Nat st i in Seg n holds
ex P being FinSequence of K st
( len P = n & Det (RLine (C,i,(Line ((A * B),i)))) = the addF of K "**" P & ( for j being Nat st j in Seg n holds
P . j = (A * (i,j)) * (Det (RLine (C,i,(Line (B,j))))) ) )
proof
let n be Nat; ::_thesis: for K being Field
for A, B, C being Matrix of n,K
for i being Nat st i in Seg n holds
ex P being FinSequence of K st
( len P = n & Det (RLine (C,i,(Line ((A * B),i)))) = the addF of K "**" P & ( for j being Nat st j in Seg n holds
P . j = (A * (i,j)) * (Det (RLine (C,i,(Line (B,j))))) ) )
let K be Field; ::_thesis: for A, B, C being Matrix of n,K
for i being Nat st i in Seg n holds
ex P being FinSequence of K st
( len P = n & Det (RLine (C,i,(Line ((A * B),i)))) = the addF of K "**" P & ( for j being Nat st j in Seg n holds
P . j = (A * (i,j)) * (Det (RLine (C,i,(Line (B,j))))) ) )
let A, B, C be Matrix of n,K; ::_thesis: for i being Nat st i in Seg n holds
ex P being FinSequence of K st
( len P = n & Det (RLine (C,i,(Line ((A * B),i)))) = the addF of K "**" P & ( for j being Nat st j in Seg n holds
P . j = (A * (i,j)) * (Det (RLine (C,i,(Line (B,j))))) ) )
let i be Nat; ::_thesis: ( i in Seg n implies ex P being FinSequence of K st
( len P = n & Det (RLine (C,i,(Line ((A * B),i)))) = the addF of K "**" P & ( for j being Nat st j in Seg n holds
P . j = (A * (i,j)) * (Det (RLine (C,i,(Line (B,j))))) ) ) )
assume A1: i in Seg n ; ::_thesis: ex P being FinSequence of K st
( len P = n & Det (RLine (C,i,(Line ((A * B),i)))) = the addF of K "**" P & ( for j being Nat st j in Seg n holds
P . j = (A * (i,j)) * (Det (RLine (C,i,(Line (B,j))))) ) )
Seg n <> {} by A1;
then A2: n <> 0 ;
set a = addFinS K;
A3: len B = n by MATRIX_1:24;
deffunc H1( Nat) -> Element of the carrier of K = (A * (i,$1)) * (Det (RLine (C,i,(Line (B,$1)))));
set aa = the addF of K;
set KK = the carrier of K;
A4: len A = n by MATRIX_1:24;
consider D being FinSequence of the carrier of K such that
A5: len D = len A and
A6: for j being Nat st j in dom D holds
D . j = H1(j) from FINSEQ_2:sch_1();
A7: n <> 0 by A1;
then len D >= 1 by A4, A5, NAT_1:14;
then consider Fd being Function of NAT, the carrier of K such that
A8: Fd . 1 = D . 1 and
A9: for k being Element of NAT st 0 <> k & k < n holds
Fd . (k + 1) = the addF of K . ((Fd . k),(D . (k + 1))) and
A10: the addF of K "**" D = Fd . n by A4, A5, FINSOP_1:def_1;
A11: dom D = Seg (len A) by A5, FINSEQ_1:def_3;
width A = n by MATRIX_1:24;
then consider P being FinSequence of the carrier of K * such that
A12: len P = n and
A13: Line ((A * B),i) = (addFinS K) "**" P and
A14: for j being Nat st j in Seg (len B) holds
P . j = (A * (i,j)) * (Line (B,j)) by A1, A7, A3, A4, Th55;
len P >= 1 by A12, A2, NAT_1:14;
then consider Fp being Function of NAT,( the carrier of K *) such that
A15: Fp . 1 = P . 1 and
A16: for k being Element of NAT st 0 <> k & k < n holds
Fp . (k + 1) = (addFinS K) . ((Fp . k),(P . (k + 1))) and
A17: Line ((A * B),i) = Fp . n by A12, A13, FINSOP_1:def_1;
defpred S1[ Nat] means ( 1 <= $1 & $1 <= n implies for pK being FinSequence of K st pK = Fp . $1 holds
( len pK = n & Fd . $1 = Det (RLine (C,i,pK)) ) );
A18: width B = n by MATRIX_1:24;
A19: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A20: S1[k] ; ::_thesis: S1[k + 1]
set k1 = k + 1;
set A9 = A * (i,(k + 1));
set L = Line (B,(k + 1));
assume that
A21: 1 <= k + 1 and
A22: k + 1 <= n ; ::_thesis: for pK being FinSequence of K st pK = Fp . (k + 1) holds
( len pK = n & Fd . (k + 1) = Det (RLine (C,i,pK)) )
A23: k + 1 in Seg n by A21, A22;
let Fpk1 be FinSequence of the carrier of K; ::_thesis: ( Fpk1 = Fp . (k + 1) implies ( len Fpk1 = n & Fd . (k + 1) = Det (RLine (C,i,Fpk1)) ) )
assume A24: Fpk1 = Fp . (k + 1) ; ::_thesis: ( len Fpk1 = n & Fd . (k + 1) = Det (RLine (C,i,Fpk1)) )
percases ( k = 0 or k > 0 ) ;
supposeA25: k = 0 ; ::_thesis: ( len Fpk1 = n & Fd . (k + 1) = Det (RLine (C,i,Fpk1)) )
A26: P . (k + 1) = (A * (i,(k + 1))) * (Line (B,(k + 1))) by A3, A14, A23;
A27: len (Line (B,(k + 1))) = n by A18, MATRIX_1:def_7;
D . (k + 1) = H1(1) by A4, A6, A11, A23, A25;
hence ( len Fpk1 = n & Fd . (k + 1) = Det (RLine (C,i,Fpk1)) ) by A1, A15, A8, A24, A25, A26, A27, Lm5, Th34; ::_thesis: verum
end;
supposeA28: k > 0 ; ::_thesis: ( len Fpk1 = n & Fd . (k + 1) = Det (RLine (C,i,Fpk1)) )
k + 1 in dom P by A12, A23, FINSEQ_1:def_3;
then A29: P . (k + 1) in rng P by FUNCT_1:def_3;
rng P c= the carrier of K * by FINSEQ_1:def_4;
then reconsider Pk1 = P . (k + 1), Fpk = Fp . k as Element of the carrier of K * by A29;
A30: k + 0 < k + 1 by XREAL_1:8;
then A31: Fd . k = Det (RLine (C,i,Fpk)) by A20, A22, A28, NAT_1:14, XXREAL_0:2;
A32: len Fpk = n by A20, A22, A28, A30, NAT_1:14, XXREAL_0:2;
A33: k < n by A22, A30, XXREAL_0:2;
then A34: Fd . (k + 1) = the addF of K . ((Fd . k),(D . (k + 1))) by A9, A28;
A35: P . (k + 1) = (A * (i,(k + 1))) * (Line (B,(k + 1))) by A3, A14, A23;
Fpk1 = (addFinS K) . (Fpk,Pk1) by A16, A24, A28, A33;
then A36: Fpk1 = Fpk + ((A * (i,(k + 1))) * (Line (B,(k + 1)))) by A35, Def5;
A37: len (Line (B,(k + 1))) = n by A18, MATRIX_1:def_7;
then A38: len ((A * (i,(k + 1))) * (Line (B,(k + 1)))) = n by Lm5;
Det (RLine (C,i,((A * (i,(k + 1))) * (Line (B,(k + 1)))))) = (A * (i,(k + 1))) * (Det (RLine (C,i,(Line (B,(k + 1)))))) by A1, A37, Th34;
then Det (RLine (C,i,Fpk1)) = (Det (RLine (C,i,Fpk))) + ((A * (i,(k + 1))) * (Det (RLine (C,i,(Line (B,(k + 1))))))) by A1, A32, A36, A38, Th36;
hence ( len Fpk1 = n & Fd . (k + 1) = Det (RLine (C,i,Fpk1)) ) by A4, A6, A11, A23, A31, A34, A32, A36, A38, Lm6; ::_thesis: verum
end;
end;
end;
take D ; ::_thesis: ( len D = n & Det (RLine (C,i,(Line ((A * B),i)))) = the addF of K "**" D & ( for j being Nat st j in Seg n holds
D . j = (A * (i,j)) * (Det (RLine (C,i,(Line (B,j))))) ) )
thus len D = n by A5, MATRIX_1:24; ::_thesis: ( Det (RLine (C,i,(Line ((A * B),i)))) = the addF of K "**" D & ( for j being Nat st j in Seg n holds
D . j = (A * (i,j)) * (Det (RLine (C,i,(Line (B,j))))) ) )
A39: S1[ 0 ] ;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A39, A19);
then S1[ len P] ;
hence ( Det (RLine (C,i,(Line ((A * B),i)))) = the addF of K "**" D & ( for j being Nat st j in Seg n holds
D . j = (A * (i,j)) * (Det (RLine (C,i,(Line (B,j))))) ) ) by A4, A12, A6, A11, A17, A10, A2, NAT_1:14; ::_thesis: verum
end;
theorem Th57: :: MATRIX11:57
for X being set
for Y being non empty set
for x being set st not x in X holds
ex BIJECT being Function of [:(Funcs (X,Y)),Y:],(Funcs ((X \/ {x}),Y)) st
( BIJECT is bijective & ( for f being Function of X,Y
for F being Function of (X \/ {x}),Y st F | X = f holds
BIJECT . (f,(F . x)) = F ) )
proof
let X be set ; ::_thesis: for Y being non empty set
for x being set st not x in X holds
ex BIJECT being Function of [:(Funcs (X,Y)),Y:],(Funcs ((X \/ {x}),Y)) st
( BIJECT is bijective & ( for f being Function of X,Y
for F being Function of (X \/ {x}),Y st F | X = f holds
BIJECT . (f,(F . x)) = F ) )
let Y be non empty set ; ::_thesis: for x being set st not x in X holds
ex BIJECT being Function of [:(Funcs (X,Y)),Y:],(Funcs ((X \/ {x}),Y)) st
( BIJECT is bijective & ( for f being Function of X,Y
for F being Function of (X \/ {x}),Y st F | X = f holds
BIJECT . (f,(F . x)) = F ) )
let x be set ; ::_thesis: ( not x in X implies ex BIJECT being Function of [:(Funcs (X,Y)),Y:],(Funcs ((X \/ {x}),Y)) st
( BIJECT is bijective & ( for f being Function of X,Y
for F being Function of (X \/ {x}),Y st F | X = f holds
BIJECT . (f,(F . x)) = F ) ) )
assume A1: not x in X ; ::_thesis: ex BIJECT being Function of [:(Funcs (X,Y)),Y:],(Funcs ((X \/ {x}),Y)) st
( BIJECT is bijective & ( for f being Function of X,Y
for F being Function of (X \/ {x}),Y st F | X = f holds
BIJECT . (f,(F . x)) = F ) )
set Xx = X \/ {x};
set FXY = Funcs (X,Y);
set FXxY = Funcs ((X \/ {x}),Y);
defpred S1[ set , set ] means for f being Function of X,Y
for F being Function of (X \/ {x}),Y
for y being set st [f,y] = $1 & F . x = y & F | X = f holds
F = $2;
A2: for x being Element of [:(Funcs (X,Y)),Y:] ex y being Element of Funcs ((X \/ {x}),Y) st S1[x,y]
proof
let x9 be Element of [:(Funcs (X,Y)),Y:]; ::_thesis: ex y being Element of Funcs ((X \/ {x}),Y) st S1[x9,y]
consider f, y being set such that
A3: f in Funcs (X,Y) and
A4: y in Y and
A5: x9 = [f,y] by ZFMISC_1:def_2;
reconsider f = f as Function of X,Y by A3, FUNCT_2:66;
Y \/ {y} = Y by A4, ZFMISC_1:40;
then consider F being Function of (X \/ {x}),Y such that
A6: F | X = f and
A7: F . x = y by A1, STIRL2_1:57;
reconsider F9 = F as Element of Funcs ((X \/ {x}),Y) by FUNCT_2:8;
take F9 ; ::_thesis: S1[x9,F9]
let g be Function of X,Y; ::_thesis: for F being Function of (X \/ {x}),Y
for y being set st [g,y] = x9 & F . x = y & F | X = g holds
F = F9
let G be Function of (X \/ {x}),Y; ::_thesis: for y being set st [g,y] = x9 & G . x = y & G | X = g holds
G = F9
let y9 be set ; ::_thesis: ( [g,y9] = x9 & G . x = y9 & G | X = g implies G = F9 )
assume that
A8: [g,y9] = x9 and
A9: G . x = y9 and
A10: G | X = g ; ::_thesis: G = F9
now__::_thesis:_for_xx_being_set_st_xx_in_X_\/_{x}_holds_
G_._xx_=_F_._xx
let xx be set ; ::_thesis: ( xx in X \/ {x} implies G . xx = F . xx )
assume xx in X \/ {x} ; ::_thesis: G . xx = F . xx
then A11: ( xx in X or xx in {x} ) by XBOOLE_0:def_3;
