:: MATRTOP3 semantic presentation
begin
theorem Th1: :: MATRTOP3:1
for n being Nat
for K being Field
for M being Matrix of n,K
for P being Permutation of (Seg n) holds
( Det ((((M * P) @) * P) @) = Det M & ( for i, j being Nat st [i,j] in Indices M holds
((((M * P) @) * P) @) * (i,j) = M * ((P . i),(P . j)) ) )
proof
let n be Nat; ::_thesis: for K being Field
for M being Matrix of n,K
for P being Permutation of (Seg n) holds
( Det ((((M * P) @) * P) @) = Det M & ( for i, j being Nat st [i,j] in Indices M holds
((((M * P) @) * P) @) * (i,j) = M * ((P . i),(P . j)) ) )
let K be Field; ::_thesis: for M being Matrix of n,K
for P being Permutation of (Seg n) holds
( Det ((((M * P) @) * P) @) = Det M & ( for i, j being Nat st [i,j] in Indices M holds
((((M * P) @) * P) @) * (i,j) = M * ((P . i),(P . j)) ) )
let M be Matrix of n,K; ::_thesis: for P being Permutation of (Seg n) holds
( Det ((((M * P) @) * P) @) = Det M & ( for i, j being Nat st [i,j] in Indices M holds
((((M * P) @) * P) @) * (i,j) = M * ((P . i),(P . j)) ) )
let P be Permutation of (Seg n); ::_thesis: ( Det ((((M * P) @) * P) @) = Det M & ( for i, j being Nat st [i,j] in Indices M holds
((((M * P) @) * P) @) * (i,j) = M * ((P . i),(P . j)) ) )
reconsider p = P as Element of Permutations n by MATRIX_2:def_9;
A1: - (- (Det M)) = Det M by RLVECT_1:17;
A2: ( ( p is even & - ((Det M),p) = Det M ) or ( p is odd & - ((Det M),p) = - (Det M) ) ) by MATRIX_2:def_13;
thus Det ((((M * P) @) * P) @) = Det (((M * P) @) * P) by MATRIXR2:43
.= - ((Det ((M * P) @)),p) by MATRIX11:46
.= - ((Det (M * P)),p) by MATRIXR2:43
.= - ((- ((Det M),p)),p) by MATRIX11:46
.= Det M by A1, A2, MATRIX_2:def_13 ; ::_thesis: for i, j being Nat st [i,j] in Indices M holds
((((M * P) @) * P) @) * (i,j) = M * ((P . i),(P . j))
let i, j be Nat; ::_thesis: ( [i,j] in Indices M implies ((((M * P) @) * P) @) * (i,j) = M * ((P . i),(P . j)) )
assume A3: [i,j] in Indices M ; ::_thesis: ((((M * P) @) * P) @) * (i,j) = M * ((P . i),(P . j))
Indices M = Indices ((((M * P) @) * P) @) by MATRIX_1:26;
then A4: [j,i] in Indices (((M * P) @) * P) by A3, MATRIX_1:def_6;
then A5: ((((M * P) @) * P) @) * (i,j) = (((M * P) @) * P) * (j,i) by MATRIX_1:def_6;
( Indices M = Indices (((M * P) @) * P) & Indices M = Indices ((M * P) @) ) by MATRIX_1:26;
then A6: ex k being Nat st
( k = P . j & [k,i] in Indices ((M * P) @) & (((M * P) @) * P) * (j,i) = ((M * P) @) * (k,i) ) by A4, MATRIX11:37;
then A7: [i,(P . j)] in Indices (M * P) by MATRIX_1:def_6;
Indices (M * P) = Indices M by MATRIX_1:26;
then (M * P) * (i,(P . j)) = M * ((P . i),(P . j)) by A7, MATRIX11:def_4;
hence ((((M * P) @) * P) @) * (i,j) = M * ((P . i),(P . j)) by A5, A6, A7, MATRIX_1:def_6; ::_thesis: verum
end;
theorem Th2: :: MATRTOP3:2
for n being Nat
for K being Field
for M being V244(b2) Matrix of n,K holds M @ = M
proof
let n be Nat; ::_thesis: for K being Field
for M being V244(b1) Matrix of n,K holds M @ = M
let K be Field; ::_thesis: for M being V244(K) Matrix of n,K holds M @ = M
let M be V244(K) Matrix of n,K; ::_thesis: M @ = M
for i, j being Nat st [i,j] in Indices M holds
M * (i,j) = (M @) * (i,j)
proof
let i, j be Nat; ::_thesis: ( [i,j] in Indices M implies M * (i,j) = (M @) * (i,j) )
assume A1: [i,j] in Indices M ; ::_thesis: M * (i,j) = (M @) * (i,j)
then A2: [j,i] in Indices M by MATRIX_1:28;
then A3: (M @) * (i,j) = M * (j,i) by MATRIX_1:def_6;
percases ( i = j or i <> j ) ;
suppose i = j ; ::_thesis: M * (i,j) = (M @) * (i,j)
hence M * (i,j) = (M @) * (i,j) by A1, MATRIX_1:def_6; ::_thesis: verum
end;
supposeA4: i <> j ; ::_thesis: M * (i,j) = (M @) * (i,j)
then M * (i,j) = 0. K by A1, MATRIX_1:def_14;
hence M * (i,j) = (M @) * (i,j) by A2, A3, A4, MATRIX_1:def_14; ::_thesis: verum
end;
end;
end;
hence M @ = M by MATRIX_1:27; ::_thesis: verum
end;
theorem Th3: :: MATRTOP3:3
for r being real number
for f being real-valued FinSequence
for i being Nat st i in dom f holds
Sum (sqr (f +* (i,r))) = ((Sum (sqr f)) - ((f . i) ^2)) + (r ^2)
proof
let r be real number ; ::_thesis: for f being real-valued FinSequence
for i being Nat st i in dom f holds
Sum (sqr (f +* (i,r))) = ((Sum (sqr f)) - ((f . i) ^2)) + (r ^2)
let f be real-valued FinSequence; ::_thesis: for i being Nat st i in dom f holds
Sum (sqr (f +* (i,r))) = ((Sum (sqr f)) - ((f . i) ^2)) + (r ^2)
let i be Nat; ::_thesis: ( i in dom f implies Sum (sqr (f +* (i,r))) = ((Sum (sqr f)) - ((f . i) ^2)) + (r ^2) )
assume A1: i in dom f ; ::_thesis: Sum (sqr (f +* (i,r))) = ((Sum (sqr f)) - ((f . i) ^2)) + (r ^2)
reconsider fi = f . i as Element of REAL ;
set F = @ (@ f);
set G = (@ (@ f)) | (i -' 1);
set H = (@ (@ f)) /^ i;
A2: sqr <*fi*> = <*(fi ^2)*> by RVSUM_1:55;
@ (@ f) = (@ (@ f)) +* (i,fi) by FUNCT_7:35
.= (((@ (@ f)) | (i -' 1)) ^ <*fi*>) ^ ((@ (@ f)) /^ i) by A1, FUNCT_7:98 ;
then sqr (@ (@ f)) = (sqr (((@ (@ f)) | (i -' 1)) ^ <*fi*>)) ^ (sqr ((@ (@ f)) /^ i)) by RVSUM_1:144
.= ((sqr ((@ (@ f)) | (i -' 1))) ^ (sqr <*fi*>)) ^ (sqr ((@ (@ f)) /^ i)) by RVSUM_1:144 ;
then A3: Sum (sqr (@ (@ f))) = (Sum ((sqr ((@ (@ f)) | (i -' 1))) ^ (sqr <*fi*>))) + (Sum (sqr ((@ (@ f)) /^ i))) by RVSUM_1:75
.= ((Sum (sqr ((@ (@ f)) | (i -' 1)))) + (fi ^2)) + (Sum (sqr ((@ (@ f)) /^ i))) by A2, RVSUM_1:74 ;
reconsider R = r as Element of REAL by XREAL_0:def_1;
A4: sqr <*R*> = <*(R ^2)*> by RVSUM_1:55;
(@ (@ f)) +* (i,R) = (((@ (@ f)) | (i -' 1)) ^ <*R*>) ^ ((@ (@ f)) /^ i) by A1, FUNCT_7:98;
then sqr ((@ (@ f)) +* (i,R)) = (sqr (((@ (@ f)) | (i -' 1)) ^ <*R*>)) ^ (sqr ((@ (@ f)) /^ i)) by RVSUM_1:144
.= ((sqr ((@ (@ f)) | (i -' 1))) ^ (sqr <*R*>)) ^ (sqr ((@ (@ f)) /^ i)) by RVSUM_1:144 ;
then Sum (sqr ((@ (@ f)) +* (i,R))) = (Sum ((sqr ((@ (@ f)) | (i -' 1))) ^ (sqr <*R*>))) + (Sum (sqr ((@ (@ f)) /^ i))) by RVSUM_1:75
.= ((Sum (sqr ((@ (@ f)) | (i -' 1)))) + (R ^2)) + (Sum (sqr ((@ (@ f)) /^ i))) by A4, RVSUM_1:74 ;
hence Sum (sqr (f +* (i,r))) = ((Sum (sqr f)) - ((f . i) ^2)) + (r ^2) by A3; ::_thesis: verum
end;
definition
let X be set ;
let F be Function-yielding Function;
attrF is X -support-yielding means :Def1: :: MATRTOP3:def 1
for f being Function
for x being set st f in dom F & (F . f) . x <> f . x holds
x in X;
end;
:: deftheorem Def1 defines -support-yielding MATRTOP3:def_1_:_
for X being set
for F being Function-yielding Function holds
( F is X -support-yielding iff for f being Function
for x being set st f in dom F & (F . f) . x <> f . x holds
x in X );
registration
let X be set ;
cluster Relation-like Function-like Function-yielding V235() X -support-yielding for set ;
existence
ex b1 being Function-yielding Function st b1 is X -support-yielding
proof
reconsider F = {} as Function-yielding Function ;
for f being Function
for x being set st f in dom F & (F . f) . x <> f . x holds
x in X ;
then F is X -support-yielding by Def1;
hence ex b1 being Function-yielding Function st b1 is X -support-yielding ; ::_thesis: verum
end;
end;
registration
let X be set ;
let Y be Subset of X;
cluster Relation-like Function-like Function-yielding Y -support-yielding -> Function-yielding X -support-yielding for set ;
coherence
for b1 being Function-yielding Function st b1 is Y -support-yielding holds
b1 is X -support-yielding
proof
let F be Function-yielding Function; ::_thesis: ( F is Y -support-yielding implies F is X -support-yielding )
assume A1: F is Y -support-yielding ; ::_thesis: F is X -support-yielding
let f be Function; :: according to MATRTOP3:def_1 ::_thesis: for x being set st f in dom F & (F . f) . x <> f . x holds
x in X
let x be set ; ::_thesis: ( f in dom F & (F . f) . x <> f . x implies x in X )
assume ( f in dom F & (F . f) . x <> f . x ) ; ::_thesis: x in X
then x in Y by A1, Def1;
hence x in X ; ::_thesis: verum
end;
end;
registration
let X, Y be set ;
cluster Relation-like Function-like Function-yielding X -support-yielding Y -support-yielding -> Function-yielding X /\ Y -support-yielding for set ;
coherence
for b1 being Function-yielding Function st b1 is X -support-yielding & b1 is Y -support-yielding holds
b1 is X /\ Y -support-yielding
proof
let F be Function-yielding Function; ::_thesis: ( F is X -support-yielding & F is Y -support-yielding implies F is X /\ Y -support-yielding )
assume A1: ( F is X -support-yielding & F is Y -support-yielding ) ; ::_thesis: F is X /\ Y -support-yielding
let f be Function; :: according to MATRTOP3:def_1 ::_thesis: for x being set st f in dom F & (F . f) . x <> f . x holds
x in X /\ Y
let x be set ; ::_thesis: ( f in dom F & (F . f) . x <> f . x implies x in X /\ Y )
assume ( f in dom F & (F . f) . x <> f . x ) ; ::_thesis: x in X /\ Y
then ( x in X & x in Y ) by A1, Def1;
hence x in X /\ Y by XBOOLE_0:def_4; ::_thesis: verum
end;
let f be Function-yielding X -support-yielding Function;
let g be Function-yielding Y -support-yielding Function;
clusterg (#) f -> X \/ Y -support-yielding ;
coherence
f * g is X \/ Y -support-yielding
proof
set fg = f * g;
let h be Function; :: according to MATRTOP3:def_1 ::_thesis: for x being set st h in dom (f * g) & ((f * g) . h) . x <> h . x holds
x in X \/ Y
let x be set ; ::_thesis: ( h in dom (f * g) & ((f * g) . h) . x <> h . x implies x in X \/ Y )
assume that
A2: h in dom (f * g) and
A3: ((f * g) . h) . x <> h . x ; ::_thesis: x in X \/ Y
A4: h in dom g by A2, FUNCT_1:11;
assume A5: not x in X \/ Y ; ::_thesis: contradiction
then not x in Y by XBOOLE_0:def_3;
then A6: (g . h) . x = h . x by A4, Def1;
A7: ( (f * g) . h = f . (g . h) & g . h in dom f ) by A2, FUNCT_1:11, FUNCT_1:12;
not x in X by A5, XBOOLE_0:def_3;
hence contradiction by A3, A6, A7, Def1; ::_thesis: verum
end;
end;
registration
let n be Nat;
cluster non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total homogeneous for Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):];
existence
ex b1 being Function of (TOP-REAL n),(TOP-REAL n) st b1 is homogeneous
proof
reconsider N = n as Element of NAT by ORDINAL1:def_12;
the homogeneous Function of (TOP-REAL N),(TOP-REAL N) is homogeneous ;
hence ex b1 being Function of (TOP-REAL n),(TOP-REAL n) st b1 is homogeneous ; ::_thesis: verum
end;
end;
registration
let n, m be Nat;
cluster Function-like quasi_total -> FinSequence-yielding for Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL m):];
coherence
for b1 being Function of (TOP-REAL n),(TOP-REAL m) holds b1 is FinSequence-yielding
proof
let F be Function of (TOP-REAL n),(TOP-REAL m); ::_thesis: F is FinSequence-yielding
now__::_thesis:_for_x_being_set_st_x_in_dom_F_holds_
F_._x_is_FinSequence
let x be set ; ::_thesis: ( x in dom F implies F . x is FinSequence )
assume x in dom F ; ::_thesis: F . x is FinSequence
then ( rng F c= the carrier of (TOP-REAL m) & F . x in rng F ) by FUNCT_1:def_3, RELAT_1:def_19;
hence F . x is FinSequence ; ::_thesis: verum
end;
hence F is FinSequence-yielding by PRE_POLY:def_3; ::_thesis: verum
end;
end;
registration
let n, m be Nat;
let A be Matrix of n,m,F_Real;
cluster Mx2Tran A -> additive ;
coherence
Mx2Tran A is additive
proof
let f1, f2 be Point of (TOP-REAL n); :: according to VECTSP_1:def_20 ::_thesis: (Mx2Tran A) . (f1 + f2) = ((Mx2Tran A) . f1) + ((Mx2Tran A) . f2)
thus (Mx2Tran A) . (f1 + f2) = ((Mx2Tran A) . f1) + ((Mx2Tran A) . f2) by MATRTOP1:22; ::_thesis: verum
end;
end;
registration
let n be Nat;
let A be Matrix of n,F_Real;
cluster Mx2Tran A -> homogeneous ;
coherence
Mx2Tran A is homogeneous
proof
let r be real number ; :: according to TOPREAL9:def_4 ::_thesis: for b1 being Element of the carrier of (TOP-REAL n) holds (Mx2Tran A) . (r * b1) = r * ((Mx2Tran A) . b1)
let p1 be Point of (TOP-REAL n); ::_thesis: (Mx2Tran A) . (r * p1) = r * ((Mx2Tran A) . p1)
thus (Mx2Tran A) . (r * p1) = r * ((Mx2Tran A) . p1) by MATRTOP1:23; ::_thesis: verum
end;
end;
registration
let n be Nat;
let f, g be homogeneous Function of (TOP-REAL n),(TOP-REAL n);
clusterg (#) f -> homogeneous for Function of (TOP-REAL n),(TOP-REAL n);
coherence
for b1 being Function of (TOP-REAL n),(TOP-REAL n) st b1 = f * g holds
b1 is homogeneous
proof
set TR = TOP-REAL n;
now__::_thesis:_for_r_being_real_number_
for_p_being_Point_of_(TOP-REAL_n)_holds_(f_*_g)_._(r_*_p)_=_r_*_((f_*_g)_._p)
let r be real number ; ::_thesis: for p being Point of (TOP-REAL n) holds (f * g) . (r * p) = r * ((f * g) . p)
let p be Point of (TOP-REAL n); ::_thesis: (f * g) . (r * p) = r * ((f * g) . p)
reconsider gp = g . p as Point of (TOP-REAL n) ;
A1: dom (f * g) = the carrier of (TOP-REAL n) by FUNCT_2:52;
hence (f * g) . (r * p) = f . (g . (r * p)) by FUNCT_1:12
.= f . (r * gp) by TOPREAL9:def_4
.= r * (f . gp) by TOPREAL9:def_4
.= r * ((f * g) . p) by A1, FUNCT_1:12 ;
::_thesis: verum
end;
hence for b1 being Function of (TOP-REAL n),(TOP-REAL n) st b1 = f * g holds
b1 is homogeneous by TOPREAL9:def_4; ::_thesis: verum
end;
end;
begin
Lm1: for i, n being Nat st i in Seg n holds
ex M being Matrix of n,F_Real st
( Det M = - (1. F_Real) & M * (i,i) = - (1. F_Real) & ( for k, m being Nat st [k,m] in Indices M holds
( ( k = m & k <> i implies M * (k,k) = 1. F_Real ) & ( k <> m implies M * (k,m) = 0. F_Real ) ) ) )
proof
let i, n be Nat; ::_thesis: ( i in Seg n implies ex M being Matrix of n,F_Real st
( Det M = - (1. F_Real) & M * (i,i) = - (1. F_Real) & ( for k, m being Nat st [k,m] in Indices M holds
( ( k = m & k <> i implies M * (k,k) = 1. F_Real ) & ( k <> m implies M * (k,m) = 0. F_Real ) ) ) ) )
set FR = the carrier of F_Real;
set mm = the multF of F_Real;
reconsider N = n as Element of NAT by ORDINAL1:def_12;
defpred S1[ set , set , set ] means ( ( $1 = $2 & $1 = i implies $3 = - (1. F_Real) ) & ( $1 = $2 & $1 <> i implies $3 = 1. F_Real ) & ( $1 <> $2 implies $3 = 0. F_Real ) );
A1: for k, m being Nat st [k,m] in [:(Seg N),(Seg N):] holds
ex x being Element of F_Real st S1[k,m,x]
proof
let k, m be Nat; ::_thesis: ( [k,m] in [:(Seg N),(Seg N):] implies ex x being Element of F_Real st S1[k,m,x] )
assume [k,m] in [:(Seg N),(Seg N):] ; ::_thesis: ex x being Element of F_Real st S1[k,m,x]
( ( k = m & k = i ) or ( k = m & k <> i ) or k <> m ) ;
hence ex x being Element of F_Real st S1[k,m,x] ; ::_thesis: verum
end;
consider M being Matrix of N,F_Real such that
A2: for n, m being Nat st [n,m] in Indices M holds
S1[n,m,M * (n,m)] from MATRIX_1:sch_2(A1);
reconsider M = M as Matrix of n,F_Real ;
A3: Indices M = [:(Seg n),(Seg n):] by MATRIX_1:24;
now__::_thesis:_for_k,_m_being_Element_of_NAT_st_k_in_Seg_n_&_m_in_Seg_n_&_k_<>_m_holds_
M_*_(k,m)_=_0._F_Real
let k, m be Element of NAT ; ::_thesis: ( k in Seg n & m in Seg n & k <> m implies M * (k,m) = 0. F_Real )
assume ( k in Seg n & m in Seg n ) ; ::_thesis: ( k <> m implies M * (k,m) = 0. F_Real )
then A4: [k,m] in Indices M by A3, ZFMISC_1:87;
assume k <> m ; ::_thesis: M * (k,m) = 0. F_Real
hence M * (k,m) = 0. F_Real by A2, A4; ::_thesis: verum
end;
then A5: M is V244( F_Real ) by MATRIX_7:def_2;
set D = diagonal_of_Matrix M;
defpred S2[ Nat] means ( $1 + i <= n implies the multF of F_Real "**" ((diagonal_of_Matrix M) | ($1 + i)) = - (1. F_Real) );
A6: len (diagonal_of_Matrix M) = n by MATRIX_3:def_10;
A7: for k being Nat st S2[k] holds
S2[k + 1]
proof
let k be Nat; ::_thesis: ( S2[k] implies S2[k + 1] )
assume A8: S2[k] ; ::_thesis: S2[k + 1]
set k1 = k + 1;
set ki = (k + 1) + i;
assume A9: (k + 1) + i <= n ; ::_thesis: the multF of F_Real "**" ((diagonal_of_Matrix M) | ((k + 1) + i)) = - (1. F_Real)
A10: (k + 1) + i = (k + i) + 1 ;
then A11: 1 <= (k + 1) + i by NAT_1:11;
then (k + 1) + i in dom (diagonal_of_Matrix M) by A6, A9, FINSEQ_3:25;
then A12: (diagonal_of_Matrix M) | ((k + 1) + i) = ((diagonal_of_Matrix M) | (k + i)) ^ <*((diagonal_of_Matrix M) . ((k + 1) + i))*> by A10, FINSEQ_5:10;
i <= k + i by NAT_1:11;
then A13: i < (k + 1) + i by A10, NAT_1:13;
A14: (k + 1) + i in Seg n by A9, A11;
then [((k + 1) + i),((k + 1) + i)] in Indices M by A3, ZFMISC_1:87;
then 1. F_Real = M * (((k + 1) + i),((k + 1) + i)) by A2, A13
.= (diagonal_of_Matrix M) . ((k + 1) + i) by A14, MATRIX_3:def_10 ;
hence the multF of F_Real "**" ((diagonal_of_Matrix M) | ((k + 1) + i)) = (- (1. F_Real)) * (1. F_Real) by A8, A9, A10, A12, FINSOP_1:4, NAT_1:13
.= - (1. F_Real) ;
::_thesis: verum
end;
defpred S3[ Nat] means ( $1 < i implies the multF of F_Real "**" ((diagonal_of_Matrix M) | $1) = 1. F_Real );
A15: S3[ 0 ]
proof
assume 0 < i ; ::_thesis: the multF of F_Real "**" ((diagonal_of_Matrix M) | 0) = 1. F_Real
( (diagonal_of_Matrix M) | 0 = <*> the carrier of F_Real & the_unity_wrt the multF of F_Real = 1. F_Real ) by FVSUM_1:5;
hence the multF of F_Real "**" ((diagonal_of_Matrix M) | 0) = 1. F_Real by FINSOP_1:10; ::_thesis: verum
end;
assume A16: i in Seg n ; ::_thesis: ex M being Matrix of n,F_Real st
( Det M = - (1. F_Real) & M * (i,i) = - (1. F_Real) & ( for k, m being Nat st [k,m] in Indices M holds
( ( k = m & k <> i implies M * (k,k) = 1. F_Real ) & ( k <> m implies M * (k,m) = 0. F_Real ) ) ) )
then A17: 1 <= i by FINSEQ_1:1;
A18: i <= n by A16, FINSEQ_1:1;
then A19: ( (n - i) + i = n & n - i is Nat ) by NAT_1:21;
take M ; ::_thesis: ( Det M = - (1. F_Real) & M * (i,i) = - (1. F_Real) & ( for k, m being Nat st [k,m] in Indices M holds
( ( k = m & k <> i implies M * (k,k) = 1. F_Real ) & ( k <> m implies M * (k,m) = 0. F_Real ) ) ) )
A20: for k being Nat st S3[k] holds
S3[k + 1]
proof
let k be Nat; ::_thesis: ( S3[k] implies S3[k + 1] )
assume A21: S3[k] ; ::_thesis: S3[k + 1]
set k1 = k + 1;
assume A22: k + 1 < i ; ::_thesis: the multF of F_Real "**" ((diagonal_of_Matrix M) | (k + 1)) = 1. F_Real
then A23: ( 1 <= k + 1 & k + 1 <= n ) by A18, NAT_1:14, XXREAL_0:2;
then k + 1 in dom (diagonal_of_Matrix M) by A6, FINSEQ_3:25;
then A24: (diagonal_of_Matrix M) | (k + 1) = ((diagonal_of_Matrix M) | k) ^ <*((diagonal_of_Matrix M) . (k + 1))*> by FINSEQ_5:10;
A25: k + 1 in Seg n by A23;
then [(k + 1),(k + 1)] in Indices M by A3, ZFMISC_1:87;
then 1. F_Real = M * ((k + 1),(k + 1)) by A2, A22
.= (diagonal_of_Matrix M) . (k + 1) by A25, MATRIX_3:def_10 ;
hence the multF of F_Real "**" ((diagonal_of_Matrix M) | (k + 1)) = (1. F_Real) * (1. F_Real) by A21, A22, A24, FINSOP_1:4, NAT_1:13
.= 1. F_Real ;
::_thesis: verum
end;
A26: for k being Nat holds S3[k] from NAT_1:sch_2(A15, A20);
A27: S2[ 0 ]
proof
reconsider I = i - 1 as Nat by A17, NAT_1:21;
assume 0 + i <= n ; ::_thesis: the multF of F_Real "**" ((diagonal_of_Matrix M) | (0 + i)) = - (1. F_Real)
A28: I + 1 = i ;
then I < i by NAT_1:13;
then A29: the multF of F_Real "**" ((diagonal_of_Matrix M) | I) = 1. F_Real by A26;
1 <= i by A16, FINSEQ_1:1;
then i in dom (diagonal_of_Matrix M) by A6, A18, FINSEQ_3:25;
then A30: (diagonal_of_Matrix M) | i = ((diagonal_of_Matrix M) | I) ^ <*((diagonal_of_Matrix M) . i)*> by A28, FINSEQ_5:10;
[i,i] in Indices M by A3, A16, ZFMISC_1:87;
then - (1. F_Real) = M * (i,i) by A2
.= (diagonal_of_Matrix M) . i by A16, MATRIX_3:def_10 ;
hence the multF of F_Real "**" ((diagonal_of_Matrix M) | (0 + i)) = (1. F_Real) * (- (1. F_Real)) by A29, A30, FINSOP_1:4
.= - (1. F_Real) ;
::_thesis: verum
end;
for k being Nat holds S2[k] from NAT_1:sch_2(A27, A7);
hence - (1. F_Real) = the multF of F_Real "**" ((diagonal_of_Matrix M) | n) by A19
.= the multF of F_Real "**" (diagonal_of_Matrix M) by A6, FINSEQ_1:58
.= Det M by A5, A17, A18, MATRIX_7:17, XXREAL_0:2 ;
::_thesis: ( M * (i,i) = - (1. F_Real) & ( for k, m being Nat st [k,m] in Indices M holds
( ( k = m & k <> i implies M * (k,k) = 1. F_Real ) & ( k <> m implies M * (k,m) = 0. F_Real ) ) ) )
[i,i] in Indices M by A3, A16, ZFMISC_1:87;
hence M * (i,i) = - (1. F_Real) by A2; ::_thesis: for k, m being Nat st [k,m] in Indices M holds
( ( k = m & k <> i implies M * (k,k) = 1. F_Real ) & ( k <> m implies M * (k,m) = 0. F_Real ) )
let k, m be Nat; ::_thesis: ( [k,m] in Indices M implies ( ( k = m & k <> i implies M * (k,k) = 1. F_Real ) & ( k <> m implies M * (k,m) = 0. F_Real ) ) )
assume [k,m] in Indices M ; ::_thesis: ( ( k = m & k <> i implies M * (k,k) = 1. F_Real ) & ( k <> m implies M * (k,m) = 0. F_Real ) )
hence ( ( k = m & k <> i implies M * (k,k) = 1. F_Real ) & ( k <> m implies M * (k,m) = 0. F_Real ) ) by A2; ::_thesis: verum
end;
definition
let n, i be Nat;
assume B1: i in Seg n ;
func AxialSymmetry (i,n) -> invertible Matrix of n,F_Real means :Def2: :: MATRTOP3:def 2
( it * (i,i) = - (1. F_Real) & ( for k, m being Nat st [k,m] in Indices it holds
( ( k = m & k <> i implies it * (k,k) = 1. F_Real ) & ( k <> m implies it * (k,m) = 0. F_Real ) ) ) );
existence
ex b1 being invertible Matrix of n,F_Real st
( b1 * (i,i) = - (1. F_Real) & ( for k, m being Nat st [k,m] in Indices b1 holds
( ( k = m & k <> i implies b1 * (k,k) = 1. F_Real ) & ( k <> m implies b1 * (k,m) = 0. F_Real ) ) ) )
proof
consider M being Matrix of n,F_Real such that
A1: Det M = - (1. F_Real) and
A2: ( M * (i,i) = - (1. F_Real) & ( for k, m being Nat st [k,m] in Indices M holds
( ( k = m & k <> i implies M * (k,k) = 1. F_Real ) & ( k <> m implies M * (k,m) = 0. F_Real ) ) ) ) by B1, Lm1;
Det M <> 0. F_Real by A1;
then M is invertible by LAPLACE:34;
hence ex b1 being invertible Matrix of n,F_Real st
( b1 * (i,i) = - (1. F_Real) & ( for k, m being Nat st [k,m] in Indices b1 holds
( ( k = m & k <> i implies b1 * (k,k) = 1. F_Real ) & ( k <> m implies b1 * (k,m) = 0. F_Real ) ) ) ) by A2; ::_thesis: verum
end;
uniqueness
for b1, b2 being invertible Matrix of n,F_Real st b1 * (i,i) = - (1. F_Real) & ( for k, m being Nat st [k,m] in Indices b1 holds
( ( k = m & k <> i implies b1 * (k,k) = 1. F_Real ) & ( k <> m implies b1 * (k,m) = 0. F_Real ) ) ) & b2 * (i,i) = - (1. F_Real) & ( for k, m being Nat st [k,m] in Indices b2 holds
( ( k = m & k <> i implies b2 * (k,k) = 1. F_Real ) & ( k <> m implies b2 * (k,m) = 0. F_Real ) ) ) holds
b1 = b2
proof
let A1, A2 be invertible Matrix of n,F_Real; ::_thesis: ( A1 * (i,i) = - (1. F_Real) & ( for k, m being Nat st [k,m] in Indices A1 holds
( ( k = m & k <> i implies A1 * (k,k) = 1. F_Real ) & ( k <> m implies A1 * (k,m) = 0. F_Real ) ) ) & A2 * (i,i) = - (1. F_Real) & ( for k, m being Nat st [k,m] in Indices A2 holds
( ( k = m & k <> i implies A2 * (k,k) = 1. F_Real ) & ( k <> m implies A2 * (k,m) = 0. F_Real ) ) ) implies A1 = A2 )
assume that
A3: A1 * (i,i) = - (1. F_Real) and
A4: for k, m being Nat st [k,m] in Indices A1 holds
( ( k = m & k <> i implies A1 * (k,k) = 1. F_Real ) & ( k <> m implies A1 * (k,m) = 0. F_Real ) ) and
A5: A2 * (i,i) = - (1. F_Real) and
A6: for k, m being Nat st [k,m] in Indices A2 holds
( ( k = m & k <> i implies A2 * (k,k) = 1. F_Real ) & ( k <> m implies A2 * (k,m) = 0. F_Real ) ) ; ::_thesis: A1 = A2
for k, m being Nat st [k,m] in Indices A1 holds
A1 * (k,m) = A2 * (k,m)
proof
let k, m be Nat; ::_thesis: ( [k,m] in Indices A1 implies A1 * (k,m) = A2 * (k,m) )
assume A7: [k,m] in Indices A1 ; ::_thesis: A1 * (k,m) = A2 * (k,m)
then A8: [k,m] in Indices A2 by MATRIX_1:26;
percases ( k <> m or ( k = m & k <> i ) or ( k = m & k = i ) ) ;
supposeA9: k <> m ; ::_thesis: A1 * (k,m) = A2 * (k,m)
then A1 * (k,m) = 0. F_Real by A4, A7;
hence A1 * (k,m) = A2 * (k,m) by A6, A8, A9; ::_thesis: verum
end;
supposeA10: ( k = m & k <> i ) ; ::_thesis: A1 * (k,m) = A2 * (k,m)
then A1 * (k,m) = 1. F_Real by A4, A7;
hence A1 * (k,m) = A2 * (k,m) by A6, A8, A10; ::_thesis: verum
end;
suppose ( k = m & k = i ) ; ::_thesis: A1 * (k,m) = A2 * (k,m)
hence A1 * (k,m) = A2 * (k,m) by A3, A5; ::_thesis: verum
end;
end;
end;
hence A1 = A2 by MATRIX_1:27; ::_thesis: verum
end;
end;
:: deftheorem Def2 defines AxialSymmetry MATRTOP3:def_2_:_
for n, i being Nat st i in Seg n holds
for b3 being invertible Matrix of n,F_Real holds
( b3 = AxialSymmetry (i,n) iff ( b3 * (i,i) = - (1. F_Real) & ( for k, m being Nat st [k,m] in Indices b3 holds
( ( k = m & k <> i implies b3 * (k,k) = 1. F_Real ) & ( k <> m implies b3 * (k,m) = 0. F_Real ) ) ) ) );
theorem Th4: :: MATRTOP3:4
for i, n being Nat st i in Seg n holds
Det (AxialSymmetry (i,n)) = - (1. F_Real)
proof
let i, n be Nat; ::_thesis: ( i in Seg n implies Det (AxialSymmetry (i,n)) = - (1. F_Real) )
assume A1: i in Seg n ; ::_thesis: Det (AxialSymmetry (i,n)) = - (1. F_Real)
then consider M being Matrix of n,F_Real such that
A2: Det M = - (1. F_Real) and
A3: ( M * (i,i) = - (1. F_Real) & ( for k, m being Nat st [k,m] in Indices M holds
( ( k = m & k <> i implies M * (k,k) = 1. F_Real ) & ( k <> m implies M * (k,m) = 0. F_Real ) ) ) ) by Lm1;
Det M <> 0. F_Real by A2;
then M is invertible by LAPLACE:34;
hence Det (AxialSymmetry (i,n)) = - (1. F_Real) by A1, A2, A3, Def2; ::_thesis: verum
end;
theorem Th5: :: MATRTOP3:5
for i, n, j being Nat
for p being Point of (TOP-REAL n) st i in Seg n & j in Seg n & i <> j holds
(@ p) "*" (Col ((AxialSymmetry (i,n)),j)) = p . j
proof
let i, n, j be Nat; ::_thesis: for p being Point of (TOP-REAL n) st i in Seg n & j in Seg n & i <> j holds
(@ p) "*" (Col ((AxialSymmetry (i,n)),j)) = p . j
let p be Point of (TOP-REAL n); ::_thesis: ( i in Seg n & j in Seg n & i <> j implies (@ p) "*" (Col ((AxialSymmetry (i,n)),j)) = p . j )
set S = Seg n;
assume that
A1: i in Seg n and
A2: j in Seg n and
A3: i <> j ; ::_thesis: (@ p) "*" (Col ((AxialSymmetry (i,n)),j)) = p . j
set A = AxialSymmetry (i,n);
set C = Col ((AxialSymmetry (i,n)),j);
A4: Indices (AxialSymmetry (i,n)) = [:(Seg n),(Seg n):] by MATRIX_1:24;
then A5: [j,j] in Indices (AxialSymmetry (i,n)) by A2, ZFMISC_1:87;
A6: len (AxialSymmetry (i,n)) = n by MATRIX_1:25;
then A7: dom (AxialSymmetry (i,n)) = Seg n by FINSEQ_1:def_3;
len (Col ((AxialSymmetry (i,n)),j)) = n by A6, MATRIX_1:def_8;
then A8: dom (Col ((AxialSymmetry (i,n)),j)) = Seg n by FINSEQ_1:def_3;
A9: now__::_thesis:_for_m_being_Nat_st_m_in_dom_(Col_((AxialSymmetry_(i,n)),j))_&_m_<>_j_holds_
(Col_((AxialSymmetry_(i,n)),j))_._m_=_0._F_Real
let m be Nat; ::_thesis: ( m in dom (Col ((AxialSymmetry (i,n)),j)) & m <> j implies (Col ((AxialSymmetry (i,n)),j)) . m = 0. F_Real )
assume that
A10: m in dom (Col ((AxialSymmetry (i,n)),j)) and
A11: m <> j ; ::_thesis: (Col ((AxialSymmetry (i,n)),j)) . m = 0. F_Real
A12: [m,j] in Indices (AxialSymmetry (i,n)) by A2, A4, A8, A10, ZFMISC_1:87;
thus (Col ((AxialSymmetry (i,n)),j)) . m = (AxialSymmetry (i,n)) * (m,j) by A7, A8, A10, MATRIX_1:def_8
.= 0. F_Real by A1, A11, A12, Def2 ; ::_thesis: verum
end;
len p = n by CARD_1:def_7;
then A13: dom p = Seg n by FINSEQ_1:def_3;
(Col ((AxialSymmetry (i,n)),j)) . j = (AxialSymmetry (i,n)) * (j,j) by A2, A7, MATRIX_1:def_8
.= 1. F_Real by A1, A3, A5, Def2 ;
hence p . j = Sum (mlt ((Col ((AxialSymmetry (i,n)),j)),(@ p))) by A2, A8, A9, A13, MATRIX_3:17
.= (@ p) "*" (Col ((AxialSymmetry (i,n)),j)) by FVSUM_1:64 ;
::_thesis: verum
end;
theorem Th6: :: MATRTOP3:6
for i, n being Nat
for p being Point of (TOP-REAL n) st i in Seg n holds
(@ p) "*" (Col ((AxialSymmetry (i,n)),i)) = - (p . i)
proof
let i, n be Nat; ::_thesis: for p being Point of (TOP-REAL n) st i in Seg n holds
(@ p) "*" (Col ((AxialSymmetry (i,n)),i)) = - (p . i)
let p be Point of (TOP-REAL n); ::_thesis: ( i in Seg n implies (@ p) "*" (Col ((AxialSymmetry (i,n)),i)) = - (p . i) )
set S = Seg n;
assume A1: i in Seg n ; ::_thesis: (@ p) "*" (Col ((AxialSymmetry (i,n)),i)) = - (p . i)
reconsider pI = (@ p) . i as Element of F_Real ;
set A = AxialSymmetry (i,n);
set C = Col ((AxialSymmetry (i,n)),i);
A2: len (AxialSymmetry (i,n)) = n by MATRIX_1:25;
then A3: dom (AxialSymmetry (i,n)) = Seg n by FINSEQ_1:def_3;
then A4: (Col ((AxialSymmetry (i,n)),i)) . i = (AxialSymmetry (i,n)) * (i,i) by A1, MATRIX_1:def_8;
( len p = n & len (Col ((AxialSymmetry (i,n)),i)) = n ) by A2, CARD_1:def_7;
then len (mlt ((@ p),(Col ((AxialSymmetry (i,n)),i)))) = n by MATRIX_3:6;
then A5: dom (mlt ((@ p),(Col ((AxialSymmetry (i,n)),i)))) = Seg n by FINSEQ_1:def_3;
A6: Indices (AxialSymmetry (i,n)) = [:(Seg n),(Seg n):] by MATRIX_1:24;
A7: for k being Nat st k in dom (mlt ((@ p),(Col ((AxialSymmetry (i,n)),i)))) & k <> i holds
(mlt ((@ p),(Col ((AxialSymmetry (i,n)),i)))) . k = 0. F_Real
proof
let k be Nat; ::_thesis: ( k in dom (mlt ((@ p),(Col ((AxialSymmetry (i,n)),i)))) & k <> i implies (mlt ((@ p),(Col ((AxialSymmetry (i,n)),i)))) . k = 0. F_Real )
assume that
A8: k in dom (mlt ((@ p),(Col ((AxialSymmetry (i,n)),i)))) and
A9: k <> i ; ::_thesis: (mlt ((@ p),(Col ((AxialSymmetry (i,n)),i)))) . k = 0. F_Real
reconsider pk = (@ p) . k as Element of F_Real ;
A10: [k,i] in Indices (AxialSymmetry (i,n)) by A1, A5, A6, A8, ZFMISC_1:87;
(Col ((AxialSymmetry (i,n)),i)) . k = (AxialSymmetry (i,n)) * (k,i) by A3, A5, A8, MATRIX_1:def_8;
hence (mlt ((@ p),(Col ((AxialSymmetry (i,n)),i)))) . k = pk * ((AxialSymmetry (i,n)) * (k,i)) by A8, FVSUM_1:60
.= pk * (0. F_Real) by A1, A9, A10, Def2
.= 0. F_Real ;
::_thesis: verum
end;
thus (@ p) "*" (Col ((AxialSymmetry (i,n)),i)) = (mlt ((@ p),(Col ((AxialSymmetry (i,n)),i)))) . i by A1, A5, A7, MATRIX_3:12
.= pI * ((AxialSymmetry (i,n)) * (i,i)) by A1, A4, A5, FVSUM_1:60
.= pI * (- (1. F_Real)) by A1, Def2
.= - (p . i) ; ::_thesis: verum
end;
theorem Th7: :: MATRTOP3:7
for i, n being Nat st i in Seg n holds
( AxialSymmetry (i,n) is V244( F_Real ) & (AxialSymmetry (i,n)) ~ = AxialSymmetry (i,n) )
proof
let i, n be Nat; ::_thesis: ( i in Seg n implies ( AxialSymmetry (i,n) is V244( F_Real ) & (AxialSymmetry (i,n)) ~ = AxialSymmetry (i,n) ) )
set S = Seg n;
set A = AxialSymmetry (i,n);
set ONE = 1. (F_Real,n);
set AA = (AxialSymmetry (i,n)) * (AxialSymmetry (i,n));
assume A1: i in Seg n ; ::_thesis: ( AxialSymmetry (i,n) is V244( F_Real ) & (AxialSymmetry (i,n)) ~ = AxialSymmetry (i,n) )
for k, m being Nat st [k,m] in Indices (AxialSymmetry (i,n)) & (AxialSymmetry (i,n)) * (k,m) <> 0. F_Real holds
k = m by A1, Def2;
hence AxialSymmetry (i,n) is V244( F_Real ) by MATRIX_1:def_14; ::_thesis: (AxialSymmetry (i,n)) ~ = AxialSymmetry (i,n)
for k, m being Nat st [k,m] in Indices ((AxialSymmetry (i,n)) * (AxialSymmetry (i,n))) holds
((AxialSymmetry (i,n)) * (AxialSymmetry (i,n))) * (k,m) = (1. (F_Real,n)) * (k,m)
proof
let k, m be Nat; ::_thesis: ( [k,m] in Indices ((AxialSymmetry (i,n)) * (AxialSymmetry (i,n))) implies ((AxialSymmetry (i,n)) * (AxialSymmetry (i,n))) * (k,m) = (1. (F_Real,n)) * (k,m) )
assume A2: [k,m] in Indices ((AxialSymmetry (i,n)) * (AxialSymmetry (i,n))) ; ::_thesis: ((AxialSymmetry (i,n)) * (AxialSymmetry (i,n))) * (k,m) = (1. (F_Real,n)) * (k,m)
A3: width (AxialSymmetry (i,n)) = n by MATRIX_1:24;
then len (@ (Line ((AxialSymmetry (i,n)),k))) = n by CARD_1:def_7;
then reconsider L = @ (Line ((AxialSymmetry (i,n)),k)) as Element of (TOP-REAL n) by TOPREAL3:46;
len (AxialSymmetry (i,n)) = n by MATRIX_1:25;
then A4: ((AxialSymmetry (i,n)) * (AxialSymmetry (i,n))) * (k,m) = (@ L) "*" (Col ((AxialSymmetry (i,n)),m)) by A2, A3, MATRIX_3:def_4;
A5: Indices ((AxialSymmetry (i,n)) * (AxialSymmetry (i,n))) = [:(Seg n),(Seg n):] by MATRIX_1:24;
then A6: m in Seg n by A2, ZFMISC_1:87;
then A7: (Line ((AxialSymmetry (i,n)),k)) . m = (AxialSymmetry (i,n)) * (k,m) by A3, MATRIX_1:def_7;
A8: Indices (AxialSymmetry (i,n)) = [:(Seg n),(Seg n):] by MATRIX_1:24;
A9: Indices (1. (F_Real,n)) = [:(Seg n),(Seg n):] by MATRIX_1:24;
percases ( m <> i or m = i ) ;
supposeA10: m <> i ; ::_thesis: ((AxialSymmetry (i,n)) * (AxialSymmetry (i,n))) * (k,m) = (1. (F_Real,n)) * (k,m)
then A11: ((AxialSymmetry (i,n)) * (AxialSymmetry (i,n))) * (k,m) = (AxialSymmetry (i,n)) * (k,m) by A1, A4, A6, A7, Th5;
percases ( k <> m or k = m ) ;
supposeA12: k <> m ; ::_thesis: ((AxialSymmetry (i,n)) * (AxialSymmetry (i,n))) * (k,m) = (1. (F_Real,n)) * (k,m)
then (1. (F_Real,n)) * (k,m) = 0. F_Real by A2, A5, A9, MATRIX_1:def_11;
hence ((AxialSymmetry (i,n)) * (AxialSymmetry (i,n))) * (k,m) = (1. (F_Real,n)) * (k,m) by A1, A2, A5, A8, A11, A12, Def2; ::_thesis: verum
end;
supposeA13: k = m ; ::_thesis: ((AxialSymmetry (i,n)) * (AxialSymmetry (i,n))) * (k,m) = (1. (F_Real,n)) * (k,m)
then (1. (F_Real,n)) * (k,m) = 1. F_Real by A2, A5, A9, MATRIX_1:def_11;
hence ((AxialSymmetry (i,n)) * (AxialSymmetry (i,n))) * (k,m) = (1. (F_Real,n)) * (k,m) by A1, A2, A5, A8, A10, A11, A13, Def2; ::_thesis: verum
end;
end;
end;
supposeA14: m = i ; ::_thesis: ((AxialSymmetry (i,n)) * (AxialSymmetry (i,n))) * (k,m) = (1. (F_Real,n)) * (k,m)
then A15: ((AxialSymmetry (i,n)) * (AxialSymmetry (i,n))) * (k,m) = - ((AxialSymmetry (i,n)) * (k,m)) by A4, A6, A7, Th6;
percases ( k <> m or k = m ) ;
supposeA16: k <> m ; ::_thesis: ((AxialSymmetry (i,n)) * (AxialSymmetry (i,n))) * (k,m) = (1. (F_Real,n)) * (k,m)
then (AxialSymmetry (i,n)) * (k,m) = 0. F_Real by A1, A2, A5, A8, Def2;
hence ((AxialSymmetry (i,n)) * (AxialSymmetry (i,n))) * (k,m) = (1. (F_Real,n)) * (k,m) by A2, A5, A9, A15, A16, MATRIX_1:def_11; ::_thesis: verum
end;
supposeA17: k = m ; ::_thesis: ((AxialSymmetry (i,n)) * (AxialSymmetry (i,n))) * (k,m) = (1. (F_Real,n)) * (k,m)
then ((AxialSymmetry (i,n)) * (AxialSymmetry (i,n))) * (k,m) = - (- (1. F_Real)) by A1, A14, A15, Def2;
hence ((AxialSymmetry (i,n)) * (AxialSymmetry (i,n))) * (k,m) = (1. (F_Real,n)) * (k,m) by A2, A5, A9, A17, MATRIX_1:def_11; ::_thesis: verum
end;
end;
end;
end;
end;
then (AxialSymmetry (i,n)) * (AxialSymmetry (i,n)) = 1. (F_Real,n) by MATRIX_1:27;
then AxialSymmetry (i,n) is_reverse_of AxialSymmetry (i,n) by MATRIX_6:def_2;
hence (AxialSymmetry (i,n)) ~ = AxialSymmetry (i,n) by MATRIX_6:def_4; ::_thesis: verum
end;
theorem Th8: :: MATRTOP3:8
for i, n, j being Nat
for p being Point of (TOP-REAL n) st i in Seg n & i <> j holds
((Mx2Tran (AxialSymmetry (i,n))) . p) . j = p . j
proof
let i, n, j be Nat; ::_thesis: for p being Point of (TOP-REAL n) st i in Seg n & i <> j holds
((Mx2Tran (AxialSymmetry (i,n))) . p) . j = p . j
let p be Point of (TOP-REAL n); ::_thesis: ( i in Seg n & i <> j implies ((Mx2Tran (AxialSymmetry (i,n))) . p) . j = p . j )
set A = AxialSymmetry (i,n);
set M = Mx2Tran (AxialSymmetry (i,n));
set Mp = (Mx2Tran (AxialSymmetry (i,n))) . p;
set S = Seg n;
assume A1: ( i in Seg n & i <> j ) ; ::_thesis: ((Mx2Tran (AxialSymmetry (i,n))) . p) . j = p . j
len ((Mx2Tran (AxialSymmetry (i,n))) . p) = n by CARD_1:def_7;
then A2: dom ((Mx2Tran (AxialSymmetry (i,n))) . p) = Seg n by FINSEQ_1:def_3;
len p = n by CARD_1:def_7;
then A3: dom p = Seg n by FINSEQ_1:def_3;
percases ( j in Seg n or not j in Seg n ) ;
supposeA4: j in Seg n ; ::_thesis: ((Mx2Tran (AxialSymmetry (i,n))) . p) . j = p . j
then ( 1 <= j & j <= n ) by FINSEQ_1:1;
hence ((Mx2Tran (AxialSymmetry (i,n))) . p) . j = (@ p) "*" (Col ((AxialSymmetry (i,n)),j)) by MATRTOP1:18
.= p . j by A1, A4, Th5 ;
::_thesis: verum
end;
supposeA5: not j in Seg n ; ::_thesis: ((Mx2Tran (AxialSymmetry (i,n))) . p) . j = p . j
then ((Mx2Tran (AxialSymmetry (i,n))) . p) . j = {} by A2, FUNCT_1:def_2;
hence ((Mx2Tran (AxialSymmetry (i,n))) . p) . j = p . j by A3, A5, FUNCT_1:def_2; ::_thesis: verum
end;
end;
end;
theorem Th9: :: MATRTOP3:9
for i, n being Nat
for p being Point of (TOP-REAL n) st i in Seg n holds
((Mx2Tran (AxialSymmetry (i,n))) . p) . i = - (p . i)
proof
let i, n be Nat; ::_thesis: for p being Point of (TOP-REAL n) st i in Seg n holds
((Mx2Tran (AxialSymmetry (i,n))) . p) . i = - (p . i)
let p be Point of (TOP-REAL n); ::_thesis: ( i in Seg n implies ((Mx2Tran (AxialSymmetry (i,n))) . p) . i = - (p . i) )
set A = AxialSymmetry (i,n);
set M = Mx2Tran (AxialSymmetry (i,n));
set Mp = (Mx2Tran (AxialSymmetry (i,n))) . p;
set S = Seg n;
assume A1: i in Seg n ; ::_thesis: ((Mx2Tran (AxialSymmetry (i,n))) . p) . i = - (p . i)
then ( 1 <= i & i <= n ) by FINSEQ_1:1;
hence ((Mx2Tran (AxialSymmetry (i,n))) . p) . i = (@ p) "*" (Col ((AxialSymmetry (i,n)),i)) by MATRTOP1:18
.= - (p . i) by A1, Th6 ;
::_thesis: verum
end;
theorem Th10: :: MATRTOP3:10
for i, n being Nat
for p being Point of (TOP-REAL n) st i in Seg n holds
(Mx2Tran (AxialSymmetry (i,n))) . p = p +* (i,(- (p . i)))
proof
let i, n be Nat; ::_thesis: for p being Point of (TOP-REAL n) st i in Seg n holds
(Mx2Tran (AxialSymmetry (i,n))) . p = p +* (i,(- (p . i)))
let p be Point of (TOP-REAL n); ::_thesis: ( i in Seg n implies (Mx2Tran (AxialSymmetry (i,n))) . p = p +* (i,(- (p . i))) )
set S = Seg n;
set Mp = (Mx2Tran (AxialSymmetry (i,n))) . p;
set p0 = p +* (i,(- (p . i)));
A1: len p = n by CARD_1:def_7;
assume A2: i in Seg n ; ::_thesis: (Mx2Tran (AxialSymmetry (i,n))) . p = p +* (i,(- (p . i)))
A3: for j being Nat st 1 <= j & j <= n holds
((Mx2Tran (AxialSymmetry (i,n))) . p) . j = (p +* (i,(- (p . i)))) . j
proof
let j be Nat; ::_thesis: ( 1 <= j & j <= n implies ((Mx2Tran (AxialSymmetry (i,n))) . p) . j = (p +* (i,(- (p . i)))) . j )
assume A4: ( 1 <= j & j <= n ) ; ::_thesis: ((Mx2Tran (AxialSymmetry (i,n))) . p) . j = (p +* (i,(- (p . i)))) . j
A5: j in Seg n by A4, FINSEQ_1:1;
A6: j in dom p by A1, A4, FINSEQ_3:25;
percases ( j <> i or j = i ) ;
supposeA7: j <> i ; ::_thesis: ((Mx2Tran (AxialSymmetry (i,n))) . p) . j = (p +* (i,(- (p . i)))) . j
then (p +* (i,(- (p . i)))) . j = p . j by FUNCT_7:32;
hence ((Mx2Tran (AxialSymmetry (i,n))) . p) . j = (p +* (i,(- (p . i)))) . j by A2, A7, Th8; ::_thesis: verum
end;
supposeA8: j = i ; ::_thesis: ((Mx2Tran (AxialSymmetry (i,n))) . p) . j = (p +* (i,(- (p . i)))) . j
then (p +* (i,(- (p . i)))) . j = - (p . i) by A6, FUNCT_7:31;
hence ((Mx2Tran (AxialSymmetry (i,n))) . p) . j = (p +* (i,(- (p . i)))) . j by A5, A8, Th9; ::_thesis: verum
end;
end;
end;
( len (p +* (i,(- (p . i)))) = len p & len ((Mx2Tran (AxialSymmetry (i,n))) . p) = n ) by CARD_1:def_7, FUNCT_7:97;
hence (Mx2Tran (AxialSymmetry (i,n))) . p = p +* (i,(- (p . i))) by A1, A3, FINSEQ_1:14; ::_thesis: verum
end;
theorem Th11: :: MATRTOP3:11
for i, n being Nat st i in Seg n holds
Mx2Tran (AxialSymmetry (i,n)) is {i} -support-yielding
proof
let i, n be Nat; ::_thesis: ( i in Seg n implies Mx2Tran (AxialSymmetry (i,n)) is {i} -support-yielding )
set M = Mx2Tran (AxialSymmetry (i,n));
assume A1: i in Seg n ; ::_thesis: Mx2Tran (AxialSymmetry (i,n)) is {i} -support-yielding
let f be Function; :: according to MATRTOP3:def_1 ::_thesis: for x being set st f in dom (Mx2Tran (AxialSymmetry (i,n))) & ((Mx2Tran (AxialSymmetry (i,n))) . f) . x <> f . x holds
x in {i}
let x be set ; ::_thesis: ( f in dom (Mx2Tran (AxialSymmetry (i,n))) & ((Mx2Tran (AxialSymmetry (i,n))) . f) . x <> f . x implies x in {i} )
assume f in dom (Mx2Tran (AxialSymmetry (i,n))) ; ::_thesis: ( not ((Mx2Tran (AxialSymmetry (i,n))) . f) . x <> f . x or x in {i} )
then reconsider F = f as Point of (TOP-REAL n) by FUNCT_2:52;
assume A2: ((Mx2Tran (AxialSymmetry (i,n))) . f) . x <> f . x ; ::_thesis: x in {i}
len ((Mx2Tran (AxialSymmetry (i,n))) . F) = n by CARD_1:def_7;
then A3: dom ((Mx2Tran (AxialSymmetry (i,n))) . F) = Seg n by FINSEQ_1:def_3;
A4: len F = n by CARD_1:def_7;
then A5: dom F = Seg n by FINSEQ_1:def_3;
percases ( not x in Seg n or x in Seg n ) ;
supposeA6: not x in Seg n ; ::_thesis: x in {i}
then ((Mx2Tran (AxialSymmetry (i,n))) . F) . x = {} by A3, FUNCT_1:def_2
.= F . x by A5, A6, FUNCT_1:def_2 ;
hence x in {i} by A2; ::_thesis: verum
end;
suppose x in Seg n ; ::_thesis: x in {i}
then x = i by A1, A2, A4, Th8;
hence x in {i} by TARSKI:def_1; ::_thesis: verum
end;
end;
end;
theorem Th12: :: MATRTOP3:12
for r being real number
for n being Nat
for a, b being Element of F_Real st a = cos r & b = sin r holds
Det (block_diagonal (<*((a,b) ][ ((- b),a)),(1. (F_Real,n))*>,(0. F_Real))) = 1. F_Real
proof
let r be real number ; ::_thesis: for n being Nat
for a, b being Element of F_Real st a = cos r & b = sin r holds
Det (block_diagonal (<*((a,b) ][ ((- b),a)),(1. (F_Real,n))*>,(0. F_Real))) = 1. F_Real
let n be Nat; ::_thesis: for a, b being Element of F_Real st a = cos r & b = sin r holds
Det (block_diagonal (<*((a,b) ][ ((- b),a)),(1. (F_Real,n))*>,(0. F_Real))) = 1. F_Real
let a, b be Element of F_Real; ::_thesis: ( a = cos r & b = sin r implies Det (block_diagonal (<*((a,b) ][ ((- b),a)),(1. (F_Real,n))*>,(0. F_Real))) = 1. F_Real )
set A = (a,b) ][ ((- b),a);
set ONE = 1. (F_Real,n);
set B = block_diagonal (<*((a,b) ][ ((- b),a)),(1. (F_Real,n))*>,(0. F_Real));
A1: ( n = 0 or n >= 1 ) by NAT_1:14;
n in NAT by ORDINAL1:def_12;
then A2: ( Det (1. (F_Real,n)) = 1_ F_Real or Det (1. (F_Real,n)) = 1. F_Real ) by A1, MATRIXR2:41, MATRIX_7:16;
assume ( a = cos r & b = sin r ) ; ::_thesis: Det (block_diagonal (<*((a,b) ][ ((- b),a)),(1. (F_Real,n))*>,(0. F_Real))) = 1. F_Real
then A3: ((cos r) * (cos r)) + ((sin r) * (sin r)) = (a * a) - (b * (- b))
.= Det ((a,b) ][ ((- b),a)) by MATRIX_9:13 ;
A4: ( cos r = cos . r & sin r = sin . r ) by SIN_COS:def_17, SIN_COS:def_19;
thus Det (block_diagonal (<*((a,b) ][ ((- b),a)),(1. (F_Real,n))*>,(0. F_Real))) = (Det ((a,b) ][ ((- b),a))) * (Det (1. (F_Real,n))) by MATRIXJ1:52
.= 1. F_Real by A2, A3, A4, SIN_COS:28 ; ::_thesis: verum
end;
Lm2: for i, j, n being Nat st 1 <= i & i < j & j <= n holds
ex P being Permutation of (Seg n) st
( P . 1 = i & P . 2 = j & ( for k being Nat st k in Seg n & k > 2 holds
( ( k <= i + 1 implies P . k = k - 2 ) & ( i + 1 < k & k <= j implies P . k = k - 1 ) & ( k > j implies P . k = k ) ) ) )
proof
let i, j, n be Nat; ::_thesis: ( 1 <= i & i < j & j <= n implies ex P being Permutation of (Seg n) st
( P . 1 = i & P . 2 = j & ( for k being Nat st k in Seg n & k > 2 holds
( ( k <= i + 1 implies P . k = k - 2 ) & ( i + 1 < k & k <= j implies P . k = k - 1 ) & ( k > j implies P . k = k ) ) ) ) )
assume that
A1: 1 <= i and
A2: i < j and
A3: j <= n ; ::_thesis: ex P being Permutation of (Seg n) st
( P . 1 = i & P . 2 = j & ( for k being Nat st k in Seg n & k > 2 holds
( ( k <= i + 1 implies P . k = k - 2 ) & ( i + 1 < k & k <= j implies P . k = k - 1 ) & ( k > j implies P . k = k ) ) ) )
i <= n by A2, A3, XXREAL_0:2;
then A4: i in Seg n by A1, FINSEQ_1:1;
1 <= j by A1, A2, XXREAL_0:2;
then A5: j in Seg n by A3, FINSEQ_1:1;
reconsider S = Seg n as non empty Subset of NAT by A1, A2, A3;
defpred S1[ Nat, Nat] means ( ( $1 = 1 implies $2 = i ) & ( $1 = 2 implies $2 = j ) & ( $1 > 2 implies ( ( $1 <= i + 1 implies $2 = $1 - 2 ) & ( i + 1 < $1 & $1 <= j implies $2 = $1 - 1 ) & ( $1 > j implies $2 = $1 ) ) ) );
A6: i + 1 < j + 1 by A2, XREAL_1:8;
A7: for k being Nat st k in Seg n holds
ex d being Element of S st S1[k,d]
proof
let k be Nat; ::_thesis: ( k in Seg n implies ex d being Element of S st S1[k,d] )
assume A8: k in Seg n ; ::_thesis: ex d being Element of S st S1[k,d]
then A9: k <= n by FINSEQ_1:1;
A10: k <> 0 by A8, FINSEQ_1:1;
percases ( k = 1 or k = 2 or ( k > 2 & k <> 1 & k <> 2 ) ) by A10, NAT_1:26;
suppose ( k = 1 or k = 2 ) ; ::_thesis: ex d being Element of S st S1[k,d]
hence ex d being Element of S st S1[k,d] by A4, A5; ::_thesis: verum
end;
supposeA11: ( k > 2 & k <> 1 & k <> 2 ) ; ::_thesis: ex d being Element of S st S1[k,d]
then reconsider k2 = k - 2 as Nat by NAT_1:21;
k2 > 2 - 2 by A11, XREAL_1:8;
then A12: k2 >= 1 by NAT_1:14;
A13: k2 <= k - 0 by XREAL_1:10;
percases ( k <= i + 1 or ( k > i + 1 & k <= j ) or ( k > i + 1 & k > j & k in S ) ) by A8;
supposeA14: k <= i + 1 ; ::_thesis: ex d being Element of S st S1[k,d]
then k < j + 1 by A6, XXREAL_0:2;
then A15: k <= j by NAT_1:13;
k2 <= n by A9, A13, XXREAL_0:2;
then k2 in S by A12, FINSEQ_1:1;
hence ex d being Element of S st S1[k,d] by A11, A14, A15; ::_thesis: verum
end;
supposeA16: ( k > i + 1 & k <= j ) ; ::_thesis: ex d being Element of S st S1[k,d]
set k1 = k2 + 1;
k2 + 1 <= (k2 + 1) + 1 by NAT_1:11;
then A17: k2 + 1 <= n by A9, XXREAL_0:2;
k2 + 1 >= 1 by NAT_1:11;
then k2 + 1 in S by A17;
hence ex d being Element of S st S1[k,d] by A11, A16; ::_thesis: verum
end;
suppose ( k > i + 1 & k > j & k in S ) ; ::_thesis: ex d being Element of S st S1[k,d]
hence ex d being Element of S st S1[k,d] by A11; ::_thesis: verum
end;
end;
end;
end;
end;
consider f being FinSequence of S such that
A18: ( len f = n & ( for k being Nat st k in Seg n holds
S1[k,f /. k] ) ) from FINSEQ_4:sch_1(A7);
A19: 1 < j by A1, A2, XXREAL_0:2;
then 1 <= n by A3, XXREAL_0:2;
then A20: 1 in S ;
then A21: S1[1,f /. 1] by A18;
1 + 1 <= j by A19, NAT_1:13;
then 2 <= n by A3, XXREAL_0:2;
then A22: 2 in S ;
then A23: S1[2,f /. 2] by A18;
A24: dom f = S by A18, FINSEQ_1:def_3;
then A25: f /. 1 = f . 1 by A20, PARTFUN1:def_6;
A26: rng f c= S by FINSEQ_1:def_4;
then reconsider F = f as Function of S,S by A24, FUNCT_2:2;
A27: f /. 2 = f . 2 by A22, A24, PARTFUN1:def_6;
S c= rng f
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in S or x in rng f )
assume A28: x in S ; ::_thesis: x in rng f
then reconsider k = x as Nat ;
set k1 = k + 1;
set k2 = (k + 1) + 1;
A29: 1 <= k by A28, FINSEQ_1:1;
percases ( k < i or k = i or ( k > i & k < j ) or k = j or k > j ) by XXREAL_0:1;
supposeA30: k < i ; ::_thesis: x in rng f
A31: k + 2 > 0 + 2 by A29, XREAL_1:8;
A32: k + 1 <= i by A30, NAT_1:13;
then k + 1 < j by A2, XXREAL_0:2;
then (k + 1) + 1 <= j by NAT_1:13;
then ( 1 <= (k + 1) + 1 & (k + 1) + 1 <= n ) by A3, NAT_1:11, XXREAL_0:2;
then A33: (k + 1) + 1 in S ;
then S1[(k + 1) + 1,f /. ((k + 1) + 1)] by A18;
then f . ((k + 1) + 1) = k by A24, A31, A32, A33, PARTFUN1:def_6, XREAL_1:6;
hence x in rng f by A24, A33, FUNCT_1:def_3; ::_thesis: verum
end;
suppose k = i ; ::_thesis: x in rng f
hence x in rng f by A20, A21, A24, A25, FUNCT_1:def_3; ::_thesis: verum
end;
supposeA34: ( k > i & k < j ) ; ::_thesis: x in rng f
then k > 1 by A1, A29, XXREAL_0:1;
then A35: k >= 1 + 1 by NAT_1:13;
k + 1 <= j by A34, NAT_1:13;
then ( 1 <= k + 1 & k + 1 <= n ) by A3, NAT_1:11, XXREAL_0:2;
then A36: k + 1 in S ;
then S1[k + 1,f /. (k + 1)] by A18;
then f . (k + 1) = k by A24, A34, A35, A36, NAT_1:13, PARTFUN1:def_6, XREAL_1:8;
hence x in rng f by A24, A36, FUNCT_1:def_3; ::_thesis: verum
end;
suppose k = j ; ::_thesis: x in rng f
hence x in rng f by A22, A23, A24, A27, FUNCT_1:def_3; ::_thesis: verum
end;
supposeA37: k > j ; ::_thesis: x in rng f
j > 1 by A1, A2, XXREAL_0:2;
then A38: j >= 1 + 1 by NAT_1:13;
S1[k,f /. k] by A18, A28;
then f . k = k by A24, A37, A38, PARTFUN1:def_6, XXREAL_0:2;
hence x in rng f by A24, A28, FUNCT_1:def_3; ::_thesis: verum
end;
end;
end;
then rng F = S by A26, XBOOLE_0:def_10;
then A39: F is onto by FUNCT_2:def_3;
card S = card S ;
then F is one-to-one by A39, STIRL2_1:60;
then reconsider F = F as Permutation of (Seg n) by A39;
take F ; ::_thesis: ( F . 1 = i & F . 2 = j & ( for k being Nat st k in Seg n & k > 2 holds
( ( k <= i + 1 implies F . k = k - 2 ) & ( i + 1 < k & k <= j implies F . k = k - 1 ) & ( k > j implies F . k = k ) ) ) )
thus ( F . 1 = i & F . 2 = j ) by A20, A21, A22, A23, A24, PARTFUN1:def_6; ::_thesis: for k being Nat st k in Seg n & k > 2 holds
( ( k <= i + 1 implies F . k = k - 2 ) & ( i + 1 < k & k <= j implies F . k = k - 1 ) & ( k > j implies F . k = k ) )
let k be Nat; ::_thesis: ( k in Seg n & k > 2 implies ( ( k <= i + 1 implies F . k = k - 2 ) & ( i + 1 < k & k <= j implies F . k = k - 1 ) & ( k > j implies F . k = k ) ) )
assume that
A40: k in Seg n and
A41: k > 2 ; ::_thesis: ( ( k <= i + 1 implies F . k = k - 2 ) & ( i + 1 < k & k <= j implies F . k = k - 1 ) & ( k > j implies F . k = k ) )
f /. k = f . k by A24, A40, PARTFUN1:def_6;
hence ( ( k <= i + 1 implies F . k = k - 2 ) & ( i + 1 < k & k <= j implies F . k = k - 1 ) & ( k > j implies F . k = k ) ) by A18, A40, A41; ::_thesis: verum
end;
Lm3: for r being real number
for i, j, n being Nat st 1 <= i & i < j & j <= n holds
ex A being Matrix of n,F_Real st
( Det A = 1. F_Real & A * (i,i) = cos r & A * (j,j) = cos r & A * (i,j) = sin r & A * (j,i) = - (sin r) & ( for k, m being Nat st [k,m] in Indices A holds
( ( k = m & k <> i & k <> j implies A * (k,k) = 1. F_Real ) & ( k <> m & {k,m} <> {i,j} implies A * (k,m) = 0. F_Real ) ) ) )
proof
let r be real number ; ::_thesis: for i, j, n being Nat st 1 <= i & i < j & j <= n holds
ex A being Matrix of n,F_Real st
( Det A = 1. F_Real & A * (i,i) = cos r & A * (j,j) = cos r & A * (i,j) = sin r & A * (j,i) = - (sin r) & ( for k, m being Nat st [k,m] in Indices A holds
( ( k = m & k <> i & k <> j implies A * (k,k) = 1. F_Real ) & ( k <> m & {k,m} <> {i,j} implies A * (k,m) = 0. F_Real ) ) ) )
let i, j, n be Nat; ::_thesis: ( 1 <= i & i < j & j <= n implies ex A being Matrix of n,F_Real st
( Det A = 1. F_Real & A * (i,i) = cos r & A * (j,j) = cos r & A * (i,j) = sin r & A * (j,i) = - (sin r) & ( for k, m being Nat st [k,m] in Indices A holds
( ( k = m & k <> i & k <> j implies A * (k,k) = 1. F_Real ) & ( k <> m & {k,m} <> {i,j} implies A * (k,m) = 0. F_Real ) ) ) ) )
A1: now__::_thesis:_for_k_being_Nat_st_k_>=_1_&_k_<>_1_&_k_<>_2_holds_
k_>_2
let k be Nat; ::_thesis: ( k >= 1 & k <> 1 & k <> 2 implies k > 2 )
assume that
A2: ( k >= 1 & k <> 1 ) and
A3: k <> 2 ; ::_thesis: k > 2
k > 1 by A2, XXREAL_0:1;
then k >= 1 + 1 by NAT_1:13;
hence k > 2 by A3, XXREAL_0:1; ::_thesis: verum
end;
reconsider s = sin r, c = cos r as Element of F_Real by XREAL_0:def_1;
set S = Seg n;
assume that
A4: 1 <= i and
A5: i < j and
A6: j <= n ; ::_thesis: ex A being Matrix of n,F_Real st
( Det A = 1. F_Real & A * (i,i) = cos r & A * (j,j) = cos r & A * (i,j) = sin r & A * (j,i) = - (sin r) & ( for k, m being Nat st [k,m] in Indices A holds
( ( k = m & k <> i & k <> j implies A * (k,k) = 1. F_Real ) & ( k <> m & {k,m} <> {i,j} implies A * (k,m) = 0. F_Real ) ) ) )
A7: 1 < j by A4, A5, XXREAL_0:2;
then A8: 1 + 1 <= j by NAT_1:13;
then reconsider n2 = n - 2 as Nat by A6, NAT_1:21, XXREAL_0:2;
consider P being Permutation of (Seg n) such that
A9: P . 1 = i and
A10: P . 2 = j and
for k being Nat st k in Seg n & k > 2 holds
( ( k <= i + 1 implies P . k = k - 2 ) & ( i + 1 < k & k <= j implies P . k = k - 1 ) & ( k > j implies P . k = k ) ) by A4, A5, A6, Lm2;
reconsider p = P as one-to-one Function ;
dom P = Seg n by FUNCT_2:52;
then A11: rng (p ") = Seg n by FUNCT_1:33;
rng P = Seg n by FUNCT_2:def_3;
then dom (p ") = Seg n by FUNCT_1:33;
then reconsider P1 = p " as one-to-one Function of (Seg n),(Seg n) by A11, FUNCT_2:1;
P1 is onto by A11, FUNCT_2:def_3;
then reconsider P1 = P1 as Permutation of (Seg n) ;
A12: dom P = Seg n by FUNCT_2:52;
1 <= n by A6, A7, XXREAL_0:2;
then A13: 1 in Seg n ;
then A14: P1 . (P . 1) = 1 by A12, FUNCT_1:34;
set A = (c,s) ][ ((- s),c);
set ONE = 1. (F_Real,n2);
set ao = <*((c,s) ][ ((- s),c)),(1. (F_Real,n2))*>;
set B = block_diagonal (<*((c,s) ][ ((- s),c)),(1. (F_Real,n2))*>,(0. F_Real));
A15: len ((c,s) ][ ((- s),c)) = 2 by MATRIX_1:25;
then Len <*((c,s) ][ ((- s),c))*> = <*2*> by MATRIXJ1:15;
then A16: Sum (Len <*((c,s) ][ ((- s),c))*>) = 2 by RVSUM_1:73;
len (1. (F_Real,n2)) = n2 by MATRIX_1:25;
then A17: Sum (Len <*((c,s) ][ ((- s),c)),(1. (F_Real,n2))*>) = 2 + n2 by A15, MATRIXJ1:16;
then reconsider B = block_diagonal (<*((c,s) ][ ((- s),c)),(1. (F_Real,n2))*>,(0. F_Real)) as Matrix of n,F_Real ;
A18: Indices B = [:(Seg n),(Seg n):] by MATRIX_1:24;
2 <= n by A6, A8, XXREAL_0:2;
then A19: 2 in Seg n ;
then A20: P1 . (P . 2) = 2 by A12, FUNCT_1:34;
set pBp = (((B * P1) @) * P1) @ ;
A21: dom P1 = Seg n by FUNCT_2:52;
i < n by A5, A6, XXREAL_0:2;
then A22: i in Seg n by A4, FINSEQ_1:1;
then [i,i] in Indices B by A18, ZFMISC_1:87;
then A23: ((((B * P1) @) * P1) @) * (i,i) = B * (1,1) by A9, A14, Th1;
take (((B * P1) @) * P1) @ ; ::_thesis: ( Det ((((B * P1) @) * P1) @) = 1. F_Real & ((((B * P1) @) * P1) @) * (i,i) = cos r & ((((B * P1) @) * P1) @) * (j,j) = cos r & ((((B * P1) @) * P1) @) * (i,j) = sin r & ((((B * P1) @) * P1) @) * (j,i) = - (sin r) & ( for k, m being Nat st [k,m] in Indices ((((B * P1) @) * P1) @) holds
( ( k = m & k <> i & k <> j implies ((((B * P1) @) * P1) @) * (k,k) = 1. F_Real ) & ( k <> m & {k,m} <> {i,j} implies ((((B * P1) @) * P1) @) * (k,m) = 0. F_Real ) ) ) )
A24: Indices ((((B * P1) @) * P1) @) = [:(Seg n),(Seg n):] by MATRIX_1:24;
A25: Det B = 1. F_Real by A17, Th12;
A26: block_diagonal (<*((c,s) ][ ((- s),c))*>,(0. F_Real)) = (c,s) ][ ((- s),c) by MATRIXJ1:34;
width ((c,s) ][ ((- s),c)) = 2 by MATRIX_1:24;
then Width <*((c,s) ][ ((- s),c))*> = <*2*> by MATRIXJ1:19;
then A27: Sum (Width <*((c,s) ][ ((- s),c))*>) = 2 by RVSUM_1:73;
1 < j by A4, A5, XXREAL_0:2;
then A28: j in Seg n by A6, FINSEQ_1:1;
then [j,j] in Indices B by A18, ZFMISC_1:87;
then A29: ((((B * P1) @) * P1) @) * (j,j) = B * (2,2) by A10, A20, Th1;
A30: block_diagonal (<*(1. (F_Real,n2))*>,(0. F_Real)) = 1. (F_Real,n2) by MATRIXJ1:34;
[1,1] in Indices ((c,s) ][ ((- s),c)) by MATRIX_2:4;
then A31: B * (1,1) = ((c,s) ][ ((- s),c)) * (1,1) by A26, A30, MATRIXJ1:26;
[2,1] in Indices ((c,s) ][ ((- s),c)) by MATRIX_2:4;
then A32: B * (2,1) = ((c,s) ][ ((- s),c)) * (2,1) by A26, A30, MATRIXJ1:26;
[1,2] in Indices ((c,s) ][ ((- s),c)) by MATRIX_2:4;
then A33: B * (1,2) = ((c,s) ][ ((- s),c)) * (1,2) by A26, A30, MATRIXJ1:26;
[2,2] in Indices ((c,s) ][ ((- s),c)) by MATRIX_2:4;
then A34: B * (2,2) = ((c,s) ][ ((- s),c)) * (2,2) by A26, A30, MATRIXJ1:26;
A35: <*((c,s) ][ ((- s),c))*> ^ <*(1. (F_Real,n2))*> = <*((c,s) ][ ((- s),c)),(1. (F_Real,n2))*> ;
[i,j] in Indices B by A18, A22, A28, ZFMISC_1:87;
then A36: ((((B * P1) @) * P1) @) * (i,j) = B * (1,2) by A9, A10, A14, A20, Th1;
[j,i] in Indices B by A18, A22, A28, ZFMISC_1:87;
then ((((B * P1) @) * P1) @) * (j,i) = B * (2,1) by A9, A10, A14, A20, Th1;
hence ( Det ((((B * P1) @) * P1) @) = 1. F_Real & ((((B * P1) @) * P1) @) * (i,i) = cos r & ((((B * P1) @) * P1) @) * (j,j) = cos r & ((((B * P1) @) * P1) @) * (i,j) = sin r & ((((B * P1) @) * P1) @) * (j,i) = - (sin r) ) by A23, A25, A29, A34, A33, A32, A31, A36, Th1, MATRIX_2:6; ::_thesis: for k, m being Nat st [k,m] in Indices ((((B * P1) @) * P1) @) holds
( ( k = m & k <> i & k <> j implies ((((B * P1) @) * P1) @) * (k,k) = 1. F_Real ) & ( k <> m & {k,m} <> {i,j} implies ((((B * P1) @) * P1) @) * (k,m) = 0. F_Real ) )