A12: dom f = X by FUNCT_2:def_1;
dom g = X by FUNCT_2:def_1;
then ( ( G . xx = g . xx & F . xx = f . xx ) or xx = x ) by A6, A10, A11, A12, FUNCT_1:47, TARSKI:def_1;
hence G . xx = F . xx by A5, A7, A8, A9, XTUPLE_0:1; ::_thesis: verum
end;
hence G = F9 by FUNCT_2:12; ::_thesis: verum
end;
consider H being Function of [:(Funcs (X,Y)),Y:],(Funcs ((X \/ {x}),Y)) such that
A13: for x being Element of [:(Funcs (X,Y)),Y:] holds S1[x,H . x] from FUNCT_2:sch_3(A2);
A14: now__::_thesis:_for_x1,_x2_being_set_st_x1_in_[:(Funcs_(X,Y)),Y:]_&_x2_in_[:(Funcs_(X,Y)),Y:]_&_H_._x1_=_H_._x2_holds_
x1_=_x2
let x1, x2 be set ; ::_thesis: ( x1 in [:(Funcs (X,Y)),Y:] & x2 in [:(Funcs (X,Y)),Y:] & H . x1 = H . x2 implies x1 = x2 )
assume that
A15: x1 in [:(Funcs (X,Y)),Y:] and
A16: x2 in [:(Funcs (X,Y)),Y:] and
A17: H . x1 = H . x2 ; ::_thesis: x1 = x2
consider f2, y2 being set such that
A18: f2 in Funcs (X,Y) and
A19: y2 in Y and
A20: x2 = [f2,y2] by A16, ZFMISC_1:def_2;
consider f1, y1 being set such that
A21: f1 in Funcs (X,Y) and
A22: y1 in Y and
A23: x1 = [f1,y1] by A15, ZFMISC_1:def_2;
reconsider f1 = f1, f2 = f2 as Function of X,Y by A21, A18, FUNCT_2:66;
Y \/ {y2} = Y by A19, ZFMISC_1:40;
then consider F2 being Function of (X \/ {x}),Y such that
A24: F2 | X = f2 and
A25: F2 . x = y2 by A1, STIRL2_1:57;
A26: H . x2 = F2 by A13, A16, A20, A24, A25;
Y \/ {y1} = Y by A22, ZFMISC_1:40;
then consider F1 being Function of (X \/ {x}),Y such that
A27: F1 | X = f1 and
A28: F1 . x = y1 by A1, STIRL2_1:57;
H . x1 = F1 by A13, A15, A23, A27, A28;
hence x1 = x2 by A17, A23, A20, A27, A28, A24, A25, A26; ::_thesis: verum
end;
take H ; ::_thesis: ( H is bijective & ( for f being Function of X,Y
for F being Function of (X \/ {x}),Y st F | X = f holds
H . (f,(F . x)) = F ) )
x in {x} by TARSKI:def_1;
then A29: x in X \/ {x} by XBOOLE_0:def_3;
A30: Funcs ((X \/ {x}),Y) c= rng H
proof
let f9 be set ; :: according to TARSKI:def_3 ::_thesis: ( not f9 in Funcs ((X \/ {x}),Y) or f9 in rng H )
assume f9 in Funcs ((X \/ {x}),Y) ; ::_thesis: f9 in rng H
then reconsider f = f9 as Function of (X \/ {x}),Y by FUNCT_2:66;
dom f = X \/ {x} by FUNCT_2:def_1;
then A31: dom (f | X) = X by RELAT_1:62, XBOOLE_1:7;
rng (f | X) c= Y by RELAT_1:def_19;
then reconsider fX = f | X as Function of X,Y by A31, FUNCT_2:2;
A32: fX in Funcs (X,Y) by FUNCT_2:8;
x in {x} by TARSKI:def_1;
then A33: x in X \/ {x} by XBOOLE_0:def_3;
X \/ {x} = dom f by FUNCT_2:def_1;
then A34: f . x in rng f by A33, FUNCT_1:def_3;
rng f c= Y by RELAT_1:def_19;
then A35: [fX,(f . x)] in [:(Funcs (X,Y)),Y:] by A34, A32, ZFMISC_1:87;
[:(Funcs (X,Y)),Y:] = dom H by FUNCT_2:def_1;
then H . [fX,(f . x)] in rng H by A35, FUNCT_1:def_3;
hence f9 in rng H by A13, A35; ::_thesis: verum
end;
rng H c= Funcs ((X \/ {x}),Y) by RELAT_1:def_19;
then Funcs ((X \/ {x}),Y) = rng H by A30, XBOOLE_0:def_10;
then ( H is one-to-one & H is onto ) by A14, FUNCT_2:19, FUNCT_2:def_3;
hence H is bijective ; ::_thesis: for f being Function of X,Y
for F being Function of (X \/ {x}),Y st F | X = f holds
H . (f,(F . x)) = F
let f be Function of X,Y; ::_thesis: for F being Function of (X \/ {x}),Y st F | X = f holds
H . (f,(F . x)) = F
let F be Function of (X \/ {x}),Y; ::_thesis: ( F | X = f implies H . (f,(F . x)) = F )
assume A36: F | X = f ; ::_thesis: H . (f,(F . x)) = F
X \/ {x} = dom F by FUNCT_2:def_1;
then A37: F . x in rng F by A29, FUNCT_1:def_3;
A38: f in Funcs (X,Y) by FUNCT_2:8;
rng F c= Y by RELAT_1:def_19;
then [f,(F . x)] in [:(Funcs (X,Y)),Y:] by A37, A38, ZFMISC_1:87;
hence H . (f,(F . x)) = F by A13, A36; ::_thesis: verum
end;
theorem Th58: :: MATRIX11:58
for D being non empty set
for X being finite set
for Y being non empty finite set
for x being set st not x in X holds
for F being BinOp of D st F is having_a_unity & F is commutative & F is associative holds
for f being Function of (Funcs (X,Y)),D
for g being Function of (Funcs ((X \/ {x}),Y)),D st ( for H being Function of X,Y
for SF being Element of Fin (Funcs ((X \/ {x}),Y)) st SF = { h where h is Function of (X \/ {x}),Y : h | X = H } holds
F $$ (SF,g) = f . H ) holds
F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g)
proof
let D be non empty set ; ::_thesis: for X being finite set
for Y being non empty finite set
for x being set st not x in X holds
for F being BinOp of D st F is having_a_unity & F is commutative & F is associative holds
for f being Function of (Funcs (X,Y)),D
for g being Function of (Funcs ((X \/ {x}),Y)),D st ( for H being Function of X,Y
for SF being Element of Fin (Funcs ((X \/ {x}),Y)) st SF = { h where h is Function of (X \/ {x}),Y : h | X = H } holds
F $$ (SF,g) = f . H ) holds
F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g)
let X be finite set ; ::_thesis: for Y being non empty finite set
for x being set st not x in X holds
for F being BinOp of D st F is having_a_unity & F is commutative & F is associative holds
for f being Function of (Funcs (X,Y)),D
for g being Function of (Funcs ((X \/ {x}),Y)),D st ( for H being Function of X,Y
for SF being Element of Fin (Funcs ((X \/ {x}),Y)) st SF = { h where h is Function of (X \/ {x}),Y : h | X = H } holds
F $$ (SF,g) = f . H ) holds
F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g)
let Y be non empty finite set ; ::_thesis: for x being set st not x in X holds
for F being BinOp of D st F is having_a_unity & F is commutative & F is associative holds
for f being Function of (Funcs (X,Y)),D
for g being Function of (Funcs ((X \/ {x}),Y)),D st ( for H being Function of X,Y
for SF being Element of Fin (Funcs ((X \/ {x}),Y)) st SF = { h where h is Function of (X \/ {x}),Y : h | X = H } holds
F $$ (SF,g) = f . H ) holds
F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g)
let x be set ; ::_thesis: ( not x in X implies for F being BinOp of D st F is having_a_unity & F is commutative & F is associative holds
for f being Function of (Funcs (X,Y)),D
for g being Function of (Funcs ((X \/ {x}),Y)),D st ( for H being Function of X,Y
for SF being Element of Fin (Funcs ((X \/ {x}),Y)) st SF = { h where h is Function of (X \/ {x}),Y : h | X = H } holds
F $$ (SF,g) = f . H ) holds
F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g) )
assume A1: not x in X ; ::_thesis: for F being BinOp of D st F is having_a_unity & F is commutative & F is associative holds
for f being Function of (Funcs (X,Y)),D
for g being Function of (Funcs ((X \/ {x}),Y)),D st ( for H being Function of X,Y
for SF being Element of Fin (Funcs ((X \/ {x}),Y)) st SF = { h where h is Function of (X \/ {x}),Y : h | X = H } holds
F $$ (SF,g) = f . H ) holds
F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g)
set Xx = X \/ {x};
set FXY = Funcs (X,Y);
set FXxY = Funcs ((X \/ {x}),Y);
consider B being Function of [:(Funcs (X,Y)),Y:],(Funcs ((X \/ {x}),Y)) such that
A2: B is bijective and
A3: for f being Function of X,Y
for F being Function of (X \/ {x}),Y st F | X = f holds
B . (f,(F . x)) = F by A1, Th57;
A4: Funcs (X,Y) is finite by FRAENKEL:6;
dom B = [:(Funcs (X,Y)),Y:] by FUNCT_2:def_1;
then reconsider domB = dom B as Element of Fin [:(Funcs (X,Y)),Y:] by A4, FINSUB_1:def_5;
Funcs (X,Y) is finite by FRAENKEL:6;
then reconsider FXY9 = Funcs (X,Y) as Element of Fin (Funcs (X,Y)) by FINSUB_1:def_5;
A5: FinOmega (Funcs ((X \/ {x}),Y)) = Funcs ((X \/ {x}),Y) by FRAENKEL:6, MATRIX_2:def_14;
reconsider Y9 = Y as Element of Fin Y by FINSUB_1:def_5;
let F be BinOp of D; ::_thesis: ( F is having_a_unity & F is commutative & F is associative implies for f being Function of (Funcs (X,Y)),D
for g being Function of (Funcs ((X \/ {x}),Y)),D st ( for H being Function of X,Y
for SF being Element of Fin (Funcs ((X \/ {x}),Y)) st SF = { h where h is Function of (X \/ {x}),Y : h | X = H } holds
F $$ (SF,g) = f . H ) holds
F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g) )
assume that
A6: F is having_a_unity and
A7: F is commutative and
A8: F is associative ; ::_thesis: for f being Function of (Funcs (X,Y)),D
for g being Function of (Funcs ((X \/ {x}),Y)),D st ( for H being Function of X,Y
for SF being Element of Fin (Funcs ((X \/ {x}),Y)) st SF = { h where h is Function of (X \/ {x}),Y : h | X = H } holds
F $$ (SF,g) = f . H ) holds
F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g)
let f be Function of (Funcs (X,Y)),D; ::_thesis: for g being Function of (Funcs ((X \/ {x}),Y)),D st ( for H being Function of X,Y
for SF being Element of Fin (Funcs ((X \/ {x}),Y)) st SF = { h where h is Function of (X \/ {x}),Y : h | X = H } holds
F $$ (SF,g) = f . H ) holds
F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g)
let g be Function of (Funcs ((X \/ {x}),Y)),D; ::_thesis: ( ( for H being Function of X,Y
for SF being Element of Fin (Funcs ((X \/ {x}),Y)) st SF = { h where h is Function of (X \/ {x}),Y : h | X = H } holds
F $$ (SF,g) = f . H ) implies F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g) )
assume A9: for H being Function of X,Y
for SF being Element of Fin (Funcs ((X \/ {x}),Y)) st SF = { h where h is Function of (X \/ {x}),Y : h | X = H } holds
F $$ (SF,g) = f . H ; ::_thesis: F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g)
reconsider gB = g * B as Function of [:(Funcs (X,Y)),Y:],D ;
for z being Element of Funcs (X,Y) holds f . z = F $$ (Y9,((curry gB) . z))
proof