let k, m be Nat; ::_thesis: ( [k,m] in Indices ((((B * P1) @) * P1) @) implies ( ( k = m & k <> i & k <> j implies ((((B * P1) @) * P1) @) * (k,k) = 1. F_Real ) & ( k <> m & {k,m} <> {i,j} implies ((((B * P1) @) * P1) @) * (k,m) = 0. F_Real ) ) )
assume A37: [k,m] in Indices ((((B * P1) @) * P1) @) ; ::_thesis: ( ( k = m & k <> i & k <> j implies ((((B * P1) @) * P1) @) * (k,k) = 1. F_Real ) & ( k <> m & {k,m} <> {i,j} implies ((((B * P1) @) * P1) @) * (k,m) = 0. F_Real ) )
A38: k in Seg n by A24, A37, ZFMISC_1:87;
set Pk = P1 . k;
set Pm = P1 . m;
A39: rng P1 = Seg n by FUNCT_2:def_3;
then A40: P1 . k in Seg n by A21, A38, FUNCT_1:def_3;
then A41: P1 . k >= 1 by FINSEQ_1:1;
A42: m in Seg n by A24, A37, ZFMISC_1:87;
then A43: P1 . m in Seg n by A21, A39, FUNCT_1:def_3;
then A44: [(P1 . k),(P1 . m)] in [:(Seg n),(Seg n):] by A40, ZFMISC_1:87;
A45: ((((B * P1) @) * P1) @) * (k,m) = B * ((P1 . k),(P1 . m)) by A18, A24, A37, Th1;
thus ( k = m & k <> i & k <> j implies ((((B * P1) @) * P1) @) * (k,k) = 1. F_Real ) ::_thesis: ( k <> m & {k,m} <> {i,j} implies ((((B * P1) @) * P1) @) * (k,m) = 0. F_Real )
proof
assume that
A46: k = m and
A47: ( k <> i & k <> j ) ; ::_thesis: ((((B * P1) @) * P1) @) * (k,k) = 1. F_Real
( P1 . k <> 1 & P1 . k <> 2 ) by A9, A10, A14, A20, A21, A22, A28, A38, A47, FUNCT_1:def_4;
then A48: P1 . k > 2 by A1, A41;
then reconsider Pk2 = (P1 . k) - 2 as Nat by NAT_1:21;
( P1 . k = Pk2 + 2 & Pk2 > 2 - 2 ) by A48, XREAL_1:8;
then A49: [Pk2,Pk2] in Indices (1. (F_Real,n2)) by A16, A18, A27, A30, A44, A46, MATRIXJ1:27;
then (1. (F_Real,n2)) * (Pk2,Pk2) = B * ((Pk2 + 2),(Pk2 + 2)) by A16, A27, A30, MATRIXJ1:28;
hence ((((B * P1) @) * P1) @) * (k,k) = 1. F_Real by A45, A46, A49, MATRIX_1:def_11; ::_thesis: verum
end;
A50: P1 . m >= 1 by A43, FINSEQ_1:1;
thus ( k <> m & {k,m} <> {i,j} implies ((((B * P1) @) * P1) @) * (k,m) = 0. F_Real ) ::_thesis: verum
proof
assume that
A51: k <> m and
A52: {k,m} <> {i,j} ; ::_thesis: ((((B * P1) @) * P1) @) * (k,m) = 0. F_Real
A53: P1 . k <> P1 . m by A21, A38, A42, A51, FUNCT_1:def_4;
percases ( ( k <> i & k <> j & m <> i & m <> j ) or ( k = i & m <> j ) or ( k = j & m <> i ) or ( m = i & k <> j ) or ( m = j & k <> i ) ) by A52;
supposeA54: ( k <> i & k <> j & m <> i & m <> j ) ; ::_thesis: ((((B * P1) @) * P1) @) * (k,m) = 0. F_Real
then ( P1 . k <> 1 & P1 . k <> 2 ) by A9, A10, A14, A20, A21, A22, A28, A38, FUNCT_1:def_4;
then A55: P1 . k > 2 by A1, A41;
( P1 . m <> 1 & P1 . m <> 2 ) by A9, A10, A14, A20, A21, A22, A28, A42, A54, FUNCT_1:def_4;
then A56: P1 . m > 2 by A1, A50;
then reconsider Pk2 = (P1 . k) - 2, Pm2 = (P1 . m) - 2 as Nat by A55, NAT_1:21;
A57: Pk2 > 2 - 2 by A55, XREAL_1:8;
A58: ( P1 . k = Pk2 + 2 & P1 . m = Pm2 + 2 ) ;
Pm2 > 2 - 2 by A56, XREAL_1:8;
then A59: [Pk2,Pm2] in Indices (1. (F_Real,n2)) by A16, A18, A27, A30, A44, A58, A57, MATRIXJ1:27;
then (1. (F_Real,n2)) * (Pk2,Pm2) = B * ((Pk2 + 2),(Pm2 + 2)) by A16, A27, A30, MATRIXJ1:28;
hence ((((B * P1) @) * P1) @) * (k,m) = 0. F_Real by A45, A53, A59, MATRIX_1:def_11; ::_thesis: verum
end;
supposeA60: ( k = i & m <> j ) ; ::_thesis: ((((B * P1) @) * P1) @) * (k,m) = 0. F_Real
then P1 . m <> 2 by A10, A20, A21, A28, A42, FUNCT_1:def_4;
then A61: P1 . m > 2 by A1, A9, A14, A50, A53, A60;
P1 . k = 1 by A9, A12, A13, A60, FUNCT_1:34;
hence ((((B * P1) @) * P1) @) * (k,m) = 0. F_Real by A16, A18, A27, A35, A44, A45, A61, MATRIXJ1:29; ::_thesis: verum
end;
supposeA62: ( k = j & m <> i ) ; ::_thesis: ((((B * P1) @) * P1) @) * (k,m) = 0. F_Real
then P1 . m <> 1 by A9, A14, A21, A22, A42, FUNCT_1:def_4;
then A63: P1 . m > 2 by A1, A10, A20, A50, A53, A62;
P1 . k = 2 by A10, A12, A19, A62, FUNCT_1:34;
hence ((((B * P1) @) * P1) @) * (k,m) = 0. F_Real by A16, A18, A27, A35, A44, A45, A63, MATRIXJ1:29; ::_thesis: verum
end;
supposeA64: ( m = i & k <> j ) ; ::_thesis: ((((B * P1) @) * P1) @) * (k,m) = 0. F_Real
then P1 . k <> 2 by A10, A20, A21, A28, A38, FUNCT_1:def_4;
then A65: P1 . k > 2 by A1, A9, A14, A41, A53, A64;
P1 . m = 1 by A9, A12, A13, A64, FUNCT_1:34;
hence ((((B * P1) @) * P1) @) * (k,m) = 0. F_Real by A16, A18, A27, A35, A44, A45, A65, MATRIXJ1:29; ::_thesis: verum
end;
supposeA66: ( m = j & k <> i ) ; ::_thesis: ((((B * P1) @) * P1) @) * (k,m) = 0. F_Real
then P1 . k <> 1 by A9, A14, A21, A22, A38, FUNCT_1:def_4;
then A67: P1 . k > 2 by A1, A10, A20, A41, A53, A66;
P1 . m = 2 by A10, A12, A19, A66, FUNCT_1:34;
hence ((((B * P1) @) * P1) @) * (k,m) = 0. F_Real by A16, A18, A27, A35, A44, A45, A67, MATRIXJ1:29; ::_thesis: verum
end;
end;
end;
end;
begin
definition
let n be Nat;
let r be real number ;
let i, j be Nat;
assume B1: ( 1 <= i & i < j & j <= n ) ;
func Rotation (i,j,n,r) -> invertible Matrix of n,F_Real means :Def3: :: MATRTOP3:def 3
( it * (i,i) = cos r & it * (j,j) = cos r & it * (i,j) = sin r & it * (j,i) = - (sin r) & ( for k, m being Nat st [k,m] in Indices it holds
( ( k = m & k <> i & k <> j implies it * (k,k) = 1. F_Real ) & ( k <> m & {k,m} <> {i,j} implies it * (k,m) = 0. F_Real ) ) ) );
existence
ex b1 being invertible Matrix of n,F_Real st
( b1 * (i,i) = cos r & b1 * (j,j) = cos r & b1 * (i,j) = sin r & b1 * (j,i) = - (sin r) & ( for k, m being Nat st [k,m] in Indices b1 holds
( ( k = m & k <> i & k <> j implies b1 * (k,k) = 1. F_Real ) & ( k <> m & {k,m} <> {i,j} implies b1 * (k,m) = 0. F_Real ) ) ) )
proof
consider A being Matrix of n,F_Real such that
A1: Det A = 1. F_Real and
A2: ( A * (i,i) = cos r & A * (j,j) = cos r & A * (i,j) = sin r & A * (j,i) = - (sin r) & ( for k, m being Nat st [k,m] in Indices A holds
( ( k = m & k <> i & k <> j implies A * (k,k) = 1. F_Real ) & ( k <> m & {k,m} <> {i,j} implies A * (k,m) = 0. F_Real ) ) ) ) by B1, Lm3;
Det A <> 0. F_Real by A1;
then A is invertible by LAPLACE:34;
hence ex b1 being invertible Matrix of n,F_Real st
( b1 * (i,i) = cos r & b1 * (j,j) = cos r & b1 * (i,j) = sin r & b1 * (j,i) = - (sin r) & ( for k, m being Nat st [k,m] in Indices b1 holds
( ( k = m & k <> i & k <> j implies b1 * (k,k) = 1. F_Real ) & ( k <> m & {k,m} <> {i,j} implies b1 * (k,m) = 0. F_Real ) ) ) ) by A2; ::_thesis: verum
end;
uniqueness
for b1, b2 being invertible Matrix of n,F_Real st b1 * (i,i) = cos r & b1 * (j,j) = cos r & b1 * (i,j) = sin r & b1 * (j,i) = - (sin r) & ( for k, m being Nat st [k,m] in Indices b1 holds
( ( k = m & k <> i & k <> j implies b1 * (k,k) = 1. F_Real ) & ( k <> m & {k,m} <> {i,j} implies b1 * (k,m) = 0. F_Real ) ) ) & b2 * (i,i) = cos r & b2 * (j,j) = cos r & b2 * (i,j) = sin r & b2 * (j,i) = - (sin r) & ( for k, m being Nat st [k,m] in Indices b2 holds
( ( k = m & k <> i & k <> j implies b2 * (k,k) = 1. F_Real ) & ( k <> m & {k,m} <> {i,j} implies b2 * (k,m) = 0. F_Real ) ) ) holds
b1 = b2
proof
let I1, I2 be invertible Matrix of n,F_Real; ::_thesis: ( I1 * (i,i) = cos r & I1 * (j,j) = cos r & I1 * (i,j) = sin r & I1 * (j,i) = - (sin r) & ( for k, m being Nat st [k,m] in Indices I1 holds
( ( k = m & k <> i & k <> j implies I1 * (k,k) = 1. F_Real ) & ( k <> m & {k,m} <> {i,j} implies I1 * (k,m) = 0. F_Real ) ) ) & I2 * (i,i) = cos r & I2 * (j,j) = cos r & I2 * (i,j) = sin r & I2 * (j,i) = - (sin r) & ( for k, m being Nat st [k,m] in Indices I2 holds
( ( k = m & k <> i & k <> j implies I2 * (k,k) = 1. F_Real ) & ( k <> m & {k,m} <> {i,j} implies I2 * (k,m) = 0. F_Real ) ) ) implies I1 = I2 )
assume that
A3: ( I1 * (i,i) = cos r & I1 * (j,j) = cos r & I1 * (i,j) = sin r & I1 * (j,i) = - (sin r) ) and
A4: for k, m being Nat st [k,m] in Indices I1 holds
( ( k = m & k <> i & k <> j implies I1 * (k,k) = 1. F_Real ) & ( k <> m & {k,m} <> {i,j} implies I1 * (k,m) = 0. F_Real ) ) and
A5: ( I2 * (i,i) = cos r & I2 * (j,j) = cos r & I2 * (i,j) = sin r & I2 * (j,i) = - (sin r) ) and
A6: for k, m being Nat st [k,m] in Indices I2 holds
( ( k = m & k <> i & k <> j implies I2 * (k,k) = 1. F_Real ) & ( k <> m & {k,m} <> {i,j} implies I2 * (k,m) = 0. F_Real ) ) ; ::_thesis: I1 = I2
for k, m being Nat st [k,m] in Indices I1 holds
I1 * (k,m) = I2 * (k,m)
proof
let k, m be Nat; ::_thesis: ( [k,m] in Indices I1 implies I1 * (k,m) = I2 * (k,m) )
assume A7: [k,m] in Indices I1 ; ::_thesis: I1 * (k,m) = I2 * (k,m)
then A8: [k,m] in Indices I2 by MATRIX_1:26;
percases ( ( k = m & ( k = i or k = j ) ) or ( k <> m & ( ( k = i & m = j ) or ( k = j & m = i ) ) ) or ( k = m & k <> i & k <> j ) or ( k <> m & ( ( k = i & m <> j ) or ( k = j & m <> i ) or ( m = i & k <> j ) or ( m = j & k <> i ) or ( k <> i & k <> j & m <> i & m <> j ) ) ) ) ;
suppose ( ( k = m & ( k = i or k = j ) ) or ( k <> m & ( ( k = i & m = j ) or ( k = j & m = i ) ) ) ) ; ::_thesis: I1 * (k,m) = I2 * (k,m)
hence I1 * (k,m) = I2 * (k,m) by A3, A5; ::_thesis: verum
end;
supposeA9: ( k = m & k <> i & k <> j ) ; ::_thesis: I1 * (k,m) = I2 * (k,m)
then I1 * (k,m) = 1. F_Real by A4, A7;
hence I1 * (k,m) = I2 * (k,m) by A6, A8, A9; ::_thesis: verum
end;
supposeA10: ( k <> m & ( ( k = i & m <> j ) or ( k = j & m <> i ) or ( m = i & k <> j ) or ( m = j & k <> i ) or ( k <> i & k <> j & m <> i & m <> j ) ) ) ; ::_thesis: I1 * (k,m) = I2 * (k,m)
then A11: {k,m} <> {i,j} by ZFMISC_1:6;
then I1 * (k,m) = 0. F_Real by A4, A7, A10;
hence I1 * (k,m) = I2 * (k,m) by A6, A8, A10, A11; ::_thesis: verum
end;
end;
end;
hence I1 = I2 by MATRIX_1:27; ::_thesis: verum
end;
end;
:: deftheorem Def3 defines Rotation MATRTOP3:def_3_:_
for n being Nat
for r being real number
for i, j being Nat st 1 <= i & i < j & j <= n holds
for b5 being invertible Matrix of n,F_Real holds
( b5 = Rotation (i,j,n,r) iff ( b5 * (i,i) = cos r & b5 * (j,j) = cos r & b5 * (i,j) = sin r & b5 * (j,i) = - (sin r) & ( for k, m being Nat st [k,m] in Indices b5 holds
( ( k = m & k <> i & k <> j implies b5 * (k,k) = 1. F_Real ) & ( k <> m & {k,m} <> {i,j} implies b5 * (k,m) = 0. F_Real ) ) ) ) );
theorem Th13: :: MATRTOP3:13
for r being real number
for i, j, n being Nat st 1 <= i & i < j & j <= n holds
Det (Rotation (i,j,n,r)) = 1. F_Real
proof
let r be real number ; ::_thesis: for i, j, n being Nat st 1 <= i & i < j & j <= n holds
Det (Rotation (i,j,n,r)) = 1. F_Real
let i, j, n be Nat; ::_thesis: ( 1 <= i & i < j & j <= n implies Det (Rotation (i,j,n,r)) = 1. F_Real )
assume A1: ( 1 <= i & i < j & j <= n ) ; ::_thesis: Det (Rotation (i,j,n,r)) = 1. F_Real
then consider A being Matrix of n,F_Real such that
A2: Det A = 1. F_Real and
A3: ( A * (i,i) = cos r & A * (j,j) = cos r & A * (i,j) = sin r & A * (j,i) = - (sin r) & ( for k, m being Nat st [k,m] in Indices A holds
( ( k = m & k <> i & k <> j implies A * (k,k) = 1. F_Real ) & ( k <> m & {k,m} <> {i,j} implies A * (k,m) = 0. F_Real ) ) ) ) by Lm3;
Det A <> 0. F_Real by A2;
then A is invertible by LAPLACE:34;
hence Det (Rotation (i,j,n,r)) = 1. F_Real by A1, A2, A3, Def3; ::_thesis: verum
end;
theorem Th14: :: MATRTOP3:14
for r being real number
for i, j, n, k being Nat
for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n & k in Seg n & k <> i & k <> j holds
(@ p) "*" (Col ((Rotation (i,j,n,r)),k)) = p . k
proof
let r be real number ; ::_thesis: for i, j, n, k being Nat
for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n & k in Seg n & k <> i & k <> j holds
(@ p) "*" (Col ((Rotation (i,j,n,r)),k)) = p . k
let i, j, n, k be Nat; ::_thesis: for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n & k in Seg n & k <> i & k <> j holds
(@ p) "*" (Col ((Rotation (i,j,n,r)),k)) = p . k
let p be Point of (TOP-REAL n); ::_thesis: ( 1 <= i & i < j & j <= n & k in Seg n & k <> i & k <> j implies (@ p) "*" (Col ((Rotation (i,j,n,r)),k)) = p . k )
set S = Seg n;
assume that
A1: ( 1 <= i & i < j & j <= n ) and
A2: k in Seg n and
A3: ( k <> i & k <> j ) ; ::_thesis: (@ p) "*" (Col ((Rotation (i,j,n,r)),k)) = p . k
set O = Rotation (i,j,n,r);
set C = Col ((Rotation (i,j,n,r)),k);
A4: Indices (Rotation (i,j,n,r)) = [:(Seg n),(Seg n):] by MATRIX_1:24;
then A5: [k,k] in Indices (Rotation (i,j,n,r)) by A2, ZFMISC_1:87;
A6: len (Rotation (i,j,n,r)) = n by MATRIX_1:25;
then A7: dom (Rotation (i,j,n,r)) = Seg n by FINSEQ_1:def_3;
len (Col ((Rotation (i,j,n,r)),k)) = n by A6, MATRIX_1:def_8;
then A8: dom (Col ((Rotation (i,j,n,r)),k)) = Seg n by FINSEQ_1:def_3;
A9: now__::_thesis:_for_m_being_Nat_st_m_in_dom_(Col_((Rotation_(i,j,n,r)),k))_&_m_<>_k_holds_
(Col_((Rotation_(i,j,n,r)),k))_._m_=_0._F_Real
let m be Nat; ::_thesis: ( m in dom (Col ((Rotation (i,j,n,r)),k)) & m <> k implies (Col ((Rotation (i,j,n,r)),k)) . m = 0. F_Real )
assume that
A10: m in dom (Col ((Rotation (i,j,n,r)),k)) and
A11: m <> k ; ::_thesis: (Col ((Rotation (i,j,n,r)),k)) . m = 0. F_Real
A12: [m,k] in Indices (Rotation (i,j,n,r)) by A2, A4, A8, A10, ZFMISC_1:87;
not k in {i,j} by A3, TARSKI:def_2;
then A13: {m,k} <> {i,j} by TARSKI:def_2;
thus (Col ((Rotation (i,j,n,r)),k)) . m = (Rotation (i,j,n,r)) * (m,k) by A7, A8, A10, MATRIX_1:def_8
.= 0. F_Real by A1, A11, A12, A13, Def3 ; ::_thesis: verum
end;
len p = n by CARD_1:def_7;
then A14: dom p = Seg n by FINSEQ_1:def_3;
(Col ((Rotation (i,j,n,r)),k)) . k = (Rotation (i,j,n,r)) * (k,k) by A2, A7, MATRIX_1:def_8
.= 1. F_Real by A1, A3, A5, Def3 ;
hence p . k = Sum (mlt ((Col ((Rotation (i,j,n,r)),k)),(@ p))) by A2, A8, A9, A14, MATRIX_3:17
.= (@ p) "*" (Col ((Rotation (i,j,n,r)),k)) by FVSUM_1:64 ;
::_thesis: verum
end;
theorem Th15: :: MATRTOP3:15
for r being real number
for i, j, n being Nat
for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n holds
(@ p) "*" (Col ((Rotation (i,j,n,r)),i)) = ((p . i) * (cos r)) + ((p . j) * (- (sin r)))
proof
let r be real number ; ::_thesis: for i, j, n being Nat
for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n holds
(@ p) "*" (Col ((Rotation (i,j,n,r)),i)) = ((p . i) * (cos r)) + ((p . j) * (- (sin r)))
let i, j, n be Nat; ::_thesis: for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n holds
(@ p) "*" (Col ((Rotation (i,j,n,r)),i)) = ((p . i) * (cos r)) + ((p . j) * (- (sin r)))
let p be Point of (TOP-REAL n); ::_thesis: ( 1 <= i & i < j & j <= n implies (@ p) "*" (Col ((Rotation (i,j,n,r)),i)) = ((p . i) * (cos r)) + ((p . j) * (- (sin r))) )
assume that
A1: 1 <= i and
A2: i < j and
A3: j <= n ; ::_thesis: (@ p) "*" (Col ((Rotation (i,j,n,r)),i)) = ((p . i) * (cos r)) + ((p . j) * (- (sin r)))
set O = Rotation (i,j,n,r);
set C = Col ((Rotation (i,j,n,r)),i);
set S = Seg n;
1 <= j by A1, A2, XXREAL_0:2;
then A4: j in Seg n by A3, FINSEQ_1:1;
A5: len (Rotation (i,j,n,r)) = n by MATRIX_1:25;
then A6: dom (Rotation (i,j,n,r)) = Seg n by FINSEQ_1:def_3;
then A7: (Col ((Rotation (i,j,n,r)),i)) . j = (Rotation (i,j,n,r)) * (j,i) by A4, MATRIX_1:def_8;
i <= n by A2, A3, XXREAL_0:2;
then A8: i in Seg n by A1, FINSEQ_1:1;
then A9: (Col ((Rotation (i,j,n,r)),i)) . i = (Rotation (i,j,n,r)) * (i,i) by A6, MATRIX_1:def_8;
( len p = n & len (Col ((Rotation (i,j,n,r)),i)) = n ) by A5, CARD_1:def_7;
then A10: len (mlt ((@ p),(Col ((Rotation (i,j,n,r)),i)))) = n by MATRIX_3:6;
then A11: dom (mlt ((@ p),(Col ((Rotation (i,j,n,r)),i)))) = Seg n by FINSEQ_1:def_3;
A12: Indices (Rotation (i,j,n,r)) = [:(Seg n),(Seg n):] by MATRIX_1:24;
for k being Nat st k in dom (mlt ((@ p),(Col ((Rotation (i,j,n,r)),i)))) & k <> i & k <> j holds
(mlt ((@ p),(Col ((Rotation (i,j,n,r)),i)))) . k = 0. F_Real
proof
let k be Nat; ::_thesis: ( k in dom (mlt ((@ p),(Col ((Rotation (i,j,n,r)),i)))) & k <> i & k <> j implies (mlt ((@ p),(Col ((Rotation (i,j,n,r)),i)))) . k = 0. F_Real )
assume that
A13: k in dom (mlt ((@ p),(Col ((Rotation (i,j,n,r)),i)))) and
A14: k <> i and
A15: k <> j ; ::_thesis: (mlt ((@ p),(Col ((Rotation (i,j,n,r)),i)))) . k = 0. F_Real
not k in {i,j} by A14, A15, TARSKI:def_2;
then A16: {k,i} <> {i,j} by TARSKI:def_2;
reconsider pk = (@ p) . k as Element of F_Real ;
A17: [k,i] in Indices (Rotation (i,j,n,r)) by A8, A11, A12, A13, ZFMISC_1:87;
(Col ((Rotation (i,j,n,r)),i)) . k = (Rotation (i,j,n,r)) * (k,i) by A6, A11, A13, MATRIX_1:def_8;
hence (mlt ((@ p),(Col ((Rotation (i,j,n,r)),i)))) . k = pk * ((Rotation (i,j,n,r)) * (k,i)) by A13, FVSUM_1:60
.= pk * (0. F_Real) by A1, A2, A3, A14, A16, A17, Def3
.= 0. F_Real ;
::_thesis: verum
end;
then A18: Sum (mlt ((@ p),(Col ((Rotation (i,j,n,r)),i)))) = ((mlt ((@ p),(Col ((Rotation (i,j,n,r)),i)))) /. i) + ((mlt ((@ p),(Col ((Rotation (i,j,n,r)),i)))) /. j) by A2, A4, A8, A11, MATRIX15:7;
A19: i in dom (mlt ((@ p),(Col ((Rotation (i,j,n,r)),i)))) by A8, A10, FINSEQ_1:def_3;
reconsider pii = (@ p) . i, pj = (@ p) . j as Element of F_Real ;
A20: (mlt ((@ p),(Col ((Rotation (i,j,n,r)),i)))) /. i = (mlt ((@ p),(Col ((Rotation (i,j,n,r)),i)))) . i by A8, A11, PARTFUN1:def_6
.= pii * ((Rotation (i,j,n,r)) * (i,i)) by A9, A19, FVSUM_1:60
.= (p . i) * (cos r) by A1, A2, A3, Def3 ;
(mlt ((@ p),(Col ((Rotation (i,j,n,r)),i)))) /. j = (mlt ((@ p),(Col ((Rotation (i,j,n,r)),i)))) . j by A4, A11, PARTFUN1:def_6
.= pj * ((Rotation (i,j,n,r)) * (j,i)) by A4, A7, A11, FVSUM_1:60
.= (p . j) * (- (sin r)) by A1, A2, A3, Def3 ;
hence (@ p) "*" (Col ((Rotation (i,j,n,r)),i)) = ((p . i) * (cos r)) + ((p . j) * (- (sin r))) by A18, A20; ::_thesis: verum
end;
theorem Th16: :: MATRTOP3:16
for r being real number
for i, j, n being Nat
for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n holds
(@ p) "*" (Col ((Rotation (i,j,n,r)),j)) = ((p . i) * (sin r)) + ((p . j) * (cos r))
proof
let r be real number ; ::_thesis: for i, j, n being Nat
for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n holds
(@ p) "*" (Col ((Rotation (i,j,n,r)),j)) = ((p . i) * (sin r)) + ((p . j) * (cos r))
let i, j, n be Nat; ::_thesis: for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n holds
(@ p) "*" (Col ((Rotation (i,j,n,r)),j)) = ((p . i) * (sin r)) + ((p . j) * (cos r))
let p be Point of (TOP-REAL n); ::_thesis: ( 1 <= i & i < j & j <= n implies (@ p) "*" (Col ((Rotation (i,j,n,r)),j)) = ((p . i) * (sin r)) + ((p . j) * (cos r)) )
assume that
A1: 1 <= i and
A2: i < j and
A3: j <= n ; ::_thesis: (@ p) "*" (Col ((Rotation (i,j,n,r)),j)) = ((p . i) * (sin r)) + ((p . j) * (cos r))
set O = Rotation (i,j,n,r);
set C = Col ((Rotation (i,j,n,r)),j);
set S = Seg n;
1 <= j by A1, A2, XXREAL_0:2;
then A4: j in Seg n by A3, FINSEQ_1:1;
A5: len (Rotation (i,j,n,r)) = n by MATRIX_1:25;
then A6: dom (Rotation (i,j,n,r)) = Seg n by FINSEQ_1:def_3;
then A7: (Col ((Rotation (i,j,n,r)),j)) . j = (Rotation (i,j,n,r)) * (j,j) by A4, MATRIX_1:def_8;
A8: i <= n by A2, A3, XXREAL_0:2;
then A9: i in Seg n by A1, FINSEQ_1:1;
then A10: (Col ((Rotation (i,j,n,r)),j)) . i = (Rotation (i,j,n,r)) * (i,j) by A6, MATRIX_1:def_8;
( len p = n & len (Col ((Rotation (i,j,n,r)),j)) = n ) by A5, CARD_1:def_7;
then len (mlt ((@ p),(Col ((Rotation (i,j,n,r)),j)))) = n by MATRIX_3:6;
then A11: dom (mlt ((@ p),(Col ((Rotation (i,j,n,r)),j)))) = Seg n by FINSEQ_1:def_3;
then A12: i in dom (mlt ((@ p),(Col ((Rotation (i,j,n,r)),j)))) by A1, A8, FINSEQ_1:1;
A13: Indices (Rotation (i,j,n,r)) = [:(Seg n),(Seg n):] by MATRIX_1:24;
for k being Nat st k in dom (mlt ((@ p),(Col ((Rotation (i,j,n,r)),j)))) & k <> i & k <> j holds
(mlt ((@ p),(Col ((Rotation (i,j,n,r)),j)))) . k = 0. F_Real
proof
let k be Nat; ::_thesis: ( k in dom (mlt ((@ p),(Col ((Rotation (i,j,n,r)),j)))) & k <> i & k <> j implies (mlt ((@ p),(Col ((Rotation (i,j,n,r)),j)))) . k = 0. F_Real )
assume that
A14: k in dom (mlt ((@ p),(Col ((Rotation (i,j,n,r)),j)))) and
A15: k <> i and
A16: k <> j ; ::_thesis: (mlt ((@ p),(Col ((Rotation (i,j,n,r)),j)))) . k = 0. F_Real
not k in {i,j} by A15, A16, TARSKI:def_2;
then A17: {k,j} <> {i,j} by TARSKI:def_2;
reconsider pk = (@ p) . k as Element of F_Real ;
A18: [k,j] in Indices (Rotation (i,j,n,r)) by A4, A11, A13, A14, ZFMISC_1:87;
(Col ((Rotation (i,j,n,r)),j)) . k = (Rotation (i,j,n,r)) * (k,j) by A6, A11, A14, MATRIX_1:def_8;
hence (mlt ((@ p),(Col ((Rotation (i,j,n,r)),j)))) . k = pk * ((Rotation (i,j,n,r)) * (k,j)) by A14, FVSUM_1:60
.= pk * (0. F_Real) by A1, A2, A3, A16, A17, A18, Def3
.= 0. F_Real ;
::_thesis: verum
end;
then A19: Sum (mlt ((@ p),(Col ((Rotation (i,j,n,r)),j)))) = ((mlt ((@ p),(Col ((Rotation (i,j,n,r)),j)))) /. i) + ((mlt ((@ p),(Col ((Rotation (i,j,n,r)),j)))) /. j) by A2, A4, A9, A11, MATRIX15:7;
reconsider pii = (@ p) . i, pj = (@ p) . j as Element of F_Real ;
A20: (mlt ((@ p),(Col ((Rotation (i,j,n,r)),j)))) /. i = (mlt ((@ p),(Col ((Rotation (i,j,n,r)),j)))) . i by A9, A11, PARTFUN1:def_6
.= pii * ((Rotation (i,j,n,r)) * (i,j)) by A10, A12, FVSUM_1:60
.= (p . i) * (sin r) by A1, A2, A3, Def3 ;
(mlt ((@ p),(Col ((Rotation (i,j,n,r)),j)))) /. j = (mlt ((@ p),(Col ((Rotation (i,j,n,r)),j)))) . j by A4, A11, PARTFUN1:def_6
.= pj * ((Rotation (i,j,n,r)) * (j,j)) by A4, A7, A11, FVSUM_1:60
.= (p . j) * (cos r) by A1, A2, A3, Def3 ;
hence (@ p) "*" (Col ((Rotation (i,j,n,r)),j)) = ((p . i) * (sin r)) + ((p . j) * (cos r)) by A19, A20; ::_thesis: verum
end;
theorem Th17: :: MATRTOP3:17
for r1, r2 being real number
for i, j, n being Nat st 1 <= i & i < j & j <= n holds
(Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2)) = Rotation (i,j,n,(r1 + r2))
proof
let r1, r2 be real number ; ::_thesis: for i, j, n being Nat st 1 <= i & i < j & j <= n holds
(Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2)) = Rotation (i,j,n,(r1 + r2))
let i, j, n be Nat; ::_thesis: ( 1 <= i & i < j & j <= n implies (Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2)) = Rotation (i,j,n,(r1 + r2)) )
assume that
A1: 1 <= i and
A2: i < j and
A3: j <= n ; ::_thesis: (Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2)) = Rotation (i,j,n,(r1 + r2))
set S = Seg n;
1 <= j by A1, A2, XXREAL_0:2;
then A4: j in Seg n by A3, FINSEQ_1:1;
set O1 = Rotation (i,j,n,r1);
set O2 = Rotation (i,j,n,r2);
set O = Rotation (i,j,n,(r1 + r2));
set O12 = (Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2));
A5: width (Rotation (i,j,n,r1)) = n by MATRIX_1:24;
i <= n by A2, A3, XXREAL_0:2;
then A6: i in Seg n by A1, FINSEQ_1:1;
A7: Indices (Rotation (i,j,n,r1)) = [:(Seg n),(Seg n):] by MATRIX_1:24;
A8: Indices ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) = [:(Seg n),(Seg n):] by MATRIX_1:24;
A9: Indices (Rotation (i,j,n,(r1 + r2))) = [:(Seg n),(Seg n):] by MATRIX_1:24;
A10: len (Rotation (i,j,n,r2)) = n by MATRIX_1:25;
for k, m being Nat st [k,m] in Indices ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) holds
((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = (Rotation (i,j,n,(r1 + r2))) * (k,m)
proof
let k, m be Nat; ::_thesis: ( [k,m] in Indices ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) implies ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = (Rotation (i,j,n,(r1 + r2))) * (k,m) )
assume A11: [k,m] in Indices ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) ; ::_thesis: ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = (Rotation (i,j,n,(r1 + r2))) * (k,m)
then A12: k in Seg n by A8, ZFMISC_1:87;
A13: ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = (Line ((Rotation (i,j,n,r1)),k)) "*" (Col ((Rotation (i,j,n,r2)),m)) by A5, A10, A11, MATRIX_3:def_4;
len (@ (Line ((Rotation (i,j,n,r1)),k))) = n by A5, CARD_1:def_7;
then reconsider L = @ (Line ((Rotation (i,j,n,r1)),k)) as Element of (TOP-REAL n) by TOPREAL3:46;
A14: m in Seg n by A8, A11, ZFMISC_1:87;
then A15: L . m = (Rotation (i,j,n,r1)) * (k,m) by A5, MATRIX_1:def_7;
A16: @ L = Line ((Rotation (i,j,n,r1)),k) ;
A17: L . i = (Rotation (i,j,n,r1)) * (k,i) by A5, A6, MATRIX_1:def_7;
A18: L . j = (Rotation (i,j,n,r1)) * (k,j) by A4, A5, MATRIX_1:def_7;
percases ( m = i or m = j or ( m <> i & m <> j ) ) ;
supposeA19: m = i ; ::_thesis: ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = (Rotation (i,j,n,(r1 + r2))) * (k,m)
then A20: (Line ((Rotation (i,j,n,r1)),k)) "*" (Col ((Rotation (i,j,n,r2)),m)) = ((L . i) * (cos r2)) + ((L . j) * (- (sin r2))) by A1, A2, A3, A16, Th15;
percases ( k = i or k = j or ( k <> j & k <> i ) ) ;
supposeA21: k = i ; ::_thesis: ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = (Rotation (i,j,n,(r1 + r2))) * (k,m)
hence ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = ((cos r1) * (cos r2)) + ((L . j) * (- (sin r2))) by A1, A2, A3, A13, A17, A20, Def3
.= ((cos r1) * (cos r2)) + ((sin r1) * (- (sin r2))) by A1, A2, A3, A18, A21, Def3
.= ((cos r1) * (cos r2)) - ((sin r1) * (sin r2))
.= cos (r1 + r2) by SIN_COS:75
.= (Rotation (i,j,n,(r1 + r2))) * (k,m) by A1, A2, A3, A19, A21, Def3 ;
::_thesis: verum
end;
supposeA22: k = j ; ::_thesis: ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = (Rotation (i,j,n,(r1 + r2))) * (k,m)
hence ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = ((- (sin r1)) * (cos r2)) + ((L . j) * (- (sin r2))) by A1, A2, A3, A13, A17, A20, Def3
.= ((- (sin r1)) * (cos r2)) + ((cos r1) * (- (sin r2))) by A1, A2, A3, A18, A22, Def3
.= - (((sin r1) * (cos r2)) + ((cos r1) * (sin r2)))
.= - (sin (r1 + r2)) by SIN_COS:75
.= (Rotation (i,j,n,(r1 + r2))) * (k,m) by A1, A2, A3, A19, A22, Def3 ;
::_thesis: verum
end;
suppose ( k <> j & k <> i ) ; ::_thesis: ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = (Rotation (i,j,n,(r1 + r2))) * (k,m)
then not k in {i,j} by TARSKI:def_2;
then A23: ( {k,i} <> {i,j} & {k,j} <> {i,j} ) by TARSKI:def_2;
A24: [k,j] in [:(Seg n),(Seg n):] by A4, A12, ZFMISC_1:87;
A25: [k,i] in [:(Seg n),(Seg n):] by A6, A12, ZFMISC_1:87;
then (Rotation (i,j,n,r1)) * (k,i) = 0. F_Real by A1, A2, A3, A7, A23, Def3;
hence ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = (0 * (cos r2)) + (0 * (- (sin r2))) by A1, A2, A3, A7, A13, A17, A18, A20, A23, A24, Def3
.= (Rotation (i,j,n,(r1 + r2))) * (k,m) by A1, A2, A3, A9, A19, A23, A25, Def3 ;
::_thesis: verum
end;
end;
end;
supposeA26: m = j ; ::_thesis: ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = (Rotation (i,j,n,(r1 + r2))) * (k,m)
then A27: (Line ((Rotation (i,j,n,r1)),k)) "*" (Col ((Rotation (i,j,n,r2)),m)) = ((L . i) * (sin r2)) + ((L . j) * (cos r2)) by A1, A2, A3, A16, Th16;
percases ( k = i or k = j or ( k <> j & k <> i ) ) ;
supposeA28: k = i ; ::_thesis: ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = (Rotation (i,j,n,(r1 + r2))) * (k,m)
hence ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = ((cos r1) * (sin r2)) + ((L . j) * (cos r2)) by A1, A2, A3, A13, A17, A27, Def3
.= ((cos r1) * (sin r2)) + ((sin r1) * (cos r2)) by A1, A2, A3, A18, A28, Def3
.= sin (r1 + r2) by SIN_COS:75
.= (Rotation (i,j,n,(r1 + r2))) * (k,m) by A1, A2, A3, A26, A28, Def3 ;
::_thesis: verum
end;
supposeA29: k = j ; ::_thesis: ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = (Rotation (i,j,n,(r1 + r2))) * (k,m)
hence ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = ((- (sin r1)) * (sin r2)) + ((L . j) * (cos r2)) by A1, A2, A3, A13, A17, A27, Def3
.= ((cos r1) * (cos r2)) - ((sin r1) * (sin r2)) by A1, A2, A3, A18, A29, Def3
.= cos (r1 + r2) by SIN_COS:75
.= (Rotation (i,j,n,(r1 + r2))) * (k,m) by A1, A2, A3, A26, A29, Def3 ;
::_thesis: verum
end;
suppose ( k <> j & k <> i ) ; ::_thesis: ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = (Rotation (i,j,n,(r1 + r2))) * (k,m)
then not k in {i,j} by TARSKI:def_2;
then A30: ( {k,i} <> {i,j} & {k,j} <> {i,j} ) by TARSKI:def_2;
A31: [k,j] in [:(Seg n),(Seg n):] by A4, A12, ZFMISC_1:87;
[k,i] in [:(Seg n),(Seg n):] by A6, A12, ZFMISC_1:87;
then (Rotation (i,j,n,r1)) * (k,i) = 0. F_Real by A1, A2, A3, A7, A30, Def3;
hence ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = (0 * (sin r2)) + (0 * (cos r2)) by A1, A2, A3, A7, A13, A17, A18, A27, A30, A31, Def3
.= (Rotation (i,j,n,(r1 + r2))) * (k,m) by A1, A2, A3, A9, A26, A30, A31, Def3 ;
::_thesis: verum
end;
end;
end;
supposeA32: ( m <> i & m <> j ) ; ::_thesis: ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = (Rotation (i,j,n,(r1 + r2))) * (k,m)
then A33: ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = L . m by A1, A2, A3, A13, A14, A16, Th14;
A34: [k,m] in [:(Seg n),(Seg n):] by A11, MATRIX_1:24;
percases ( k = m or k <> m ) ;
supposeA35: k = m ; ::_thesis: ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = (Rotation (i,j,n,(r1 + r2))) * (k,m)
then (Rotation (i,j,n,r1)) * (k,m) = 1. F_Real by A1, A2, A3, A7, A8, A11, A32, Def3;
hence ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = (Rotation (i,j,n,(r1 + r2))) * (k,m) by A1, A2, A3, A9, A15, A32, A33, A34, A35, Def3; ::_thesis: verum
end;
supposeA36: k <> m ; ::_thesis: ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = (Rotation (i,j,n,(r1 + r2))) * (k,m)
not m in {i,j} by A32, TARSKI:def_2;
then A37: {k,m} <> {i,j} by TARSKI:def_2;
then (Rotation (i,j,n,r1)) * (k,m) = 0. F_Real by A1, A2, A3, A7, A8, A11, A36, Def3;
hence ((Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2))) * (k,m) = (Rotation (i,j,n,(r1 + r2))) * (k,m) by A1, A2, A3, A9, A15, A33, A34, A36, A37, Def3; ::_thesis: verum
end;
end;
end;
end;
end;
hence (Rotation (i,j,n,r1)) * (Rotation (i,j,n,r2)) = Rotation (i,j,n,(r1 + r2)) by MATRIX_1:27; ::_thesis: verum
end;
Lm4: for r being real number
for i, j, n being Nat st 1 <= i & i < j & j <= n holds
(Rotation (i,j,n,r)) @ = Rotation (i,j,n,(- r))
proof
let r be real number ; ::_thesis: for i, j, n being Nat st 1 <= i & i < j & j <= n holds
(Rotation (i,j,n,r)) @ = Rotation (i,j,n,(- r))
let i, j, n be Nat; ::_thesis: ( 1 <= i & i < j & j <= n implies (Rotation (i,j,n,r)) @ = Rotation (i,j,n,(- r)) )
set O1 = Rotation (i,j,n,r);
set O2 = Rotation (i,j,n,(- r));
set S = Seg n;
assume A1: ( 1 <= i & i < j & j <= n ) ; ::_thesis: (Rotation (i,j,n,r)) @ = Rotation (i,j,n,(- r))
A2: Indices (Rotation (i,j,n,(- r))) = [:(Seg n),(Seg n):] by MATRIX_1:24;
A3: Indices (Rotation (i,j,n,r)) = [:(Seg n),(Seg n):] by MATRIX_1:24;
A4: Indices ((Rotation (i,j,n,r)) @) = [:(Seg n),(Seg n):] by MATRIX_1:24;
for k, m being Nat st [k,m] in [:(Seg n),(Seg n):] holds
((Rotation (i,j,n,r)) @) * (k,m) = (Rotation (i,j,n,(- r))) * (k,m)
proof
A5: cos r = cos (- r) by SIN_COS:31;
let k, m be Nat; ::_thesis: ( [k,m] in [:(Seg n),(Seg n):] implies ((Rotation (i,j,n,r)) @) * (k,m) = (Rotation (i,j,n,(- r))) * (k,m) )
A6: - (sin r) = sin (- r) by SIN_COS:31;
assume A7: [k,m] in [:(Seg n),(Seg n):] ; ::_thesis: ((Rotation (i,j,n,r)) @) * (k,m) = (Rotation (i,j,n,(- r))) * (k,m)
then A8: [m,k] in [:(Seg n),(Seg n):] by A3, A4, MATRIX_1:def_6;
then A9: ((Rotation (i,j,n,r)) @) * (k,m) = (Rotation (i,j,n,r)) * (m,k) by A3, MATRIX_1:def_6;
percases ( ( k = m & k = i ) or ( k = m & k = j ) or ( k = m & k <> i & k <> j ) or ( k <> m & k = i & m = j ) or ( k <> m & k = i & m <> j ) or ( k <> m & k = j & m = i ) or ( k <> m & k = j & m <> i ) or ( k <> m & k <> j & k <> i ) or ( k <> m & m <> j & m <> i ) ) ;
supposeA10: ( k = m & k = i ) ; ::_thesis: ((Rotation (i,j,n,r)) @) * (k,m) = (Rotation (i,j,n,(- r))) * (k,m)
then (Rotation (i,j,n,r)) * (m,k) = cos r by A1, Def3;
hence ((Rotation (i,j,n,r)) @) * (k,m) = (Rotation (i,j,n,(- r))) * (k,m) by A1, A5, A9, A10, Def3; ::_thesis: verum
end;
supposeA11: ( k = m & k = j ) ; ::_thesis: ((Rotation (i,j,n,r)) @) * (k,m) = (Rotation (i,j,n,(- r))) * (k,m)
then (Rotation (i,j,n,r)) * (m,k) = cos r by A1, Def3;
hence ((Rotation (i,j,n,r)) @) * (k,m) = (Rotation (i,j,n,(- r))) * (k,m) by A1, A5, A9, A11, Def3; ::_thesis: verum
end;
supposeA12: ( k = m & k <> i & k <> j ) ; ::_thesis: ((Rotation (i,j,n,r)) @) * (k,m) = (Rotation (i,j,n,(- r))) * (k,m)
then (Rotation (i,j,n,r)) * (m,k) = 1. F_Real by A1, A3, A7, Def3;
hence ((Rotation (i,j,n,r)) @) * (k,m) = (Rotation (i,j,n,(- r))) * (k,m) by A1, A2, A7, A9, A12, Def3; ::_thesis: verum
end;
supposeA13: ( k <> m & k = i & m = j ) ; ::_thesis: ((Rotation (i,j,n,r)) @) * (k,m) = (Rotation (i,j,n,(- r))) * (k,m)
then (Rotation (i,j,n,r)) * (m,k) = - (sin r) by A1, Def3;
hence ((Rotation (i,j,n,r)) @) * (k,m) = (Rotation (i,j,n,(- r))) * (k,m) by A1, A6, A9, A13, Def3; ::_thesis: verum
end;
supposeA14: ( k <> m & k = i & m <> j ) ; ::_thesis: ((Rotation (i,j,n,r)) @) * (k,m) = (Rotation (i,j,n,(- r))) * (k,m)
then not m in {i,j} by TARSKI:def_2;
then A15: {m,k} <> {i,j} by TARSKI:def_2;
then (Rotation (i,j,n,r)) * (m,k) = 0. F_Real by A1, A3, A8, A14, Def3;
hence ((Rotation (i,j,n,r)) @) * (k,m) = (Rotation (i,j,n,(- r))) * (k,m) by A1, A2, A7, A9, A14, A15, Def3; ::_thesis: verum
end;
supposeA16: ( k <> m & k = j & m = i ) ; ::_thesis: ((Rotation (i,j,n,r)) @) * (k,m) = (Rotation (i,j,n,(- r))) * (k,m)
then (Rotation (i,j,n,(- r))) * (k,m) = - (sin (- r)) by A1, Def3;
hence ((Rotation (i,j,n,r)) @) * (k,m) = (Rotation (i,j,n,(- r))) * (k,m) by A1, A6, A9, A16, Def3; ::_thesis: verum
end;
supposeA17: ( k <> m & k = j & m <> i ) ; ::_thesis: ((Rotation (i,j,n,r)) @) * (k,m) = (Rotation (i,j,n,(- r))) * (k,m)
then not m in {i,j} by TARSKI:def_2;
then A18: {m,k} <> {i,j} by TARSKI:def_2;
then (Rotation (i,j,n,r)) * (m,k) = 0. F_Real by A1, A3, A8, A17, Def3;
hence ((Rotation (i,j,n,r)) @) * (k,m) = (Rotation (i,j,n,(- r))) * (k,m) by A1, A2, A7, A9, A17, A18, Def3; ::_thesis: verum
end;
supposeA19: ( k <> m & k <> j & k <> i ) ; ::_thesis: ((Rotation (i,j,n,r)) @) * (k,m) = (Rotation (i,j,n,(- r))) * (k,m)
then not k in {i,j} by TARSKI:def_2;
then A20: {m,k} <> {i,j} by TARSKI:def_2;
then (Rotation (i,j,n,r)) * (m,k) = 0. F_Real by A1, A3, A8, A19, Def3;
hence ((Rotation (i,j,n,r)) @) * (k,m) = (Rotation (i,j,n,(- r))) * (k,m) by A1, A2, A7, A9, A19, A20, Def3; ::_thesis: verum
end;
supposeA21: ( k <> m & m <> j & m <> i ) ; ::_thesis: ((Rotation (i,j,n,r)) @) * (k,m) = (Rotation (i,j,n,(- r))) * (k,m)
then not m in {i,j} by TARSKI:def_2;
then A22: {m,k} <> {i,j} by TARSKI:def_2;
then (Rotation (i,j,n,r)) * (m,k) = 0. F_Real by A1, A3, A8, A21, Def3;
hence ((Rotation (i,j,n,r)) @) * (k,m) = (Rotation (i,j,n,(- r))) * (k,m) by A1, A2, A7, A9, A21, A22, Def3; ::_thesis: verum
end;
end;
end;
hence (Rotation (i,j,n,r)) @ = Rotation (i,j,n,(- r)) by A4, MATRIX_1:27; ::_thesis: verum
end;
theorem Th18: :: MATRTOP3:18
for i, j, n being Nat st 1 <= i & i < j & j <= n holds
Rotation (i,j,n,0) = 1. (F_Real,n)
proof
let i, j, n be Nat; ::_thesis: ( 1 <= i & i < j & j <= n implies Rotation (i,j,n,0) = 1. (F_Real,n) )
set O = Rotation (i,j,n,0);
assume A1: ( 1 <= i & i < j & j <= n ) ; ::_thesis: Rotation (i,j,n,0) = 1. (F_Real,n)
A2: for k, m being Nat st [k,m] in Indices (Rotation (i,j,n,0)) & k <> m holds
(Rotation (i,j,n,0)) * (k,m) = 0. F_Real
proof
let k, m be Nat; ::_thesis: ( [k,m] in Indices (Rotation (i,j,n,0)) & k <> m implies (Rotation (i,j,n,0)) * (k,m) = 0. F_Real )
assume that
A3: [k,m] in Indices (Rotation (i,j,n,0)) and
A4: k <> m ; ::_thesis: (Rotation (i,j,n,0)) * (k,m) = 0. F_Real
percases ( ( k = i & m = j ) or ( k = j & m = i ) or ( k = i & m <> j ) or ( k = j & m <> i ) or ( m = i & k <> j ) or ( m = j & k <> i ) or ( k <> i & k <> j & m <> i & m <> j ) ) ;
suppose ( ( k = i & m = j ) or ( k = j & m = i ) ) ; ::_thesis: (Rotation (i,j,n,0)) * (k,m) = 0. F_Real
then ( (Rotation (i,j,n,0)) * (k,m) = sin 0 or (Rotation (i,j,n,0)) * (k,m) = - (sin 0) ) by A1, Def3;
hence (Rotation (i,j,n,0)) * (k,m) = 0. F_Real by SIN_COS:31; ::_thesis: verum
end;
suppose ( k = i & m <> j ) ; ::_thesis: (Rotation (i,j,n,0)) * (k,m) = 0. F_Real
then not m in {i,j} by A4, TARSKI:def_2;
then {k,m} <> {i,j} by TARSKI:def_2;
hence (Rotation (i,j,n,0)) * (k,m) = 0. F_Real by A1, A3, A4, Def3; ::_thesis: verum
end;
suppose ( k = j & m <> i ) ; ::_thesis: (Rotation (i,j,n,0)) * (k,m) = 0. F_Real
then not m in {i,j} by A4, TARSKI:def_2;
then {k,m} <> {i,j} by TARSKI:def_2;
hence (Rotation (i,j,n,0)) * (k,m) = 0. F_Real by A1, A3, A4, Def3; ::_thesis: verum
end;
suppose ( m = i & k <> j ) ; ::_thesis: (Rotation (i,j,n,0)) * (k,m) = 0. F_Real
then not k in {i,j} by A4, TARSKI:def_2;
then {k,m} <> {i,j} by TARSKI:def_2;
hence (Rotation (i,j,n,0)) * (k,m) = 0. F_Real by A1, A3, A4, Def3; ::_thesis: verum
end;
suppose ( m = j & k <> i ) ; ::_thesis: (Rotation (i,j,n,0)) * (k,m) = 0. F_Real
then not k in {i,j} by A4, TARSKI:def_2;
then {k,m} <> {i,j} by TARSKI:def_2;
hence (Rotation (i,j,n,0)) * (k,m) = 0. F_Real by A1, A3, A4, Def3; ::_thesis: verum
end;
suppose ( k <> i & k <> j & m <> i & m <> j ) ; ::_thesis: (Rotation (i,j,n,0)) * (k,m) = 0. F_Real
then not m in {i,j} by TARSKI:def_2;
then {k,m} <> {i,j} by TARSKI:def_2;
hence (Rotation (i,j,n,0)) * (k,m) = 0. F_Real by A1, A3, A4, Def3; ::_thesis: verum
end;
end;
end;
for k being Nat st [k,k] in Indices (Rotation (i,j,n,0)) holds
(Rotation (i,j,n,0)) * (k,k) = 1. F_Real
proof
let k be Nat; ::_thesis: ( [k,k] in Indices (Rotation (i,j,n,0)) implies (Rotation (i,j,n,0)) * (k,k) = 1. F_Real )
assume A5: [k,k] in Indices (Rotation (i,j,n,0)) ; ::_thesis: (Rotation (i,j,n,0)) * (k,k) = 1. F_Real
( k = i or k = j or ( k <> i & k <> j ) ) ;
hence (Rotation (i,j,n,0)) * (k,k) = 1. F_Real by A1, A5, Def3, SIN_COS:31; ::_thesis: verum
end;
hence Rotation (i,j,n,0) = 1. (F_Real,n) by A2, MATRIX_1:def_11; ::_thesis: verum
end;
Lm5: for r being real number
for i, j, n being Nat st 1 <= i & i < j & j <= n holds
(Rotation (i,j,n,r)) ~ = Rotation (i,j,n,(- r))
proof
let r be real number ; ::_thesis: for i, j, n being Nat st 1 <= i & i < j & j <= n holds
(Rotation (i,j,n,r)) ~ = Rotation (i,j,n,(- r))
let i, j, n be Nat; ::_thesis: ( 1 <= i & i < j & j <= n implies (Rotation (i,j,n,r)) ~ = Rotation (i,j,n,(- r)) )
assume A1: ( 1 <= i & i < j & j <= n ) ; ::_thesis: (Rotation (i,j,n,r)) ~ = Rotation (i,j,n,(- r))
then A2: (Rotation (i,j,n,r)) * (Rotation (i,j,n,(- r))) = Rotation (i,j,n,(r + (- r))) by Th17
.= 1. (F_Real,n) by A1, Th18 ;
(Rotation (i,j,n,(- r))) * (Rotation (i,j,n,r)) = Rotation (i,j,n,((- r) + r)) by A1, Th17
.= 1. (F_Real,n) by A1, Th18 ;
then Rotation (i,j,n,r) is_reverse_of Rotation (i,j,n,(- r)) by A2, MATRIX_6:def_2;
hence (Rotation (i,j,n,r)) ~ = Rotation (i,j,n,(- r)) by MATRIX_6:def_4; ::_thesis: verum
end;
theorem Th19: :: MATRTOP3:19
for r being real number
for i, j, n being Nat st 1 <= i & i < j & j <= n holds
( Rotation (i,j,n,r) is Orthogonal & (Rotation (i,j,n,r)) ~ = Rotation (i,j,n,(- r)) )
proof
let r be real number ; ::_thesis: for i, j, n being Nat st 1 <= i & i < j & j <= n holds
( Rotation (i,j,n,r) is Orthogonal & (Rotation (i,j,n,r)) ~ = Rotation (i,j,n,(- r)) )
let i, j, n be Nat; ::_thesis: ( 1 <= i & i < j & j <= n implies ( Rotation (i,j,n,r) is Orthogonal & (Rotation (i,j,n,r)) ~ = Rotation (i,j,n,(- r)) ) )
assume ( 1 <= i & i < j & j <= n ) ; ::_thesis: ( Rotation (i,j,n,r) is Orthogonal & (Rotation (i,j,n,r)) ~ = Rotation (i,j,n,(- r)) )
then ( (Rotation (i,j,n,r)) ~ = Rotation (i,j,n,(- r)) & (Rotation (i,j,n,r)) @ = Rotation (i,j,n,(- r)) ) by Lm4, Lm5;
hence ( Rotation (i,j,n,r) is Orthogonal & (Rotation (i,j,n,r)) ~ = Rotation (i,j,n,(- r)) ) by MATRIX_6:def_7; ::_thesis: verum
end;
theorem Th20: :: MATRTOP3:20
for r being real number
for i, j, n, k being Nat
for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n & k <> i & k <> j holds
((Mx2Tran (Rotation (i,j,n,r))) . p) . k = p . k
proof
let r be real number ; ::_thesis: for i, j, n, k being Nat
for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n & k <> i & k <> j holds
((Mx2Tran (Rotation (i,j,n,r))) . p) . k = p . k
let i, j, n, k be Nat; ::_thesis: for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n & k <> i & k <> j holds
((Mx2Tran (Rotation (i,j,n,r))) . p) . k = p . k
let p be Point of (TOP-REAL n); ::_thesis: ( 1 <= i & i < j & j <= n & k <> i & k <> j implies ((Mx2Tran (Rotation (i,j,n,r))) . p) . k = p . k )
set O = Rotation (i,j,n,r);
set M = Mx2Tran (Rotation (i,j,n,r));
set Mp = (Mx2Tran (Rotation (i,j,n,r))) . p;
set S = Seg n;
assume A1: ( 1 <= i & i < j & j <= n & k <> i & k <> j ) ; ::_thesis: ((Mx2Tran (Rotation (i,j,n,r))) . p) . k = p . k
len ((Mx2Tran (Rotation (i,j,n,r))) . p) = n by CARD_1:def_7;
then A2: dom ((Mx2Tran (Rotation (i,j,n,r))) . p) = Seg n by FINSEQ_1:def_3;
len p = n by CARD_1:def_7;
then A3: dom p = Seg n by FINSEQ_1:def_3;
percases ( k in Seg n or not k in Seg n ) ;
supposeA4: k in Seg n ; ::_thesis: ((Mx2Tran (Rotation (i,j,n,r))) . p) . k = p . k
then ( 1 <= k & k <= n ) by FINSEQ_1:1;
hence ((Mx2Tran (Rotation (i,j,n,r))) . p) . k = (@ p) "*" (Col ((Rotation (i,j,n,r)),k)) by MATRTOP1:18
.= p . k by A1, A4, Th14 ;
::_thesis: verum
end;
supposeA5: not k in Seg n ; ::_thesis: ((Mx2Tran (Rotation (i,j,n,r))) . p) . k = p . k
then ((Mx2Tran (Rotation (i,j,n,r))) . p) . k = {} by A2, FUNCT_1:def_2;
hence ((Mx2Tran (Rotation (i,j,n,r))) . p) . k = p . k by A3, A5, FUNCT_1:def_2; ::_thesis: verum
end;
end;
end;
theorem Th21: :: MATRTOP3:21
for r being real number
for i, j, n being Nat
for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n holds
((Mx2Tran (Rotation (i,j,n,r))) . p) . i = ((p . i) * (cos r)) + ((p . j) * (- (sin r)))
proof
let r be real number ; ::_thesis: for i, j, n being Nat
for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n holds
((Mx2Tran (Rotation (i,j,n,r))) . p) . i = ((p . i) * (cos r)) + ((p . j) * (- (sin r)))
let i, j, n be Nat; ::_thesis: for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n holds
((Mx2Tran (Rotation (i,j,n,r))) . p) . i = ((p . i) * (cos r)) + ((p . j) * (- (sin r)))
let p be Point of (TOP-REAL n); ::_thesis: ( 1 <= i & i < j & j <= n implies ((Mx2Tran (Rotation (i,j,n,r))) . p) . i = ((p . i) * (cos r)) + ((p . j) * (- (sin r))) )
set O = Rotation (i,j,n,r);
set M = Mx2Tran (Rotation (i,j,n,r));
set Mp = (Mx2Tran (Rotation (i,j,n,r))) . p;
set S = Seg n;
assume that
A1: 1 <= i and
A2: ( i < j & j <= n ) ; ::_thesis: ((Mx2Tran (Rotation (i,j,n,r))) . p) . i = ((p . i) * (cos r)) + ((p . j) * (- (sin r)))
i <= n by A2, XXREAL_0:2;
hence ((Mx2Tran (Rotation (i,j,n,r))) . p) . i = (@ p) "*" (Col ((Rotation (i,j,n,r)),i)) by A1, MATRTOP1:18
.= ((p . i) * (cos r)) + ((p . j) * (- (sin r))) by A1, A2, Th15 ;
::_thesis: verum
end;
theorem Th22: :: MATRTOP3:22
for r being real number
for i, j, n being Nat
for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n holds
((Mx2Tran (Rotation (i,j,n,r))) . p) . j = ((p . i) * (sin r)) + ((p . j) * (cos r))
proof
let r be real number ; ::_thesis: for i, j, n being Nat
for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n holds
((Mx2Tran (Rotation (i,j,n,r))) . p) . j = ((p . i) * (sin r)) + ((p . j) * (cos r))
let i, j, n be Nat; ::_thesis: for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n holds
((Mx2Tran (Rotation (i,j,n,r))) . p) . j = ((p . i) * (sin r)) + ((p . j) * (cos r))
let p be Point of (TOP-REAL n); ::_thesis: ( 1 <= i & i < j & j <= n implies ((Mx2Tran (Rotation (i,j,n,r))) . p) . j = ((p . i) * (sin r)) + ((p . j) * (cos r)) )
set O = Rotation (i,j,n,r);
set M = Mx2Tran (Rotation (i,j,n,r));
set Mp = (Mx2Tran (Rotation (i,j,n,r))) . p;
set S = Seg n;
assume that
A1: ( 1 <= i & i < j ) and
A2: j <= n ; ::_thesis: ((Mx2Tran (Rotation (i,j,n,r))) . p) . j = ((p . i) * (sin r)) + ((p . j) * (cos r))
1 <= j by A1, XXREAL_0:2;
hence ((Mx2Tran (Rotation (i,j,n,r))) . p) . j = (@ p) "*" (Col ((Rotation (i,j,n,r)),j)) by A2, MATRTOP1:18
.= ((p . i) * (sin r)) + ((p . j) * (cos r)) by A1, A2, Th16 ;
::_thesis: verum
end;
theorem Th23: :: MATRTOP3:23
for r being real number
for i, j, n being Nat
for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n holds
(Mx2Tran (Rotation (i,j,n,r))) . p = ((((p | (i -' 1)) ^ <*(((p . i) * (cos r)) + ((p . j) * (- (sin r))))*>) ^ ((p /^ i) | ((j -' i) -' 1))) ^ <*(((p . i) * (sin r)) + ((p . j) * (cos r)))*>) ^ (p /^ j)
proof
let r be real number ; ::_thesis: for i, j, n being Nat
for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n holds
(Mx2Tran (Rotation (i,j,n,r))) . p = ((((p | (i -' 1)) ^ <*(((p . i) * (cos r)) + ((p . j) * (- (sin r))))*>) ^ ((p /^ i) | ((j -' i) -' 1))) ^ <*(((p . i) * (sin r)) + ((p . j) * (cos r)))*>) ^ (p /^ j)
let i, j, n be Nat; ::_thesis: for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n holds
(Mx2Tran (Rotation (i,j,n,r))) . p = ((((p | (i -' 1)) ^ <*(((p . i) * (cos r)) + ((p . j) * (- (sin r))))*>) ^ ((p /^ i) | ((j -' i) -' 1))) ^ <*(((p . i) * (sin r)) + ((p . j) * (cos r)))*>) ^ (p /^ j)
let p be Point of (TOP-REAL n); ::_thesis: ( 1 <= i & i < j & j <= n implies (Mx2Tran (Rotation (i,j,n,r))) . p = ((((p | (i -' 1)) ^ <*(((p . i) * (cos r)) + ((p . j) * (- (sin r))))*>) ^ ((p /^ i) | ((j -' i) -' 1))) ^ <*(((p . i) * (sin r)) + ((p . j) * (cos r)))*>) ^ (p /^ j) )
assume that
A1: 1 <= i and
A2: i < j and
A3: j <= n ; ::_thesis: (Mx2Tran (Rotation (i,j,n,r))) . p = ((((p | (i -' 1)) ^ <*(((p . i) * (cos r)) + ((p . j) * (- (sin r))))*>) ^ ((p /^ i) | ((j -' i) -' 1))) ^ <*(((p . i) * (sin r)) + ((p . j) * (cos r)))*>) ^ (p /^ j)
set M = Mx2Tran (Rotation (i,j,n,r));
set Mp = (Mx2Tran (Rotation (i,j,n,r))) . p;
set i1 = i -' 1;
set ji = j -' i;
A4: i < n by A2, A3, XXREAL_0:2;
A5: ( i -' 1 < (i -' 1) + 1 & i -' 1 = i - 1 ) by A1, NAT_1:13, XREAL_1:233;
A6: now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_k_<=_i_-'_1_holds_
(((Mx2Tran_(Rotation_(i,j,n,r)))_._p)_|_(i_-'_1))_._k_=_(p_|_(i_-'_1))_._k
let k be Nat; ::_thesis: ( 1 <= k & k <= i -' 1 implies (((Mx2Tran (Rotation (i,j,n,r))) . p) | (i -' 1)) . k = (p | (i -' 1)) . k )
assume that
1 <= k and
A7: k <= i -' 1 ; ::_thesis: (((Mx2Tran (Rotation (i,j,n,r))) . p) | (i -' 1)) . k = (p | (i -' 1)) . k
A8: ( (((Mx2Tran (Rotation (i,j,n,r))) . p) | (i -' 1)) . k = ((Mx2Tran (Rotation (i,j,n,r))) . p) . k & (p | (i -' 1)) . k = p . k ) by A7, FINSEQ_3:112;
k < i by A5, A7, XXREAL_0:2;
hence (((Mx2Tran (Rotation (i,j,n,r))) . p) | (i -' 1)) . k = (p | (i -' 1)) . k by A1, A2, A3, A8, Th20; ::_thesis: verum
end;
A9: len ((Mx2Tran (Rotation (i,j,n,r))) . p) = n by CARD_1:def_7;
then A10: len (((Mx2Tran (Rotation (i,j,n,r))) . p) | (i -' 1)) = i -' 1 by A5, A4, FINSEQ_1:59, XXREAL_0:2;
A11: len p = n by CARD_1:def_7;
then A12: len (p /^ i) = n - i by A4, RFINSEQ:def_1;
A13: j -' i = j - i by A2, XREAL_1:233;
then A14: ( (j -' i) -' 1 < ((j -' i) -' 1) + 1 & j -' i <= n - i ) by A3, NAT_1:13, XREAL_1:6;
j - i > i - i by A2, XREAL_1:8;
then A15: (j -' i) -' 1 = (j -' i) - 1 by A13, NAT_1:14, XREAL_1:233;
A16: len (p /^ j) = n - j by A3, A11, RFINSEQ:def_1;
A17: len (((Mx2Tran (Rotation (i,j,n,r))) . p) /^ i) = n - i by A9, A4, RFINSEQ:def_1;
then A18: len ((((Mx2Tran (Rotation (i,j,n,r))) . p) /^ i) | ((j -' i) -' 1)) = (j -' i) -' 1 by A14, A15, FINSEQ_1:59, XXREAL_0:2;
A19: (j -' i) -' 1 < n - i by A14, A15, XXREAL_0:2;
A20: now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_k_<=_(j_-'_i)_-'_1_holds_
((((Mx2Tran_(Rotation_(i,j,n,r)))_._p)_/^_i)_|_((j_-'_i)_-'_1))_._k_=_((p_/^_i)_|_((j_-'_i)_-'_1))_._k
let k be Nat; ::_thesis: ( 1 <= k & k <= (j -' i) -' 1 implies ((((Mx2Tran (Rotation (i,j,n,r))) . p) /^ i) | ((j -' i) -' 1)) . k = ((p /^ i) | ((j -' i) -' 1)) . k )
assume that
A21: 1 <= k and
A22: k <= (j -' i) -' 1 ; ::_thesis: ((((Mx2Tran (Rotation (i,j,n,r))) . p) /^ i) | ((j -' i) -' 1)) . k = ((p /^ i) | ((j -' i) -' 1)) . k
A23: ((p /^ i) | ((j -' i) -' 1)) . k = (p /^ i) . k by A22, FINSEQ_3:112;
A24: k <= n - i by A19, A22, XXREAL_0:2;
then k in dom (((Mx2Tran (Rotation (i,j,n,r))) . p) /^ i) by A17, A21, FINSEQ_3:25;
then A25: (((Mx2Tran (Rotation (i,j,n,r))) . p) /^ i) . k = ((Mx2Tran (Rotation (i,j,n,r))) . p) . (i + k) by A9, A4, RFINSEQ:def_1;
k < ((j -' i) -' 1) + 1 by A22, NAT_1:13;
then A26: i + k < i + (j -' i) by A15, XREAL_1:8;
k in dom (p /^ i) by A12, A21, A24, FINSEQ_3:25;
then A27: (p /^ i) . k = p . (i + k) by A11, A4, RFINSEQ:def_1;
( k + i <> i & ((((Mx2Tran (Rotation (i,j,n,r))) . p) /^ i) | ((j -' i) -' 1)) . k = (((Mx2Tran (Rotation (i,j,n,r))) . p) /^ i) . k ) by A21, A22, FINSEQ_3:112, NAT_1:14;
hence ((((Mx2Tran (Rotation (i,j,n,r))) . p) /^ i) | ((j -' i) -' 1)) . k = ((p /^ i) | ((j -' i) -' 1)) . k by A1, A2, A3, A13, A25, A26, A27, A23, Th20; ::_thesis: verum
end;
len ((p /^ i) | ((j -' i) -' 1)) = (j -' i) -' 1 by A14, A15, A12, FINSEQ_1:59, XXREAL_0:2;
then A28: (((Mx2Tran (Rotation (i,j,n,r))) . p) /^ i) | ((j -' i) -' 1) = (p /^ i) | ((j -' i) -' 1) by A20, A18, FINSEQ_1:14;
A29: len (((Mx2Tran (Rotation (i,j,n,r))) . p) /^ j) = n - j by A3, A9, RFINSEQ:def_1;
now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_k_<=_n_-_j_holds_
(((Mx2Tran_(Rotation_(i,j,n,r)))_._p)_/^_j)_._k_=_(p_/^_j)_._k
let k be Nat; ::_thesis: ( 1 <= k & k <= n - j implies (((Mx2Tran (Rotation (i,j,n,r))) . p) /^ j) . k = (p /^ j) . k )
assume that
A30: 1 <= k and
A31: k <= n - j ; ::_thesis: (((Mx2Tran (Rotation (i,j,n,r))) . p) /^ j) . k = (p /^ j) . k
k in dom (((Mx2Tran (Rotation (i,j,n,r))) . p) /^ j) by A29, A30, A31, FINSEQ_3:25;
then A32: (((Mx2Tran (Rotation (i,j,n,r))) . p) /^ j) . k = ((Mx2Tran (Rotation (i,j,n,r))) . p) . (j + k) by A3, A9, RFINSEQ:def_1;
k in dom (p /^ j) by A16, A30, A31, FINSEQ_3:25;
then A33: (p /^ j) . k = p . (j + k) by A3, A11, RFINSEQ:def_1;
( j + k >= j & j + k <> j ) by A30, NAT_1:11, NAT_1:14;
hence (((Mx2Tran (Rotation (i,j,n,r))) . p) /^ j) . k = (p /^ j) . k by A1, A2, A3, A32, A33, Th20; ::_thesis: verum
end;
then A34: ((Mx2Tran (Rotation (i,j,n,r))) . p) /^ j = p /^ j by A16, A29, FINSEQ_1:14;
len (p | (i -' 1)) = i -' 1 by A5, A11, A4, FINSEQ_1:59, XXREAL_0:2;
then A35: ((Mx2Tran (Rotation (i,j,n,r))) . p) | (i -' 1) = p | (i -' 1) by A6, A10, FINSEQ_1:14;
A36: ((Mx2Tran (Rotation (i,j,n,r))) . p) . i = ((p . i) * (cos r)) + ((p . j) * (- (sin r))) by A1, A2, A3, Th21;
(Mx2Tran (Rotation (i,j,n,r))) . p = @ (@ ((Mx2Tran (Rotation (i,j,n,r))) . p)) ;
then (Mx2Tran (Rotation (i,j,n,r))) . p = ((((((Mx2Tran (Rotation (i,j,n,r))) . p) | (i -' 1)) ^ <*(((Mx2Tran (Rotation (i,j,n,r))) . p) . i)*>) ^ ((((Mx2Tran (Rotation (i,j,n,r))) . p) /^ i) | ((j -' i) -' 1))) ^ <*(((Mx2Tran (Rotation (i,j,n,r))) . p) . j)*>) ^ (((Mx2Tran (Rotation (i,j,n,r))) . p) /^ j) by A1, A2, A3, A9, FINSEQ_7:1;
hence (Mx2Tran (Rotation (i,j,n,r))) . p = ((((p | (i -' 1)) ^ <*(((p . i) * (cos r)) + ((p . j) * (- (sin r))))*>) ^ ((p /^ i) | ((j -' i) -' 1))) ^ <*(((p . i) * (sin r)) + ((p . j) * (cos r)))*>) ^ (p /^ j) by A1, A2, A3, A34, A28, A35, A36, Th22; ::_thesis: verum
end;
theorem Th24: :: MATRTOP3:24
for s being real number
for i, j, n being Nat
for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n & s ^2 <= ((p . i) ^2) + ((p . j) ^2) holds
ex r being real number st ((Mx2Tran (Rotation (i,j,n,r))) . p) . i = s
proof
let s be real number ; ::_thesis: for i, j, n being Nat
for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n & s ^2 <= ((p . i) ^2) + ((p . j) ^2) holds
ex r being real number st ((Mx2Tran (Rotation (i,j,n,r))) . p) . i = s
let i, j, n be Nat; ::_thesis: for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n & s ^2 <= ((p . i) ^2) + ((p . j) ^2) holds
ex r being real number st ((Mx2Tran (Rotation (i,j,n,r))) . p) . i = s
let p be Point of (TOP-REAL n); ::_thesis: ( 1 <= i & i < j & j <= n & s ^2 <= ((p . i) ^2) + ((p . j) ^2) implies ex r being real number st ((Mx2Tran (Rotation (i,j,n,r))) . p) . i = s )
set pk = p . i;
set pj = p . j;
set pkj = ((p . i) ^2) + ((p . j) ^2);
set P = sqrt (((p . i) ^2) + ((p . j) ^2));
assume that
A1: ( 1 <= i & i < j & j <= n ) and
A2: s ^2 <= ((p . i) ^2) + ((p . j) ^2) ; ::_thesis: ex r being real number st ((Mx2Tran (Rotation (i,j,n,r))) . p) . i = s
A3: ( 0 <= (p . i) * (p . i) & 0 <= (p . j) * (p . j) ) by XREAL_1:63;
percases ( p . i <> 0 or p . j <> 0 or ( p . i = 0 & p . j = 0 ) ) ;
supposeA4: ( p . i <> 0 or p . j <> 0 ) ; ::_thesis: ex r being real number st ((Mx2Tran (Rotation (i,j,n,r))) . p) . i = s
A5: 0 <= (p . i) * (p . i) by XREAL_1:63;
A6: sqrt (((p . i) ^2) + ((p . j) ^2)) > 0 by A3, A4, SQUARE_1:25;
then A7: (s / (sqrt (((p . i) ^2) + ((p . j) ^2)))) * (sqrt (((p . i) ^2) + ((p . j) ^2))) = s by XCMPLX_1:87;
A8: 0 <= (p . j) * (p . j) by XREAL_1:63;
then ((p . i) ^2) + 0 <= ((p . i) ^2) + ((p . j) ^2) by XREAL_1:6;
then A9: sqrt ((p . i) ^2) <= sqrt (((p . i) ^2) + ((p . j) ^2)) by A5, SQUARE_1:26;
now__::_thesis:_(p_._i)_/_(sqrt_(((p_._i)_^2)_+_((p_._j)_^2)))_in_[.(-_1),1.]
percases ( p . i >= 0 or p . i <= 0 ) ;
supposeA10: p . i >= 0 ; ::_thesis: (p . i) / (sqrt (((p . i) ^2) + ((p . j) ^2))) in [.(- 1),1.]
then A11: p . i <= sqrt (((p . i) ^2) + ((p . j) ^2)) by A9, SQUARE_1:22;
then (p . i) / (sqrt (((p . i) ^2) + ((p . j) ^2))) <= 1 by A6, XREAL_1:185;
hence (p . i) / (sqrt (((p . i) ^2) + ((p . j) ^2))) in [.(- 1),1.] by A10, A11, XXREAL_1:1; ::_thesis: verum
end;
supposeA12: p . i <= 0 ; ::_thesis: (p . i) / (sqrt (((p . i) ^2) + ((p . j) ^2))) in [.(- 1),1.]
then - (p . i) <= sqrt (((p . i) ^2) + ((p . j) ^2)) by A9, SQUARE_1:23;
then - 1 <= (p . i) / (sqrt (((p . i) ^2) + ((p . j) ^2))) by A6, XREAL_1:194;
hence (p . i) / (sqrt (((p . i) ^2) + ((p . j) ^2))) in [.(- 1),1.] by A6, A12, XXREAL_1:1; ::_thesis: verum
end;
end;
end;
then consider x being set such that
A13: x in dom sin and
x in [.(- (PI / 2)),(PI / 2).] and
A14: sin . x = (p . i) / (sqrt (((p . i) ^2) + ((p . j) ^2))) by FUNCT_1:def_6, SIN_COS6:45;
A15: (sqrt (((p . i) ^2) + ((p . j) ^2))) * (sqrt (((p . i) ^2) + ((p . j) ^2))) = (sqrt (((p . i) ^2) + ((p . j) ^2))) ^2
.= ((p . i) ^2) + ((p . j) ^2) by A5, A8, SQUARE_1:def_2 ;
0 <= s * s by XREAL_1:63;
then A16: sqrt (s ^2) <= sqrt (((p . i) ^2) + ((p . j) ^2)) by A2, SQUARE_1:26;
now__::_thesis:_s_/_(sqrt_(((p_._i)_^2)_+_((p_._j)_^2)))_in_[.(-_1),1.]
percases ( s >= 0 or s <= 0 ) ;
supposeA17: s >= 0 ; ::_thesis: s / (sqrt (((p . i) ^2) + ((p . j) ^2))) in [.(- 1),1.]
then A18: s <= sqrt (((p . i) ^2) + ((p . j) ^2)) by A16, SQUARE_1:22;
then s / (sqrt (((p . i) ^2) + ((p . j) ^2))) <= 1 by A6, XREAL_1:185;
hence s / (sqrt (((p . i) ^2) + ((p . j) ^2))) in [.(- 1),1.] by A17, A18, XXREAL_1:1; ::_thesis: verum
end;
supposeA19: s <= 0 ; ::_thesis: s / (sqrt (((p . i) ^2) + ((p . j) ^2))) in [.(- 1),1.]