let z be Element of Funcs (X,Y); ::_thesis: f . z = F $$ (Y9,((curry gB) . z))
reconsider Z = z as Function of X,Y ;
set SF = { h where h is Function of (X \/ {x}),Y : h | X = Z } ;
deffunc H1( set ) -> set = [z,$1];
consider q being Function such that
A10: ( dom q = Y & ( for x being set st x in Y holds
q . x = H1(x) ) ) from FUNCT_1:sch_3();
A11: {z} c= Funcs (X,Y) by ZFMISC_1:31;
then [:{Z},Y:] c= [:(Funcs (X,Y)),Y:] by ZFMISC_1:95;
then reconsider ZY = [:{Z},Y:] as Element of Fin [:(Funcs (X,Y)),Y:] by FINSUB_1:def_5;
for x9 being set holds
( x9 in ZY iff x9 in q .: Y )
proof
let x9 be set ; ::_thesis: ( x9 in ZY iff x9 in q .: Y )
thus ( x9 in ZY implies x9 in q .: Y ) ::_thesis: ( x9 in q .: Y implies x9 in ZY )
proof
assume x9 in ZY ; ::_thesis: x9 in q .: Y
then consider z9, y9 being set such that
A12: z9 in {Z} and
A13: y9 in Y and
A14: x9 = [z9,y9] by ZFMISC_1:def_2;
A15: z = z9 by A12, TARSKI:def_1;
q . y9 = [z,y9] by A10, A13;
hence x9 in q .: Y by A10, A13, A14, A15, FUNCT_1:def_6; ::_thesis: verum
end;
assume x9 in q .: Y ; ::_thesis: x9 in ZY
then consider y9 being set such that
A16: y9 in dom q and
A17: y9 in Y and
A18: x9 = q . y9 by FUNCT_1:def_6;
x9 = [z,y9] by A10, A16, A18;
hence x9 in ZY by A17, ZFMISC_1:105; ::_thesis: verum
end;
then A19: q .: Y = ZY by TARSKI:1;
then A20: rng q = ZY by A10, RELAT_1:113;
now__::_thesis:_for_x1,_x2_being_set_st_x1_in_dom_q_&_x2_in_dom_q_&_q_._x1_=_q_._x2_holds_
x1_=_x2
let x1, x2 be set ; ::_thesis: ( x1 in dom q & x2 in dom q & q . x1 = q . x2 implies x1 = x2 )
assume that
A21: x1 in dom q and
A22: x2 in dom q and
A23: q . x1 = q . x2 ; ::_thesis: x1 = x2
A24: q . x2 = [z,x2] by A10, A22;
[z,x1] = q . x1 by A10, A21;
hence x1 = x2 by A23, A24, XTUPLE_0:1; ::_thesis: verum
end;
then A25: q is one-to-one by FUNCT_1:def_4;
ZY c= [:(Funcs (X,Y)),Y:] by A11, ZFMISC_1:95;
then reconsider q = q as Function of Y,[:(Funcs (X,Y)),Y:] by A10, A20, FUNCT_2:2;
reconsider gBq = gB * q as Function of Y,D ;
dom gB = [:(Funcs (X,Y)),Y:] by FUNCT_2:def_1;
then consider C being Function such that
A26: (curry gB) . z = C and
dom C = Y and
rng C c= rng gB and
A27: for y9 being set st y9 in Y holds
C . y9 = gB . (z,y9) by FUNCT_5:29;
reconsider C = C as Function of Y,D by A26;
now__::_thesis:_for_x9_being_set_st_x9_in_Y_holds_
C_._x9_=_gBq_._x9
let x9 be set ; ::_thesis: ( x9 in Y implies C . x9 = gBq . x9 )
assume A28: x9 in Y ; ::_thesis: C . x9 = gBq . x9
A29: q . x9 = [z,x9] by A10, A28;
C . x9 = gB . (z,x9) by A27, A28;
hence C . x9 = gBq . x9 by A28, A29, FUNCT_2:15; ::_thesis: verum
end;
then A30: C = gBq by FUNCT_2:12;
A31: B .: ZY c= { h where h is Function of (X \/ {x}),Y : h | X = Z }
proof
let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in B .: ZY or b in { h where h is Function of (X \/ {x}),Y : h | X = Z } )
assume b in B .: ZY ; ::_thesis: b in { h where h is Function of (X \/ {x}),Y : h | X = Z }
then consider zy being set such that
zy in dom B and
A32: zy in ZY and
A33: b = B . zy by FUNCT_1:def_6;
consider z9, y9 being set such that
A34: z9 in {Z} and
A35: y9 in Y and
A36: zy = [z9,y9] by A32, ZFMISC_1:def_2;
Y \/ {y9} = Y by A35, ZFMISC_1:40;
then consider F1 being Function of (X \/ {x}),Y such that
A37: F1 | X = Z and
A38: F1 . x = y9 by A1, STIRL2_1:57;
z9 = Z by A34, TARSKI:def_1;
then B . (z9,y9) = F1 by A3, A37, A38;
hence b in { h where h is Function of (X \/ {x}),Y : h | X = Z } by A33, A36, A37; ::_thesis: verum
end;
A39: { h where h is Function of (X \/ {x}),Y : h | X = Z } c= B .: ZY
proof
x in {x} by TARSKI:def_1;
then A40: x in X \/ {x} by XBOOLE_0:def_3;
let sf be set ; :: according to TARSKI:def_3 ::_thesis: ( not sf in { h where h is Function of (X \/ {x}),Y : h | X = Z } or sf in B .: ZY )
assume sf in { h where h is Function of (X \/ {x}),Y : h | X = Z } ; ::_thesis: sf in B .: ZY
then consider h being Function of (X \/ {x}),Y such that
A41: h = sf and
A42: h | X = Z ;
A43: [:(Funcs (X,Y)),Y:] = dom B by FUNCT_2:def_1;
dom h = X \/ {x} by FUNCT_2:def_1;
then A44: h . x in rng h by A40, FUNCT_1:def_3;
A45: rng h c= Y by RELAT_1:def_19;
then A46: [z,(h . x)] in [:(Funcs (X,Y)),Y:] by A44, ZFMISC_1:87;
z in {Z} by TARSKI:def_1;
then [z,(h . x)] in ZY by A44, A45, ZFMISC_1:87;
then B . (z,(h . x)) in B .: ZY by A46, A43, FUNCT_1:def_6;
hence sf in B .: ZY by A3, A41, A42; ::_thesis: verum
end;
then reconsider SF = { h where h is Function of (X \/ {x}),Y : h | X = Z } as Element of Fin (Funcs ((X \/ {x}),Y)) by A31, XBOOLE_0:def_10;
B .: ZY = SF by A31, A39, XBOOLE_0:def_10;
then A47: F $$ ((B .: ZY),g) = f . Z by A9;
F $$ ((B .: ZY),g) = F $$ (ZY,gB) by A7, A8, A2, SETWOP_2:6;
hence f . z = F $$ (Y9,((curry gB) . z)) by A7, A8, A47, A25, A19, A26, A30, SETWOP_2:6; ::_thesis: verum
end;
then F $$ ([:FXY9,Y9:],gB) = F $$ (FXY9,f) by A6, A7, A8, MATRIX_3:30;
then A48: F $$ (domB,gB) = F $$ (FXY9,f) by FUNCT_2:def_1;
A49: rng B = Funcs ((X \/ {x}),Y) by A2, FUNCT_2:def_3;
F $$ ((B .: domB),g) = F $$ (domB,(g * B)) by A7, A8, A2, SETWOP_2:6;
then F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g) = F $$ (domB,(g * B)) by A49, A5, RELAT_1:113;
hence F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g) by A48, MATRIX_2:def_14; ::_thesis: verum
end;
theorem Th59: :: MATRIX11:59
for n, m being Nat
for D being non empty set
for A, B being Matrix of n,m,D
for i being Nat st i <= n & 0 < n holds
for F being Function of (Seg i),(Seg n) ex M being Matrix of n,m,D st
( M = A +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds
( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = A . j ) ) ) )
proof
let n, m be Nat; ::_thesis: for D being non empty set
for A, B being Matrix of n,m,D
for i being Nat st i <= n & 0 < n holds
for F being Function of (Seg i),(Seg n) ex M being Matrix of n,m,D st
( M = A +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds
( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = A . j ) ) ) )
let D be non empty set ; ::_thesis: for A, B being Matrix of n,m,D
for i being Nat st i <= n & 0 < n holds
for F being Function of (Seg i),(Seg n) ex M being Matrix of n,m,D st
( M = A +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds
( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = A . j ) ) ) )
let A, B be Matrix of n,m,D; ::_thesis: for i being Nat st i <= n & 0 < n holds
for F being Function of (Seg i),(Seg n) ex M being Matrix of n,m,D st
( M = A +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds
( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = A . j ) ) ) )
let i be Nat; ::_thesis: ( i <= n & 0 < n implies for F being Function of (Seg i),(Seg n) ex M being Matrix of n,m,D st
( M = A +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds
( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = A . j ) ) ) ) )
assume that
A1: i <= n and
A2: 0 < n ; ::_thesis: for F being Function of (Seg i),(Seg n) ex M being Matrix of n,m,D st
( M = A +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds
( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = A . j ) ) ) )
set I = idseq n;
let F be Function of (Seg i),(Seg n); ::_thesis: ex M being Matrix of n,m,D st
( M = A +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds
( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = A . j ) ) ) )
set IF = (idseq n) +* F;
A3: dom (idseq n) = Seg n by RELAT_1:45;
A4: rng (idseq n) = Seg n by RELAT_1:45;
rng F c= Seg n by RELAT_1:def_19;
then (rng F) \/ (Seg n) = Seg n by XBOOLE_1:12;
then A5: rng ((idseq n) +* F) c= Seg n by A4, FUNCT_4:17;
A6: Seg i c= Seg n by A1, FINSEQ_1:5;
then A7: (Seg i) \/ (Seg n) = Seg n by XBOOLE_1:12;
A8: dom F = Seg i by A2, FUNCT_2:def_1;
then dom F c= Seg n by A1, FINSEQ_1:5;
then (dom F) \/ (Seg n) = Seg n by XBOOLE_1:12;
then dom ((idseq n) +* F) = Seg n by A3, FUNCT_4:def_1;
then reconsider IF = (idseq n) +* F as Function of (Seg n),(Seg n) by A5, FUNCT_2:2;
reconsider BIF = B * IF as Matrix of n,m,D ;
set BIFi = BIF | (Seg i);
set M = A +* (BIF | (Seg i));
A9: len B = n by A2, MATRIX_1:23;
len BIF = len B by Def4;
then dom BIF = Seg n by A9, FINSEQ_1:def_3;
then A10: dom (BIF | (Seg i)) = Seg i by A6, RELAT_1:62;
len A = n by A2, MATRIX_1:23;
then dom A = Seg n by FINSEQ_1:def_3;
then A11: dom (A +* (BIF | (Seg i))) = (Seg i) \/ (Seg n) by A10, FUNCT_4:def_1;
then reconsider M = A +* (BIF | (Seg i)) as FinSequence by A7, FINSEQ_1:def_2;
A12: for j being Nat holds
( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = A . j ) )
proof
let j be Nat; ::_thesis: ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = A . j ) )
thus ( j in Seg i implies M . j = B . (F . j) ) ::_thesis: ( not j in Seg i implies M . j = A . j )
proof
A13: Seg i c= Seg n by A1, FINSEQ_1:5;
assume A14: j in Seg i ; ::_thesis: M . j = B . (F . j)
then A15: (BIF | (Seg i)) . j = BIF . j by A10, FUNCT_1:47;
IF . j = F . j by A8, A14, FUNCT_4:13;
then A16: Line (BIF,j) = B . (F . j) by A14, A13, Th38;
Line (BIF,j) = BIF . j by A14, A13, MATRIX_2:8;
hence M . j = B . (F . j) by A10, A14, A16, A15, FUNCT_4:13; ::_thesis: verum
end;
assume not j in Seg i ; ::_thesis: M . j = A . j
hence M . j = A . j by A10, FUNCT_4:11; ::_thesis: verum
end;
A17: for x being set st x in rng M holds
ex p being FinSequence of D st
( x = p & len p = m )
proof
let x be set ; ::_thesis: ( x in rng M implies ex p being FinSequence of D st
( x = p & len p = m ) )
assume x in rng M ; ::_thesis: ex p being FinSequence of D st
( x = p & len p = m )
then consider k being set such that
A18: k in dom M and
A19: M . k = x by FUNCT_1:def_3;
reconsider k = k as Nat by A18;
percases ( k in Seg i or not k in Seg i ) ;
supposeA20: k in Seg i ; ::_thesis: ex p being FinSequence of D st
( x = p & len p = m )
A21: rng F c= Seg n by RELAT_1:def_19;
A22: F . k in rng F by A8, A20, FUNCT_1:def_3;
then reconsider Fk = F . k as Element of NAT by A21, TARSKI:def_3;
take L = Line (B,Fk); ::_thesis: ( x = L & len L = m )
A23: len L = width B by MATRIX_1:def_7;
B . (F . k) = L by A22, A21, MATRIX_2:8;
hence ( x = L & len L = m ) by A2, A12, A19, A20, A23, MATRIX_1:23; ::_thesis: verum
end;
supposeA24: not k in Seg i ; ::_thesis: ex p being FinSequence of D st
( x = p & len p = m )
take L = Line (A,k); ::_thesis: ( x = L & len L = m )
A25: len L = width A by MATRIX_1:def_7;
M . k = A . k by A12, A24;
hence ( x = L & len L = m ) by A2, A11, A7, A18, A19, A25, MATRIX_1:23, MATRIX_2:8; ::_thesis: verum
end;
end;
end;
then reconsider M = M as Matrix of D by MATRIX_1:9;
n is Element of NAT by ORDINAL1:def_12;
then A26: len M = n by A11, A7, FINSEQ_1:def_3;
now__::_thesis:_for_p_being_FinSequence_of_D_st_p_in_rng_M_holds_
len_p_=_m
let p be FinSequence of D; ::_thesis: ( p in rng M implies len p = m )
assume p in rng M ; ::_thesis: len p = m
then ex q being FinSequence of D st
( p = q & len q = m ) by A17;
hence len p = m ; ::_thesis: verum
end;
then reconsider M = M as Matrix of n,m,D by A26, MATRIX_1:def_2;
take M ; ::_thesis: ( M = A +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds
( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = A . j ) ) ) )
thus ( M = A +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds
( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = A . j ) ) ) ) by A12; ::_thesis: verum
end;
Lm10: for n being Nat
for K being Field
for A, B being Matrix of n,n,K
for i being Nat st i <= n & 0 < n holds
ex P being Function of (Funcs ((Seg i),(Seg n))), the carrier of K st
for F being Function of (Seg i),(Seg n)
for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds
( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) holds
ex Path being FinSequence of K st
( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds
Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det M) )
proof
let n be Nat; ::_thesis: for K being Field
for A, B being Matrix of n,n,K
for i being Nat st i <= n & 0 < n holds
ex P being Function of (Funcs ((Seg i),(Seg n))), the carrier of K st
for F being Function of (Seg i),(Seg n)
for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds
( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) holds
ex Path being FinSequence of K st
( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds
Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det M) )
let K be Field; ::_thesis: for A, B being Matrix of n,n,K
for i being Nat st i <= n & 0 < n holds
ex P being Function of (Funcs ((Seg i),(Seg n))), the carrier of K st
for F being Function of (Seg i),(Seg n)
for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds
( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) holds
ex Path being FinSequence of K st
( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds
Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det M) )
let A, B be Matrix of n,n,K; ::_thesis: for i being Nat st i <= n & 0 < n holds
ex P being Function of (Funcs ((Seg i),(Seg n))), the carrier of K st
for F being Function of (Seg i),(Seg n)
for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds
( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) holds
ex Path being FinSequence of K st
( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds
Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det M) )
let i be Nat; ::_thesis: ( i <= n & 0 < n implies ex P being Function of (Funcs ((Seg i),(Seg n))), the carrier of K st
for F being Function of (Seg i),(Seg n)
for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds
( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) holds
ex Path being FinSequence of K st
( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds
Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det M) ) )
assume that
A1: i <= n and
A2: 0 < n ; ::_thesis: ex P being Function of (Funcs ((Seg i),(Seg n))), the carrier of K st
for F being Function of (Seg i),(Seg n)
for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds
( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) holds
ex Path being FinSequence of K st
( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds
Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det M) )
set KK = the carrier of K;
set I = idseq n;
set Sn = Seg n;
set Si = Seg i;
set mm = the multF of K;
set FF = Funcs ((Seg i),(Seg n));
reconsider Sn = Seg n as non empty set by A2;
set AB = A * B;
reconsider AB = A * B as Matrix of n,K ;
defpred S1[ set , set ] means for F being Function of (Seg i),(Seg n) st F = $1 holds
for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds
( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) holds
ex Path being FinSequence of K st
( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds
Path . j = A * (j,Fj) ) & $2 = ( the multF of K $$ Path) * (Det M) );
ex f being Function st
( dom f = Seg i & rng f c= Sn ) by FUNCT_1:8;
then reconsider FF = Funcs ((Seg i),(Seg n)) as non empty set ;
A3: for x being Element of FF ex y being Element of the carrier of K st S1[x,y]
proof
let x be Element of FF; ::_thesis: ex y being Element of the carrier of K st S1[x,y]
reconsider F = x as Function of (Seg i),Sn by FUNCT_2:66;
defpred S2[ set , set ] means for Fj, j being Nat st j = $1 & Fj = F . j holds
$2 = A * (j,Fj);
A4: i is Element of NAT by ORDINAL1:def_12;
A5: for x9 being set st x9 in Seg i holds
ex y being set st
( y in the carrier of K & S2[x9,y] )
proof
let x9 be set ; ::_thesis: ( x9 in Seg i implies ex y being set st
( y in the carrier of K & S2[x9,y] ) )
assume A6: x9 in Seg i ; ::_thesis: ex y being set st
( y in the carrier of K & S2[x9,y] )
reconsider i = x9 as Nat by A6;
A7: rng F c= Seg n by RELAT_1:def_19;
Seg i = dom F by FUNCT_2:def_1;
then F . i in rng F by A6, FUNCT_1:def_3;
then F . i in Sn by A7;
then reconsider Fi = F . i as Nat ;
take A * (i,Fi) ; ::_thesis: ( A * (i,Fi) in the carrier of K & S2[x9,A * (i,Fi)] )
thus ( A * (i,Fi) in the carrier of K & S2[x9,A * (i,Fi)] ) ; ::_thesis: verum
end;
consider path being Function of (Seg i), the carrier of K such that
A8: for x being set st x in Seg i holds
S2[x,path . x] from FUNCT_2:sch_1(A5);
dom path = Seg i by FUNCT_2:def_1;
then reconsider p = path as FinSequence by FINSEQ_1:def_2;
rng path c= the carrier of K by RELAT_1:def_19;
then reconsider p = p as FinSequence of K by FINSEQ_1:def_4;
consider M being Matrix of n,K such that
M = AB +* ((B * ((idseq n) +* F)) | (Seg i)) and
A9: for j being Nat holds
( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = AB . j ) ) by A1, A2, Th59;
take ( the multF of K $$ p) * (Det M) ; ::_thesis: S1[x,( the multF of K $$ p) * (Det M)]
let F9 be Function of (Seg i),(Seg n); ::_thesis: ( F9 = x implies for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F9)) | (Seg i)) & ( for j being Nat holds
( ( j in Seg i implies M . j = B . (F9 . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) holds
ex Path being FinSequence of K st
( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F9 . j holds
Path . j = A * (j,Fj) ) & ( the multF of K $$ p) * (Det M) = ( the multF of K $$ Path) * (Det M) ) )
assume A10: F9 = x ; ::_thesis: for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F9)) | (Seg i)) & ( for j being Nat holds
( ( j in Seg i implies M . j = B . (F9 . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) holds
ex Path being FinSequence of K st
( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F9 . j holds
Path . j = A * (j,Fj) ) & ( the multF of K $$ p) * (Det M) = ( the multF of K $$ Path) * (Det M) )
let M9 be Matrix of n,n,K; ::_thesis: ( M9 = (A * B) +* ((B * ((idseq n) +* F9)) | (Seg i)) & ( for j being Nat holds
( ( j in Seg i implies M9 . j = B . (F9 . j) ) & ( not j in Seg i implies M9 . j = (A * B) . j ) ) ) implies ex Path being FinSequence of K st
( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F9 . j holds
Path . j = A * (j,Fj) ) & ( the multF of K $$ p) * (Det M) = ( the multF of K $$ Path) * (Det M9) ) )
assume that
M9 = (A * B) +* ((B * ((idseq n) +* F9)) | (Seg i)) and
A11: for j being Nat holds
( ( j in Seg i implies M9 . j = B . (F9 . j) ) & ( not j in Seg i implies M9 . j = (A * B) . j ) ) ; ::_thesis: ex Path being FinSequence of K st
( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F9 . j holds
Path . j = A * (j,Fj) ) & ( the multF of K $$ p) * (Det M) = ( the multF of K $$ Path) * (Det M9) )
take p ; ::_thesis: ( len p = i & ( for Fj, j being Nat st j in Seg i & Fj = F9 . j holds
p . j = A * (j,Fj) ) & ( the multF of K $$ p) * (Det M) = ( the multF of K $$ p) * (Det M9) )
dom path = Seg i by FUNCT_2:def_1;
hence ( len p = i & ( for Fj, j being Nat st j in Seg i & Fj = F9 . j holds