then - s <= sqrt (((p . i) ^2) + ((p . j) ^2)) by A16, SQUARE_1:23;
then - 1 <= s / (sqrt (((p . i) ^2) + ((p . j) ^2))) by A6, XREAL_1:194;
hence s / (sqrt (((p . i) ^2) + ((p . j) ^2))) in [.(- 1),1.] by A6, A19, XXREAL_1:1; ::_thesis: verum
end;
end;
end;
then consider y being set such that
y in dom sin and
A20: y in [.(- (PI / 2)),(PI / 2).] and
A21: sin . y = s / (sqrt (((p . i) ^2) + ((p . j) ^2))) by FUNCT_1:def_6, SIN_COS6:45;
reconsider y = y as Real by A20;
A22: ((sqrt (((p . i) ^2) + ((p . j) ^2))) * (p . i)) / (sqrt (((p . i) ^2) + ((p . j) ^2))) = p . i by A6, XCMPLX_1:89;
reconsider x = x as Real by A13, FUNCT_2:def_1;
A23: sin . x = sin x by SIN_COS:def_17;
((p . i) / (sqrt (((p . i) ^2) + ((p . j) ^2)))) * ((p . i) / (sqrt (((p . i) ^2) + ((p . j) ^2)))) = ((p . i) * (p . i)) / (((p . i) ^2) + ((p . j) ^2)) by A15, XCMPLX_1:76;
then ((cos x) * (cos x)) + (((p . i) * (p . i)) / (((p . i) ^2) + ((p . j) ^2))) = 1 by A14, A23, SIN_COS:29;
then (cos x) * (cos x) = 1 - (((p . i) * (p . i)) / (((p . i) ^2) + ((p . j) ^2)))
.= ((((p . i) ^2) + ((p . j) ^2)) / (((p . i) ^2) + ((p . j) ^2))) - (((p . i) * (p . i)) / (((p . i) ^2) + ((p . j) ^2))) by A3, A4, XCMPLX_1:60
.= ((p . j) * (p . j)) / (((p . i) ^2) + ((p . j) ^2))
.= ((p . j) / (sqrt (((p . i) ^2) + ((p . j) ^2)))) ^2 by A15, XCMPLX_1:76 ;
then A24: (cos x) ^2 = ((p . j) / (sqrt (((p . i) ^2) + ((p . j) ^2)))) ^2 ;
percases ( cos x = (p . j) / (sqrt (((p . i) ^2) + ((p . j) ^2))) or cos x = - ((p . j) / (sqrt (((p . i) ^2) + ((p . j) ^2)))) ) by A24, SQUARE_1:40;
supposeA25: cos x = (p . j) / (sqrt (((p . i) ^2) + ((p . j) ^2))) ; ::_thesis: ex r being real number st ((Mx2Tran (Rotation (i,j,n,r))) . p) . i = s
take r = x - y; ::_thesis: ((Mx2Tran (Rotation (i,j,n,r))) . p) . i = s
- (sin y) = sin ((- x) + r) by SIN_COS:31
.= ((sin (- x)) * (cos r)) + ((cos (- x)) * (sin r)) by SIN_COS:75
.= ((- (sin x)) * (cos r)) + ((cos (- x)) * (sin r)) by SIN_COS:31
.= (- ((sin x) * (cos r))) + ((cos x) * (sin r)) by SIN_COS:31
.= (- ((sin x) * (cos r))) + (- ((cos x) * (- (sin r)))) ;
then sin y = ((sin x) * (cos r)) + ((cos x) * (- (sin r))) ;
hence s = (sqrt (((p . i) ^2) + ((p . j) ^2))) * ((((p . i) / (sqrt (((p . i) ^2) + ((p . j) ^2)))) * (cos r)) + (((p . j) / (sqrt (((p . i) ^2) + ((p . j) ^2)))) * (- (sin r)))) by A7, A14, A21, A23, A25, SIN_COS:def_17
.= ((((sqrt (((p . i) ^2) + ((p . j) ^2))) * (p . i)) / (sqrt (((p . i) ^2) + ((p . j) ^2)))) * (cos r)) + ((((sqrt (((p . i) ^2) + ((p . j) ^2))) * (p . j)) / (sqrt (((p . i) ^2) + ((p . j) ^2)))) * (- (sin r)))
.= ((p . i) * (cos r)) + ((p . j) * (- (sin r))) by A6, A22, XCMPLX_1:89
.= ((Mx2Tran (Rotation (i,j,n,r))) . p) . i by A1, Th21 ;
::_thesis: verum
end;
supposeA26: cos x = - ((p . j) / (sqrt (((p . i) ^2) + ((p . j) ^2)))) ; ::_thesis: ex r being real number st ((Mx2Tran (Rotation (i,j,n,r))) . p) . i = s
take r = y - x; ::_thesis: ((Mx2Tran (Rotation (i,j,n,r))) . p) . i = s
sin y = sin (x + r)
.= ((sin x) * (cos r)) + ((cos x) * (sin r)) by SIN_COS:75
.= (((p . i) / (sqrt (((p . i) ^2) + ((p . j) ^2)))) * (cos r)) + (((p . j) / (sqrt (((p . i) ^2) + ((p . j) ^2)))) * (- (sin r))) by A14, A26, SIN_COS:def_17 ;
hence s = (sqrt (((p . i) ^2) + ((p . j) ^2))) * ((((p . i) / (sqrt (((p . i) ^2) + ((p . j) ^2)))) * (cos r)) + (((p . j) / (sqrt (((p . i) ^2) + ((p . j) ^2)))) * (- (sin r)))) by A7, A21, SIN_COS:def_17
.= ((((sqrt (((p . i) ^2) + ((p . j) ^2))) * (p . i)) / (sqrt (((p . i) ^2) + ((p . j) ^2)))) * (cos r)) + ((((sqrt (((p . i) ^2) + ((p . j) ^2))) * (p . j)) / (sqrt (((p . i) ^2) + ((p . j) ^2)))) * (- (sin r)))
.= ((p . i) * (cos r)) + ((p . j) * (- (sin r))) by A6, A22, XCMPLX_1:89
.= ((Mx2Tran (Rotation (i,j,n,r))) . p) . i by A1, Th21 ;
::_thesis: verum
end;
end;
end;
supposeA27: ( p . i = 0 & p . j = 0 ) ; ::_thesis: ex r being real number st ((Mx2Tran (Rotation (i,j,n,r))) . p) . i = s
take r = 0 ; ::_thesis: ((Mx2Tran (Rotation (i,j,n,r))) . p) . i = s
set M = Mx2Tran (Rotation (i,j,n,r));
Mx2Tran (Rotation (i,j,n,r)) = Mx2Tran (1. (F_Real,n)) by A1, Th18;
then A28: Mx2Tran (Rotation (i,j,n,r)) = id (TOP-REAL n) by MATRTOP1:33;
s = 0 by A2, A27, XREAL_1:63;
hence ((Mx2Tran (Rotation (i,j,n,r))) . p) . i = s by A27, A28, FUNCT_1:17; ::_thesis: verum
end;
end;
end;
Lm6: for r being real number
for i, j, n being Nat
for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n holds
((((Mx2Tran (Rotation (i,j,n,r))) . p) . i) * (((Mx2Tran (Rotation (i,j,n,r))) . p) . i)) + ((((Mx2Tran (Rotation (i,j,n,r))) . p) . j) * (((Mx2Tran (Rotation (i,j,n,r))) . p) . j)) = ((p . i) * (p . i)) + ((p . j) * (p . j))
proof
let r be real number ; ::_thesis: for i, j, n being Nat
for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n holds
((((Mx2Tran (Rotation (i,j,n,r))) . p) . i) * (((Mx2Tran (Rotation (i,j,n,r))) . p) . i)) + ((((Mx2Tran (Rotation (i,j,n,r))) . p) . j) * (((Mx2Tran (Rotation (i,j,n,r))) . p) . j)) = ((p . i) * (p . i)) + ((p . j) * (p . j))
let i, j, n be Nat; ::_thesis: for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n holds
((((Mx2Tran (Rotation (i,j,n,r))) . p) . i) * (((Mx2Tran (Rotation (i,j,n,r))) . p) . i)) + ((((Mx2Tran (Rotation (i,j,n,r))) . p) . j) * (((Mx2Tran (Rotation (i,j,n,r))) . p) . j)) = ((p . i) * (p . i)) + ((p . j) * (p . j))
let p be Point of (TOP-REAL n); ::_thesis: ( 1 <= i & i < j & j <= n implies ((((Mx2Tran (Rotation (i,j,n,r))) . p) . i) * (((Mx2Tran (Rotation (i,j,n,r))) . p) . i)) + ((((Mx2Tran (Rotation (i,j,n,r))) . p) . j) * (((Mx2Tran (Rotation (i,j,n,r))) . p) . j)) = ((p . i) * (p . i)) + ((p . j) * (p . j)) )
set pk = p . i;
set pj = p . j;
set pkj = ((p . i) ^2) + ((p . j) ^2);
set M = Mx2Tran (Rotation (i,j,n,r));
set S = sin r;
set C = cos r;
set Mp = (Mx2Tran (Rotation (i,j,n,r))) . p;
A1: ((cos r) * (cos r)) + ((sin r) * (sin r)) = 1 by SIN_COS:29;
assume A2: ( 1 <= i & i < j & j <= n ) ; ::_thesis: ((((Mx2Tran (Rotation (i,j,n,r))) . p) . i) * (((Mx2Tran (Rotation (i,j,n,r))) . p) . i)) + ((((Mx2Tran (Rotation (i,j,n,r))) . p) . j) * (((Mx2Tran (Rotation (i,j,n,r))) . p) . j)) = ((p . i) * (p . i)) + ((p . j) * (p . j))
then ((Mx2Tran (Rotation (i,j,n,r))) . p) . i = ((p . i) * (cos r)) + ((p . j) * (- (sin r))) by Th21;
then A3: (((Mx2Tran (Rotation (i,j,n,r))) . p) . i) * (((Mx2Tran (Rotation (i,j,n,r))) . p) . i) = (((((p . i) * (p . i)) * (cos r)) * (cos r)) - ((((2 * (p . i)) * (p . j)) * (cos r)) * (sin r))) + ((((p . j) * (p . j)) * (sin r)) * (sin r)) ;
((Mx2Tran (Rotation (i,j,n,r))) . p) . j = ((p . i) * (sin r)) + ((p . j) * (cos r)) by A2, Th22;
then (((Mx2Tran (Rotation (i,j,n,r))) . p) . j) * (((Mx2Tran (Rotation (i,j,n,r))) . p) . j) = (((((p . i) * (p . i)) * (sin r)) * (sin r)) + ((2 * ((p . i) * (p . j))) * ((cos r) * (sin r)))) + (((p . j) * (p . j)) * ((cos r) * (cos r))) ;
hence ((((Mx2Tran (Rotation (i,j,n,r))) . p) . i) * (((Mx2Tran (Rotation (i,j,n,r))) . p) . i)) + ((((Mx2Tran (Rotation (i,j,n,r))) . p) . j) * (((Mx2Tran (Rotation (i,j,n,r))) . p) . j)) = (((p . i) * (p . i)) * (((cos r) * (cos r)) + ((sin r) * (sin r)))) + (((p . j) * (p . j)) * (((cos r) * (cos r)) + ((sin r) * (sin r)))) by A3
.= ((p . i) * (p . i)) + ((p . j) * (p . j)) by A1 ;
::_thesis: verum
end;
theorem Th25: :: MATRTOP3:25
for s being real number
for i, j, n being Nat
for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n & s ^2 <= ((p . i) ^2) + ((p . j) ^2) holds
ex r being real number st ((Mx2Tran (Rotation (i,j,n,r))) . p) . j = s
proof
let s be real number ; ::_thesis: for i, j, n being Nat
for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n & s ^2 <= ((p . i) ^2) + ((p . j) ^2) holds
ex r being real number st ((Mx2Tran (Rotation (i,j,n,r))) . p) . j = s
let i, j, n be Nat; ::_thesis: for p being Point of (TOP-REAL n) st 1 <= i & i < j & j <= n & s ^2 <= ((p . i) ^2) + ((p . j) ^2) holds
ex r being real number st ((Mx2Tran (Rotation (i,j,n,r))) . p) . j = s
let p be Point of (TOP-REAL n); ::_thesis: ( 1 <= i & i < j & j <= n & s ^2 <= ((p . i) ^2) + ((p . j) ^2) implies ex r being real number st ((Mx2Tran (Rotation (i,j,n,r))) . p) . j = s )
set pk = p . i;
set pj = p . j;
set pkj = ((p . i) ^2) + ((p . j) ^2);
set ps = (((p . i) ^2) + ((p . j) ^2)) - (s ^2);
assume that
A1: ( 1 <= i & i < j & j <= n ) and
A2: s ^2 <= ((p . i) ^2) + ((p . j) ^2) ; ::_thesis: ex r being real number st ((Mx2Tran (Rotation (i,j,n,r))) . p) . j = s
0 <= s * s by XREAL_1:63;
then A3: (((p . i) ^2) + ((p . j) ^2)) - (s ^2) <= (((p . i) ^2) + ((p . j) ^2)) - 0 by XREAL_1:6;
A4: (s ^2) - (s ^2) <= (((p . i) ^2) + ((p . j) ^2)) - (s ^2) by A2, XREAL_1:6;
then (sqrt ((((p . i) ^2) + ((p . j) ^2)) - (s ^2))) ^2 = (((p . i) ^2) + ((p . j) ^2)) - (s ^2) by SQUARE_1:def_2;
then consider r being real number such that
A5: ((Mx2Tran (Rotation (i,j,n,r))) . p) . i = sqrt ((((p . i) ^2) + ((p . j) ^2)) - (s ^2)) by A1, A3, Th24;
set M = Mx2Tran (Rotation (i,j,n,r));
set Mp = (Mx2Tran (Rotation (i,j,n,r))) . p;
((p . i) ^2) + ((p . j) ^2) = ((sqrt ((((p . i) ^2) + ((p . j) ^2)) - (s ^2))) ^2) + ((((Mx2Tran (Rotation (i,j,n,r))) . p) . j) * (((Mx2Tran (Rotation (i,j,n,r))) . p) . j)) by A5, A1, Lm6
.= ((((p . i) ^2) + ((p . j) ^2)) - (s ^2)) + ((((Mx2Tran (Rotation (i,j,n,r))) . p) . j) * (((Mx2Tran (Rotation (i,j,n,r))) . p) . j)) by A4, SQUARE_1:def_2 ;
then A6: s ^2 = (((Mx2Tran (Rotation (i,j,n,r))) . p) . j) ^2 ;
percases ( ((Mx2Tran (Rotation (i,j,n,r))) . p) . j = s or ((Mx2Tran (Rotation (i,j,n,r))) . p) . j = - s ) by A6, SQUARE_1:40;
suppose ((Mx2Tran (Rotation (i,j,n,r))) . p) . j = s ; ::_thesis: ex r being real number st ((Mx2Tran (Rotation (i,j,n,r))) . p) . j = s
hence ex r being real number st ((Mx2Tran (Rotation (i,j,n,r))) . p) . j = s ; ::_thesis: verum
end;
supposeA7: ((Mx2Tran (Rotation (i,j,n,r))) . p) . j = - s ; ::_thesis: ex r being real number st ((Mx2Tran (Rotation (i,j,n,r))) . p) . j = s
take R = r + PI; ::_thesis: ((Mx2Tran (Rotation (i,j,n,R))) . p) . j = s
thus ((Mx2Tran (Rotation (i,j,n,R))) . p) . j = ((p . i) * (sin R)) + ((p . j) * (cos R)) by A1, Th22
.= ((p . i) * (- (sin r))) + ((p . j) * (cos R)) by SIN_COS:79
.= ((p . i) * (- (sin r))) + ((p . j) * (- (cos r))) by SIN_COS:79
.= - (((p . i) * (sin r)) + ((p . j) * (cos r)))
.= - (((Mx2Tran (Rotation (i,j,n,r))) . p) . j) by A1, Th22
.= s by A7 ; ::_thesis: verum
end;
end;
end;
theorem Th26: :: MATRTOP3:26
for r being real number
for i, j, n being Nat st 1 <= i & i < j & j <= n holds
Mx2Tran (Rotation (i,j,n,r)) is {i,j} -support-yielding
proof
let r be real number ; ::_thesis: for i, j, n being Nat st 1 <= i & i < j & j <= n holds
Mx2Tran (Rotation (i,j,n,r)) is {i,j} -support-yielding
let i, j, n be Nat; ::_thesis: ( 1 <= i & i < j & j <= n implies Mx2Tran (Rotation (i,j,n,r)) is {i,j} -support-yielding )
set M = Mx2Tran (Rotation (i,j,n,r));
assume A1: ( 1 <= i & i < j & j <= n ) ; ::_thesis: Mx2Tran (Rotation (i,j,n,r)) is {i,j} -support-yielding
let f be Function; :: according to MATRTOP3:def_1 ::_thesis: for x being set st f in dom (Mx2Tran (Rotation (i,j,n,r))) & ((Mx2Tran (Rotation (i,j,n,r))) . f) . x <> f . x holds
x in {i,j}
let x be set ; ::_thesis: ( f in dom (Mx2Tran (Rotation (i,j,n,r))) & ((Mx2Tran (Rotation (i,j,n,r))) . f) . x <> f . x implies x in {i,j} )
assume that
A2: f in dom (Mx2Tran (Rotation (i,j,n,r))) and
A3: ((Mx2Tran (Rotation (i,j,n,r))) . f) . x <> f . x ; ::_thesis: x in {i,j}
reconsider p = f as Point of (TOP-REAL n) by A2, FUNCT_2:52;
len p = n by CARD_1:def_7;
then A4: dom p = Seg n by FINSEQ_1:def_3;
len ((Mx2Tran (Rotation (i,j,n,r))) . p) = n by CARD_1:def_7;
then A5: dom ((Mx2Tran (Rotation (i,j,n,r))) . p) = Seg n by FINSEQ_1:def_3;
percases ( not x in Seg n or x in Seg n ) ;
supposeA6: not x in Seg n ; ::_thesis: x in {i,j}
then ((Mx2Tran (Rotation (i,j,n,r))) . p) . x = {} by A5, FUNCT_1:def_2;
hence x in {i,j} by A3, A4, A6, FUNCT_1:def_2; ::_thesis: verum
end;
supposeA7: x in Seg n ; ::_thesis: x in {i,j}
((Mx2Tran (Rotation (i,j,n,r))) . p) . x <> p . x by A3;
then ( x = i or x = j ) by A1, A7, Th20;
hence x in {i,j} by TARSKI:def_2; ::_thesis: verum
end;
end;
end;
begin
definition
let n be Nat;
let f be Function of (TOP-REAL n),(TOP-REAL n);
attrf is rotation means :Def4: :: MATRTOP3:def 4
for p being Point of (TOP-REAL n) holds |.p.| = |.(f . p).|;
end;
:: deftheorem Def4 defines rotation MATRTOP3:def_4_:_
for n being Nat
for f being Function of (TOP-REAL n),(TOP-REAL n) holds
( f is rotation iff for p being Point of (TOP-REAL n) holds |.p.| = |.(f . p).| );
theorem Th27: :: MATRTOP3:27
for i, n being Nat st i in Seg n holds
Mx2Tran (AxialSymmetry (i,n)) is rotation
proof
let i, n be Nat; ::_thesis: ( i in Seg n implies Mx2Tran (AxialSymmetry (i,n)) is rotation )
set S = Seg n;
set M = Mx2Tran (AxialSymmetry (i,n));
assume A1: i in Seg n ; ::_thesis: Mx2Tran (AxialSymmetry (i,n)) is rotation
let p be Point of (TOP-REAL n); :: according to MATRTOP3:def_4 ::_thesis: |.p.| = |.((Mx2Tran (AxialSymmetry (i,n))) . p).|
len p = n by CARD_1:def_7;
then A2: i in dom p by A1, FINSEQ_1:def_3;
thus |.((Mx2Tran (AxialSymmetry (i,n))) . p).| = sqrt (Sum (sqr (p +* (i,(- (p . i)))))) by A1, Th10
.= sqrt (((Sum (sqr p)) - ((p . i) ^2)) + ((- (p . i)) ^2)) by A2, Th3
.= |.p.| ; ::_thesis: verum
end;
definition
let n be Nat;
let f be Function of (TOP-REAL n),(TOP-REAL n);
attrf is base_rotation means :Def5: :: MATRTOP3:def 5
ex F being FinSequence of (GFuncs the carrier of (TOP-REAL n)) st
( f = Product F & ( for k being Nat st k in dom F holds
ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) ) ) );
end;
:: deftheorem Def5 defines base_rotation MATRTOP3:def_5_:_
for n being Nat
for f being Function of (TOP-REAL n),(TOP-REAL n) holds
( f is base_rotation iff ex F being FinSequence of (GFuncs the carrier of (TOP-REAL n)) st
( f = Product F & ( for k being Nat st k in dom F holds
ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) ) ) ) );
registration
let n be Nat;
cluster id (TOP-REAL n) -> base_rotation ;
coherence
id (TOP-REAL n) is base_rotation
proof
set S = the carrier of (TOP-REAL n);
set G = GFuncs the carrier of (TOP-REAL n);
take F = <*> the carrier of (GFuncs the carrier of (TOP-REAL n)); :: according to MATRTOP3:def_5 ::_thesis: ( id (TOP-REAL n) = Product F & ( for k being Nat st k in dom F holds
ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) ) ) )
thus Product F = 1_ (GFuncs the carrier of (TOP-REAL n)) by GROUP_4:8
.= the_unity_wrt the multF of (GFuncs the carrier of (TOP-REAL n)) by GROUP_1:22
.= id (TOP-REAL n) by MONOID_0:75 ; ::_thesis: for k being Nat st k in dom F holds
ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) )
let k be Nat; ::_thesis: ( k in dom F implies ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) ) )
assume k in dom F ; ::_thesis: ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) )
hence ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) ) ; ::_thesis: verum
end;
end;
registration
let n be Nat;
cluster non empty Relation-like the carrier of (TOP-REAL n) -defined the carrier of (TOP-REAL n) -valued Function-like total quasi_total FinSequence-yielding Function-yielding V235() base_rotation for Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):];
existence
ex b1 being Function of (TOP-REAL n),(TOP-REAL n) st b1 is base_rotation
proof
id (TOP-REAL n) is base_rotation ;
hence ex b1 being Function of (TOP-REAL n),(TOP-REAL n) st b1 is base_rotation ; ::_thesis: verum
end;
end;
registration
let n be Nat;
let f, g be base_rotation Function of (TOP-REAL n),(TOP-REAL n);
clusterg (#) f -> base_rotation for Function of (TOP-REAL n),(TOP-REAL n);
coherence
for b1 being Function of (TOP-REAL n),(TOP-REAL n) st b1 = f * g holds
b1 is base_rotation
proof
consider F being FinSequence of (GFuncs the carrier of (TOP-REAL n)) such that
A1: f = Product F and
A2: for k being Nat st k in dom F holds
ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) ) by Def5;
consider G being FinSequence of (GFuncs the carrier of (TOP-REAL n)) such that
A3: g = Product G and
A4: for k being Nat st k in dom G holds
ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & G . k = Mx2Tran (Rotation (i,j,n,r)) ) by Def5;
f * g is base_rotation
proof
take GF = G ^ F; :: according to MATRTOP3:def_5 ::_thesis: ( f * g = Product GF & ( for k being Nat st k in dom GF holds
ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & GF . k = Mx2Tran (Rotation (i,j,n,r)) ) ) )
thus Product GF = (Product G) * (Product F) by GROUP_4:5
.= f * g by A3, A1, MONOID_0:70 ; ::_thesis: for k being Nat st k in dom GF holds
ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & GF . k = Mx2Tran (Rotation (i,j,n,r)) )
let k be Nat; ::_thesis: ( k in dom GF implies ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & GF . k = Mx2Tran (Rotation (i,j,n,r)) ) )
assume A5: k in dom GF ; ::_thesis: ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & GF . k = Mx2Tran (Rotation (i,j,n,r)) )
percases ( k in dom G or ex m being Nat st
( m in dom F & k = (len G) + m ) ) by A5, FINSEQ_1:25;
supposeA6: k in dom G ; ::_thesis: ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & GF . k = Mx2Tran (Rotation (i,j,n,r)) )
then G . k = GF . k by FINSEQ_1:def_7;
hence ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & GF . k = Mx2Tran (Rotation (i,j,n,r)) ) by A4, A6; ::_thesis: verum
end;
suppose ex m being Nat st
( m in dom F & k = (len G) + m ) ; ::_thesis: ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & GF . k = Mx2Tran (Rotation (i,j,n,r)) )
then consider m being Nat such that
A7: m in dom F and
A8: k = (len G) + m ;
GF . k = F . m by A7, A8, FINSEQ_1:def_7;
hence ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & GF . k = Mx2Tran (Rotation (i,j,n,r)) ) by A2, A7; ::_thesis: verum
end;
end;
end;
hence for b1 being Function of (TOP-REAL n),(TOP-REAL n) st b1 = f * g holds
b1 is base_rotation ; ::_thesis: verum
end;
end;
Lm7: for r being real number
for i, j, n being Nat st 1 <= i & i < j & j <= n holds
Mx2Tran (Rotation (i,j,n,r)) is rotation
proof
let r be real number ; ::_thesis: for i, j, n being Nat st 1 <= i & i < j & j <= n holds
Mx2Tran (Rotation (i,j,n,r)) is rotation
let i, j, n be Nat; ::_thesis: ( 1 <= i & i < j & j <= n implies Mx2Tran (Rotation (i,j,n,r)) is rotation )
assume A1: ( 1 <= i & i < j & j <= n ) ; ::_thesis: Mx2Tran (Rotation (i,j,n,r)) is rotation
let p be Point of (TOP-REAL n); :: according to MATRTOP3:def_4 ::_thesis: |.p.| = |.((Mx2Tran (Rotation (i,j,n,r))) . p).|
p = @ (@ p) ;
then reconsider P = p as FinSequence of REAL ;
set M = Mx2Tran (Rotation (i,j,n,r));
set Mp = (Mx2Tran (Rotation (i,j,n,r))) . p;
set p1 = P | (i -' 1);
set p2 = (P /^ i) | ((j -' i) -' 1);
set p3 = P /^ j;
set s = sin r;
set c = cos r;
reconsider pk = P . i, pj = P . j as Element of REAL ;
A2: sqr <*pk*> = <*(pk ^2)*> by RVSUM_1:55;
set pij1 = (pk * (cos r)) + (pj * (- (sin r)));
set pij2 = (pk * (sin r)) + (pj * (cos r));
A3: sqr <*((pk * (cos r)) + (pj * (- (sin r))))*> = <*(((pk * (cos r)) + (pj * (- (sin r)))) ^2)*> by RVSUM_1:55;
A4: sqr <*((pk * (sin r)) + (pj * (cos r)))*> = <*(((pk * (sin r)) + (pj * (cos r))) ^2)*> by RVSUM_1:55;
A5: sqr <*pj*> = <*(pj ^2)*> by RVSUM_1:55;
len p = n by CARD_1:def_7;
then p = ((((P | (i -' 1)) ^ <*pk*>) ^ ((P /^ i) | ((j -' i) -' 1))) ^ <*pj*>) ^ (P /^ j) by A1, FINSEQ_7:1;
then sqr p = (sqr ((((P | (i -' 1)) ^ <*pk*>) ^ ((P /^ i) | ((j -' i) -' 1))) ^ <*pj*>)) ^ (sqr (P /^ j)) by RVSUM_1:144
.= ((sqr (((P | (i -' 1)) ^ <*pk*>) ^ ((P /^ i) | ((j -' i) -' 1)))) ^ (sqr <*pj*>)) ^ (sqr (P /^ j)) by RVSUM_1:144
.= (((sqr ((P | (i -' 1)) ^ <*pk*>)) ^ (sqr ((P /^ i) | ((j -' i) -' 1)))) ^ (sqr <*pj*>)) ^ (sqr (P /^ j)) by RVSUM_1:144
.= ((((sqr (P | (i -' 1))) ^ (sqr <*pk*>)) ^ (sqr ((P /^ i) | ((j -' i) -' 1)))) ^ (sqr <*pj*>)) ^ (sqr (P /^ j)) by RVSUM_1:144 ;
then A6: Sum (sqr p) = (Sum ((((sqr (P | (i -' 1))) ^ (sqr <*pk*>)) ^ (sqr ((P /^ i) | ((j -' i) -' 1)))) ^ (sqr <*pj*>))) + (Sum (sqr (P /^ j))) by RVSUM_1:75
.= ((Sum (((sqr (P | (i -' 1))) ^ (sqr <*pk*>)) ^ (sqr ((P /^ i) | ((j -' i) -' 1))))) + (pj ^2)) + (Sum (sqr (P /^ j))) by A5, RVSUM_1:74
.= (((Sum ((sqr (P | (i -' 1))) ^ (sqr <*pk*>))) + (Sum (sqr ((P /^ i) | ((j -' i) -' 1))))) + (pj ^2)) + (Sum (sqr (P /^ j))) by RVSUM_1:75
.= ((((Sum (sqr (P | (i -' 1)))) + (pk ^2)) + (Sum (sqr ((P /^ i) | ((j -' i) -' 1))))) + (pj ^2)) + (Sum (sqr (P /^ j))) by A2, RVSUM_1:74 ;
A7: ((cos r) * (cos r)) + ((sin r) * (sin r)) = 1 by SIN_COS:29;
A8: (((pk * (cos r)) + (pj * (- (sin r)))) ^2) + (((pk * (sin r)) + (pj * (cos r))) ^2) = ((pk * pk) * (((cos r) * (cos r)) + ((sin r) * (sin r)))) + ((pj * pj) * (((cos r) * (cos r)) + ((sin r) * (sin r))))
.= (pk ^2) + (pj ^2) by A7 ;
(Mx2Tran (Rotation (i,j,n,r))) . p = ((((P | (i -' 1)) ^ <*((pk * (cos r)) + (pj * (- (sin r))))*>) ^ ((P /^ i) | ((j -' i) -' 1))) ^ <*((pk * (sin r)) + (pj * (cos r)))*>) ^ (P /^ j) by A1, Th23;
then sqr ((Mx2Tran (Rotation (i,j,n,r))) . p) = (sqr ((((P | (i -' 1)) ^ <*((pk * (cos r)) + (pj * (- (sin r))))*>) ^ ((P /^ i) | ((j -' i) -' 1))) ^ <*((pk * (sin r)) + (pj * (cos r)))*>)) ^ (sqr (P /^ j)) by RVSUM_1:144
.= ((sqr (((P | (i -' 1)) ^ <*((pk * (cos r)) + (pj * (- (sin r))))*>) ^ ((P /^ i) | ((j -' i) -' 1)))) ^ (sqr <*((pk * (sin r)) + (pj * (cos r)))*>)) ^ (sqr (P /^ j)) by RVSUM_1:144
.= (((sqr ((P | (i -' 1)) ^ <*((pk * (cos r)) + (pj * (- (sin r))))*>)) ^ (sqr ((P /^ i) | ((j -' i) -' 1)))) ^ (sqr <*((pk * (sin r)) + (pj * (cos r)))*>)) ^ (sqr (P /^ j)) by RVSUM_1:144
.= ((((sqr (P | (i -' 1))) ^ (sqr <*((pk * (cos r)) + (pj * (- (sin r))))*>)) ^ (sqr ((P /^ i) | ((j -' i) -' 1)))) ^ (sqr <*((pk * (sin r)) + (pj * (cos r)))*>)) ^ (sqr (P /^ j)) by RVSUM_1:144 ;
then Sum (sqr ((Mx2Tran (Rotation (i,j,n,r))) . p)) = (Sum ((((sqr (P | (i -' 1))) ^ (sqr <*((pk * (cos r)) + (pj * (- (sin r))))*>)) ^ (sqr ((P /^ i) | ((j -' i) -' 1)))) ^ (sqr <*((pk * (sin r)) + (pj * (cos r)))*>))) + (Sum (sqr (P /^ j))) by RVSUM_1:75
.= ((Sum (((sqr (P | (i -' 1))) ^ (sqr <*((pk * (cos r)) + (pj * (- (sin r))))*>)) ^ (sqr ((P /^ i) | ((j -' i) -' 1))))) + (((pk * (sin r)) + (pj * (cos r))) ^2)) + (Sum (sqr (P /^ j))) by A4, RVSUM_1:74
.= (((Sum ((sqr (P | (i -' 1))) ^ (sqr <*((pk * (cos r)) + (pj * (- (sin r))))*>))) + (Sum (sqr ((P /^ i) | ((j -' i) -' 1))))) + (((pk * (sin r)) + (pj * (cos r))) ^2)) + (Sum (sqr (P /^ j))) by RVSUM_1:75
.= ((((Sum (sqr (P | (i -' 1)))) + (((pk * (cos r)) + (pj * (- (sin r)))) ^2)) + (Sum (sqr ((P /^ i) | ((j -' i) -' 1))))) + (((pk * (sin r)) + (pj * (cos r))) ^2)) + (Sum (sqr (P /^ j))) by A3, RVSUM_1:74 ;
hence |.p.| = |.((Mx2Tran (Rotation (i,j,n,r))) . p).| by A6, A8; ::_thesis: verum
end;
theorem Th28: :: MATRTOP3:28
for r being real number
for i, j, n being Nat st 1 <= i & i < j & j <= n holds
Mx2Tran (Rotation (i,j,n,r)) is base_rotation
proof
let r be real number ; ::_thesis: for i, j, n being Nat st 1 <= i & i < j & j <= n holds
Mx2Tran (Rotation (i,j,n,r)) is base_rotation
let i, j, n be Nat; ::_thesis: ( 1 <= i & i < j & j <= n implies Mx2Tran (Rotation (i,j,n,r)) is base_rotation )
assume A1: ( 1 <= i & i < j & j <= n ) ; ::_thesis: Mx2Tran (Rotation (i,j,n,r)) is base_rotation
set S = the carrier of (TOP-REAL n);
set G = GFuncs the carrier of (TOP-REAL n);
reconsider M = Mx2Tran (Rotation (i,j,n,r)) as Element of (GFuncs the carrier of (TOP-REAL n)) by MONOID_0:73;
take F = <*M*>; :: according to MATRTOP3:def_5 ::_thesis: ( Mx2Tran (Rotation (i,j,n,r)) = Product F & ( for k being Nat st k in dom F holds
ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) ) ) )
thus Product F = Mx2Tran (Rotation (i,j,n,r)) by GROUP_4:9; ::_thesis: for k being Nat st k in dom F holds
ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) )
let k be Nat; ::_thesis: ( k in dom F implies ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) ) )
assume k in dom F ; ::_thesis: ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) )
then k in {1} by FINSEQ_1:2, FINSEQ_1:38;
then A2: k = 1 by TARSKI:def_1;
F . 1 = M by FINSEQ_1:40;
hence ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) ) by A1, A2; ::_thesis: verum
end;
Lm8: for n being Nat
for f, g being Function of (TOP-REAL n),(TOP-REAL n) st f is rotation & g is rotation holds
f * g is rotation
proof
let n be Nat; ::_thesis: for f, g being Function of (TOP-REAL n),(TOP-REAL n) st f is rotation & g is rotation holds
f * g is rotation
set TR = TOP-REAL n;
let f, g be Function of (TOP-REAL n),(TOP-REAL n); ::_thesis: ( f is rotation & g is rotation implies f * g is rotation )
assume that
A1: f is rotation and
A2: g is rotation ; ::_thesis: f * g is rotation
let p be Point of (TOP-REAL n); :: according to MATRTOP3:def_4 ::_thesis: |.p.| = |.((f * g) . p).|
dom (f * g) = the carrier of (TOP-REAL n) by FUNCT_2:52;
hence |.((f * g) . p).| = |.(f . (g . p)).| by FUNCT_1:12
.= |.(g . p).| by A1, Def4
.= |.p.| by A2, Def4 ;
::_thesis: verum
end;
registration
let n be Nat;
cluster Function-like quasi_total base_rotation -> additive homogeneous being_homeomorphism rotation for Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):];
coherence
for b1 being Function of (TOP-REAL n),(TOP-REAL n) st b1 is base_rotation holds
( b1 is homogeneous & b1 is additive & b1 is rotation & b1 is being_homeomorphism )
proof
set TR = TOP-REAL n;
set G = GFuncs the carrier of (TOP-REAL n);
let f be Function of (TOP-REAL n),(TOP-REAL n); ::_thesis: ( f is base_rotation implies ( f is homogeneous & f is additive & f is rotation & f is being_homeomorphism ) )
assume f is base_rotation ; ::_thesis: ( f is homogeneous & f is additive & f is rotation & f is being_homeomorphism )
then consider F being FinSequence of (GFuncs the carrier of (TOP-REAL n)) such that
A1: f = Product F and
A2: for k being Nat st k in dom F holds
ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) ) by Def5;
defpred S1[ Nat] means ( n <= len F implies for g being Function of (TOP-REAL n),(TOP-REAL n) st g = Product (F | n) holds
( g is homogeneous & g is additive & g is being_homeomorphism & g is rotation ) );
A3: for m being Nat st S1[m] holds
S1[m + 1]
proof
let m be Nat; ::_thesis: ( S1[m] implies S1[m + 1] )
assume A4: S1[m] ; ::_thesis: S1[m + 1]
set m1 = m + 1;
reconsider P = Product (F | m) as Function of (TOP-REAL n),(TOP-REAL n) by MONOID_0:73;
assume A5: m + 1 <= len F ; ::_thesis: for g being Function of (TOP-REAL n),(TOP-REAL n) st g = Product (F | (m + 1)) holds
( g is homogeneous & g is additive & g is being_homeomorphism & g is rotation )
1 <= m + 1 by NAT_1:11;
then A6: m + 1 in dom F by A5, FINSEQ_3:25;
then consider i, j being Nat, r being real number such that
A7: ( 1 <= i & i < j & j <= n ) and
A8: F . (m + 1) = Mx2Tran (Rotation (i,j,n,r)) by A2;
reconsider M = Mx2Tran (Rotation (i,j,n,r)) as Element of (GFuncs the carrier of (TOP-REAL n)) by MONOID_0:73;
A9: F | (m + 1) = (F | m) ^ <*M*> by A6, A8, FINSEQ_5:10;
let g be Function of (TOP-REAL n),(TOP-REAL n); ::_thesis: ( g = Product (F | (m + 1)) implies ( g is homogeneous & g is additive & g is being_homeomorphism & g is rotation ) )
assume A10: g = Product (F | (m + 1)) ; ::_thesis: ( g is homogeneous & g is additive & g is being_homeomorphism & g is rotation )
A11: g = (Product (F | m)) * M by A9, A10, GROUP_4:6
.= (Mx2Tran (Rotation (i,j,n,r))) * P by MONOID_0:70 ;
A12: Mx2Tran (Rotation (i,j,n,r)) is rotation by A7, Lm7;
( P is homogeneous & P is additive & P is being_homeomorphism & P is rotation ) by A4, A5, NAT_1:13;
hence ( g is homogeneous & g is additive & g is being_homeomorphism & g is rotation ) by A11, A12, Lm8, TOPS_2:57; ::_thesis: verum
end;
A13: F | (len F) = F by FINSEQ_1:58;
A14: S1[ 0 ]
proof
A15: id (TOP-REAL n) is rotation
proof
let p be Point of (TOP-REAL n); :: according to MATRTOP3:def_4 ::_thesis: |.p.| = |.((id (TOP-REAL n)) . p).|
thus |.p.| = |.((id (TOP-REAL n)) . p).| by FUNCT_1:17; ::_thesis: verum
end;
assume 0 <= len F ; ::_thesis: for g being Function of (TOP-REAL n),(TOP-REAL n) st g = Product (F | 0) holds
( g is homogeneous & g is additive & g is being_homeomorphism & g is rotation )
let g be Function of (TOP-REAL n),(TOP-REAL n); ::_thesis: ( g = Product (F | 0) implies ( g is homogeneous & g is additive & g is being_homeomorphism & g is rotation ) )
A16: id (TOP-REAL n) is homogeneous
proof
let r be real number ; :: according to TOPREAL9:def_4 ::_thesis: for b1 being Element of the carrier of (TOP-REAL n) holds (id (TOP-REAL n)) . (r * b1) = r * ((id (TOP-REAL n)) . b1)
let p be Point of (TOP-REAL n); ::_thesis: (id (TOP-REAL n)) . (r * p) = r * ((id (TOP-REAL n)) . p)
thus (id (TOP-REAL n)) . (r * p) = r * p by FUNCT_1:17
.= r * ((id (TOP-REAL n)) . p) by FUNCT_1:17 ; ::_thesis: verum
end;
assume A17: g = Product (F | 0) ; ::_thesis: ( g is homogeneous & g is additive & g is being_homeomorphism & g is rotation )
F | 0 = <*> the carrier of (GFuncs the carrier of (TOP-REAL n)) ;
then g = 1_ (GFuncs the carrier of (TOP-REAL n)) by A17, GROUP_4:8
.= the_unity_wrt the multF of (GFuncs the carrier of (TOP-REAL n)) by GROUP_1:22
.= id (TOP-REAL n) by MONOID_0:75 ;
hence ( g is homogeneous & g is additive & g is being_homeomorphism & g is rotation ) by A16, A15; ::_thesis: verum
end;
for m being Nat holds S1[m] from NAT_1:sch_2(A14, A3);
hence ( f is homogeneous & f is additive & f is rotation & f is being_homeomorphism ) by A1, A13; ::_thesis: verum
end;
end;
registration
let n be Nat;
let f be base_rotation Function of (TOP-REAL n),(TOP-REAL n);
clusterf /" -> base_rotation ;
coherence
f " is base_rotation
proof
set TR = TOP-REAL n;
set G = GFuncs the carrier of (TOP-REAL n);
defpred S1[ Nat] means for F being FinSequence of (GFuncs the carrier of (TOP-REAL n))
for f being Function of (TOP-REAL n),(TOP-REAL n) st len F = n & Product F = f & ( for k being Nat st k in dom F holds
ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) ) ) holds
f " is base_rotation ;
consider F being FinSequence of (GFuncs the carrier of (TOP-REAL n)) such that
A1: ( f = Product F & ( for k being Nat st k in dom F holds
ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) ) ) ) by Def5;
A2: for i being Nat st S1[i] holds
S1[i + 1]
proof
let z be Nat; ::_thesis: ( S1[z] implies S1[z + 1] )
assume A3: S1[z] ; ::_thesis: S1[z + 1]
set z1 = z + 1;
let F be FinSequence of (GFuncs the carrier of (TOP-REAL n)); ::_thesis: for f being Function of (TOP-REAL n),(TOP-REAL n) st len F = z + 1 & Product F = f & ( for k being Nat st k in dom F holds
ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) ) ) holds
f " is base_rotation
let f be Function of (TOP-REAL n),(TOP-REAL n); ::_thesis: ( len F = z + 1 & Product F = f & ( for k being Nat st k in dom F holds
ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) ) ) implies f " is base_rotation )
assume that
A4: len F = z + 1 and
A5: Product F = f and
A6: for k being Nat st k in dom F holds
ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) ) ; ::_thesis: f " is base_rotation
set Fz = F | z;
reconsider fz = Product (F | z) as Function of (TOP-REAL n),(TOP-REAL n) by MONOID_0:73;
1 <= z + 1 by NAT_1:11;
then z + 1 in dom F by A4, FINSEQ_3:25;
then consider i, j being Nat, r being real number such that
A7: ( 1 <= i & i < j & j <= n ) and
A8: F . (z + 1) = Mx2Tran (Rotation (i,j,n,r)) by A6;
set m = Mx2Tran (Rotation (i,j,n,r));
reconsider M = Mx2Tran (Rotation (i,j,n,r)) as Element of (GFuncs the carrier of (TOP-REAL n)) by MONOID_0:73;
F = (F | z) ^ <*M*> by A4, A8, FINSEQ_3:55;
then A9: f = (Product (F | z)) * M by A5, GROUP_4:6
.= (Mx2Tran (Rotation (i,j,n,r))) * fz by MONOID_0:70 ;
A10: dom (F | z) c= dom F by RELAT_1:60;
A11: now__::_thesis:_for_k_being_Nat_st_k_in_dom_(F_|_z)_holds_
ex_i,_j_being_Nat_ex_r_being_real_number_st_
(_1_<=_i_&_i_<_j_&_j_<=_n_&_(F_|_z)_._k_=_Mx2Tran_(Rotation_(i,j,n,r))_)
let k be Nat; ::_thesis: ( k in dom (F | z) implies ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & (F | z) . k = Mx2Tran (Rotation (i,j,n,r)) ) )
assume A12: k in dom (F | z) ; ::_thesis: ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & (F | z) . k = Mx2Tran (Rotation (i,j,n,r)) )
then (F | z) . k = F . k by FUNCT_1:47;
hence ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & (F | z) . k = Mx2Tran (Rotation (i,j,n,r)) ) by A6, A10, A12; ::_thesis: verum
end;
then A13: fz is base_rotation by Def5;
( Det (Rotation (i,j,n,r)) <> 0. F_Real & (Rotation (i,j,n,r)) ~ = Rotation (i,j,n,(- r)) ) by A7, Lm5, Th13;
then A14: (Mx2Tran (Rotation (i,j,n,r))) " = Mx2Tran (Rotation (i,j,n,(- r))) by MATRTOP1:43;
A15: ( rng (Mx2Tran (Rotation (i,j,n,r))) = [#] (TOP-REAL n) & Mx2Tran (Rotation (i,j,n,r)) is one-to-one & dom (Mx2Tran (Rotation (i,j,n,r))) = [#] (TOP-REAL n) ) by TOPS_2:def_5;
then Mx2Tran (Rotation (i,j,n,r)) is onto by FUNCT_2:def_3;
then A16: (Mx2Tran (Rotation (i,j,n,r))) " = (Mx2Tran (Rotation (i,j,n,r))) " by A15, TOPS_2:def_4;
len (F | z) = z by A4, FINSEQ_1:59, NAT_1:11;
then A17: fz " is base_rotation by A3, A11;
( fz is one-to-one & dom fz = [#] (TOP-REAL n) & rng fz = [#] (TOP-REAL n) ) by A13, TOPS_2:def_5;
then A18: f " = (fz ") * ((Mx2Tran (Rotation (i,j,n,r))) ") by A9, A15, TOPS_2:53;
Mx2Tran (Rotation (i,j,n,(- r))) is base_rotation by A7, Th28;
hence f " is base_rotation by A14, A16, A17, A18; ::_thesis: verum
end;
A19: S1[ 0 ]
proof
let F be FinSequence of (GFuncs the carrier of (TOP-REAL n)); ::_thesis: for f being Function of (TOP-REAL n),(TOP-REAL n) st len F = 0 & Product F = f & ( for k being Nat st k in dom F holds
ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) ) ) holds
f " is base_rotation
let f be Function of (TOP-REAL n),(TOP-REAL n); ::_thesis: ( len F = 0 & Product F = f & ( for k being Nat st k in dom F holds
ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) ) ) implies f " is base_rotation )
assume that
A20: len F = 0 and
A21: Product F = f ; ::_thesis: ( ex k being Nat st
( k in dom F & ( for i, j being Nat
for r being real number holds
( not 1 <= i or not i < j or not j <= n or not F . k = Mx2Tran (Rotation (i,j,n,r)) ) ) ) or f " is base_rotation )
F = <*> the carrier of (GFuncs the carrier of (TOP-REAL n)) by A20;
then A22: f = 1_ (GFuncs the carrier of (TOP-REAL n)) by A21, GROUP_4:8
.= the_unity_wrt the multF of (GFuncs the carrier of (TOP-REAL n)) by GROUP_1:22
.= id (TOP-REAL n) by MONOID_0:75 ;
then rng f = [#] (TOP-REAL n) by TOPS_2:def_5;
then f is onto by FUNCT_2:def_3;
then f /" = f " by A22, TOPS_2:def_4;
hence ( ex k being Nat st
( k in dom F & ( for i, j being Nat
for r being real number holds
( not 1 <= i or not i < j or not j <= n or not F . k = Mx2Tran (Rotation (i,j,n,r)) ) ) ) or f " is base_rotation ) by A22, FUNCT_1:45; ::_thesis: verum
end;
for i being Nat holds S1[i] from NAT_1:sch_2(A19, A2);
then S1[ len F] ;
hence f " is base_rotation by A1; ::_thesis: verum
end;
end;
registration
let n be Nat;
let f, g be rotation Function of (TOP-REAL n),(TOP-REAL n);
clusterg (#) f -> rotation for Function of (TOP-REAL n),(TOP-REAL n);
coherence
for b1 being Function of (TOP-REAL n),(TOP-REAL n) st b1 = f * g holds
b1 is rotation by Lm8;
end;
definition
let n be Nat;
let f be additive homogeneous Function of (TOP-REAL n),(TOP-REAL n);
func AutMt f -> Matrix of n,F_Real means :Def6: :: MATRTOP3:def 6
f = Mx2Tran it;
existence
ex b1 being Matrix of n,F_Real st f = Mx2Tran b1
proof
set T = n -VectSp_over F_Real;
set TR = TOP-REAL n;
reconsider B = MX2FinS (1. (F_Real,n)) as OrdBasis of n -VectSp_over F_Real by MATRLIN2:45;
A1: the carrier of (n -VectSp_over F_Real) = REAL n by MATRIX13:102
.= the carrier of (TOP-REAL n) by EUCLID:22 ;
then reconsider F = f as Function of (n -VectSp_over F_Real),(n -VectSp_over F_Real) ;
now__::_thesis:_for_v1,_v2_being_Vector_of_(n_-VectSp_over_F_Real)_holds_F_._(v1_+_v2)_=_(F_._v1)_+_(F_._v2)
let v1, v2 be Vector of (n -VectSp_over F_Real); ::_thesis: F . (v1 + v2) = (F . v1) + (F . v2)
reconsider P1 = v1, P2 = v2, FP1 = F . v1, FP2 = F . v2 as Element of n -tuples_on the carrier of F_Real by MATRIX13:102;
A2: ( @ (@ FP1) = FP1 & @ (@ FP2) = FP2 ) ;
reconsider p1 = v1, p2 = v2 as Point of (TOP-REAL n) by A1;
A3: ( @ (@ P1) = P1 & @ (@ P2) = P2 ) ;
v1 + v2 = P1 + P2 by MATRIX13:102
.= p1 + p2 by A3, MATRTOP1:1 ;
hence F . (v1 + v2) = (f . p1) + (f . p2) by VECTSP_1:def_20
.= FP1 + FP2 by A2, MATRTOP1:1
.= (F . v1) + (F . v2) by MATRIX13:102 ;
::_thesis: verum
end;
then A4: F is additive by VECTSP_1:def_20;
len B = n by MATRTOP1:19;
then reconsider A = AutMt (F,B,B) as Matrix of n,F_Real ;
take A ; ::_thesis: f = Mx2Tran A
now__::_thesis:_for_r_being_Scalar_of_F_Real
for_v_being_Vector_of_(n_-VectSp_over_F_Real)_holds_F_._(r_*_v)_=_r_*_(F_._v)
let r be Scalar of F_Real; ::_thesis: for v being Vector of (n -VectSp_over F_Real) holds F . (r * v) = r * (F . v)
let v be Vector of (n -VectSp_over F_Real); ::_thesis: F . (r * v) = r * (F . v)
reconsider p = v as Point of (TOP-REAL n) by A1;
reconsider P = v, FP = F . v as Element of n -tuples_on the carrier of F_Real by MATRIX13:102;
r * v = r * P by MATRIX13:102
.= r * p by MATRIXR1:17 ;
hence F . (r * v) = r * (f . p) by TOPREAL9:def_4
.= r * FP by MATRIXR1:17
.= r * (F . v) by MATRIX13:102 ;
::_thesis: verum
end;
then A5: F is homogeneous by MOD_2:def_2;
Mx2Tran A = Mx2Tran ((AutMt (F,B,B)),B,B) by MATRTOP1:20
.= f by A5, A4, MATRLIN2:34 ;
hence f = Mx2Tran A ; ::_thesis: verum
end;
uniqueness
for b1, b2 being Matrix of n,F_Real st f = Mx2Tran b1 & f = Mx2Tran b2 holds
b1 = b2 by MATRTOP1:34;
end;
:: deftheorem Def6 defines AutMt MATRTOP3:def_6_:_
for n being Nat
for f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n)
for b3 being Matrix of n,F_Real holds
( b3 = AutMt f iff f = Mx2Tran b3 );
theorem Th29: :: MATRTOP3:29
for n being Nat
for f1, f2 being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) holds AutMt (f1 * f2) = (AutMt f2) * (AutMt f1)
proof
let n be Nat; ::_thesis: for f1, f2 being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) holds AutMt (f1 * f2) = (AutMt f2) * (AutMt f1)
let f1, f2 be additive homogeneous Function of (TOP-REAL n),(TOP-REAL n); ::_thesis: AutMt (f1 * f2) = (AutMt f2) * (AutMt f1)
set A1 = AutMt f1;
set A2 = AutMt f2;
A1: ( width (AutMt f1) = n & width (AutMt f2) = n & len (AutMt f2) = n ) by MATRIX_1:24;
( n = 0 implies n = 0 ) ;
then Mx2Tran ((AutMt f2) * (AutMt f1)) = (Mx2Tran (AutMt f1)) * (Mx2Tran (AutMt f2)) by A1, MATRTOP1:32
.= f1 * (Mx2Tran (AutMt f2)) by Def6
.= f1 * f2 by Def6 ;
hence AutMt (f1 * f2) = (AutMt f2) * (AutMt f1) by Def6; ::_thesis: verum
end;
theorem Th30: :: MATRTOP3:30
for X being set
for k, n being Nat
for p being Point of (TOP-REAL n) st k in X & k in Seg n holds
ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is X -support-yielding & f is base_rotation & ( card (X /\ (Seg n)) > 1 implies (f . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg n) & i <> k holds
(f . p) . i = 0 ) )
proof
let X be set ; ::_thesis: for k, n being Nat
for p being Point of (TOP-REAL n) st k in X & k in Seg n holds
ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is X -support-yielding & f is base_rotation & ( card (X /\ (Seg n)) > 1 implies (f . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg n) & i <> k holds
(f . p) . i = 0 ) )
let k, n be Nat; ::_thesis: for p being Point of (TOP-REAL n) st k in X & k in Seg n holds
ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is X -support-yielding & f is base_rotation & ( card (X /\ (Seg n)) > 1 implies (f . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg n) & i <> k holds
(f . p) . i = 0 ) )
let p be Point of (TOP-REAL n); ::_thesis: ( k in X & k in Seg n implies ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is X -support-yielding & f is base_rotation & ( card (X /\ (Seg n)) > 1 implies (f . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg n) & i <> k holds
(f . p) . i = 0 ) ) )
assume that
A1: k in X and
A2: k in Seg n ; ::_thesis: ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is X -support-yielding & f is base_rotation & ( card (X /\ (Seg n)) > 1 implies (f . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg n) & i <> k holds
(f . p) . i = 0 ) )
set TR = TOP-REAL n;
defpred S1[ Nat] means ( $1 <= n implies ex f being base_rotation Function of (TOP-REAL n),(TOP-REAL n) st
( f is (X /\ (Seg $1)) \/ {k} -support-yielding & ( card ((X /\ (Seg $1)) \/ {k}) > 1 implies (f . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg $1) & i <> k holds
(f . p) . i = 0 ) ) );
A3: for z being Nat st S1[z] holds
S1[z + 1]
proof
let z be Nat; ::_thesis: ( S1[z] implies S1[z + 1] )
set z1 = z + 1;
assume A4: S1[z] ; ::_thesis: S1[z + 1]
A5: Seg (z + 1) = (Seg z) \/ {(z + 1)} by FINSEQ_1:9;
A6: Seg (z + 1) = (Seg z) \/ {(z + 1)} by FINSEQ_1:9;
A7: ( z + 1 in X implies ((X /\ (Seg z)) \/ {k}) \/ {(z + 1),k} = (X /\ (Seg (z + 1))) \/ {k} )
proof
assume z + 1 in X ; ::_thesis: ((X /\ (Seg z)) \/ {k}) \/ {(z + 1),k} = (X /\ (Seg (z + 1))) \/ {k}
then A8: X \/ {(z + 1)} = X by ZFMISC_1:40;
{(z + 1),k} = {(z + 1)} \/ {k} by ENUMSET1:1;
hence ((X /\ (Seg z)) \/ {k}) \/ {(z + 1),k} = (X /\ (Seg z)) \/ ({k} \/ ({k} \/ {(z + 1)})) by XBOOLE_1:4
.= (X /\ (Seg z)) \/ (({k} \/ {k}) \/ {(z + 1)}) by XBOOLE_1:4
.= ((X /\ (Seg z)) \/ {(z + 1)}) \/ {k} by XBOOLE_1:4
.= (X /\ (Seg (z + 1))) \/ {k} by A6, A8, XBOOLE_1:24 ;
::_thesis: verum
end;
assume A9: z + 1 <= n ; ::_thesis: ex f being base_rotation Function of (TOP-REAL n),(TOP-REAL n) st
( f is (X /\ (Seg (z + 1))) \/ {k} -support-yielding & ( card ((X /\ (Seg (z + 1))) \/ {k}) > 1 implies (f . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg (z + 1)) & i <> k holds
(f . p) . i = 0 ) )
then consider f being base_rotation Function of (TOP-REAL n),(TOP-REAL n) such that
A10: f is (X /\ (Seg z)) \/ {k} -support-yielding and
A11: ( card ((X /\ (Seg z)) \/ {k}) > 1 implies (f . p) . k >= 0 ) and
A12: for m being Nat st m in X /\ (Seg z) & m <> k holds
(f . p) . m = 0 by A4, NAT_1:13;
set z1 = z + 1;
percases ( z + 1 = k or not z + 1 in X or ( z + 1 < k & z + 1 in X ) or ( z + 1 > k & z + 1 in X ) ) by XXREAL_0:1;
supposeA13: z + 1 = k ; ::_thesis: ex f being base_rotation Function of (TOP-REAL n),(TOP-REAL n) st
( f is (X /\ (Seg (z + 1))) \/ {k} -support-yielding & ( card ((X /\ (Seg (z + 1))) \/ {k}) > 1 implies (f . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg (z + 1)) & i <> k holds
(f . p) . i = 0 ) )
take f ; ::_thesis: ( f is (X /\ (Seg (z + 1))) \/ {k} -support-yielding & ( card ((X /\ (Seg (z + 1))) \/ {k}) > 1 implies (f . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg (z + 1)) & i <> k holds
(f . p) . i = 0 ) )
(Seg (z + 1)) \/ {(z + 1)} = (Seg z) \/ ({(z + 1)} \/ {(z + 1)}) by A5, XBOOLE_1:4
.= (Seg z) \/ {(z + 1)} ;
then A14: (X /\ (Seg (z + 1))) \/ {k} = (X \/ {k}) /\ ((Seg z) \/ {k}) by A13, XBOOLE_1:24
.= (X /\ (Seg z)) \/ {k} by XBOOLE_1:24 ;
hence f is (X /\ (Seg (z + 1))) \/ {k} -support-yielding by A10; ::_thesis: ( ( card ((X /\ (Seg (z + 1))) \/ {k}) > 1 implies (f . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg (z + 1)) & i <> k holds
(f . p) . i = 0 ) )
thus ( card ((X /\ (Seg (z + 1))) \/ {k}) > 1 implies (f . p) . k >= 0 ) by A11, A14; ::_thesis: for i being Nat st i in X /\ (Seg (z + 1)) & i <> k holds
(f . p) . i = 0
let m be Nat; ::_thesis: ( m in X /\ (Seg (z + 1)) & m <> k implies (f . p) . m = 0 )
assume that
A15: m in X /\ (Seg (z + 1)) and
A16: m <> k ; ::_thesis: (f . p) . m = 0
A17: m in Seg (z + 1) by A15, XBOOLE_0:def_4;
A18: m in X by A15, XBOOLE_0:def_4;
not m in {(z + 1)} by A13, A16, TARSKI:def_1;
then m in Seg z by A5, A17, XBOOLE_0:def_3;
then m in X /\ (Seg z) by A18, XBOOLE_0:def_4;
hence (f . p) . m = 0 by A12, A16; ::_thesis: verum
end;
supposeA19: not z + 1 in X ; ::_thesis: ex f being base_rotation Function of (TOP-REAL n),(TOP-REAL n) st
( f is (X /\ (Seg (z + 1))) \/ {k} -support-yielding & ( card ((X /\ (Seg (z + 1))) \/ {k}) > 1 implies (f . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg (z + 1)) & i <> k holds
(f . p) . i = 0 ) )
take f ; ::_thesis: ( f is (X /\ (Seg (z + 1))) \/ {k} -support-yielding & ( card ((X /\ (Seg (z + 1))) \/ {k}) > 1 implies (f . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg (z + 1)) & i <> k holds
(f . p) . i = 0 ) )
A20: {(z + 1)} misses X by A19, ZFMISC_1:50;
A21: X /\ (Seg (z + 1)) = (X /\ (Seg z)) \/ (X /\ {(z + 1)}) by A5, XBOOLE_1:23
.= (X /\ (Seg z)) \/ {} by A20, XBOOLE_0:def_7
.= X /\ (Seg z) ;
hence f is (X /\ (Seg (z + 1))) \/ {k} -support-yielding by A10; ::_thesis: ( ( card ((X /\ (Seg (z + 1))) \/ {k}) > 1 implies (f . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg (z + 1)) & i <> k holds
(f . p) . i = 0 ) )
thus ( card ((X /\ (Seg (z + 1))) \/ {k}) > 1 implies (f . p) . k >= 0 ) by A11, A21; ::_thesis: for i being Nat st i in X /\ (Seg (z + 1)) & i <> k holds
(f . p) . i = 0
let m be Nat; ::_thesis: ( m in X /\ (Seg (z + 1)) & m <> k implies (f . p) . m = 0 )
assume that
A22: m in X /\ (Seg (z + 1)) and
A23: m <> k ; ::_thesis: (f . p) . m = 0
A24: m in Seg (z + 1) by A22, XBOOLE_0:def_4;
A25: m in X by A22, XBOOLE_0:def_4;
then not m in {(z + 1)} by A19, TARSKI:def_1;
then m in Seg z by A5, A24, XBOOLE_0:def_3;
then m in X /\ (Seg z) by A25, XBOOLE_0:def_4;
hence (f . p) . m = 0 by A12, A23; ::_thesis: verum
end;
supposeA26: ( z + 1 < k & z + 1 in X ) ; ::_thesis: ex f being base_rotation Function of (TOP-REAL n),(TOP-REAL n) st
( f is (X /\ (Seg (z + 1))) \/ {k} -support-yielding & ( card ((X /\ (Seg (z + 1))) \/ {k}) > 1 implies (f . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg (z + 1)) & i <> k holds
(f . p) . i = 0 ) )
set fp = f . p;
set S = (((f . p) . (z + 1)) ^2) + (((f . p) . k) ^2);
A27: ( z + 1 >= 1 & k <= n ) by A2, FINSEQ_1:1, NAT_1:11;
A28: ( ((f . p) . k) ^2 >= 0 & ((f . p) . (z + 1)) ^2 >= 0 ) by XREAL_1:63;
then A29: (sqrt ((((f . p) . (z + 1)) ^2) + (((f . p) . k) ^2))) ^2 = (((f . p) . (z + 1)) ^2) + (((f . p) . k) ^2) by SQUARE_1:def_2;
then consider r being real number such that
A30: ((Mx2Tran (Rotation ((z + 1),k,n,r))) . (f . p)) . k = sqrt ((((f . p) . (z + 1)) ^2) + (((f . p) . k) ^2)) by A26, A27, Th25;
reconsider M = Mx2Tran (Rotation ((z + 1),k,n,r)) as base_rotation Function of (TOP-REAL n),(TOP-REAL n) by A26, A27, Th28;
take Mf = M * f; ::_thesis: ( Mf is (X /\ (Seg (z + 1))) \/ {k} -support-yielding & ( card ((X /\ (Seg (z + 1))) \/ {k}) > 1 implies (Mf . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg (z + 1)) & i <> k holds
(Mf . p) . i = 0 ) )
A31: M is {(z + 1),k} -support-yielding by A26, A27, Th26;
hence Mf is (X /\ (Seg (z + 1))) \/ {k} -support-yielding by A7, A10, A26; ::_thesis: ( ( card ((X /\ (Seg (z + 1))) \/ {k}) > 1 implies (Mf . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg (z + 1)) & i <> k holds
(Mf . p) . i = 0 ) )
A32: dom Mf = the carrier of (TOP-REAL n) by FUNCT_2:52;
then A33: f . p in dom M by FUNCT_1:11;
A34: Mf . p = M . (f . p) by A32, FUNCT_1:12;
hence ( card ((X /\ (Seg (z + 1))) \/ {k}) > 1 implies (Mf . p) . k >= 0 ) by A28, A30, SQUARE_1:def_2; ::_thesis: for i being Nat st i in X /\ (Seg (z + 1)) & i <> k holds
(Mf . p) . i = 0
let i be Nat; ::_thesis: ( i in X /\ (Seg (z + 1)) & i <> k implies (Mf . p) . i = 0 )
assume that
A35: i in X /\ (Seg (z + 1)) and
A36: i <> k ; ::_thesis: (Mf . p) . i = 0
A37: i in X by A35, XBOOLE_0:def_4;
i in Seg (z + 1) by A35, XBOOLE_0:def_4;
then A38: ( i in Seg z or i in {(z + 1)} ) by A5, XBOOLE_0:def_3;
percases ( i in Seg z or i = z + 1 ) by A38, TARSKI:def_1;
supposeA39: i in Seg z ; ::_thesis: (Mf . p) . i = 0
then A40: i in X /\ (Seg z) by A37, XBOOLE_0:def_4;
i <= z by A39, FINSEQ_1:1;
then i < z + 1 by NAT_1:13;
then not i in {(z + 1),k} by A36, TARSKI:def_2;
hence (Mf . p) . i = (f . p) . i by A31, A33, A34, Def1
.= 0 by A12, A36, A40 ;
::_thesis: verum
end;
suppose i = z + 1 ; ::_thesis: (Mf . p) . i = 0
then A41: (((M . (f . p)) . i) * ((M . (f . p)) . i)) + ((((f . p) . (z + 1)) ^2) + (((f . p) . k) ^2)) = (((f . p) . (z + 1)) ^2) + (((f . p) . k) ^2) by A26, A27, A29, A30, Lm6;
thus (Mf . p) . i = (M . (f . p)) . i by A32, FUNCT_1:12
.= 0 by A41 ; ::_thesis: verum
end;
end;
end;
supposeA42: ( z + 1 > k & z + 1 in X ) ; ::_thesis: ex f being base_rotation Function of (TOP-REAL n),(TOP-REAL n) st
( f is (X /\ (Seg (z + 1))) \/ {k} -support-yielding & ( card ((X /\ (Seg (z + 1))) \/ {k}) > 1 implies (f . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg (z + 1)) & i <> k holds
(f . p) . i = 0 ) )
set fp = f . p;
set S = (((f . p) . (z + 1)) ^2) + (((f . p) . k) ^2);
A43: 1 <= k by A2, FINSEQ_1:1;
A44: ( ((f . p) . k) ^2 >= 0 & ((f . p) . (z + 1)) ^2 >= 0 ) by XREAL_1:63;
then A45: (sqrt ((((f . p) . (z + 1)) ^2) + (((f . p) . k) ^2))) ^2 = (((f . p) . (z + 1)) ^2) + (((f . p) . k) ^2) by SQUARE_1:def_2;
then consider r being real number such that
A46: ((Mx2Tran (Rotation (k,(z + 1),n,r))) . (f . p)) . k = sqrt ((((f . p) . (z + 1)) ^2) + (((f . p) . k) ^2)) by A9, A42, A43, Th24;
reconsider M = Mx2Tran (Rotation (k,(z + 1),n,r)) as base_rotation Function of (TOP-REAL n),(TOP-REAL n) by A9, A42, A43, Th28;
take Mf = M * f; ::_thesis: ( Mf is (X /\ (Seg (z + 1))) \/ {k} -support-yielding & ( card ((X /\ (Seg (z + 1))) \/ {k}) > 1 implies (Mf . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg (z + 1)) & i <> k holds
(Mf . p) . i = 0 ) )
A47: M is {k,(z + 1)} -support-yielding by A9, A42, A43, Th26;
hence Mf is (X /\ (Seg (z + 1))) \/ {k} -support-yielding by A7, A10, A42; ::_thesis: ( ( card ((X /\ (Seg (z + 1))) \/ {k}) > 1 implies (Mf . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg (z + 1)) & i <> k holds
(Mf . p) . i = 0 ) )
A48: dom Mf = the carrier of (TOP-REAL n) by FUNCT_2:52;
then A49: Mf . p = M . (f . p) by FUNCT_1:12;
hence ( card ((X /\ (Seg (z + 1))) \/ {k}) > 1 implies (Mf . p) . k >= 0 ) by A44, A46, SQUARE_1:def_2; ::_thesis: for i being Nat st i in X /\ (Seg (z + 1)) & i <> k holds
(Mf . p) . i = 0
let i be Nat; ::_thesis: ( i in X /\ (Seg (z + 1)) & i <> k implies (Mf . p) . i = 0 )
assume that
A50: i in X /\ (Seg (z + 1)) and
A51: i <> k ; ::_thesis: (Mf . p) . i = 0
A52: i in X by A50, XBOOLE_0:def_4;
i in Seg (z + 1) by A50, XBOOLE_0:def_4;
then A53: ( i in Seg z or i in {(z + 1)} ) by A5, XBOOLE_0:def_3;
percases ( i in Seg z or i = z + 1 ) by A53, TARSKI:def_1;
supposeA54: i in Seg z ; ::_thesis: (Mf . p) . i = 0
then i <= z by FINSEQ_1:1;
then i < z + 1 by NAT_1:13;
then A55: not i in {(z + 1),k} by A51, TARSKI:def_2;
A56: i in X /\ (Seg z) by A52, A54, XBOOLE_0:def_4;
f . p in dom M by A48, FUNCT_1:11;
hence (Mf . p) . i = (f . p) . i by A47, A49, A55, Def1
.= 0 by A12, A51, A56 ;
::_thesis: verum
end;
suppose i = z + 1 ; ::_thesis: (Mf . p) . i = 0
then A57: (((M . (f . p)) . i) * ((M . (f . p)) . i)) + ((((f . p) . (z + 1)) ^2) + (((f . p) . k) ^2)) = (((f . p) . (z + 1)) ^2) + (((f . p) . k) ^2) by A9, A42, A43, A45, A46, Lm6;
thus (Mf . p) . i = (M . (f . p)) . i by A48, FUNCT_1:12
.= 0 by A57 ; ::_thesis: verum
end;
end;
end;
end;
end;
A58: S1[ 0 ]
proof
assume 0 <= n ; ::_thesis: ex f being base_rotation Function of (TOP-REAL n),(TOP-REAL n) st
( f is (X /\ (Seg 0)) \/ {k} -support-yielding & ( card ((X /\ (Seg 0)) \/ {k}) > 1 implies (f . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg 0) & i <> k holds
(f . p) . i = 0 ) )
take f = id (TOP-REAL n); ::_thesis: ( f is (X /\ (Seg 0)) \/ {k} -support-yielding & ( card ((X /\ (Seg 0)) \/ {k}) > 1 implies (f . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg 0) & i <> k holds
(f . p) . i = 0 ) )
A59: f is {} -support-yielding
proof
let F be Function; :: according to MATRTOP3:def_1 ::_thesis: for x being set st F in dom f & (f . F) . x <> F . x holds
x in {}
let x be set ; ::_thesis: ( F in dom f & (f . F) . x <> F . x implies x in {} )
assume that
A60: F in dom f and
A61: (f . F) . x <> F . x ; ::_thesis: x in {}
F in the carrier of (TOP-REAL n) by A60, FUNCT_1:17;
hence x in {} by A61, FUNCT_1:17; ::_thesis: verum
end;
{} c= (X /\ (Seg 0)) \/ {k} by XBOOLE_1:2;
hence f is (X /\ (Seg 0)) \/ {k} -support-yielding by A59; ::_thesis: ( ( card ((X /\ (Seg 0)) \/ {k}) > 1 implies (f . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg 0) & i <> k holds
(f . p) . i = 0 ) )
thus ( card ((X /\ (Seg 0)) \/ {k}) > 1 implies (f . p) . k >= 0 ) by CARD_2:42; ::_thesis: for i being Nat st i in X /\ (Seg 0) & i <> k holds
(f . p) . i = 0
let m be Nat; ::_thesis: ( m in X /\ (Seg 0) & m <> k implies (f . p) . m = 0 )
assume m in X /\ (Seg 0) ; ::_thesis: ( not m <> k or (f . p) . m = 0 )
hence ( not m <> k or (f . p) . m = 0 ) ; ::_thesis: verum
end;
for z being Nat holds S1[z] from NAT_1:sch_2(A58, A3);
then consider f being base_rotation Function of (TOP-REAL n),(TOP-REAL n) such that
A62: ( f is (X /\ (Seg n)) \/ {k} -support-yielding & ( card ((X /\ (Seg n)) \/ {k}) > 1 implies (f . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg n) & i <> k holds
(f . p) . i = 0 ) ) ;
take f ; ::_thesis: ( f is X -support-yielding & f is base_rotation & ( card (X /\ (Seg n)) > 1 implies (f . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg n) & i <> k holds
(f . p) . i = 0 ) )
A63: X /\ (Seg n) c= X by XBOOLE_1:17;
( {k} c= X & {k} c= Seg n ) by A1, A2, ZFMISC_1:31;
then (X /\ (Seg n)) \/ {k} = X /\ (Seg n) by XBOOLE_1:12, XBOOLE_1:19;
hence ( f is X -support-yielding & f is base_rotation & ( card (X /\ (Seg n)) > 1 implies (f . p) . k >= 0 ) & ( for i being Nat st i in X /\ (Seg n) & i <> k holds
(f . p) . i = 0 ) ) by A62, A63; ::_thesis: verum
end;
theorem Th31: :: MATRTOP3:31
for n being Nat
for f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n)
for A being Subset of (TOP-REAL n) st f | A = id A holds
f | (Lin A) = id (Lin A)
proof
let n be Nat; ::_thesis: for f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n)
for A being Subset of (TOP-REAL n) st f | A = id A holds
f | (Lin A) = id (Lin A)
let f be additive homogeneous Function of (TOP-REAL n),(TOP-REAL n); ::_thesis: for A being Subset of (TOP-REAL n) st f | A = id A holds
f | (Lin A) = id (Lin A)
set TR = TOP-REAL n;
let A be Subset of (TOP-REAL n); ::_thesis: ( f | A = id A implies f | (Lin A) = id (Lin A) )
assume A1: f | A = id A ; ::_thesis: f | (Lin A) = id (Lin A)
defpred S1[ Nat] means for L being Linear_Combination of A st card (Carrier L) <= $1 holds
f . (Sum L) = Sum L;
A2: for i being Nat st S1[i] holds
S1[i + 1]
proof
let i be Nat; ::_thesis: ( S1[i] implies S1[i + 1] )
assume A3: S1[i] ; ::_thesis: S1[i + 1]
set i1 = i + 1;
let L be Linear_Combination of A; ::_thesis: ( card (Carrier L) <= i + 1 implies f . (Sum L) = Sum L )
assume A4: card (Carrier L) <= i + 1 ; ::_thesis: f . (Sum L) = Sum L
percases ( card (Carrier L) <= i or card (Carrier L) = i + 1 ) by A4, NAT_1:8;
suppose card (Carrier L) <= i ; ::_thesis: f . (Sum L) = Sum L
hence f . (Sum L) = Sum L by A3; ::_thesis: verum
end;
supposeA5: card (Carrier L) = i + 1 ; ::_thesis: f . (Sum L) = Sum L
then not Carrier L is empty ;
then consider x being set such that
A6: x in Carrier L by XBOOLE_0:def_1;
reconsider x = x as Point of (TOP-REAL n) by A6;
reconsider X = {x} as Subset of (TOP-REAL n) by ZFMISC_1:31;
consider K being Linear_Combination of X such that
A7: K . x = L . x by RLVECT_4:37;
L . x <> 0 by A6, RLVECT_2:19;
then ( Carrier K c= {x} & x in Carrier K ) by A7, RLVECT_2:19, RLVECT_2:def_6;
then A8: Carrier K = {x} by ZFMISC_1:33;
{x} \/ (Carrier L) = Carrier L by A6, ZFMISC_1:40;
then A9: Carrier (L - K) c= Carrier L by A8, RLVECT_2:55;
(L - K) . x = (L . x) - (K . x) by RLVECT_2:54
.= 0 by A7 ;
then not x in Carrier (L - K) by RLVECT_2:19;
then Carrier (L - K) c< Carrier L by A6, A9, XBOOLE_0:def_8;
then card (Carrier (L - K)) < i + 1 by A5, CARD_2:48;
then A10: card (Carrier (L - K)) <= i by NAT_1:13;
ZeroLC (TOP-REAL n) = (- K) - (- K) by RLVECT_2:57
.= (- K) + (- (- K)) by RLVECT_2:def_13
.= (- K) + K by RLVECT_2:53 ;
then L = L + ((- K) + K) by RLVECT_2:41
.= (L + (- K)) + K by RLVECT_2:40
.= (L - K) + K by RLVECT_2:def_13 ;
then A11: Sum L = (Sum (L - K)) + (Sum K) by RLVECT_3:1;
A12: Carrier L c= A by RLVECT_2:def_6;
then (f | A) . x = f . x by A6, FUNCT_1:49;
then A13: f . x = x by A1, A6, A12, FUNCT_1:17;
Carrier (L - K) c= A by A9, A12, XBOOLE_1:1;
then L - K is Linear_Combination of A by RLVECT_2:def_6;
then A14: f . (Sum (L - K)) = Sum (L - K) by A3, A10;
Sum K = (L . x) * x by A7, RLVECT_2:32;
then f . (Sum K) = Sum K by A13, TOPREAL9:def_4;
hence f . (Sum L) = Sum L by A11, A14, VECTSP_1:def_20; ::_thesis: verum
end;
end;
end;
set L = Lin A;
set cL = the carrier of (Lin A);
the carrier of (Lin A) c= the carrier of (TOP-REAL n) by RLSUB_1:def_2;
then A15: f | (Lin A) = f | the carrier of (Lin A) by TMAP_1:def_3;
A16: S1[ 0 ]
proof
let L be Linear_Combination of A; ::_thesis: ( card (Carrier L) <= 0 implies f . (Sum L) = Sum L )
assume card (Carrier L) <= 0 ; ::_thesis: f . (Sum L) = Sum L
then Carrier L = {} ;
then L is Linear_Combination of {} the carrier of (TOP-REAL n) by RLVECT_2:def_6;
then A17: Sum L = 0. (TOP-REAL n) by RLVECT_2:31;
f . (0. (TOP-REAL n)) = f . (0 * (0. (TOP-REAL n))) by RLVECT_1:10
.= 0 * (f . (0. (TOP-REAL n))) by TOPREAL9:def_4
.= 0. (TOP-REAL n) by RLVECT_1:10 ;
hence f . (Sum L) = Sum L by A17; ::_thesis: verum
end;
A18: for i being Nat holds S1[i] from NAT_1:sch_2(A16, A2);
A19: for x being set st x in the carrier of (Lin A) holds
(f | (Lin A)) . x = (id (Lin A)) . x
proof
let x be set ; ::_thesis: ( x in the carrier of (Lin A) implies (f | (Lin A)) . x = (id (Lin A)) . x )
assume A20: x in the carrier of (Lin A) ; ::_thesis: (f | (Lin A)) . x = (id (Lin A)) . x
then x in Lin A by STRUCT_0:def_5;
then consider K being Linear_Combination of A such that
A21: Sum K = x by RLVECT_3:14;
S1[ card (Carrier K)] by A18;
then A22: f . x = x by A21;
(f | (Lin A)) . x = f . x by A15, A20, FUNCT_1:49;
hence (f | (Lin A)) . x = (id (Lin A)) . x by A20, A22, FUNCT_1:17; ::_thesis: verum
end;
( dom (f | (Lin A)) = the carrier of (Lin A) & dom (id (Lin A)) = the carrier of (Lin A) ) by FUNCT_2:def_1;
hence f | (Lin A) = id (Lin A) by A19, FUNCT_1:2; ::_thesis: verum
end;
theorem Th32: :: MATRTOP3:32
for n being Nat
for p being Point of (TOP-REAL n)
for f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n)
for A being Subset of (TOP-REAL n) st f is rotation & f | A = id A holds
for i being Nat st i in Seg n & Base_FinSeq (n,i) in Lin A holds
(f . p) . i = p . i
proof
let n be Nat; ::_thesis: for p being Point of (TOP-REAL n)
for f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n)
for A being Subset of (TOP-REAL n) st f is rotation & f | A = id A holds
for i being Nat st i in Seg n & Base_FinSeq (n,i) in Lin A holds
(f . p) . i = p . i
let p be Point of (TOP-REAL n); ::_thesis: for f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n)
for A being Subset of (TOP-REAL n) st f is rotation & f | A = id A holds
for i being Nat st i in Seg n & Base_FinSeq (n,i) in Lin A holds
(f . p) . i = p . i
let f be additive homogeneous Function of (TOP-REAL n),(TOP-REAL n); ::_thesis: for A being Subset of (TOP-REAL n) st f is rotation & f | A = id A holds
for i being Nat st i in Seg n & Base_FinSeq (n,i) in Lin A holds
(f . p) . i = p . i
set TR = TOP-REAL n;
let A be Subset of (TOP-REAL n); ::_thesis: ( f is rotation & f | A = id A implies for i being Nat st i in Seg n & Base_FinSeq (n,i) in Lin A holds
(f . p) . i = p . i )
assume that
A1: f is rotation and
A2: f | A = id A ; ::_thesis: for i being Nat st i in Seg n & Base_FinSeq (n,i) in Lin A holds
(f . p) . i = p . i
set L = Lin A;
set n0 = 0* n;
A3: f | (Lin A) = id (Lin A) by A2, Th31;
A4: len (0* n) = n by CARD_1:def_7;
then A5: dom (0* n) = Seg n by FINSEQ_1:def_3;
let k be Nat; ::_thesis: ( k in Seg n & Base_FinSeq (n,k) in Lin A implies (f . p) . k = p . k )
assume that
A6: k in Seg n and
A7: Base_FinSeq (n,k) in Lin A ; ::_thesis: (f . p) . k = p . k
set n0k = (0* n) +* (k,1);
A8: (0* n) +* (k,1) = Base_FinSeq (n,k) by MATRIXR2:def_4;
set pk = p . k;
the carrier of (Lin A) c= the carrier of (TOP-REAL n) by RLSUB_1:def_2;
then A9: f | (Lin A) = f | the carrier of (Lin A) by TMAP_1:def_3;
dom ((0* n) +* (k,1)) = dom (0* n) by FUNCT_7:30;
then A10: len ((0* n) +* (k,1)) = n by A4, FINSEQ_3:29;
A11: (0* n) +* (k,1) = @ (@ ((0* n) +* (k,1))) ;
then reconsider N0k = (0* n) +* (k,1) as Point of (TOP-REAL n) by A10, TOPREAL3:46;
A12: (0* n) +* (k,1) is Element of n -tuples_on REAL by A10, A11, FINSEQ_2:92;
A13: for p being Point of (TOP-REAL n) st p . k <> 0 holds
(f . p) . k = p . k
proof
let p be Point of (TOP-REAL n); ::_thesis: ( p . k <> 0 implies (f . p) . k = p . k )
assume A14: p . k <> 0 ; ::_thesis: (f . p) . k = p . k
set fp = f . p;
set pk = p . k;
set pN = (p . k) * N0k;
set ppN = p - ((p . k) * N0k);
A15: (f . (p - ((p . k) * N0k))) + (f . ((p . k) * N0k)) = f . ((p - ((p . k) * N0k)) + ((p . k) * N0k)) by VECTSP_1:def_20;
( len (f . (p - ((p . k) * N0k))) = n & len (f . ((p . k) * N0k)) = n ) by CARD_1:def_7;
then A16: |.(f . ((p - ((p . k) * N0k)) + ((p . k) * N0k))).| ^2 = ((|.(f . (p - ((p . k) * N0k))).| ^2) + (2 * |((f . ((p . k) * N0k)),(f . (p - ((p . k) * N0k))))|)) + (|.(f . ((p . k) * N0k)).| ^2) by A15, EUCLID_2:11;
A17: (n |-> (p . k)) . k = p . k by A6, FINSEQ_2:57;
A18: (p . k) * ((0* n) +* (k,1)) = mlt ((n |-> (p . k)),((0* n) +* (k,1))) by A12, RVSUM_1:63