p . j = A * (j,Fj) ) ) by A8, A10, A4, FINSEQ_1:def_3; ::_thesis: ( the multF of K $$ p) * (Det M) = ( the multF of K $$ p) * (Det M9)
A12: len M9 = n by MATRIX_1:24;
A13: now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_k_<=_len_M_holds_
M_._k_=_M9_._k
let k be Nat; ::_thesis: ( 1 <= k & k <= len M implies M . b1 = M9 . b1 )
assume that
1 <= k and
k <= len M ; ::_thesis: M . b1 = M9 . b1
percases ( k in Seg i or not k in Seg i ) ;
supposeA14: k in Seg i ; ::_thesis: M . b1 = M9 . b1
then M . k = B . (F . k) by A9;
hence M . k = M9 . k by A10, A11, A14; ::_thesis: verum
end;
supposeA15: not k in Seg i ; ::_thesis: M . b1 = M9 . b1
then M . k = AB . k by A9;
hence M . k = M9 . k by A11, A15; ::_thesis: verum
end;
end;
end;
len M = n by MATRIX_1:24;
hence ( the multF of K $$ p) * (Det M) = ( the multF of K $$ p) * (Det M9) by A12, A13, FINSEQ_1:14; ::_thesis: verum
end;
consider P being Function of FF, the carrier of K such that
A16: for x being Element of FF holds S1[x,P . x] from FUNCT_2:sch_3(A3);
reconsider P = P as Function of (Funcs ((Seg i),(Seg n))), the carrier of K ;
take P ; ::_thesis: for F being Function of (Seg i),(Seg n)
for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds
( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) holds
ex Path being FinSequence of K st
( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds
Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det M) )
let F be Function of (Seg i),(Seg n); ::_thesis: for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds
( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) holds
ex Path being FinSequence of K st
( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds
Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det M) )
reconsider F9 = F as Function of (Seg i),Sn ;
let M be Matrix of n,K; ::_thesis: ( M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds
( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) implies ex Path being FinSequence of K st
( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds
Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det M) ) )
assume that
A17: M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg i)) and
A18: for j being Nat holds
( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ; ::_thesis: ex Path being FinSequence of K st
( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds
Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det M) )
F9 in FF by FUNCT_2:8;
hence ex Path being FinSequence of K st
( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds
Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det M) ) by A16, A17, A18; ::_thesis: verum
end;
theorem Th60: :: MATRIX11:60
for n being Nat
for K being Field
for A, B being Matrix of n,K st 0 < n holds
ex P being Function of (Funcs ((Seg n),(Seg n))), the carrier of K st
( ( for F being Function of (Seg n),(Seg n) ex Path being FinSequence of K st
( len Path = n & ( for Fj, j being Nat st j in Seg n & Fj = F . j holds
Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det (B * F)) ) ) & Det (A * B) = the addF of K $$ ((FinOmega (Funcs ((Seg n),(Seg n)))),P) )
proof
let n be Nat; ::_thesis: for K being Field
for A, B being Matrix of n,K st 0 < n holds
ex P being Function of (Funcs ((Seg n),(Seg n))), the carrier of K st
( ( for F being Function of (Seg n),(Seg n) ex Path being FinSequence of K st
( len Path = n & ( for Fj, j being Nat st j in Seg n & Fj = F . j holds
Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det (B * F)) ) ) & Det (A * B) = the addF of K $$ ((FinOmega (Funcs ((Seg n),(Seg n)))),P) )
let K be Field; ::_thesis: for A, B being Matrix of n,K st 0 < n holds
ex P being Function of (Funcs ((Seg n),(Seg n))), the carrier of K st
( ( for F being Function of (Seg n),(Seg n) ex Path being FinSequence of K st
( len Path = n & ( for Fj, j being Nat st j in Seg n & Fj = F . j holds
Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det (B * F)) ) ) & Det (A * B) = the addF of K $$ ((FinOmega (Funcs ((Seg n),(Seg n)))),P) )
let A, B be Matrix of n,K; ::_thesis: ( 0 < n implies ex P being Function of (Funcs ((Seg n),(Seg n))), the carrier of K st
( ( for F being Function of (Seg n),(Seg n) ex Path being FinSequence of K st
( len Path = n & ( for Fj, j being Nat st j in Seg n & Fj = F . j holds
Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det (B * F)) ) ) & Det (A * B) = the addF of K $$ ((FinOmega (Funcs ((Seg n),(Seg n)))),P) ) )
assume A1: 0 < n ; ::_thesis: ex P being Function of (Funcs ((Seg n),(Seg n))), the carrier of K st
( ( for F being Function of (Seg n),(Seg n) ex Path being FinSequence of K st
( len Path = n & ( for Fj, j being Nat st j in Seg n & Fj = F . j holds
Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det (B * F)) ) ) & Det (A * B) = the addF of K $$ ((FinOmega (Funcs ((Seg n),(Seg n)))),P) )
set AB = A * B;
set aa = the addF of K;
set I = idseq n;
set mm = the multF of K;
set KK = the carrier of K;
defpred S1[ Function, Nat] means for F being Function of (Seg $2),(Seg n)
for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg $2)) & ( for j being Nat holds
( ( j in Seg $2 implies M . j = B . (F . j) ) & ( not j in Seg $2 implies M . j = (A * B) . j ) ) ) holds
ex Path being FinSequence of K st
( len Path = $2 & ( for Fj, j being Nat st j in Seg $2 & Fj = F . j holds
Path . j = A * (j,Fj) ) & $1 . F = ( the multF of K $$ Path) * (Det M) );
defpred S2[ Nat] means ( $1 <= n implies for FUNC being non empty set st FUNC = Funcs ((Seg $1),(Seg n)) holds
ex P being Function of FUNC, the carrier of K st
( S1[P,$1] & Det (A * B) = the addF of K $$ ((FinOmega FUNC),P) ) );
A2: for k being Element of NAT st S2[k] holds
S2[k + 1]
proof
set Y = Seg n;
let k be Element of NAT ; ::_thesis: ( S2[k] implies S2[k + 1] )
assume A3: S2[k] ; ::_thesis: S2[k + 1]
set X = Seg k;
reconsider FUNC = Funcs ((Seg k),(Seg n)) as non empty set by A1;
set k1 = k + 1;
assume A4: k + 1 <= n ; ::_thesis: for FUNC being non empty set st FUNC = Funcs ((Seg (k + 1)),(Seg n)) holds
ex P being Function of FUNC, the carrier of K st
( S1[P,k + 1] & Det (A * B) = the addF of K $$ ((FinOmega FUNC),P) )
set Xx = (Seg k) \/ {(k + 1)};
let FUNC1 be non empty set ; ::_thesis: ( FUNC1 = Funcs ((Seg (k + 1)),(Seg n)) implies ex P being Function of FUNC1, the carrier of K st
( S1[P,k + 1] & Det (A * B) = the addF of K $$ ((FinOmega FUNC1),P) ) )
assume A5: FUNC1 = Funcs ((Seg (k + 1)),(Seg n)) ; ::_thesis: ex P being Function of FUNC1, the carrier of K st
( S1[P,k + 1] & Det (A * B) = the addF of K $$ ((FinOmega FUNC1),P) )
consider P1 being Function of FUNC1, the carrier of K such that
A6: S1[P1,k + 1] by A4, A5, Lm10;
reconsider FUNC19 = Funcs (((Seg k) \/ {(k + 1)}),(Seg n)) as non empty set by A5, FINSEQ_1:9;
A7: FUNC1 = Funcs (((Seg k) \/ {(k + 1)}),(Seg n)) by A5, FINSEQ_1:9;
then reconsider P19 = P1 as Function of FUNC19, the carrier of K ;
A8: k + 0 <= k + 1 by XREAL_1:8;
then consider P being Function of FUNC, the carrier of K such that
A9: S1[P,k] and
A10: Det (A * B) = the addF of K $$ ((FinOmega FUNC),P) by A3, A4, XXREAL_0:2;
A11: not k + 1 in Seg k by FINSEQ_3:8;
A12: for H being Function of (Seg k),(Seg n)
for SF being Element of Fin FUNC19 st SF = { h where h is Function of ((Seg k) \/ {(k + 1)}),(Seg n) : h | (Seg k) = H } holds
the addF of K $$ (SF,P19) = P . H
proof
reconsider YY = Seg n as non empty set by A1;
let H be Function of (Seg k),(Seg n); ::_thesis: for SF being Element of Fin FUNC19 st SF = { h where h is Function of ((Seg k) \/ {(k + 1)}),(Seg n) : h | (Seg k) = H } holds
the addF of K $$ (SF,P19) = P . H
let SF be Element of Fin FUNC19; ::_thesis: ( SF = { h where h is Function of ((Seg k) \/ {(k + 1)}),(Seg n) : h | (Seg k) = H } implies the addF of K $$ (SF,P19) = P . H )
assume A13: SF = { h where h is Function of ((Seg k) \/ {(k + 1)}),(Seg n) : h | (Seg k) = H } ; ::_thesis: the addF of K $$ (SF,P19) = P . H
defpred S3[ set , set ] means for h being Function of ((Seg k) \/ {(k + 1)}),(Seg n) st h | (Seg k) = H & h . (k + 1) = $1 holds
h = $2;
A14: for y being set st y in YY holds
ex f9 being set st
( f9 in SF & S3[y,f9] )
proof
let y be set ; ::_thesis: ( y in YY implies ex f9 being set st
( f9 in SF & S3[y,f9] ) )
assume y in YY ; ::_thesis: ex f9 being set st
( f9 in SF & S3[y,f9] )
then (Seg n) \/ {y} = Seg n by ZFMISC_1:40;
then consider q being Function of ((Seg k) \/ {(k + 1)}),(Seg n) such that
A15: q | (Seg k) = H and
A16: q . (k + 1) = y by A11, STIRL2_1:57;
take q ; ::_thesis: ( q in SF & S3[y,q] )
thus q in SF by A13, A15; ::_thesis: S3[y,q]
let h be Function of ((Seg k) \/ {(k + 1)}),(Seg n); ::_thesis: ( h | (Seg k) = H & h . (k + 1) = y implies h = q )
assume that
A17: h | (Seg k) = H and
A18: h . (k + 1) = y ; ::_thesis: h = q
now__::_thesis:_for_x_being_set_st_x_in_(Seg_k)_\/_{(k_+_1)}_holds_
h_._x_=_q_._x
let x be set ; ::_thesis: ( x in (Seg k) \/ {(k + 1)} implies h . x = q . x )
assume x in (Seg k) \/ {(k + 1)} ; ::_thesis: h . x = q . x
then A19: ( x in Seg k or x in {(k + 1)} ) by XBOOLE_0:def_3;
dom H = Seg k by A1, FUNCT_2:def_1;
then ( ( q . x = H . x & h . x = H . x ) or x = k + 1 ) by A15, A17, A19, FUNCT_1:47, TARSKI:def_1;
hence h . x = q . x by A16, A18; ::_thesis: verum
end;
hence h = q by FUNCT_2:12; ::_thesis: verum
end;
consider QQ being Function of YY,SF such that