.= (0* n) +* (k,((p . k) * 1)) by A17, TOPREALC:15
.= (0* n) +* (k,(p . k)) ;
len (f . p) = n by CARD_1:def_7;
then A19: dom (f . p) = Seg n by FINSEQ_1:def_3;
A20: len (p - ((p . k) * N0k)) = n by CARD_1:def_7;
then dom (p - ((p . k) * N0k)) = Seg n by FINSEQ_1:def_3;
then A21: (p - ((p . k) * N0k)) . k = (p . k) - (((p . k) * N0k) . k) by A6, VALUED_1:13;
len ((p . k) * N0k) = n by CARD_1:def_7;
then A22: |.((p - ((p . k) * N0k)) + ((p . k) * N0k)).| ^2 = ((|.(p - ((p . k) * N0k)).| ^2) + (2 * |(((p . k) * N0k),(p - ((p . k) * N0k)))|)) + (|.((p . k) * N0k).| ^2) by A20, EUCLID_2:11;
(p . k) * N0k in Lin A by A7, A8, RLSUB_1:21;
then A23: (p . k) * N0k in the carrier of (Lin A) by STRUCT_0:def_5;
then A24: (p . k) * N0k = (f | (Lin A)) . ((p . k) * N0k) by A3, FUNCT_1:17
.= f . ((p . k) * N0k) by A9, A23, FUNCT_1:49 ;
( |.((p - ((p . k) * N0k)) + ((p . k) * N0k)).| = |.(f . ((p - ((p . k) * N0k)) + ((p . k) * N0k))).| & |.(p - ((p . k) * N0k)).| = |.(f . (p - ((p . k) * N0k))).| ) by A1, Def4;
then |(((p . k) * N0k),(p - ((p . k) * N0k)))| = (p . k) * ((f . (p - ((p . k) * N0k))) . k) by A18, A16, A22, A24, TOPREALC:16;
then A25: (p . k) * ((p - ((p . k) * N0k)) . k) = (p . k) * ((f . (p - ((p . k) * N0k))) . k) by A18, TOPREALC:16;
((p . k) * N0k) . k = p . k by A6, A5, A18, FUNCT_7:31;
then A26: (f . (p - ((p . k) * N0k))) . k = 0 by A14, A21, A25;
(p - ((p . k) * N0k)) + ((p . k) * N0k) = p - (((p . k) * N0k) - ((p . k) * N0k)) by EUCLID:47
.= p - (0. (TOP-REAL n)) by EUCLID:42
.= p by RLVECT_1:13 ;
then (f . p) . k = ((f . (p - ((p . k) * N0k))) . k) + ((f . ((p . k) * N0k)) . k) by A6, A15, A19, VALUED_1:def_1
.= ((f . (p - ((p . k) * N0k))) . k) + (p . k) by A6, A5, A18, A24, FUNCT_7:31 ;
hence (f . p) . k = p . k by A26; ::_thesis: verum
end;
percases ( p . k <> 0 or p . k = 0 ) ;
suppose p . k <> 0 ; ::_thesis: (f . p) . k = p . k
hence (f . p) . k = p . k by A13; ::_thesis: verum
end;
suppose p . k = 0 ; ::_thesis: (f . p) . k = p . k
len (f . p) = n by CARD_1:def_7;
then A27: |.((f . p) + N0k).| ^2 = ((|.(f . p).| ^2) + (2 * |(N0k,(f . p))|)) + (|.N0k.| ^2) by A10, EUCLID_2:11;
len p = n by CARD_1:def_7;
then A28: |.(p + N0k).| ^2 = ((|.p.| ^2) + (2 * |(N0k,p)|)) + (|.N0k.| ^2) by A10, EUCLID_2:11;
A29: N0k in the carrier of (Lin A) by A7, A8, STRUCT_0:def_5;
then N0k = (f | (Lin A)) . N0k by A3, FUNCT_1:17
.= f . N0k by A9, A29, FUNCT_1:49 ;
then A30: f . (p + N0k) = (f . p) + N0k by VECTSP_1:def_20;
( |.(p + N0k).| = |.(f . (p + N0k)).| & |.(f . p).| = |.p.| ) by A1, Def4;
then |(N0k,(f . p))| = 1 * (p . k) by A27, A28, A30, TOPREALC:16;
then p . k = 1 * ((f . p) . k) by TOPREALC:16;
hence (f . p) . k = p . k ; ::_thesis: verum
end;
end;
end;
theorem Th33: :: MATRTOP3:33
for X being set
for n being Nat
for p being Point of (TOP-REAL n)
for f being rotation Function of (TOP-REAL n),(TOP-REAL n) st f is X -support-yielding & ( for i being Nat st i in X /\ (Seg n) holds
p . i = 0 ) holds
f . p = p
proof
let X be set ; ::_thesis: for n being Nat
for p being Point of (TOP-REAL n)
for f being rotation Function of (TOP-REAL n),(TOP-REAL n) st f is X -support-yielding & ( for i being Nat st i in X /\ (Seg n) holds
p . i = 0 ) holds
f . p = p
let n be Nat; ::_thesis: for p being Point of (TOP-REAL n)
for f being rotation Function of (TOP-REAL n),(TOP-REAL n) st f is X -support-yielding & ( for i being Nat st i in X /\ (Seg n) holds
p . i = 0 ) holds
f . p = p
let p be Point of (TOP-REAL n); ::_thesis: for f being rotation Function of (TOP-REAL n),(TOP-REAL n) st f is X -support-yielding & ( for i being Nat st i in X /\ (Seg n) holds
p . i = 0 ) holds
f . p = p
set TR = TOP-REAL n;
let f be rotation Function of (TOP-REAL n),(TOP-REAL n); ::_thesis: ( f is X -support-yielding & ( for i being Nat st i in X /\ (Seg n) holds
p . i = 0 ) implies f . p = p )
assume that
A1: f is X -support-yielding and
A2: for i being Nat st i in X /\ (Seg n) holds
p . i = 0 ; ::_thesis: f . p = p
set sp = sqr p;
set sfp = sqr (f . p);
A3: Sum (sqr p) >= 0 by RVSUM_1:86;
( Sum (sqr (f . p)) >= 0 & |.(f . p).| = |.p.| ) by Def4, RVSUM_1:86;
then A4: Sum (sqr p) = Sum (sqr (f . p)) by A3, SQUARE_1:28;
A5: len p = n by CARD_1:def_7;
p = @ (@ p) ;
then A6: len (sqr p) = n by A5, RVSUM_1:143;
A7: len (f . p) = n by CARD_1:def_7;
@ (@ (f . p)) = f . p ;
then len (sqr (f . p)) = n by A7, RVSUM_1:143;
then reconsider sp = sqr p, sfp = sqr (f . p) as Element of n -tuples_on REAL by A6, FINSEQ_2:92;
A8: dom f = the carrier of (TOP-REAL n) by FUNCT_2:52;
A9: for i being Nat st i in Seg n holds
sp . i <= sfp . i
proof
let i be Nat; ::_thesis: ( i in Seg n implies sp . i <= sfp . i )
A10: ( sp . i = (p . i) ^2 & sfp . i = ((f . p) . i) ^2 ) by VALUED_1:11;
assume A11: i in Seg n ; ::_thesis: sp . i <= sfp . i
percases ( i in X or not i in X ) ;
suppose i in X ; ::_thesis: sp . i <= sfp . i
then i in X /\ (Seg n) by A11, XBOOLE_0:def_4;
then p . i = 0 by A2;
hence sp . i <= sfp . i by A10, XREAL_1:63; ::_thesis: verum
end;
suppose not i in X ; ::_thesis: sp . i <= sfp . i
hence sp . i <= sfp . i by A1, A8, A10, Def1; ::_thesis: verum
end;
end;
end;
for i being Nat st 1 <= i & i <= n holds
p . i = (f . p) . i
proof
let i be Nat; ::_thesis: ( 1 <= i & i <= n implies p . i = (f . p) . i )
A12: sp . i = (p . i) ^2 by VALUED_1:11;
assume ( 1 <= i & i <= n ) ; ::_thesis: p . i = (f . p) . i
then A13: i in Seg n by FINSEQ_1:1;
then A14: ( sp . i >= sfp . i & sp . i <= sfp . i ) by A4, A9, RVSUM_1:83;
percases ( i in X or not i in X ) ;
suppose i in X ; ::_thesis: p . i = (f . p) . i
then A15: i in X /\ (Seg n) by A13, XBOOLE_0:def_4;
then p . i = 0 by A2;
then ((f . p) . i) ^2 = 0 by A12, A14, VALUED_1:11;
then (f . p) . i = 0 ;
hence p . i = (f . p) . i by A2, A15; ::_thesis: verum
end;
suppose not i in X ; ::_thesis: p . i = (f . p) . i
hence p . i = (f . p) . i by A1, A8, Def1; ::_thesis: verum
end;
end;
end;
hence f . p = p by A5, A7, FINSEQ_1:14; ::_thesis: verum
end;
theorem Th34: :: MATRTOP3:34
for i, n being Nat
for p being Point of (TOP-REAL n) st i in Seg n & n >= 2 holds
ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is base_rotation & f . p = p +* (i,(- (p . i))) )
proof
let i, n be Nat; ::_thesis: for p being Point of (TOP-REAL n) st i in Seg n & n >= 2 holds
ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is base_rotation & f . p = p +* (i,(- (p . i))) )
let p be Point of (TOP-REAL n); ::_thesis: ( i in Seg n & n >= 2 implies ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is base_rotation & f . p = p +* (i,(- (p . i))) ) )
set TR = TOP-REAL n;
assume that
A1: i in Seg n and
A2: n >= 2 ; ::_thesis: ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is base_rotation & f . p = p +* (i,(- (p . i))) )
A3: {i} c= Seg n by A1, ZFMISC_1:31;
A4: 1 <= i by A1, FINSEQ_1:1;
( card (Seg n) = n & card {i} = 1 ) by CARD_2:42, FINSEQ_1:57;
then {i} <> Seg n by A2;
then {i} c< Seg n by A3, XBOOLE_0:def_8;
then consider j being set such that
A5: j in Seg n and
A6: not j in {i} by XBOOLE_0:6;
reconsider j = j as Nat by A5;
A7: j <> i by A6, TARSKI:def_1;
A8: 1 <= j by A5, FINSEQ_1:1;
set p0 = p +* (i,(- (p . i)));
A9: len (p +* (i,(- (p . i)))) = len p by FUNCT_7:97;
A10: len p = n by CARD_1:def_7;
then A11: dom p = Seg n by FINSEQ_1:def_3;
A12: i <= n by A1, FINSEQ_1:1;
( (p . i) * (p . i) >= 0 & (p . j) * (p . j) >= 0 ) by XREAL_1:63;
then A13: ( 0 ^2 = 0* 0 & 0 <= ((p . i) ^2) + ((p . j) ^2) ) ;
A14: j <= n by A5, FINSEQ_1:1;
percases ( i < j or j < i ) by A7, XXREAL_0:1;
supposeA15: i < j ; ::_thesis: ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is base_rotation & f . p = p +* (i,(- (p . i))) )
then consider r being real number such that
A16: ((Mx2Tran (Rotation (i,j,n,r))) . p) . i = 0 by A4, A13, A14, Th24;
set s = sin r;
set c = cos r;
A17: 0 = ((p . i) * (cos r)) + ((p . j) * (- (sin r))) by A4, A14, A15, A16, Th21;
reconsider M = Mx2Tran (Rotation (i,j,n,(r + r))) as base_rotation Function of (TOP-REAL n),(TOP-REAL n) by A4, A14, A15, Th28;
set Mp = M . p;
A18: ( cos (r + r) = ((cos r) * (cos r)) - ((sin r) * (sin r)) & sin (r + r) = ((sin r) * (cos r)) + ((sin r) * (cos r)) ) by SIN_COS:75;
A19: M is {i,j} -support-yielding by A4, A14, A15, Th26;
A20: for k being Nat st 1 <= k & k <= n holds
(p +* (i,(- (p . i)))) . k = (M . p) . k
proof
let k be Nat; ::_thesis: ( 1 <= k & k <= n implies (p +* (i,(- (p . i)))) . k = (M . p) . k )
assume ( 1 <= k & k <= n ) ; ::_thesis: (p +* (i,(- (p . i)))) . k = (M . p) . k
then A21: k in Seg n by FINSEQ_1:1;
percases ( i = k or j = k or ( i <> k & j <> k ) ) ;
supposeA22: i = k ; ::_thesis: (p +* (i,(- (p . i)))) . k = (M . p) . k
hence (p +* (i,(- (p . i)))) . k = - ((p . i) * 1) by A11, A21, FUNCT_7:31
.= - ((p . i) * (((sin r) * (sin r)) + ((cos r) * (cos r)))) by SIN_COS:29
.= (((p . i) * (((cos r) * (cos r)) - ((sin r) * (sin r)))) - (((p . i) * (cos r)) * (cos r))) - (((p . i) * (cos r)) * (cos r))
.= (((p . i) * (((cos r) * (cos r)) - ((sin r) * (sin r)))) - (((p . j) * (sin r)) * (cos r))) - (((p . j) * (sin r)) * (cos r)) by A17
.= ((p . i) * (((cos r) * (cos r)) - ((sin r) * (sin r)))) + ((p . j) * (- (((sin r) * (cos r)) + ((sin r) * (cos r)))))
.= (M . p) . k by A4, A14, A15, A18, A22, Th21 ;
::_thesis: verum
end;
supposeA23: j = k ; ::_thesis: (p +* (i,(- (p . i)))) . k = (M . p) . k
hence (p +* (i,(- (p . i)))) . k = (p . j) * 1 by A15, FUNCT_7:32
.= (p . j) * (((sin r) * (sin r)) + ((cos r) * (cos r))) by SIN_COS:29
.= ((((p . j) * (sin r)) * (sin r)) + (((p . j) * (sin r)) * (sin r))) + ((p . j) * (((cos r) * (cos r)) - ((sin r) * (sin r))))
.= ((((p . i) * (cos r)) * (sin r)) + (((p . i) * (cos r)) * (sin r))) + ((p . j) * (((cos r) * (cos r)) - ((sin r) * (sin r)))) by A17
.= ((p . i) * (((sin r) * (cos r)) + ((sin r) * (cos r)))) + ((p . j) * (((cos r) * (cos r)) - ((sin r) * (sin r))))
.= (M . p) . k by A4, A14, A15, A18, A23, Th22 ;
::_thesis: verum
end;
supposeA24: ( i <> k & j <> k ) ; ::_thesis: (p +* (i,(- (p . i)))) . k = (M . p) . k
A25: dom M = the carrier of (TOP-REAL n) by FUNCT_2:52;
( (p +* (i,(- (p . i)))) . k = p . k & not k in {i,j} ) by A24, FUNCT_7:32, TARSKI:def_2;
hence (p +* (i,(- (p . i)))) . k = (M . p) . k by A19, A25, Def1; ::_thesis: verum
end;
end;
end;
take M ; ::_thesis: ( M is base_rotation & M . p = p +* (i,(- (p . i))) )
len (M . p) = n by CARD_1:def_7;
hence ( M is base_rotation & M . p = p +* (i,(- (p . i))) ) by A9, A10, A20, FINSEQ_1:14; ::_thesis: verum
end;
supposeA26: j < i ; ::_thesis: ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is base_rotation & f . p = p +* (i,(- (p . i))) )
then consider r being real number such that
A27: ((Mx2Tran (Rotation (j,i,n,r))) . p) . i = 0 by A8, A12, A13, Th25;
set s = sin r;
set c = cos r;
A28: 0 = ((p . j) * (sin r)) + ((p . i) * (cos r)) by A8, A12, A26, A27, Th22;
reconsider M = Mx2Tran (Rotation (j,i,n,(r + r))) as base_rotation Function of (TOP-REAL n),(TOP-REAL n) by A8, A12, A26, Th28;
set Mp = M . p;
A29: ( cos (r + r) = ((cos r) * (cos r)) - ((sin r) * (sin r)) & sin (r + r) = ((sin r) * (cos r)) + ((sin r) * (cos r)) ) by SIN_COS:75;
A30: M is {i,j} -support-yielding by A8, A12, A26, Th26;
A31: for k being Nat st 1 <= k & k <= n holds
(p +* (i,(- (p . i)))) . k = (M . p) . k
proof
let k be Nat; ::_thesis: ( 1 <= k & k <= n implies (p +* (i,(- (p . i)))) . k = (M . p) . k )
assume ( 1 <= k & k <= n ) ; ::_thesis: (p +* (i,(- (p . i)))) . k = (M . p) . k
then A32: k in Seg n by FINSEQ_1:1;
percases ( i = k or j = k or ( i <> k & j <> k ) ) ;
supposeA33: i = k ; ::_thesis: (p +* (i,(- (p . i)))) . k = (M . p) . k
hence (p +* (i,(- (p . i)))) . k = - ((p . i) * 1) by A11, A32, FUNCT_7:31
.= - ((p . i) * (((sin r) * (sin r)) + ((cos r) * (cos r)))) by SIN_COS:29
.= ((- (((p . i) * (cos r)) * (cos r))) + ((- ((p . i) * (cos r))) * (cos r))) + ((p . i) * (((cos r) * (cos r)) - ((sin r) * (sin r))))
.= ((((p . j) * (sin r)) * (cos r)) + (((p . j) * (sin r)) * (cos r))) + ((p . i) * (((cos r) * (cos r)) - ((sin r) * (sin r)))) by A28
.= ((p . j) * (((sin r) * (cos r)) + ((sin r) * (cos r)))) + ((p . i) * (((cos r) * (cos r)) - ((sin r) * (sin r))))
.= (M . p) . k by A8, A12, A26, A29, A33, Th22 ;
::_thesis: verum
end;
supposeA34: j = k ; ::_thesis: (p +* (i,(- (p . i)))) . k = (M . p) . k
hence (p +* (i,(- (p . i)))) . k = (p . j) * 1 by A26, FUNCT_7:32
.= (p . j) * (((sin r) * (sin r)) + ((cos r) * (cos r))) by SIN_COS:29
.= (((p . j) * (((cos r) * (cos r)) - ((sin r) * (sin r)))) + (((p . j) * (sin r)) * (sin r))) + (((p . j) * (sin r)) * (sin r))
.= (((p . j) * (((cos r) * (cos r)) - ((sin r) * (sin r)))) + ((- ((p . i) * (cos r))) * (sin r))) + ((- ((p . i) * (cos r))) * (sin r)) by A28
.= ((p . j) * (((cos r) * (cos r)) - ((sin r) * (sin r)))) + ((p . i) * (- (((sin r) * (cos r)) + ((sin r) * (cos r)))))
.= (M . p) . k by A8, A12, A26, A29, A34, Th21 ;
::_thesis: verum
end;
supposeA35: ( i <> k & j <> k ) ; ::_thesis: (p +* (i,(- (p . i)))) . k = (M . p) . k
A36: dom M = the carrier of (TOP-REAL n) by FUNCT_2:52;
( (p +* (i,(- (p . i)))) . k = p . k & not k in {i,j} ) by A35, FUNCT_7:32, TARSKI:def_2;
hence (p +* (i,(- (p . i)))) . k = (M . p) . k by A30, A36, Def1; ::_thesis: verum
end;
end;
end;
take M ; ::_thesis: ( M is base_rotation & M . p = p +* (i,(- (p . i))) )
len (M . p) = n by CARD_1:def_7;
hence ( M is base_rotation & M . p = p +* (i,(- (p . i))) ) by A9, A10, A31, FINSEQ_1:14; ::_thesis: verum
end;
end;
end;
Lm9: for X being set
for n, i being Nat
for f being additive homogeneous rotation Function of (TOP-REAL n),(TOP-REAL n) st f is X -support-yielding & not i in X holds
f . (Base_FinSeq (n,i)) = Base_FinSeq (n,i)
proof
let X be set ; ::_thesis: for n, i being Nat
for f being additive homogeneous rotation Function of (TOP-REAL n),(TOP-REAL n) st f is X -support-yielding & not i in X holds
f . (Base_FinSeq (n,i)) = Base_FinSeq (n,i)
let n, i be Nat; ::_thesis: for f being additive homogeneous rotation Function of (TOP-REAL n),(TOP-REAL n) st f is X -support-yielding & not i in X holds
f . (Base_FinSeq (n,i)) = Base_FinSeq (n,i)
let f be additive homogeneous rotation Function of (TOP-REAL n),(TOP-REAL n); ::_thesis: ( f is X -support-yielding & not i in X implies f . (Base_FinSeq (n,i)) = Base_FinSeq (n,i) )
set B = Base_FinSeq (n,i);
assume that
A1: f is X -support-yielding and
A2: not i in X ; ::_thesis: f . (Base_FinSeq (n,i)) = Base_FinSeq (n,i)
A3: now__::_thesis:_for_j_being_Nat_st_j_in_X_/\_(Seg_n)_holds_
(Base_FinSeq_(n,i))_._j_=_0
let j be Nat; ::_thesis: ( j in X /\ (Seg n) implies (Base_FinSeq (n,i)) . j = 0 )
assume A4: j in X /\ (Seg n) ; ::_thesis: (Base_FinSeq (n,i)) . j = 0
then j in Seg n by XBOOLE_0:def_4;
then A5: ( 1 <= j & j <= n ) by FINSEQ_1:1;
j <> i by A2, A4, XBOOLE_0:def_4;
hence (Base_FinSeq (n,i)) . j = 0 by A5, MATRIXR2:76; ::_thesis: verum
end;
len (Base_FinSeq (n,i)) = n by MATRIXR2:74;
then Base_FinSeq (n,i) is Point of (TOP-REAL n) by TOPREAL3:46;
hence f . (Base_FinSeq (n,i)) = Base_FinSeq (n,i) by A1, A3, Th33; ::_thesis: verum
end;
theorem Th35: :: MATRTOP3:35
for i, n being Nat
for f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st f is {i} -support-yielding & f is rotation & not AutMt f = AxialSymmetry (i,n) holds
AutMt f = 1. (F_Real,n)
proof
let i, n be Nat; ::_thesis: for f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st f is {i} -support-yielding & f is rotation & not AutMt f = AxialSymmetry (i,n) holds
AutMt f = 1. (F_Real,n)
let f be additive homogeneous Function of (TOP-REAL n),(TOP-REAL n); ::_thesis: ( f is {i} -support-yielding & f is rotation & not AutMt f = AxialSymmetry (i,n) implies AutMt f = 1. (F_Real,n) )
set TR = TOP-REAL n;
set S = { (Base_FinSeq (n,j)) where j is Element of NAT : ( 1 <= j & j <= n ) } ;
{ (Base_FinSeq (n,j)) where j is Element of NAT : ( 1 <= j & j <= n ) } c= the carrier of (TOP-REAL n)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (Base_FinSeq (n,j)) where j is Element of NAT : ( 1 <= j & j <= n ) } or x in the carrier of (TOP-REAL n) )
assume x in { (Base_FinSeq (n,j)) where j is Element of NAT : ( 1 <= j & j <= n ) } ; ::_thesis: x in the carrier of (TOP-REAL n)
then consider j being Element of NAT such that
A1: x = Base_FinSeq (n,j) and
( 1 <= j & j <= n ) ;
len (Base_FinSeq (n,j)) = n by MATRIXR2:74;
hence x in the carrier of (TOP-REAL n) by A1, TOPREAL3:46; ::_thesis: verum
end;
then reconsider S = { (Base_FinSeq (n,j)) where j is Element of NAT : ( 1 <= j & j <= n ) } as Subset of (TOP-REAL n) ;
set M = Mx2Tran (AxialSymmetry (n,i));
assume A2: ( f is {i} -support-yielding & f is rotation ) ; ::_thesis: ( AutMt f = AxialSymmetry (i,n) or AutMt f = 1. (F_Real,n) )
A3: id (TOP-REAL n) = Mx2Tran (1. (F_Real,n)) by MATRTOP1:33;
then A4: AutMt (id (TOP-REAL n)) = 1. (F_Real,n) by Def6;
A5: dom f = the carrier of (TOP-REAL n) by FUNCT_2:52;
percases ( not i in Seg n or i in Seg n ) ;
supposeA6: not i in Seg n ; ::_thesis: ( AutMt f = AxialSymmetry (i,n) or AutMt f = 1. (F_Real,n) )
now__::_thesis:_for_p_being_Point_of_(TOP-REAL_n)_holds_f_._p_=_p
let p be Point of (TOP-REAL n); ::_thesis: f . p = p
A7: now__::_thesis:_for_j_being_Nat_st_1_<=_j_&_j_<=_n_holds_
(f_._p)_._j_=_p_._j
let j be Nat; ::_thesis: ( 1 <= j & j <= n implies (f . p) . j = p . j )
assume ( 1 <= j & j <= n ) ; ::_thesis: (f . p) . j = p . j
then j <> i by A6, FINSEQ_1:1;
then not j in {i} by TARSKI:def_1;
hence (f . p) . j = p . j by A2, A5, Def1; ::_thesis: verum
end;
( len (f . p) = n & len p = n ) by CARD_1:def_7;
hence f . p = p by A7, FINSEQ_1:14; ::_thesis: verum
end;
then f = id (TOP-REAL n) by FUNCT_2:124;
hence ( AutMt f = AxialSymmetry (i,n) or AutMt f = 1. (F_Real,n) ) by A3, Def6; ::_thesis: verum
end;
supposeA8: i in Seg n ; ::_thesis: ( AutMt f = AxialSymmetry (i,n) or AutMt f = 1. (F_Real,n) )
then ( 1 <= i & i <= n ) by FINSEQ_1:1;
then Base_FinSeq (n,i) in S by A8;
then reconsider B = Base_FinSeq (n,i) as Point of (TOP-REAL n) ;
B = (0* n) +* (i,1) by MATRIXR2:def_4;
then A9: |.B.| = abs 1 by A8, TOPREALC:13
.= 1 by ABSVALUE:def_1 ;
set B0 = (0* n) +* (i,((f . B) . i));
A10: len (0* n) = n by CARD_1:def_7;
A11: for j being Nat st 1 <= j & j <= n holds
(f . B) . j = ((0* n) +* (i,((f . B) . i))) . j
proof
let j be Nat; ::_thesis: ( 1 <= j & j <= n implies (f . B) . j = ((0* n) +* (i,((f . B) . i))) . j )
assume A12: ( 1 <= j & j <= n ) ; ::_thesis: (f . B) . j = ((0* n) +* (i,((f . B) . i))) . j
A13: j in dom (0* n) by A10, A12, FINSEQ_3:25;
percases ( j = i or j <> i ) ;
suppose j = i ; ::_thesis: (f . B) . j = ((0* n) +* (i,((f . B) . i))) . j
hence ((0* n) +* (i,((f . B) . i))) . j = (f . B) . j by A13, FUNCT_7:31; ::_thesis: verum
end;
supposeA14: j <> i ; ::_thesis: (f . B) . j = ((0* n) +* (i,((f . B) . i))) . j
then A15: not j in {i} by TARSKI:def_1;
thus ((0* n) +* (i,((f . B) . i))) . j = (0* n) . j by A14, FUNCT_7:32
.= 0
.= B . j by A12, A14, MATRIXR2:76
.= (f . B) . j by A2, A5, A15, Def1 ; ::_thesis: verum
end;
end;
end;
( len ((0* n) +* (i,((f . B) . i))) = len (0* n) & len (f . B) = n ) by CARD_1:def_7, FUNCT_7:97;
then A16: f . B = (0* n) +* (i,((f . B) . i)) by A10, A11, FINSEQ_1:14;
then A17: |.B.| = |.((0* n) +* (i,((f . B) . i))).| by A2, Def4
.= abs ((f . B) . i) by A8, TOPREALC:13 ;
A18: for h being additive homogeneous rotation Function of (TOP-REAL n),(TOP-REAL n) st h | S = id S holds
h = id (TOP-REAL n)
proof
let h be additive homogeneous rotation Function of (TOP-REAL n),(TOP-REAL n); ::_thesis: ( h | S = id S implies h = id (TOP-REAL n) )
A19: dom (id (TOP-REAL n)) = the carrier of (TOP-REAL n) by FUNCT_2:def_1;
assume A20: h | S = id S ; ::_thesis: h = id (TOP-REAL n)
A21: for x being set st x in dom (id (TOP-REAL n)) holds
(id (TOP-REAL n)) . x = h . x
proof
let x be set ; ::_thesis: ( x in dom (id (TOP-REAL n)) implies (id (TOP-REAL n)) . x = h . x )
assume A22: x in dom (id (TOP-REAL n)) ; ::_thesis: (id (TOP-REAL n)) . x = h . x
then reconsider p = x as Point of (TOP-REAL n) by FUNCT_2:def_1;
set hp = h . p;
A23: ( len p = n & len (h . p) = n ) by CARD_1:def_7;
A24: now__::_thesis:_for_j_being_Nat_st_1_<=_j_&_j_<=_n_holds_
p_._j_=_(h_._p)_._j
let j be Nat; ::_thesis: ( 1 <= j & j <= n implies p . j = (h . p) . j )
assume A25: ( 1 <= j & j <= n ) ; ::_thesis: p . j = (h . p) . j
then A26: j in Seg n by FINSEQ_1:1;
then Base_FinSeq (n,j) in S by A25;
then Base_FinSeq (n,j) in Lin S by RLVECT_3:15;
hence p . j = (h . p) . j by A20, A26, Th32; ::_thesis: verum
end;
(id (TOP-REAL n)) . x = x by A19, A22, FUNCT_1:17;
hence (id (TOP-REAL n)) . x = h . x by A23, A24, FINSEQ_1:14; ::_thesis: verum
end;
dom h = the carrier of (TOP-REAL n) by FUNCT_2:def_1;
hence h = id (TOP-REAL n) by A19, A21, FUNCT_1:2; ::_thesis: verum
end;
percases ( (f . B) . i >= 0 or (f . B) . i < 0 ) ;
supposeA27: (f . B) . i >= 0 ; ::_thesis: ( AutMt f = AxialSymmetry (i,n) or AutMt f = 1. (F_Real,n) )
A28: dom (f | S) = S by A5, RELAT_1:62;
A29: (f . B) . i = 1 by A9, A17, A27, ABSVALUE:def_1;
A30: for x being set st x in S holds
(f | S) . x = (id S) . x
proof
let x be set ; ::_thesis: ( x in S implies (f | S) . x = (id S) . x )
assume A31: x in S ; ::_thesis: (f | S) . x = (id S) . x
then consider j being Element of NAT such that
A32: x = Base_FinSeq (n,j) and
1 <= j and
j <= n ;
A33: ( (f | S) . x = f . x & (id S) . x = x ) by A28, A31, FUNCT_1:17, FUNCT_1:47;
percases ( j = i or j <> i ) ;
suppose j = i ; ::_thesis: (f | S) . x = (id S) . x
hence (f | S) . x = (id S) . x by A16, A29, A32, A33, MATRIXR2:def_4; ::_thesis: verum
end;
suppose j <> i ; ::_thesis: (f | S) . x = (id S) . x
then not j in {i} by TARSKI:def_1;
hence (f | S) . x = (id S) . x by A2, A32, A33, Lm9; ::_thesis: verum
end;
end;
end;
dom (id S) = S ;
hence ( AutMt f = AxialSymmetry (i,n) or AutMt f = 1. (F_Real,n) ) by A2, A4, A18, A28, A30, FUNCT_1:2; ::_thesis: verum
end;
supposeA34: (f . B) . i < 0 ; ::_thesis: ( AutMt f = AxialSymmetry (i,n) or AutMt f = 1. (F_Real,n) )
set MA = Mx2Tran (AxialSymmetry (i,n));
Mx2Tran (AxialSymmetry (i,n)) is rotation by A8, Th27;
then reconsider MAf = (Mx2Tran (AxialSymmetry (i,n))) * f as additive homogeneous rotation Function of (TOP-REAL n),(TOP-REAL n) by A2;
A35: dom MAf = the carrier of (TOP-REAL n) by FUNCT_2:52;
then A36: dom (MAf | S) = S by RELAT_1:62;
A37: ( Mx2Tran (AxialSymmetry (i,n)) is {i} -support-yielding & {i} \/ {i} = {i} ) by A8, Th11;
A38: for x being set st x in S holds
(MAf | S) . x = (id S) . x
proof
let x be set ; ::_thesis: ( x in S implies (MAf | S) . x = (id S) . x )
assume A39: x in S ; ::_thesis: (MAf | S) . x = (id S) . x
then consider j being Element of NAT such that
A40: x = Base_FinSeq (n,j) and
( 1 <= j & j <= n ) ;
A41: ( (MAf | S) . x = MAf . x & (id S) . x = x ) by A36, A39, FUNCT_1:17, FUNCT_1:47;
percases ( j = i or j <> i ) ;
supposeA42: j = i ; ::_thesis: (MAf | S) . x = (id S) . x
A43: for k being Nat st 1 <= k & k <= n holds
(MAf . B) . k = B . k
proof
let k be Nat; ::_thesis: ( 1 <= k & k <= n implies (MAf . B) . k = B . k )
assume A44: ( 1 <= k & k <= n ) ; ::_thesis: (MAf . B) . k = B . k
then A45: k in Seg n by FINSEQ_1:1;
percases ( k = i or k <> i ) ;
supposeA46: k = i ; ::_thesis: (MAf . B) . k = B . k
thus (MAf . B) . k = ((Mx2Tran (AxialSymmetry (i,n))) . (f . B)) . k by A35, FUNCT_1:12
.= - ((f . B) . k) by A45, A46, Th9
.= - (- 1) by A9, A17, A34, A46, ABSVALUE:def_1
.= B . k by A44, A46, MATRIXR2:75 ; ::_thesis: verum
end;
supposeA47: k <> i ; ::_thesis: (MAf . B) . k = B . k
then A48: not k in {i} by TARSKI:def_1;
thus (MAf . B) . k = ((Mx2Tran (AxialSymmetry (i,n))) . (f . B)) . k by A35, FUNCT_1:12
.= (f . B) . k by A8, A47, Th8
.= B . k by A2, A5, A48, Def1 ; ::_thesis: verum
end;
end;
end;
( len (MAf . B) = n & len B = n ) by CARD_1:def_7;
hence (MAf | S) . x = (id S) . x by A40, A41, A42, A43, FINSEQ_1:14; ::_thesis: verum
end;
suppose j <> i ; ::_thesis: (MAf | S) . x = (id S) . x
then not j in {i} by TARSKI:def_1;
hence (MAf | S) . x = (id S) . x by A2, A37, A40, A41, Lm9; ::_thesis: verum
end;
end;
end;
dom (id S) = S ;
then A49: MAf = id (TOP-REAL n) by A18, A36, A38, FUNCT_1:2;
A50: dom (Mx2Tran (AxialSymmetry (i,n))) = [#] (TOP-REAL n) by TOPS_2:def_5;
set R = AutMt f;
A51: rng (Mx2Tran (AxialSymmetry (i,n))) = [#] (TOP-REAL n) by TOPS_2:def_5;
A52: Mx2Tran (AxialSymmetry (i,n)) is one-to-one by TOPS_2:def_5;
A53: the carrier of (TOP-REAL n) c= rng f
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in the carrier of (TOP-REAL n) or x in rng f )
assume A54: x in the carrier of (TOP-REAL n) ; ::_thesis: x in rng f
then A55: (Mx2Tran (AxialSymmetry (i,n))) . x in rng (Mx2Tran (AxialSymmetry (i,n))) by A50, FUNCT_1:def_3;
then A56: MAf . ((Mx2Tran (AxialSymmetry (i,n))) . x) = (Mx2Tran (AxialSymmetry (i,n))) . (f . ((Mx2Tran (AxialSymmetry (i,n))) . x)) by A35, A51, FUNCT_1:12;
( f . ((Mx2Tran (AxialSymmetry (i,n))) . x) in dom (Mx2Tran (AxialSymmetry (i,n))) & MAf . ((Mx2Tran (AxialSymmetry (i,n))) . x) = (Mx2Tran (AxialSymmetry (i,n))) . x ) by A35, A49, A51, A55, FUNCT_1:11, FUNCT_1:18;
then x = f . ((Mx2Tran (AxialSymmetry (i,n))) . x) by A50, A52, A54, A56, FUNCT_1:def_4;
hence x in rng f by A5, A51, A55, FUNCT_1:def_3; ::_thesis: verum
end;
rng f c= the carrier of (TOP-REAL n) by RELAT_1:def_19;
then rng f = the carrier of (TOP-REAL n) by A53, XBOOLE_0:def_10;
then A57: f = (Mx2Tran (AxialSymmetry (i,n))) " by A49, A51, A50, A52, FUNCT_1:42;
( f = Mx2Tran (AutMt f) & Det (AxialSymmetry (i,n)) <> 0. F_Real ) by A8, Def6, Th4;
then AutMt f = (AxialSymmetry (i,n)) ~ by A57, MATRTOP1:43
.= AxialSymmetry (i,n) by A8, Th7 ;
hence ( AutMt f = AxialSymmetry (i,n) or AutMt f = 1. (F_Real,n) ) ; ::_thesis: verum
end;
end;
end;
end;
end;
theorem Th36: :: MATRTOP3:36
for n being Nat
for f1 being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st f1 is rotation holds
ex f2 being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f2 is base_rotation & f2 * f1 is {n} -support-yielding )
proof
let n be Nat; ::_thesis: for f1 being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st f1 is rotation holds
ex f2 being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f2 is base_rotation & f2 * f1 is {n} -support-yielding )
let f1 be additive homogeneous Function of (TOP-REAL n),(TOP-REAL n); ::_thesis: ( f1 is rotation implies ex f2 being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f2 is base_rotation & f2 * f1 is {n} -support-yielding ) )
set TR = TOP-REAL n;
assume A1: f1 is rotation ; ::_thesis: ex f2 being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f2 is base_rotation & f2 * f1 is {n} -support-yielding )
set cTR = the carrier of (TOP-REAL n);
set f = f1;
A2: dom f1 = the carrier of (TOP-REAL n) by FUNCT_2:52;
A3: rng f1 c= the carrier of (TOP-REAL n) by RELAT_1:def_19;
percases ( n = 0 or n > 0 ) ;
supposeA4: n = 0 ; ::_thesis: ex f2 being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f2 is base_rotation & f2 * f1 is {n} -support-yielding )
take I = id (TOP-REAL n); ::_thesis: ( I is base_rotation & I * f1 is {n} -support-yielding )
A5: dom (id the carrier of (TOP-REAL n)) = the carrier of (TOP-REAL n) ;
A6: for h being Function
for x being set st h in dom I & (I . h) . x <> h . x holds
x in {n} by A5, FUNCT_1:17;
A7: the carrier of (TOP-REAL n) = {(0. (TOP-REAL n))} by A4, EUCLID:22, EUCLID:77;
then ( rng (id the carrier of (TOP-REAL n)) = the carrier of (TOP-REAL n) & rng f1 = the carrier of (TOP-REAL n) ) by A3, ZFMISC_1:33;
then f1 = id the carrier of (TOP-REAL n) by A2, A5, A7, FUNCT_1:7;
then I * f1 = I by A2, RELAT_1:52;
hence ( I is base_rotation & I * f1 is {n} -support-yielding ) by A6, Def1; ::_thesis: verum
end;
suppose n > 0 ; ::_thesis: ex f2 being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f2 is base_rotation & f2 * f1 is {n} -support-yielding )
then reconsider n1 = n - 1 as Nat by NAT_1:20;
defpred S1[ Nat] means ( $1 <= n - 1 implies for S being Subset of (TOP-REAL n) st S = { (Base_FinSeq (n,i)) where i is Element of NAT : ( 1 <= i & i <= $1 ) } holds
ex g being base_rotation Function of (TOP-REAL n),(TOP-REAL n) st (g * f1) | S = id S );
set S = { (Base_FinSeq (n,i)) where i is Element of NAT : ( 1 <= i & i <= n1 ) } ;
{ (Base_FinSeq (n,i)) where i is Element of NAT : ( 1 <= i & i <= n1 ) } c= the carrier of (TOP-REAL n)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (Base_FinSeq (n,i)) where i is Element of NAT : ( 1 <= i & i <= n1 ) } or x in the carrier of (TOP-REAL n) )
assume x in { (Base_FinSeq (n,i)) where i is Element of NAT : ( 1 <= i & i <= n1 ) } ; ::_thesis: x in the carrier of (TOP-REAL n)
then consider j being Element of NAT such that
A8: x = Base_FinSeq (n,j) and
1 <= j and
j <= n1 ;
len (Base_FinSeq (n,j)) = n by MATRIXR2:74;
hence x in the carrier of (TOP-REAL n) by A8, TOPREAL3:46; ::_thesis: verum
end;
then reconsider S = { (Base_FinSeq (n,i)) where i is Element of NAT : ( 1 <= i & i <= n1 ) } as Subset of (TOP-REAL n) ;
A9: for k being Nat st S1[k] holds
S1[k + 1]
proof
let z be Nat; ::_thesis: ( S1[z] implies S1[z + 1] )
assume A10: S1[z] ; ::_thesis: S1[z + 1]
set z1 = z + 1;
set SS = { (Base_FinSeq (n,i)) where i is Element of NAT : ( 1 <= i & i <= z ) } ;
assume A11: z + 1 <= n - 1 ; ::_thesis: for S being Subset of (TOP-REAL n) st S = { (Base_FinSeq (n,i)) where i is Element of NAT : ( 1 <= i & i <= z + 1 ) } holds
ex g being base_rotation Function of (TOP-REAL n),(TOP-REAL n) st (g * f1) | S = id S
then reconsider n1 = n - 1 as Element of NAT by INT_1:3;
A12: n1 < n1 + 1 by NAT_1:13;
then A13: z + 1 < n by A11, XXREAL_0:2;
set B = Base_FinSeq (n,(z + 1));
set X = { i where i is Element of NAT : ( z + 1 <= i & i <= n ) } ;
let S be Subset of (TOP-REAL n); ::_thesis: ( S = { (Base_FinSeq (n,i)) where i is Element of NAT : ( 1 <= i & i <= z + 1 ) } implies ex g being base_rotation Function of (TOP-REAL n),(TOP-REAL n) st (g * f1) | S = id S )
assume A14: S = { (Base_FinSeq (n,i)) where i is Element of NAT : ( 1 <= i & i <= z + 1 ) } ; ::_thesis: ex g being base_rotation Function of (TOP-REAL n),(TOP-REAL n) st (g * f1) | S = id S
{ (Base_FinSeq (n,i)) where i is Element of NAT : ( 1 <= i & i <= z ) } c= S
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { (Base_FinSeq (n,i)) where i is Element of NAT : ( 1 <= i & i <= z ) } or x in S )
assume x in { (Base_FinSeq (n,i)) where i is Element of NAT : ( 1 <= i & i <= z ) } ; ::_thesis: x in S
then consider i being Element of NAT such that
A15: ( x = Base_FinSeq (n,i) & 1 <= i ) and
A16: i <= z ;
i < z + 1 by A16, NAT_1:13;
hence x in S by A14, A15; ::_thesis: verum
end;