A20: for y being set st y in YY holds
S3[y,QQ . y] from FUNCT_2:sch_1(A14);
A21: SF c= rng QQ
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in SF or y in rng QQ )
k + 1 in {(k + 1)} by TARSKI:def_1;
then A22: k + 1 in (Seg k) \/ {(k + 1)} by XBOOLE_0:def_3;
assume A23: y in SF ; ::_thesis: y in rng QQ
then consider h being Function of ((Seg k) \/ {(k + 1)}),(Seg n) such that
A24: y = h and
A25: h | (Seg k) = H by A13;
dom h = (Seg k) \/ {(k + 1)} by A1, FUNCT_2:def_1;
then A26: h . (k + 1) in rng h by A22, FUNCT_1:def_3;
A27: rng h c= Seg n by RELAT_1:def_19;
dom QQ = YY by A23, FUNCT_2:def_1;
then QQ . (h . (k + 1)) in rng QQ by A26, A27, FUNCT_1:def_3;
hence y in rng QQ by A20, A24, A25, A26, A27; ::_thesis: verum
end;
rng QQ c= SF by RELAT_1:def_19;
then A28: rng QQ = SF by A21, XBOOLE_0:def_10;
(Seg n) \/ {n} = Seg n by A1, FINSEQ_1:3, ZFMISC_1:40;
then consider h being Function of ((Seg k) \/ {(k + 1)}),(Seg n) such that
A29: h | (Seg k) = H and
h . (k + 1) = n by A11, STIRL2_1:57;
A30: SF c= FUNC19 by FINSUB_1:def_5;
k <= n by A4, A8, XXREAL_0:2;
then consider Mh being Matrix of n,K such that
A31: Mh = (A * B) +* ((B * ((idseq n) +* H)) | (Seg k)) and
A32: for j being Nat holds
( ( j in Seg k implies Mh . j = B . (H . j) ) & ( not j in Seg k implies Mh . j = (A * B) . j ) ) by A1, Th59;
consider Path being FinSequence of K such that
A33: len Path = k and
A34: for Hj, j being Nat st j in Seg k & Hj = H . j holds
Path . j = A * (j,Hj) and
A35: P . H = ( the multF of K $$ Path) * (Det Mh) by A9, A31, A32;
A36: Mh . (k + 1) = (A * B) . (k + 1) by A11, A32;
h in SF by A13, A29;
then reconsider QQ = QQ as Function of YY,FUNC19 by A28, A30, FUNCT_2:6;
A37: dom (P19 * QQ) = Seg n by FUNCT_2:def_1;
A38: QQ .: (dom QQ) = SF by A28, RELAT_1:113;
1 + 0 <= k + 1 by XREAL_1:7;
then A39: k + 1 in Seg n by A4;
then A40: (A * B) . (k + 1) = Line ((A * B),(k + 1)) by MATRIX_2:8;
Mh . (k + 1) = Line (Mh,(k + 1)) by A39, MATRIX_2:8;
then Mh = RLine (Mh,(k + 1),(Line ((A * B),(k + 1)))) by A36, A40, Th30;
then consider SUM1 being FinSequence of the carrier of K such that
A41: len SUM1 = n and
A42: Det Mh = the addF of K "**" SUM1 and
A43: for j being Nat st j in Seg n holds
SUM1 . j = (A * ((k + 1),j)) * (Det (RLine (Mh,(k + 1),(Line (B,j))))) by A39, Th56;
A44: dom (id (Seg n)) = Seg n ;
set PA = the multF of K "**" Path;
set PS = ( the multF of K "**" Path) * SUM1;
len (( the multF of K "**" Path) * SUM1) = n by A41, Lm5;
then A45: dom (( the multF of K "**" Path) * SUM1) = Seg n by FINSEQ_1:def_3;
set PSaa = [#] ((( the multF of K "**" Path) * SUM1),(the_unity_wrt the addF of K));
A46: for j being Nat st j in Seg n holds
(P19 * QQ) . j = (( the multF of K "**" Path) * (A * ((k + 1),j))) * (Det (RLine (Mh,(k + 1),(Line (B,j)))))
proof
A47: width B = n by MATRIX_1:24;
A48: len Mh = n by MATRIX_1:24;
A49: dom (P19 * QQ) = Seg n by FUNCT_2:def_1;
A50: width Mh = n by MATRIX_1:24;
let j be Nat; ::_thesis: ( j in Seg n implies (P19 * QQ) . j = (( the multF of K "**" Path) * (A * ((k + 1),j))) * (Det (RLine (Mh,(k + 1),(Line (B,j))))) )
assume A51: j in Seg n ; ::_thesis: (P19 * QQ) . j = (( the multF of K "**" Path) * (A * ((k + 1),j))) * (Det (RLine (Mh,(k + 1),(Line (B,j)))))
(Seg n) \/ {j} = Seg n by A51, ZFMISC_1:40;
then consider hj being Function of ((Seg k) \/ {(k + 1)}),(Seg n) such that
A52: hj | (Seg k) = H and
A53: hj . (k + 1) = j by A11, STIRL2_1:57;
set L = Line (B,j);
set R = RLine (Mh,(k + 1),(Line (B,j)));
(Seg k) \/ {(k + 1)} = Seg (k + 1) by FINSEQ_1:9;
then reconsider hj9 = hj as Function of (Seg (k + 1)),(Seg n) ;
consider Mhj being Matrix of n,K such that
A54: Mhj = (A * B) +* ((B * ((idseq n) +* hj9)) | (Seg (k + 1))) and
A55: for i being Nat holds
( ( i in Seg (k + 1) implies Mhj . i = B . (hj9 . i) ) & ( not i in Seg (k + 1) implies Mhj . i = (A * B) . i ) ) by A4, Th59;
A56: len Mhj = n by MATRIX_1:24;
A57: len (Line (B,j)) = width B by MATRIX_1:def_7;
A58: now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_len_Mhj_holds_
Mhj_._i_=_(RLine_(Mh,(k_+_1),(Line_(B,j))))_._i
A59: k + 0 < k + 1 by XREAL_1:8;
let i be Nat; ::_thesis: ( 1 <= i & i <= len Mhj implies Mhj . b1 = (RLine (Mh,(k + 1),(Line (B,j)))) . b1 )
assume that
A60: 1 <= i and
A61: i <= len Mhj ; ::_thesis: Mhj . b1 = (RLine (Mh,(k + 1),(Line (B,j)))) . b1
A62: i in NAT by ORDINAL1:def_12;
then A63: i in Seg n by A56, A60, A61;
percases ( i <= k or ( i > k & i <= k + 1 ) or ( i > k & i > k + 1 ) ) ;
supposeA64: i <= k ; ::_thesis: Mhj . b1 = (RLine (Mh,(k + 1),(Line (B,j)))) . b1
then i <= k + 1 by A59, XXREAL_0:2;
then A65: i in Seg (k + 1) by A60, A62;
A66: Line ((RLine (Mh,(k + 1),(Line (B,j)))),i) = (RLine (Mh,(k + 1),(Line (B,j)))) . i by A63, MATRIX_2:8;
A67: i in Seg k by A60, A62, A64;
then A68: Mh . i = B . (H . i) by A32;
dom H = Seg k by A56, A60, A61, FUNCT_2:def_1;
then A69: H . i = hj . i by A52, A67, FUNCT_1:47;
A70: Line (Mh,i) = Mh . i by A63, MATRIX_2:8;
Line ((RLine (Mh,(k + 1),(Line (B,j)))),i) = Line (Mh,i) by A63, A59, A64, Th28;
hence Mhj . i = (RLine (Mh,(k + 1),(Line (B,j)))) . i by A55, A66, A70, A68, A69, A65; ::_thesis: verum
end;
supposeA71: ( i > k & i <= k + 1 ) ; ::_thesis: Mhj . b1 = (RLine (Mh,(k + 1),(Line (B,j)))) . b1
A72: k + 1 in Seg (k + 1) by FINSEQ_1:4;
A73: Line (B,j) = B . j by A51, MATRIX_2:8;
A74: (RLine (Mh,(k + 1),(Line (B,j)))) . i = Line ((RLine (Mh,(k + 1),(Line (B,j)))),i) by A63, MATRIX_2:8;
A75: i = k + 1 by A71, NAT_1:9;
then Line (B,j) = Line ((RLine (Mh,(k + 1),(Line (B,j)))),i) by A57, A47, A50, A63, Th28;
hence Mhj . i = (RLine (Mh,(k + 1),(Line (B,j)))) . i by A53, A55, A75, A72, A74, A73; ::_thesis: verum
end;
supposeA76: ( i > k & i > k + 1 ) ; ::_thesis: Mhj . b1 = (RLine (Mh,(k + 1),(Line (B,j)))) . b1
then not i in Seg (k + 1) by FINSEQ_1:1;
then A77: Mhj . i = (A * B) . i by A55;
A78: Line ((RLine (Mh,(k + 1),(Line (B,j)))),i) = (RLine (Mh,(k + 1),(Line (B,j)))) . i by A63, MATRIX_2:8;
A79: not i in Seg k by A76, FINSEQ_1:1;
A80: Line (Mh,i) = Mh . i by A63, MATRIX_2:8;
Line ((RLine (Mh,(k + 1),(Line (B,j)))),i) = Line (Mh,i) by A63, A76, Th28;
hence Mhj . i = (RLine (Mh,(k + 1),(Line (B,j)))) . i by A32, A78, A80, A79, A77; ::_thesis: verum
end;
end;
end;
len (RLine (Mh,(k + 1),(Line (B,j)))) = len Mh by Lm4;
then RLine (Mh,(k + 1),(Line (B,j))) = Mhj by A48, A56, A58, FINSEQ_1:14;
then consider Pathj being FinSequence of K such that
A81: len Pathj = k + 1 and
A82: for m, j being Nat st j in Seg (k + 1) & m = hj . j holds
Pathj . j = A * (j,m) and
A83: P1 . hj = ( the multF of K "**" Pathj) * (Det (RLine (Mh,(k + 1),(Line (B,j))))) by A6, A54, A55;
A84: Pathj . (k + 1) = A * ((k + 1),j) by A53, A82, FINSEQ_1:4;
A85: now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_len_Path_holds_
Path_._i_=_(Pathj_|_(Seg_k))_._i
A86: rng H c= Seg n by RELAT_1:def_19;
A87: Seg k = dom H by A51, FUNCT_2:def_1;
let i be Nat; ::_thesis: ( 1 <= i & i <= len Path implies Path . i = (Pathj | (Seg k)) . i )
assume that
A88: 1 <= i and
A89: i <= len Path ; ::_thesis: Path . i = (Pathj | (Seg k)) . i
A90: (Pathj | k) . i = Pathj . i by A33, A89, FINSEQ_3:112;
A91: i in NAT by ORDINAL1:def_12;
then A92: i in Seg k by A33, A88, A89;
then H . i in rng H by A87, FUNCT_1:def_3;
then H . i in Seg n by A86;
then reconsider Hi = H . i as Nat ;
i <= k + 1 by A8, A33, A89, XXREAL_0:2;
then A93: i in Seg (k + 1) by A88, A91;
i in Seg k by A33, A88, A89, A91;
then A94: Path . i = A * (i,Hi) by A34;
H . i = hj . i by A52, A92, A87, FUNCT_1:47;
hence Path . i = (Pathj | (Seg k)) . i by A82, A93, A94, A90; ::_thesis: verum
end;
len (Pathj | k) = k by A8, A81, FINSEQ_1:59;
then Pathj = Path ^ <*(Pathj . (k + 1))*> by A33, A81, A85, FINSEQ_1:14, FINSEQ_3:55;
then A95: the multF of K "**" Pathj = ( the multF of K "**" Path) * (A * ((k + 1),j)) by A84, FINSOP_1:4;
QQ . j = hj by A20, A51, A52, A53;
hence (P19 * QQ) . j = (( the multF of K "**" Path) * (A * ((k + 1),j))) * (Det (RLine (Mh,(k + 1),(Line (B,j))))) by A51, A83, A49, A95, FUNCT_1:12; ::_thesis: verum
end;
now__::_thesis:_for_y_being_set_st_y_in_dom_((_the_multF_of_K_"**"_Path)_*_SUM1)_holds_
((_the_multF_of_K_"**"_Path)_*_SUM1)_._y_=_(P19_*_QQ)_._y
let y be set ; ::_thesis: ( y in dom (( the multF of K "**" Path) * SUM1) implies (( the multF of K "**" Path) * SUM1) . y = (P19 * QQ) . y )
assume A96: y in dom (( the multF of K "**" Path) * SUM1) ; ::_thesis: (( the multF of K "**" Path) * SUM1) . y = (P19 * QQ) . y
reconsider j = y as Nat by A96;
SUM1 . j = (A * ((k + 1),j)) * (Det (RLine (Mh,(k + 1),(Line (B,j))))) by A43, A45, A96;
hence (( the multF of K "**" Path) * SUM1) . y = ( the multF of K "**" Path) * ((A * ((k + 1),j)) * (Det (RLine (Mh,(k + 1),(Line (B,j)))))) by A96, FVSUM_1:50
.= (( the multF of K "**" Path) * (A * ((k + 1),j))) * (Det (RLine (Mh,(k + 1),(Line (B,j))))) by GROUP_1:def_3
.= (P19 * QQ) . y by A46, A45, A96 ;
::_thesis: verum
end;
then ( the multF of K "**" Path) * SUM1 = P19 * QQ by A37, A45, FUNCT_1:2;
then A97: ([#] ((( the multF of K "**" Path) * SUM1),(the_unity_wrt the addF of K))) | (dom (( the multF of K "**" Path) * SUM1)) = P19 * QQ by SETWOP_2:21;
now__::_thesis:_for_x1,_x2_being_set_st_x1_in_Seg_n_&_x2_in_Seg_n_&_QQ_._x1_=_QQ_._x2_holds_
x1_=_x2
let x1, x2 be set ; ::_thesis: ( x1 in Seg n & x2 in Seg n & QQ . x1 = QQ . x2 implies x1 = x2 )
assume that
A98: x1 in Seg n and
A99: x2 in Seg n and
A100: QQ . x1 = QQ . x2 ; ::_thesis: x1 = x2
(Seg n) \/ {x2} = Seg n by A99, ZFMISC_1:40;
then A101: ex h2 being Function of ((Seg k) \/ {(k + 1)}),(Seg n) st
( h2 | (Seg k) = H & h2 . (k + 1) = x2 ) by A11, STIRL2_1:57;
(Seg n) \/ {x1} = Seg n by A98, ZFMISC_1:40;
then consider h1 being Function of ((Seg k) \/ {(k + 1)}),(Seg n) such that
A102: h1 | (Seg k) = H and
A103: h1 . (k + 1) = x1 by A11, STIRL2_1:57;
QQ . x1 = h1 by A20, A98, A102, A103;
hence x1 = x2 by A20, A99, A100, A103, A101; ::_thesis: verum
end;
then A104: QQ is one-to-one by FUNCT_2:19;
reconsider Y9 = Seg n as Element of Fin YY by FINSUB_1:def_5;
A105: dom QQ = Y9 by FUNCT_2:def_1;
A106: rng (id (Seg n)) = Seg n ;
(P19 * QQ) * (id (Seg n)) = P19 * QQ by A37, RELAT_1:52;
then the addF of K $$ (Y9,(P19 * QQ)) = the addF of K $$ ((findom (( the multF of K "**" Path) * SUM1)),([#] ((( the multF of K "**" Path) * SUM1),(the_unity_wrt the addF of K)))) by A45, A44, A106, A97, SETWOP_2:5
.= Sum (( the multF of K "**" Path) * SUM1) by FVSUM_1:8, SETWOP_2:def_2
.= ( the multF of K "**" Path) * (Sum SUM1) by FVSUM_1:73
.= P . H by A35, A42 ;
hence the addF of K $$ (SF,P19) = P . H by A104, A38, A105, SETWOP_2:6; ::_thesis: verum
end;
the addF of K is having_a_unity by FVSUM_1:8;
then Det (A * B) = the addF of K $$ ((FinOmega FUNC19),P19) by A1, A10, A11, A12, Th58;
hence ex P being Function of FUNC1, the carrier of K st
( S1[P,k + 1] & Det (A * B) = the addF of K $$ ((FinOmega FUNC1),P) ) by A6, A7; ::_thesis: verum
end;
set FUN = Funcs ((Seg n),(Seg n));
A107: n is Element of NAT by ORDINAL1:def_12;
A108: S2[ 0 ]
proof
reconsider E = {} as Function of (Seg 0),(Seg n) by XBOOLE_1:2;
assume 0 <= n ; ::_thesis: for FUNC being non empty set st FUNC = Funcs ((Seg 0),(Seg n)) holds
ex P being Function of FUNC, the carrier of K st
( S1[P, 0 ] & Det (A * B) = the addF of K $$ ((FinOmega FUNC),P) )
A109: the_unity_wrt the multF of K = 1_ K by GROUP_1:22;
let FUNC be non empty set ; ::_thesis: ( FUNC = Funcs ((Seg 0),(Seg n)) implies ex P being Function of FUNC, the carrier of K st
( S1[P, 0 ] & Det (A * B) = the addF of K $$ ((FinOmega FUNC),P) ) )
assume A110: FUNC = Funcs ((Seg 0),(Seg n)) ; ::_thesis: ex P being Function of FUNC, the carrier of K st
( S1[P, 0 ] & Det (A * B) = the addF of K $$ ((FinOmega FUNC),P) )
consider P being Function of FUNC, the carrier of K such that
A111: S1[P, 0 ] by A1, A110, Lm10;
A112: FUNC = {E} by A110, FUNCT_5:57;
then A113: E in FUNC by TARSKI:def_1;
FinOmega FUNC = {E} by A112, MATRIX_2:def_14;
then A114: the addF of K $$ ((FinOmega FUNC),P) = P . E by A113, SETWISEO:17;
consider M being Matrix of n,K such that
A115: M = (A * B) +* ((B * ((idseq n) +* E)) | (Seg 0)) and
A116: for j being Nat holds
( ( j in Seg 0 implies M . j = B . (E . j) ) & ( not j in Seg 0 implies M . j = (A * B) . j ) ) by A1, Th59;
A117: M = (A * B) +* {} by A115;
consider Path being FinSequence of K such that
A118: len Path = 0 and
for Fj, j being Nat st j in Seg 0 & Fj = E . j holds
Path . j = A * (j,Fj) and
A119: P . E = ( the multF of K $$ Path) * (Det M) by A111, A115, A116;
Path = <*> the carrier of K by A118;
then the multF of K "**" Path = 1_ K by A109, FINSOP_1:10;
then P . E = Det (A * B) by A119, A117, VECTSP_1:def_4;
hence ex P being Function of FUNC, the carrier of K st
( S1[P, 0 ] & Det (A * B) = the addF of K $$ ((FinOmega FUNC),P) ) by A111, A114; ::_thesis: verum
end;
for k being Element of NAT holds S2[k] from NAT_1:sch_1(A108, A2);
then consider P being Function of (Funcs ((Seg n),(Seg n))), the carrier of K such that
A120: S1[P,n] and
A121: Det (A * B) = the addF of K $$ ((FinOmega (Funcs ((Seg n),(Seg n)))),P) by A107;
take P ; ::_thesis: ( ( for F being Function of (Seg n),(Seg n) ex Path being FinSequence of K st
( len Path = n & ( for Fj, j being Nat st j in Seg n & Fj = F . j holds
Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det (B * F)) ) ) & Det (A * B) = the addF of K $$ ((FinOmega (Funcs ((Seg n),(Seg n)))),P) )
thus for F being Function of (Seg n),(Seg n) ex Path being FinSequence of K st
( len Path = n & ( for Fj, j being Nat st j in Seg n & Fj = F . j holds
Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det (B * F)) ) ::_thesis: Det (A * B) = the addF of K $$ ((FinOmega (Funcs ((Seg n),(Seg n)))),P)
proof
len (A * B) = n by MATRIX_1:24;
then A122: dom (A * B) = Seg n by FINSEQ_1:def_3;
let F be Function of (Seg n),(Seg n); ::_thesis: ex Path being FinSequence of K st
( len Path = n & ( for Fj, j being Nat st j in Seg n & Fj = F . j holds
Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det (B * F)) )
A123: dom (idseq n) = Seg n by FUNCT_2:52;
dom F = Seg n by FUNCT_2:52;
then A124: (idseq n) +* F = F by A123, FUNCT_4:19;
A125: len B = n by MATRIX_1:24;
len (B * F) = len B by Def4;
then A126: dom (B * F) = Seg n by A125, FINSEQ_1:def_3;
consider M being Matrix of n,K such that
A127: M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg n)) and
A128: for j being Nat holds
( ( j in Seg n implies M . j = B . (F . j) ) & ( not j in Seg n implies M . j = (A * B) . j ) ) by A1, Th59;
(B * F) | n = B * F ;
then M = B * F by A127, A124, A126, A122, FUNCT_4:19;
hence ex Path being FinSequence of K st
( len Path = n & ( for Fj, j being Nat st j in Seg n & Fj = F . j holds
Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det (B * F)) ) by A120, A127, A128; ::_thesis: verum
end;
thus Det (A * B) = the addF of K $$ ((FinOmega (Funcs ((Seg n),(Seg n)))),P) by A121; ::_thesis: verum
end;
theorem Th61: :: MATRIX11:61
for n being Nat
for K being Field
for A, B being Matrix of n,K st 0 < n holds
ex P being Function of (Permutations n), the carrier of K st
( Det (A * B) = the addF of K $$ ((FinOmega (Permutations n)),P) & ( for perm being Element of Permutations n holds P . perm = ( the multF of K $$ (Path_matrix (perm,A))) * (- ((Det B),perm)) ) )
proof
let n be Nat; ::_thesis: for K being Field
for A, B being Matrix of n,K st 0 < n holds
ex P being Function of (Permutations n), the carrier of K st
( Det (A * B) = the addF of K $$ ((FinOmega (Permutations n)),P) & ( for perm being Element of Permutations n holds P . perm = ( the multF of K $$ (Path_matrix (perm,A))) * (- ((Det B),perm)) ) )
let K be Field; ::_thesis: for A, B being Matrix of n,K st 0 < n holds
ex P being Function of (Permutations n), the carrier of K st
( Det (A * B) = the addF of K $$ ((FinOmega (Permutations n)),P) & ( for perm being Element of Permutations n holds P . perm = ( the multF of K $$ (Path_matrix (perm,A))) * (- ((Det B),perm)) ) )
let A, B be Matrix of n,K; ::_thesis: ( 0 < n implies ex P being Function of (Permutations n), the carrier of K st
( Det (A * B) = the addF of K $$ ((FinOmega (Permutations n)),P) & ( for perm being Element of Permutations n holds P . perm = ( the multF of K $$ (Path_matrix (perm,A))) * (- ((Det B),perm)) ) ) )
assume A1: 0 < n ; ::_thesis: ex P being Function of (Permutations n), the carrier of K st
( Det (A * B) = the addF of K $$ ((FinOmega (Permutations n)),P) & ( for perm being Element of Permutations n holds P . perm = ( the multF of K $$ (Path_matrix (perm,A))) * (- ((Det B),perm)) ) )
set P = Permutations n;
A2: dom (id (Permutations n)) = Permutations n ;
set KK = the carrier of K;
set mm = the multF of K;
set aa = the addF of K;
set AB = A * B;
set X = Seg n;
set F = Funcs ((Seg n),(Seg n));
consider SUM1 being Function of (Funcs ((Seg n),(Seg n))), the carrier of K such that
A3: for F being Function of (Seg n),(Seg n) ex Path being FinSequence of K st
( len Path = n & ( for Fj, j being Nat st j in Seg n & Fj = F . j holds
Path . j = A * (j,Fj) ) & SUM1 . F = ( the multF of K $$ Path) * (Det (B * F)) ) and
A4: Det (A * B) = the addF of K $$ ((FinOmega (Funcs ((Seg n),(Seg n)))),SUM1) by A1, Th60;
A5: Funcs ((Seg n),(Seg n)) is finite by FRAENKEL:6;
then reconsider FP = (Funcs ((Seg n),(Seg n))) \ (Permutations n) as Element of Fin (Funcs ((Seg n),(Seg n))) by FINSUB_1:def_5;
A6: Permutations n c= Funcs ((Seg n),(Seg n))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Permutations n or x in Funcs ((Seg n),(Seg n)) )
assume x in Permutations n ; ::_thesis: x in Funcs ((Seg n),(Seg n))
then reconsider p = x as Permutation of (Seg n) by MATRIX_2:def_9;
p is Element of Funcs ((Seg n),(Seg n)) by FUNCT_2:9;
hence x in Funcs ((Seg n),(Seg n)) ; ::_thesis: verum
end;
then reconsider P9 = Permutations n as Element of Fin (Funcs ((Seg n),(Seg n))) by A5, FINSUB_1:def_5;
A7: Permutations n = FinOmega (Permutations n) by MATRIX_2:26, MATRIX_2:def_14;
A8: FinOmega (Funcs ((Seg n),(Seg n))) = Funcs ((Seg n),(Seg n)) by FRAENKEL:6, MATRIX_2:def_14;
A9: now__::_thesis:_Det_(A_*_B)_=_the_addF_of_K_$$_(P9,SUM1)
percases ( FP = {} or FP <> {} ) ;
suppose FP = {} ; ::_thesis: Det (A * B) = the addF of K $$ (P9,SUM1)
then Funcs ((Seg n),(Seg n)) c= Permutations n by XBOOLE_1:37;
hence Det (A * B) = the addF of K $$ (P9,SUM1) by A4, A8, A6, XBOOLE_0:def_10; ::_thesis: verum
end;
supposeA10: FP <> {} ; ::_thesis: Det (A * B) = the addF of K $$ (P9,SUM1)
A11: 0. K = the_unity_wrt the addF of K by FVSUM_1:7;
A12: SUM1 .: FP c= {(0. K)}
proof
let s be set ; :: according to TARSKI:def_3 ::_thesis: ( not s in SUM1 .: FP or s in {(0. K)} )
assume s in SUM1 .: FP ; ::_thesis: s in {(0. K)}
then consider x being set such that
x in dom SUM1 and
A13: x in FP and
A14: s = SUM1 . x by FUNCT_1:def_6;
reconsider f = x as Function of (Seg n),(Seg n) by A13, FUNCT_2:66;
not f in Permutations n by A13, XBOOLE_0:def_5;
then A15: Det (B * f) = 0. K by Th54;
ex Path being FinSequence of K st
( len Path = n & ( for Fj, j being Nat st j in Seg n & Fj = f . j holds
Path . j = A * (j,Fj) ) & SUM1 . f = ( the multF of K $$ Path) * (Det (B * f)) ) by A3;
then SUM1 . f = 0. K by A15, VECTSP_1:6;
hence s in {(0. K)} by A14, TARSKI:def_1; ::_thesis: verum
end;
dom SUM1 = Funcs ((Seg n),(Seg n)) by FUNCT_2:def_1;
then SUM1 .: FP = {(0. K)} by A10, A12, ZFMISC_1:33;
then A16: the addF of K $$ (FP,SUM1) = 0. K by A11, FVSUM_1:8, SETWOP_2:8;
A17: FP misses Permutations n by XBOOLE_1:79;
A18: FP \/ (Permutations n) = (Funcs ((Seg n),(Seg n))) \/ (Permutations n) by XBOOLE_1:39;
(Funcs ((Seg n),(Seg n))) \/ (Permutations n) = Funcs ((Seg n),(Seg n)) by A6, XBOOLE_1:12;
hence Det (A * B) = ( the addF of K $$ (P9,SUM1)) + (0. K) by A4, A8, A16, A17, A18, FVSUM_1:8, SETWOP_2:4
.= the addF of K $$ (P9,SUM1) by RLVECT_1:4 ;
::_thesis: verum
end;
end;
end;
dom SUM1 = Funcs ((Seg n),(Seg n)) by FUNCT_2:def_1;
then A19: dom (SUM1 | (Permutations n)) = Permutations n by A6, RELAT_1:62;
rng (SUM1 | (Permutations n)) c= the carrier of K by RELAT_1:def_19;
then reconsider SP = SUM1 | (Permutations n) as Function of (Permutations n), the carrier of K by A19, FUNCT_2:2;
take SP ; ::_thesis: ( Det (A * B) = the addF of K $$ ((FinOmega (Permutations n)),SP) & ( for perm being Element of Permutations n holds SP . perm = ( the multF of K $$ (Path_matrix (perm,A))) * (- ((Det B),perm)) ) )
A20: rng (id (Permutations n)) = Permutations n ;
SP * (id (Permutations n)) = SP by A19, RELAT_1:52;
hence Det (A * B) = the addF of K $$ ((FinOmega (Permutations n)),SP) by A9, A2, A20, A7, SETWOP_2:5; ::_thesis: for perm being Element of Permutations n holds SP . perm = ( the multF of K $$ (Path_matrix (perm,A))) * (- ((Det B),perm))
let perm be Element of Permutations n; ::_thesis: SP . perm = ( the multF of K $$ (Path_matrix (perm,A))) * (- ((Det B),perm))
reconsider Perm = perm as Permutation of (Seg n) by MATRIX_2:def_9;
SUM1 . Perm = SP . Perm by A19, FUNCT_1:47;
then consider Path being FinSequence of K such that
A21: len Path = n and
A22: for Fj, j being Nat st j in Seg n & Fj = Perm . j holds
Path . j = A * (j,Fj) and
A23: SP . Perm = ( the multF of K $$ Path) * (Det (B * Perm)) by A3;
set PM = Path_matrix (perm,A);
A24: len (Path_matrix (perm,A)) = n by MATRIX_3:def_7;
now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_len_Path_holds_
Path_._i_=_(Path_matrix_(perm,A))_._i
A25: Seg n = dom Perm by FUNCT_2:52;
let i be Nat; ::_thesis: ( 1 <= i & i <= len Path implies Path . i = (Path_matrix (perm,A)) . i )
assume that
A26: 1 <= i and
A27: i <= len Path ; ::_thesis: Path . i = (Path_matrix (perm,A)) . i
A28: i in NAT by ORDINAL1:def_12;
then A29: i in Seg n by A21, A26, A27;
i in Seg n by A21, A26, A27, A28;
then A30: Perm . i in rng Perm by A25, FUNCT_1:def_3;
rng Perm c= Seg n by RELAT_1:def_19;
then Perm . i in Seg n by A30;
then reconsider Pi = Perm . i as Element of NAT ;
dom (Path_matrix (perm,A)) = Seg n by A24, FINSEQ_1:def_3;
then (Path_matrix (perm,A)) . i = A * (i,Pi) by A29, MATRIX_3:def_7;
hence Path . i = (Path_matrix (perm,A)) . i by A22, A29; ::_thesis: verum
end;
then Path = Path_matrix (perm,A) by A21, A24, FINSEQ_1:14;
hence SP . perm = ( the multF of K $$ (Path_matrix (perm,A))) * (- ((Det B),perm)) by A23, Th46; ::_thesis: verum
end;
theorem :: MATRIX11:62
for n being Nat
for K being Field
for A, B being Matrix of n,K st 0 < n holds
Det (A * B) = (Det A) * (Det B)
proof
let n be Nat; ::_thesis: for K being Field
for A, B being Matrix of n,K st 0 < n holds
Det (A * B) = (Det A) * (Det B)
let K be Field; ::_thesis: for A, B being Matrix of n,K st 0 < n holds
Det (A * B) = (Det A) * (Det B)
let A, B be Matrix of n,K; ::_thesis: ( 0 < n implies Det (A * B) = (Det A) * (Det B) )
assume A1: 0 < n ; ::_thesis: Det (A * B) = (Det A) * (Det B)
set P = Permutations n;
set KK = the carrier of K;
set mm = the multF of K;
set aa = the addF of K;
set AB = A * B;
consider SUM1 being Function of (Permutations n), the carrier of K such that
A2: Det (A * B) = the addF of K $$ ((FinOmega (Permutations n)),SUM1) and
A3: for perm being Element of Permutations n holds SUM1 . perm = ( the multF of K $$ (Path_matrix (perm,A))) * (- ((Det B),perm)) by A1, Th61;
set Path = Path_product A;
set F = FinOmega (Permutations n);
A4: FinOmega (Permutations n) = Permutations n by MATRIX_2:26, MATRIX_2:def_14;
then consider Ga being Function of (Fin (Permutations n)), the carrier of K such that
A5: Det A = Ga . (FinOmega (Permutations n)) and
A6: for e being Element of the carrier of K st e is_a_unity_wrt the addF of K holds
Ga . {} = e and
A7: for x being Element of Permutations n holds Ga . {x} = (Path_product A) . x and
A8: for B9 being Element of Fin (Permutations n) st B9 c= FinOmega (Permutations n) & B9 <> {} holds
for x being Element of Permutations n st x in (FinOmega (Permutations n)) \ B9 holds
Ga . (B9 \/ {x}) = the addF of K . ((Ga . B9),((Path_product A) . x)) by SETWISEO:def_3;
A9: Ga . {} = 0. K by A6, FVSUM_1:6;
consider Gs being Function of (Fin (Permutations n)), the carrier of K such that
A10: Det (A * B) = Gs . (FinOmega (Permutations n)) and
A11: for e being Element of the carrier of K st e is_a_unity_wrt the addF of K holds
Gs . {} = e and
A12: for x being Element of Permutations n holds Gs . {x} = SUM1 . x and
A13: for B9 being Element of Fin (Permutations n) st B9 c= FinOmega (Permutations n) & B9 <> {} holds
for x being Element of Permutations n st x in (FinOmega (Permutations n)) \ B9 holds
Gs . (B9 \/ {x}) = the addF of K . ((Gs . B9),(SUM1 . x)) by A2, A4, SETWISEO:def_3;
defpred S1[ set ] means for B9 being Element of Fin (Permutations n) st B9 = $1 holds
Gs . B9 = (Ga . B9) * (Det B);
A14: for B9 being Element of Fin (Permutations n)
for b being Element of Permutations n st S1[B9] & not b in B9 holds
S1[B9 \/ {b}]
proof
let B9 be Element of Fin (Permutations n); ::_thesis: for b being Element of Permutations n st S1[B9] & not b in B9 holds
S1[B9 \/ {b}]
let b be Element of Permutations n; ::_thesis: ( S1[B9] & not b in B9 implies S1[B9 \/ {b}] )
assume that
A15: S1[B9] and
A16: not b in B9 ; ::_thesis: S1[B9 \/ {b}]
set mA = the multF of K $$ (Path_matrix (b,A));
let Bb be Element of Fin (Permutations n); ::_thesis: ( Bb = B9 \/ {b} implies Gs . Bb = (Ga . Bb) * (Det B) )
assume A17: Bb = B9 \/ {b} ; ::_thesis: Gs . Bb = (Ga . Bb) * (Det B)
A18: now__::_thesis:_SUM1_._b_=_((Path_product_A)_._b)_*_(Det_B)
percases ( b is even or b is odd ) ;
supposeA19: b is even ; ::_thesis: SUM1 . b = ((Path_product A) . b) * (Det B)
then A20: - (( the multF of K $$ (Path_matrix (b,A))),b) = the multF of K $$ (Path_matrix (b,A)) by MATRIX_2:def_13;
- ((Det B),b) = Det B by A19, MATRIX_2:def_13;
hence SUM1 . b = (- (( the multF of K $$ (Path_matrix (b,A))),b)) * (Det B) by A3, A20
.= ((Path_product A) . b) * (Det B) by MATRIX_3:def_8 ;
::_thesis: verum
end;
supposeA21: b is odd ; ::_thesis: SUM1 . b = ((Path_product A) . b) * (Det B)
then A22: - (( the multF of K $$ (Path_matrix (b,A))),b) = - ( the multF of K $$ (Path_matrix (b,A))) by MATRIX_2:def_13;
- ((Det B),b) = - (Det B) by A21, MATRIX_2:def_13;
then - ((- (( the multF of K $$ (Path_matrix (b,A))),b)) * (Det B)) = (- ( the multF of K $$ (Path_matrix (b,A)))) * (- ((Det B),b)) by A22, VECTSP_1:9
.= - (( the multF of K $$ (Path_matrix (b,A))) * (- ((Det B),b))) by VECTSP_1:9 ;
then (( the multF of K $$ (Path_matrix (b,A))) * (- ((Det B),b))) - ((- (( the multF of K $$ (Path_matrix (b,A))),b)) * (Det B)) = 0. K by VECTSP_1:16;
then A23: (- (( the multF of K $$ (Path_matrix (b,A))),b)) * (Det B) = ( the multF of K $$ (Path_matrix (b,A))) * (- ((Det B),b)) by VECTSP_1:19;
- (( the multF of K $$ (Path_matrix (b,A))),b) = (Path_product A) . b by MATRIX_3:def_8;
hence SUM1 . b = ((Path_product A) . b) * (Det B) by A3, A23; ::_thesis: verum
end;
end;
end;
percases ( B9 = {} or B9 <> {} ) ;
supposeA24: B9 = {} ; ::_thesis: Gs . Bb = (Ga . Bb) * (Det B)
then Ga . Bb = (Path_product A) . b by A7, A17;
hence Gs . Bb = (Ga . Bb) * (Det B) by A12, A17, A18, A24; ::_thesis: verum
end;
supposeA25: B9 <> {} ; ::_thesis: Gs . Bb = (Ga . Bb) * (Det B)
A26: B9 c= Permutations n by FINSUB_1:def_5;
A27: b in (FinOmega (Permutations n)) \ B9 by A4, A16, XBOOLE_0:def_5;
then Gs . Bb = the addF of K . ((Gs . B9),(SUM1 . b)) by A4, A13, A17, A25, A26;
then A28: Gs . Bb = ((Ga . B9) * (Det B)) + (((Path_product A) . b) * (Det B)) by A15, A18;
Ga . Bb = (Ga . B9) + ((Path_product A) . b) by A4, A8, A17, A25, A27, A26;
hence Gs . Bb = (Ga . Bb) * (Det B) by A28, VECTSP_1:def_7; ::_thesis: verum
end;
end;
end;
Gs . {} = 0. K by A11, FVSUM_1:6;
then A29: S1[ {}. (Permutations n)] by A9, VECTSP_1:7;
for B being Element of Fin (Permutations n) holds S1[B] from SETWISEO:sch_2(A29, A14);
hence Det (A * B) = (Det A) * (Det B) by A10, A5; ::_thesis: verum
end;