then reconsider SS = { (Base_FinSeq (n,i)) where i is Element of NAT : ( 1 <= i & i <= z ) } as Subset of (TOP-REAL n) by XBOOLE_1:1;
z < n1 by A11, NAT_1:13;
then consider g being base_rotation Function of (TOP-REAL n),(TOP-REAL n) such that
A17: (g * f1) | SS = id SS by A10;
A18: n in NAT by ORDINAL1:def_12;
then A19: n in { i where i is Element of NAT : ( z + 1 <= i & i <= n ) } by A13;
n >= 1 by A12, NAT_1:14;
then n in Seg n by A18;
then A20: n in { i where i is Element of NAT : ( z + 1 <= i & i <= n ) } /\ (Seg n) by A19, XBOOLE_0:def_4;
A21: card {(z + 1),n} = 2 by A11, A12, CARD_2:57;
A22: 1 <= z + 1 by NAT_1:11;
then A23: z + 1 in Seg n by A13;
Base_FinSeq (n,(z + 1)) in S by A14, A22;
then reconsider B = Base_FinSeq (n,(z + 1)) as Point of (TOP-REAL n) ;
set gfB = (g * f1) . B;
A24: z + 1 in { i where i is Element of NAT : ( z + 1 <= i & i <= n ) } by A13;
then consider h being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) such that
A25: ( h is { i where i is Element of NAT : ( z + 1 <= i & i <= n ) } -support-yielding & h is base_rotation ) and
A26: ( card ( { i where i is Element of NAT : ( z + 1 <= i & i <= n ) } /\ (Seg n)) > 1 implies (h . ((g * f1) . B)) . (z + 1) >= 0 ) and
A27: for i being Nat st i in { i where i is Element of NAT : ( z + 1 <= i & i <= n ) } /\ (Seg n) & i <> z + 1 holds
(h . ((g * f1) . B)) . i = 0 by A23, Th30;
reconsider hg = h * g as base_rotation Function of (TOP-REAL n),(TOP-REAL n) by A25;
A28: dom (hg * f1) = the carrier of (TOP-REAL n) by FUNCT_2:52;
A29: for x being set st x in SS holds
( ((h * g) * f1) . x = x & h . x = x )
proof
let x be set ; ::_thesis: ( x in SS implies ( ((h * g) * f1) . x = x & h . x = x ) )
assume A30: x in SS ; ::_thesis: ( ((h * g) * f1) . x = x & h . x = x )
reconsider B = x as Point of (TOP-REAL n) by A30;
((g * f1) | SS) . x = (g * f1) . x by A30, FUNCT_1:49;
then A31: (g * f1) . x = x by A17, A30, FUNCT_1:17;
A32: ex i being Element of NAT st
( x = Base_FinSeq (n,i) & 1 <= i & i <= z ) by A30;
A33: for j being Nat st j in { i where i is Element of NAT : ( z + 1 <= i & i <= n ) } /\ (Seg n) holds
B . j = 0
proof
let j be Nat; ::_thesis: ( j in { i where i is Element of NAT : ( z + 1 <= i & i <= n ) } /\ (Seg n) implies B . j = 0 )
assume A34: j in { i where i is Element of NAT : ( z + 1 <= i & i <= n ) } /\ (Seg n) ; ::_thesis: B . j = 0
then j in { i where i is Element of NAT : ( z + 1 <= i & i <= n ) } by XBOOLE_0:def_4;
then ex I being Element of NAT st
( I = j & z + 1 <= I & I <= n ) ;
then A35: z < j by NAT_1:13;
j in Seg n by A34, XBOOLE_0:def_4;
then ( 1 <= j & j <= n ) by FINSEQ_1:1;
hence B . j = 0 by A32, A35, MATRIXR2:76; ::_thesis: verum
end;
A36: hg * f1 = h * (g * f1) by RELAT_1:36;
then (h * (g * f1)) . x = h . ((g * f1) . x) by A28, A30, FUNCT_1:12;
hence ( ((h * g) * f1) . x = x & h . x = x ) by A25, A31, A33, A36, Th33; ::_thesis: verum
end;
z + 1 in { i where i is Element of NAT : ( z + 1 <= i & i <= n ) } /\ (Seg n) by A23, A24, XBOOLE_0:def_4;
then {(z + 1),n} c= { i where i is Element of NAT : ( z + 1 <= i & i <= n ) } /\ (Seg n) by A20, ZFMISC_1:32;
then A37: 1 + 1 <= card ( { i where i is Element of NAT : ( z + 1 <= i & i <= n ) } /\ (Seg n)) by A21, NAT_1:43;
A38: for x being set st x in S holds
((hg * f1) | S) . x = (id S) . x
proof
let x be set ; ::_thesis: ( x in S implies ((hg * f1) | S) . x = (id S) . x )
assume A39: x in S ; ::_thesis: ((hg * f1) | S) . x = (id S) . x
A40: (id S) . x = x by A39, FUNCT_1:17;
A41: hg * f1 = h * (g * f1) by RELAT_1:36;
A42: ((hg * f1) | S) . x = (hg * f1) . x by A39, FUNCT_1:49;
consider i being Element of NAT such that
A43: x = Base_FinSeq (n,i) and
A44: 1 <= i and
A45: i <= z + 1 by A14, A39;
percases ( i <= z or i = z + 1 ) by A45, NAT_1:8;
suppose i <= z ; ::_thesis: ((hg * f1) | S) . x = (id S) . x
then x in SS by A43, A44;
hence ((hg * f1) | S) . x = (id S) . x by A29, A40, A42; ::_thesis: verum
end;
supposeA46: i = z + 1 ; ::_thesis: ((hg * f1) | S) . x = (id S) . x
set H = h . ((g * f1) . B);
A47: (h * (g * f1)) . x = h . ((g * f1) . B) by A28, A41, A43, A46, FUNCT_1:12;
A48: len (h . ((g * f1) . B)) = n by CARD_1:def_7;
A49: for j being Nat st j in Seg n & j < z + 1 holds
(h . ((g * f1) . B)) . j = 0
proof
let j be Nat; ::_thesis: ( j in Seg n & j < z + 1 implies (h . ((g * f1) . B)) . j = 0 )
assume that
A50: j in Seg n and
A51: j < z + 1 ; ::_thesis: (h . ((g * f1) . B)) . j = 0
A52: 1 <= j by A50, FINSEQ_1:1;
j <= z by A51, NAT_1:13;
then A53: Base_FinSeq (n,j) in SS by A50, A52;
then reconsider Bnj = Base_FinSeq (n,j) as Point of (TOP-REAL n) ;
((h * g) * f1) . Bnj = Bnj by A29, A53;
then A54: Bnj + (h . ((g * f1) . B)) = ((h * g) * f1) . (Bnj + B) by A41, A43, A46, A47, VECTSP_1:def_20;
A55: len Bnj = n by CARD_1:def_7;
|.(((h * g) * f1) . (Bnj + B)).| = |.(Bnj + B).| by A1, A25, Def4;
then A56: |.(Bnj + B).| ^2 = ((|.Bnj.| ^2) + (2 * |((h . ((g * f1) . B)),Bnj)|)) + (|.(h . ((g * f1) . B)).| ^2) by A48, A54, A55, EUCLID_2:11;
A57: Bnj = (0* n) +* (j,1) by MATRIXR2:def_4;
len B = n by CARD_1:def_7;
then A58: |.(Bnj + B).| ^2 = ((|.Bnj.| ^2) + (2 * |(B,Bnj)|)) + (|.B.| ^2) by A55, EUCLID_2:11;
A59: j <= n by A50, FINSEQ_1:1;
|.(h . ((g * f1) . B)).| = |.B.| by A1, A25, A43, A46, A47, Def4;
then (B . j) * 1 = |((h . ((g * f1) . B)),Bnj)| by A56, A57, A58, TOPREALC:16
.= ((h . ((g * f1) . B)) . j) * 1 by A57, TOPREALC:16 ;
hence (h . ((g * f1) . B)) . j = 0 by A51, A52, A59, MATRIXR2:76; ::_thesis: verum
end;
set H = h . ((g * f1) . B);
set 0H = (0* n) +* ((z + 1),((h . ((g * f1) . B)) . (z + 1)));
A60: len (0* n) = n by CARD_1:def_7;
A61: for j being Nat st j in Seg n & j > z + 1 holds
(h . ((g * f1) . B)) . j = 0
proof
let j be Nat; ::_thesis: ( j in Seg n & j > z + 1 implies (h . ((g * f1) . B)) . j = 0 )
assume that
A62: j in Seg n and
A63: j > z + 1 ; ::_thesis: (h . ((g * f1) . B)) . j = 0
j <= n by A62, FINSEQ_1:1;
then j in { i where i is Element of NAT : ( z + 1 <= i & i <= n ) } by A62, A63;
then j in { i where i is Element of NAT : ( z + 1 <= i & i <= n ) } /\ (Seg n) by A62, XBOOLE_0:def_4;
hence (h . ((g * f1) . B)) . j = 0 by A27, A63; ::_thesis: verum
end;
A64: for j being Nat st 1 <= j & j <= n holds
(h . ((g * f1) . B)) . j = ((0* n) +* ((z + 1),((h . ((g * f1) . B)) . (z + 1)))) . j
proof
let j be Nat; ::_thesis: ( 1 <= j & j <= n implies (h . ((g * f1) . B)) . j = ((0* n) +* ((z + 1),((h . ((g * f1) . B)) . (z + 1)))) . j )
assume ( 1 <= j & j <= n ) ; ::_thesis: (h . ((g * f1) . B)) . j = ((0* n) +* ((z + 1),((h . ((g * f1) . B)) . (z + 1)))) . j
then A65: j in Seg n by FINSEQ_1:1;
then A66: j in dom (0* n) by A60, FINSEQ_1:def_3;
percases ( j = z + 1 or j <> z + 1 ) ;
suppose j = z + 1 ; ::_thesis: (h . ((g * f1) . B)) . j = ((0* n) +* ((z + 1),((h . ((g * f1) . B)) . (z + 1)))) . j
hence (h . ((g * f1) . B)) . j = ((0* n) +* ((z + 1),((h . ((g * f1) . B)) . (z + 1)))) . j by A66, FUNCT_7:31; ::_thesis: verum
end;
supposeA67: j <> z + 1 ; ::_thesis: (h . ((g * f1) . B)) . j = ((0* n) +* ((z + 1),((h . ((g * f1) . B)) . (z + 1)))) . j
then ( j > z + 1 or j < z + 1 ) by XXREAL_0:1;
then A68: (h . ((g * f1) . B)) . j = 0 by A49, A61, A65;
((0* n) +* ((z + 1),((h . ((g * f1) . B)) . (z + 1)))) . j = (0* n) . j by A67, FUNCT_7:32;
hence (h . ((g * f1) . B)) . j = ((0* n) +* ((z + 1),((h . ((g * f1) . B)) . (z + 1)))) . j by A68; ::_thesis: verum
end;
end;
end;
( len (h . ((g * f1) . B)) = n & len ((0* n) +* ((z + 1),((h . ((g * f1) . B)) . (z + 1)))) = len (0* n) ) by CARD_1:def_7, FUNCT_7:97;
then A69: (0* n) +* ((z + 1),((h . ((g * f1) . B)) . (z + 1))) = h . ((g * f1) . B) by A60, A64, FINSEQ_1:14;
A70: |.((g * f1) . B).| = |.B.| by A1, Def4;
A71: B = (0* n) +* ((z + 1),1) by MATRIXR2:def_4;
then A72: |.B.| = abs 1 by A23, TOPREALC:13
.= 1 by ABSVALUE:def_1 ;
( |.(h . ((g * f1) . B)).| = |.((g * f1) . B).| & abs ((h . ((g * f1) . B)) . (z + 1)) = (h . ((g * f1) . B)) . (z + 1) ) by A25, A26, A37, Def4, ABSVALUE:def_1, NAT_1:13;
then h . ((g * f1) . B) = B by A23, A69, A71, A72, A70, TOPREALC:13;
hence ((hg * f1) | S) . x = (id S) . x by A39, A40, A41, A43, A46, A47, FUNCT_1:49; ::_thesis: verum
end;
end;
end;
take hg ; ::_thesis: (hg * f1) | S = id S
( dom (id S) = S & dom ((hg * f1) | S) = S ) by A28, RELAT_1:62;
hence (hg * f1) | S = id S by A38, FUNCT_1:2; ::_thesis: verum
end;
A73: S1[ 0 ]
proof
assume 0 <= n - 1 ; ::_thesis: for S being Subset of (TOP-REAL n) st S = { (Base_FinSeq (n,i)) where i is Element of NAT : ( 1 <= i & i <= 0 ) } holds
ex g being base_rotation Function of (TOP-REAL n),(TOP-REAL n) st (g * f1) | S = id S
let S be Subset of (TOP-REAL n); ::_thesis: ( S = { (Base_FinSeq (n,i)) where i is Element of NAT : ( 1 <= i & i <= 0 ) } implies ex g being base_rotation Function of (TOP-REAL n),(TOP-REAL n) st (g * f1) | S = id S )
assume A74: S = { (Base_FinSeq (n,i)) where i is Element of NAT : ( 1 <= i & i <= 0 ) } ; ::_thesis: ex g being base_rotation Function of (TOP-REAL n),(TOP-REAL n) st (g * f1) | S = id S
A75: S = {}
proof
assume S <> {} ; ::_thesis: contradiction
then consider x being set such that
A76: x in S by XBOOLE_0:def_1;
ex i being Element of NAT st
( x = Base_FinSeq (n,i) & 1 <= i & i <= 0 ) by A74, A76;
hence contradiction ; ::_thesis: verum
end;
take g = id (TOP-REAL n); ::_thesis: (g * f1) | S = id S
(g * f1) | S = {} by A75;
hence (g * f1) | S = id S by A75; ::_thesis: verum
end;
for k being Nat holds S1[k] from NAT_1:sch_2(A73, A9);
then consider g being base_rotation Function of (TOP-REAL n),(TOP-REAL n) such that
A77: (g * f1) | S = id S ;
take g ; ::_thesis: ( g is base_rotation & g * f1 is {n} -support-yielding )
set gf = g * f1;
thus g is base_rotation ; ::_thesis: g * f1 is {n} -support-yielding
let p be Function; :: according to MATRTOP3:def_1 ::_thesis: for x being set st p in dom (g * f1) & ((g * f1) . p) . x <> p . x holds
x in {n}
let x be set ; ::_thesis: ( p in dom (g * f1) & ((g * f1) . p) . x <> p . x implies x in {n} )
assume that
A78: p in dom (g * f1) and
A79: ((g * f1) . p) . x <> p . x ; ::_thesis: x in {n}
reconsider p = p as Point of (TOP-REAL n) by A78, FUNCT_2:52;
len ((g * f1) . p) = n by CARD_1:def_7;
then dom ((g * f1) . p) = Seg n by FINSEQ_1:def_3;
then A80: ( not x in Seg n implies ((g * f1) . p) . x = {} ) by FUNCT_1:def_2;
len p = n by CARD_1:def_7;
then dom p = Seg n by FINSEQ_1:def_3;
then A81: x in Seg n by A79, A80, FUNCT_1:def_2;
assume A82: not x in {n} ; ::_thesis: contradiction
reconsider x = x as Nat by A81;
A83: x <= n by A81, FINSEQ_1:1;
x <> n by A82, TARSKI:def_1;
then x < n1 + 1 by A83, XXREAL_0:1;
then A84: x <= n1 by NAT_1:13;
( x in NAT & 1 <= x ) by A81, FINSEQ_1:1;
then Base_FinSeq (n,x) in S by A84;
then Base_FinSeq (n,x) in Lin S by RLVECT_3:15;
then ((g * f1) . p) . x = p . x by A1, A77, A81, Th32;
hence contradiction by A79; ::_thesis: verum
end;
end;
end;
Lm10: for n being Nat
for M being Matrix of n,F_Real st Mx2Tran M is base_rotation holds
Det M = 1. F_Real
proof
let n be Nat; ::_thesis: for M being Matrix of n,F_Real st Mx2Tran M is base_rotation holds
Det M = 1. F_Real
let M be Matrix of n,F_Real; ::_thesis: ( Mx2Tran M is base_rotation implies Det M = 1. F_Real )
set TR = TOP-REAL n;
set G = GFuncs the carrier of (TOP-REAL n);
assume Mx2Tran M is base_rotation ; ::_thesis: Det M = 1. F_Real
then consider F being FinSequence of (GFuncs the carrier of (TOP-REAL n)) such that
A1: Mx2Tran M = Product F and
A2: for k being Nat st k in dom F holds
ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) ) by Def5;
defpred S1[ Nat] means ( $1 <= len F implies ( Product (F | $1) is base_rotation Function of (TOP-REAL n),(TOP-REAL n) & ( for M being Matrix of n,F_Real st Mx2Tran M = Product (F | $1) holds
Det M = 1. F_Real ) ) );
A3: for m being Nat st S1[m] holds
S1[m + 1]
proof
let m be Nat; ::_thesis: ( S1[m] implies S1[m + 1] )
A4: ( n = 0 implies n = 0 ) ;
set m1 = m + 1;
assume A5: S1[m] ; ::_thesis: S1[m + 1]
assume A6: m + 1 <= len F ; ::_thesis: ( Product (F | (m + 1)) is base_rotation Function of (TOP-REAL n),(TOP-REAL n) & ( for M being Matrix of n,F_Real st Mx2Tran M = Product (F | (m + 1)) holds
Det M = 1. F_Real ) )
then reconsider P = Product (F | m) as base_rotation Function of (TOP-REAL n),(TOP-REAL n) by A5, NAT_1:13;
set R = AutMt P;
A7: ( width (AutMt P) = n & len (AutMt P) = n ) by MATRIX_1:24;
1 <= m + 1 by NAT_1:11;
then A8: m + 1 in dom F by A6, FINSEQ_3:25;
then consider i, j being Nat, r being real number such that
A9: ( 1 <= i & i < j & j <= n ) and
A10: F . (m + 1) = Mx2Tran (Rotation (i,j,n,r)) by A2;
set O = Rotation (i,j,n,r);
reconsider MO = Mx2Tran (Rotation (i,j,n,r)) as Element of (GFuncs the carrier of (TOP-REAL n)) by MONOID_0:73;
F | (m + 1) = (F | m) ^ <*MO*> by A8, A10, FINSEQ_5:10;
then A11: Product (F | (m + 1)) = (Product (F | m)) * MO by GROUP_4:6
.= (Mx2Tran (Rotation (i,j,n,r))) * P by MONOID_0:70 ;
Mx2Tran (Rotation (i,j,n,r)) is base_rotation by A9, Th28;
hence Product (F | (m + 1)) is base_rotation Function of (TOP-REAL n),(TOP-REAL n) by A11; ::_thesis: for M being Matrix of n,F_Real st Mx2Tran M = Product (F | (m + 1)) holds
Det M = 1. F_Real
A12: ( width (Rotation (i,j,n,r)) = n & len (Rotation (i,j,n,r)) = n ) by MATRIX_1:24;
let M be Matrix of n,F_Real; ::_thesis: ( Mx2Tran M = Product (F | (m + 1)) implies Det M = 1. F_Real )
assume A13: Mx2Tran M = Product (F | (m + 1)) ; ::_thesis: Det M = 1. F_Real
Mx2Tran M = (Mx2Tran (Rotation (i,j,n,r))) * (Mx2Tran (AutMt P)) by A11, A13, Def6
.= Mx2Tran ((AutMt P) * (Rotation (i,j,n,r))) by A4, A12, A7, MATRTOP1:32 ;
then A14: M = (AutMt P) * (Rotation (i,j,n,r)) by MATRTOP1:34;
P = Mx2Tran (AutMt P) by Def6;
then A15: Det (AutMt P) = 1. F_Real by A5, A6, NAT_1:13;
( len (AutMt P) = n & Det (Rotation (i,j,n,r)) = 1. F_Real ) by A9, Th13, MATRIX_1:25;
hence Det M = (1. F_Real) * (1. F_Real) by A14, A15, MATRIXR2:45
.= 1. F_Real ;
::_thesis: verum
end;
A16: F | (len F) = F by FINSEQ_1:58;
A17: S1[ 0 ]
proof
assume 0 <= len F ; ::_thesis: ( Product (F | 0) is base_rotation Function of (TOP-REAL n),(TOP-REAL n) & ( for M being Matrix of n,F_Real st Mx2Tran M = Product (F | 0) holds
Det M = 1. F_Real ) )
A18: Mx2Tran (1. (F_Real,n)) = id (TOP-REAL n) by MATRTOP1:33;
F | 0 = <*> the carrier of (GFuncs the carrier of (TOP-REAL n)) ;
then A19: Product (F | 0) = 1_ (GFuncs the carrier of (TOP-REAL n)) by GROUP_4:8
.= the_unity_wrt the multF of (GFuncs the carrier of (TOP-REAL n)) by GROUP_1:22
.= id (TOP-REAL n) by MONOID_0:75 ;
hence Product (F | 0) is base_rotation Function of (TOP-REAL n),(TOP-REAL n) ; ::_thesis: for M being Matrix of n,F_Real st Mx2Tran M = Product (F | 0) holds
Det M = 1. F_Real
A20: n in NAT by ORDINAL1:def_12;
( n = 0 or n >= 1 ) by NAT_1:14;
then A21: ( ( Det (1. (F_Real,n)) = 1_ F_Real & n >= 1 ) or ( Det (1. (F_Real,n)) = 1. F_Real & n = 0 ) ) by A20, MATRIXR2:41, MATRIX_7:16;
let M be Matrix of n,F_Real; ::_thesis: ( Mx2Tran M = Product (F | 0) implies Det M = 1. F_Real )
thus ( Mx2Tran M = Product (F | 0) implies Det M = 1. F_Real ) by A18, A19, A21, MATRTOP1:34; ::_thesis: verum
end;
for m being Nat holds S1[m] from NAT_1:sch_2(A17, A3);
hence Det M = 1. F_Real by A1, A16; ::_thesis: verum
end;
begin
theorem Th37: :: MATRTOP3:37
for n being Nat
for f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st f is rotation holds
( Det (AutMt f) = 1. F_Real iff f is base_rotation )
proof
let n be Nat; ::_thesis: for f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st f is rotation holds
( Det (AutMt f) = 1. F_Real iff f is base_rotation )
let f be additive homogeneous Function of (TOP-REAL n),(TOP-REAL n); ::_thesis: ( f is rotation implies ( Det (AutMt f) = 1. F_Real iff f is base_rotation ) )
set TR = TOP-REAL n;
set cTR = the carrier of (TOP-REAL n);
set M = AutMt f;
set MM = Mx2Tran (AutMt f);
A1: ( len (AutMt f) = n & width (AutMt f) = n ) by MATRIX_1:24;
A2: ( n = 0 implies n = 0 ) ;
A3: Mx2Tran (AutMt f) = f by Def6;
assume A4: f is rotation ; ::_thesis: ( Det (AutMt f) = 1. F_Real iff f is base_rotation )
then consider h being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) such that
A5: h is base_rotation and
A6: h * (Mx2Tran (AutMt f)) is {n} -support-yielding by A3, Th36;
set R = AutMt h;
A7: width (AutMt h) = n by MATRIX_1:24;
A8: h = Mx2Tran (AutMt h) by Def6;
A9: ( AutMt (h * (Mx2Tran (AutMt f))) = 1. (F_Real,n) or AutMt (h * (Mx2Tran (AutMt f))) = AxialSymmetry (n,n) ) by A4, A3, A5, A6, Th35;
( Det (AutMt f) = 1. F_Real implies Mx2Tran (AutMt f) is base_rotation )
proof
assume A10: Det (AutMt f) = 1. F_Real ; ::_thesis: Mx2Tran (AutMt f) is base_rotation
( Det (AutMt h) = 1. F_Real & n in NAT ) by A5, A8, Lm10, ORDINAL1:def_12;
then A11: Det ((AutMt f) * (AutMt h)) = (1. F_Real) * (1. F_Real) by A10, MATRIXR2:45
.= 1. F_Real ;
A12: rng (Mx2Tran (AutMt f)) c= the carrier of (TOP-REAL n) by RELAT_1:def_19;
A13: rng h = [#] (TOP-REAL n) by A5, TOPS_2:def_5;
A14: ( dom h = [#] (TOP-REAL n) & h is one-to-one ) by A5, TOPS_2:def_5;
A15: dom (h ") = [#] (TOP-REAL n) by A5, TOPS_2:def_5;
A16: id (TOP-REAL n) = Mx2Tran (1. (F_Real,n)) by MATRTOP1:33;
h * (Mx2Tran (AutMt f)) = id the carrier of (TOP-REAL n)
proof
assume A17: h * (Mx2Tran (AutMt f)) <> id the carrier of (TOP-REAL n) ; ::_thesis: contradiction
n <> 0
proof
A18: ( dom (h * (Mx2Tran (AutMt f))) = the carrier of (TOP-REAL n) & dom (id the carrier of (TOP-REAL n)) = the carrier of (TOP-REAL n) ) by FUNCT_2:52;
assume n = 0 ; ::_thesis: contradiction
then A19: the carrier of (TOP-REAL n) = {(0. (TOP-REAL n))} by EUCLID:22, EUCLID:77;
rng (h * (Mx2Tran (AutMt f))) c= the carrier of (TOP-REAL n) by RELAT_1:def_19;
then ( rng (id the carrier of (TOP-REAL n)) = the carrier of (TOP-REAL n) & rng (h * (Mx2Tran (AutMt f))) = the carrier of (TOP-REAL n) ) by A19, ZFMISC_1:33;
hence contradiction by A17, A18, A19, FUNCT_1:7; ::_thesis: verum
end;
then A20: n in Seg n by FINSEQ_1:3;
Mx2Tran (AutMt (h * (Mx2Tran (AutMt f)))) = h * (Mx2Tran (AutMt f)) by Def6;
then Mx2Tran (AxialSymmetry (n,n)) = Mx2Tran ((AutMt f) * (AutMt h)) by A1, A2, A9, A8, A7, A16, A17, MATRTOP1:32;
then AxialSymmetry (n,n) = (AutMt f) * (AutMt h) by MATRTOP1:34;
hence contradiction by A11, A20, Th4; ::_thesis: verum
end;
then (h ") * (h * (Mx2Tran (AutMt f))) = h " by A15, RELAT_1:52;
then h " = ((h ") * h) * (Mx2Tran (AutMt f)) by RELAT_1:36
.= (id the carrier of (TOP-REAL n)) * (Mx2Tran (AutMt f)) by A14, A13, TOPS_2:52
.= Mx2Tran (AutMt f) by A12, RELAT_1:53 ;
hence Mx2Tran (AutMt f) is base_rotation by A5; ::_thesis: verum
end;
hence ( Det (AutMt f) = 1. F_Real iff f is base_rotation ) by A3, Lm10; ::_thesis: verum
end;
theorem Th38: :: MATRTOP3:38
for n being Nat
for f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) holds
( not f is rotation or Det (AutMt f) = 1. F_Real or Det (AutMt f) = - (1. F_Real) )
proof
let n be Nat; ::_thesis: for f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) holds
( not f is rotation or Det (AutMt f) = 1. F_Real or Det (AutMt f) = - (1. F_Real) )
let f be additive homogeneous Function of (TOP-REAL n),(TOP-REAL n); ::_thesis: ( not f is rotation or Det (AutMt f) = 1. F_Real or Det (AutMt f) = - (1. F_Real) )
set M = AutMt f;
set MM = Mx2Tran (AutMt f);
set TR = TOP-REAL n;
A1: ( len (AutMt f) = n & width (AutMt f) = n ) by MATRIX_1:24;
A2: ( n = 0 implies n = 0 ) ;
A3: n in NAT by ORDINAL1:def_12;
( n = 0 or n >= 1 ) by NAT_1:14;
then A4: ( ( Det (1. (F_Real,n)) = 1_ F_Real & n >= 1 ) or ( Det (1. (F_Real,n)) = 1. F_Real & n = 0 ) ) by A3, MATRIXR2:41, MATRIX_7:16;
assume f is rotation ; ::_thesis: ( Det (AutMt f) = 1. F_Real or Det (AutMt f) = - (1. F_Real) )
then A5: Mx2Tran (AutMt f) is rotation by Def6;
then consider h being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) such that
A6: h is base_rotation and
A7: h * (Mx2Tran (AutMt f)) is {n} -support-yielding by Th36;
set R = AutMt h;
A8: h = Mx2Tran (AutMt h) by Def6;
then ( n in NAT & Det (AutMt h) = 1. F_Real ) by A6, Lm10, ORDINAL1:def_12;
then A9: Det ((AutMt f) * (AutMt h)) = (Det (AutMt f)) * (1. F_Real) by MATRIXR2:45
.= Det (AutMt f) ;
width (AutMt h) = n by MATRIX_1:24;
then A10: h * (Mx2Tran (AutMt f)) = Mx2Tran ((AutMt f) * (AutMt h)) by A1, A2, A8, MATRTOP1:32;
percases ( AutMt (h * (Mx2Tran (AutMt f))) = 1. (F_Real,n) or ( AutMt (h * (Mx2Tran (AutMt f))) <> 1. (F_Real,n) & AutMt (h * (Mx2Tran (AutMt f))) = AxialSymmetry (n,n) ) ) by A5, A6, A7, Th35;
suppose AutMt (h * (Mx2Tran (AutMt f))) = 1. (F_Real,n) ; ::_thesis: ( Det (AutMt f) = 1. F_Real or Det (AutMt f) = - (1. F_Real) )
hence ( Det (AutMt f) = 1. F_Real or Det (AutMt f) = - (1. F_Real) ) by A4, A9, A10, Def6; ::_thesis: verum
end;
supposeA11: ( AutMt (h * (Mx2Tran (AutMt f))) <> 1. (F_Real,n) & AutMt (h * (Mx2Tran (AutMt f))) = AxialSymmetry (n,n) ) ; ::_thesis: ( Det (AutMt f) = 1. F_Real or Det (AutMt f) = - (1. F_Real) )
set cTR = the carrier of (TOP-REAL n);
n <> 0
proof
A12: ( dom (h * (Mx2Tran (AutMt f))) = the carrier of (TOP-REAL n) & dom (id the carrier of (TOP-REAL n)) = the carrier of (TOP-REAL n) ) by FUNCT_2:52;
assume n = 0 ; ::_thesis: contradiction
then A13: the carrier of (TOP-REAL n) = {(0. (TOP-REAL n))} by EUCLID:22, EUCLID:77;
rng (h * (Mx2Tran (AutMt f))) c= the carrier of (TOP-REAL n) by RELAT_1:def_19;
then ( rng (id the carrier of (TOP-REAL n)) = the carrier of (TOP-REAL n) & rng (h * (Mx2Tran (AutMt f))) = the carrier of (TOP-REAL n) ) by A13, ZFMISC_1:33;
then h * (Mx2Tran (AutMt f)) = id (TOP-REAL n) by A12, A13, FUNCT_1:7
.= Mx2Tran (1. (F_Real,n)) by MATRTOP1:33 ;
hence contradiction by A11, Def6; ::_thesis: verum
end;
then n in Seg n by FINSEQ_1:3;
then Det (AxialSymmetry (n,n)) = - (1. F_Real) by Th4;
hence ( Det (AutMt f) = 1. F_Real or Det (AutMt f) = - (1. F_Real) ) by A9, A10, A11, Def6; ::_thesis: verum
end;
end;
end;
theorem Th39: :: MATRTOP3:39
for i, n being Nat
for f1, f2 being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st f1 is rotation & Det (AutMt f1) = - (1. F_Real) & i in Seg n & AutMt f2 = AxialSymmetry (i,n) holds
f1 * f2 is base_rotation
proof
let i, n be Nat; ::_thesis: for f1, f2 being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st f1 is rotation & Det (AutMt f1) = - (1. F_Real) & i in Seg n & AutMt f2 = AxialSymmetry (i,n) holds
f1 * f2 is base_rotation
let f1, f2 be additive homogeneous Function of (TOP-REAL n),(TOP-REAL n); ::_thesis: ( f1 is rotation & Det (AutMt f1) = - (1. F_Real) & i in Seg n & AutMt f2 = AxialSymmetry (i,n) implies f1 * f2 is base_rotation )
set M = AutMt f1;
set A = AutMt f2;
assume that
A1: f1 is rotation and
A2: Det (AutMt f1) = - (1. F_Real) and
A3: i in Seg n and
A4: AutMt f2 = AxialSymmetry (i,n) ; ::_thesis: f1 * f2 is base_rotation
A5: f2 = Mx2Tran (AxialSymmetry (i,n)) by A4, Def6;
reconsider MF = (Mx2Tran (AutMt f1)) * f2 as additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) ;
set A = AxialSymmetry (i,n);
set R = AutMt MF;
A6: ( n = 0 implies n = 0 ) ;
A7: ( MF = Mx2Tran (AutMt MF) & width (AutMt f1) = n ) by Def6, MATRIX_1:24;
( len (AxialSymmetry (i,n)) = n & width (AxialSymmetry (i,n)) = n ) by MATRIX_1:24;
then Mx2Tran (AutMt MF) = Mx2Tran ((AxialSymmetry (i,n)) * (AutMt f1)) by A5, A6, A7, MATRTOP1:32;
then A8: AutMt MF = (AxialSymmetry (i,n)) * (AutMt f1) by MATRTOP1:34;
( n in NAT & Det (AxialSymmetry (i,n)) = - (1. F_Real) ) by A3, Th4, ORDINAL1:def_12;
then A9: Det (AutMt MF) = (- (1. F_Real)) * (- (1. F_Real)) by A2, A8, MATRIXR2:45
.= 1. F_Real ;
A10: Mx2Tran (AutMt f1) = f1 by Def6;
f2 is rotation by A3, A5, Th27;
hence f1 * f2 is base_rotation by A10, A1, A9, Th37; ::_thesis: verum
end;
Lm11: for n being Nat
for f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st f is base_rotation holds
AutMt f is Orthogonal
proof
let n be Nat; ::_thesis: for f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st f is base_rotation holds
AutMt f is Orthogonal
let f be additive homogeneous Function of (TOP-REAL n),(TOP-REAL n); ::_thesis: ( f is base_rotation implies AutMt f is Orthogonal )
set TR = TOP-REAL n;
set G = GFuncs the carrier of (TOP-REAL n);
assume f is base_rotation ; ::_thesis: AutMt f is Orthogonal
then consider F being FinSequence of (GFuncs the carrier of (TOP-REAL n)) such that
A1: f = Product F and
A2: for k being Nat st k in dom F holds
ex i, j being Nat ex r being real number st
( 1 <= i & i < j & j <= n & F . k = Mx2Tran (Rotation (i,j,n,r)) ) by Def5;
A3: f = Mx2Tran (AutMt f) by Def6;
defpred S1[ Nat] means ( $1 <= len F implies ( Product (F | $1) is base_rotation Function of (TOP-REAL n),(TOP-REAL n) & ( for M being Matrix of n,F_Real st Mx2Tran M = Product (F | $1) holds
M is Orthogonal ) ) );
A4: for m being Nat st S1[m] holds
S1[m + 1]
proof
let m be Nat; ::_thesis: ( S1[m] implies S1[m + 1] )
A5: ( n = 0 implies n = 0 ) ;
set m1 = m + 1;
assume A6: S1[m] ; ::_thesis: S1[m + 1]
assume A7: m + 1 <= len F ; ::_thesis: ( Product (F | (m + 1)) is base_rotation Function of (TOP-REAL n),(TOP-REAL n) & ( for M being Matrix of n,F_Real st Mx2Tran M = Product (F | (m + 1)) holds
M is Orthogonal ) )
then reconsider P = Product (F | m) as base_rotation Function of (TOP-REAL n),(TOP-REAL n) by A6, NAT_1:13;
set R = AutMt P;
A8: ( width (AutMt P) = n & len (AutMt P) = n ) by MATRIX_1:24;
1 <= m + 1 by NAT_1:11;
then A9: m + 1 in dom F by A7, FINSEQ_3:25;
then consider i, j being Nat, r being real number such that
A10: ( 1 <= i & i < j & j <= n ) and
A11: F . (m + 1) = Mx2Tran (Rotation (i,j,n,r)) by A2;
set O = Rotation (i,j,n,r);
reconsider MO = Mx2Tran (Rotation (i,j,n,r)) as Element of (GFuncs the carrier of (TOP-REAL n)) by MONOID_0:73;
F | (m + 1) = (F | m) ^ <*MO*> by A9, A11, FINSEQ_5:10;
then A12: Product (F | (m + 1)) = (Product (F | m)) * MO by GROUP_4:6
.= (Mx2Tran (Rotation (i,j,n,r))) * P by MONOID_0:70 ;
Mx2Tran (Rotation (i,j,n,r)) is base_rotation by A10, Th28;
hence Product (F | (m + 1)) is base_rotation Function of (TOP-REAL n),(TOP-REAL n) by A12; ::_thesis: for M being Matrix of n,F_Real st Mx2Tran M = Product (F | (m + 1)) holds
M is Orthogonal
A13: ( width (Rotation (i,j,n,r)) = n & len (Rotation (i,j,n,r)) = n ) by MATRIX_1:24;
let M be Matrix of n,F_Real; ::_thesis: ( Mx2Tran M = Product (F | (m + 1)) implies M is Orthogonal )
assume A14: Mx2Tran M = Product (F | (m + 1)) ; ::_thesis: M is Orthogonal
Mx2Tran M = (Mx2Tran (Rotation (i,j,n,r))) * (Mx2Tran (AutMt P)) by A12, A14, Def6
.= Mx2Tran ((AutMt P) * (Rotation (i,j,n,r))) by A5, A13, A8, MATRTOP1:32 ;
then A15: M = (AutMt P) * (Rotation (i,j,n,r)) by MATRTOP1:34;
P = Mx2Tran (AutMt P) by Def6;
then A16: ( AutMt P is Orthogonal & n > i ) by A10, A6, A7, NAT_1:13, XXREAL_0:2;
( len (AutMt P) = n & Rotation (i,j,n,r) is Orthogonal & n > 0 ) by A10, Th19, MATRIX_1:25;
hence M is Orthogonal by A15, A16, MATRIX_6:46; ::_thesis: verum
end;
A17: F | (len F) = F by FINSEQ_1:58;
A18: S1[ 0 ]
proof
assume 0 <= len F ; ::_thesis: ( Product (F | 0) is base_rotation Function of (TOP-REAL n),(TOP-REAL n) & ( for M being Matrix of n,F_Real st Mx2Tran M = Product (F | 0) holds
M is Orthogonal ) )
A19: Mx2Tran (1. (F_Real,n)) = id (TOP-REAL n) by MATRTOP1:33;
F | 0 = <*> the carrier of (GFuncs the carrier of (TOP-REAL n)) ;
then A20: Product (F | 0) = 1_ (GFuncs the carrier of (TOP-REAL n)) by GROUP_4:8
.= the_unity_wrt the multF of (GFuncs the carrier of (TOP-REAL n)) by GROUP_1:22
.= id (TOP-REAL n) by MONOID_0:75 ;
hence Product (F | 0) is base_rotation Function of (TOP-REAL n),(TOP-REAL n) ; ::_thesis: for M being Matrix of n,F_Real st Mx2Tran M = Product (F | 0) holds
M is Orthogonal
A21: 1. (F_Real,n) is Orthogonal by MATRIX_6:58;
let M be Matrix of n,F_Real; ::_thesis: ( Mx2Tran M = Product (F | 0) implies M is Orthogonal )
thus ( Mx2Tran M = Product (F | 0) implies M is Orthogonal ) by A19, A20, A21, MATRTOP1:34; ::_thesis: verum
end;
for m being Nat holds S1[m] from NAT_1:sch_2(A18, A4);
hence AutMt f is Orthogonal by A1, A17, A3; ::_thesis: verum
end;
registration
let n be Nat;
let f be additive homogeneous rotation Function of (TOP-REAL n),(TOP-REAL n);
cluster AutMt f -> Orthogonal ;
coherence
AutMt f is Orthogonal
proof
set TR = TOP-REAL n;
set cTR = the carrier of (TOP-REAL n);
set M = AutMt f;
percases ( Det (AutMt f) = 1. F_Real or ( Det (AutMt f) = - (1. F_Real) & not f is base_rotation ) ) by Th37, Th38;
suppose Det (AutMt f) = 1. F_Real ; ::_thesis: AutMt f is Orthogonal
then f is base_rotation by Th37;
hence AutMt f is Orthogonal by Lm11; ::_thesis: verum
end;
supposeA1: ( Det (AutMt f) = - (1. F_Real) & not f is base_rotation ) ; ::_thesis: AutMt f is Orthogonal
A2: n > 0
proof
A3: ( dom f = the carrier of (TOP-REAL n) & dom (id the carrier of (TOP-REAL n)) = the carrier of (TOP-REAL n) ) by FUNCT_2:52;
assume n <= 0 ; ::_thesis: contradiction
then n = 0 ;
then A4: the carrier of (TOP-REAL n) = {(0. (TOP-REAL n))} by EUCLID:22, EUCLID:77;
rng f c= the carrier of (TOP-REAL n) by RELAT_1:def_19;
then ( rng (id the carrier of (TOP-REAL n)) = the carrier of (TOP-REAL n) & rng f = the carrier of (TOP-REAL n) ) by A4, ZFMISC_1:33;
then f = id (TOP-REAL n) by A3, A4, FUNCT_1:7;
hence contradiction by A1; ::_thesis: verum
end;
then A5: n in Seg n by FINSEQ_1:3;
set A = AxialSymmetry (n,n);
set MA = Mx2Tran (AxialSymmetry (n,n));
AutMt (Mx2Tran (AxialSymmetry (n,n))) = AxialSymmetry (n,n) by Def6;
then A6: f * (Mx2Tran (AxialSymmetry (n,n))) is base_rotation by A1, A5, Th39;
( AxialSymmetry (n,n) is V244( F_Real ) & (AxialSymmetry (n,n)) ~ = AxialSymmetry (n,n) ) by A5, Th7;
then (AxialSymmetry (n,n)) @ = (AxialSymmetry (n,n)) ~ by Th2;
then A7: ( AxialSymmetry (n,n) is Orthogonal & AutMt (f * (Mx2Tran (AxialSymmetry (n,n)))) is Orthogonal ) by A6, Lm11, MATRIX_6:def_7;
A8: AutMt (f * (Mx2Tran (AxialSymmetry (n,n)))) = (AutMt (Mx2Tran (AxialSymmetry (n,n)))) * (AutMt f) by Th29
.= (AxialSymmetry (n,n)) * (AutMt f) by Def6 ;
(AxialSymmetry (n,n)) ~ is_reverse_of AxialSymmetry (n,n) by MATRIX_6:def_4;
then A9: ((AxialSymmetry (n,n)) ~) * (AxialSymmetry (n,n)) = 1. (F_Real,n) by MATRIX_6:def_2;
( width ((AxialSymmetry (n,n)) ~) = n & len (AxialSymmetry (n,n)) = n & width (AxialSymmetry (n,n)) = n & len (AutMt f) = n ) by MATRIX_1:24;
then ((AxialSymmetry (n,n)) ~) * ((AxialSymmetry (n,n)) * (AutMt f)) = (((AxialSymmetry (n,n)) ~) * (AxialSymmetry (n,n))) * (AutMt f) by MATRIX_3:33
.= AutMt f by A9, MATRIX_3:18 ;
hence AutMt f is Orthogonal by A2, A7, A8, MATRIX_6:59; ::_thesis: verum
end;
end;
end;
end;
registration
let n be Nat;
cluster Function-like quasi_total additive homogeneous rotation -> being_homeomorphism for Element of bool [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):];
coherence
for b1 being Function of (TOP-REAL n),(TOP-REAL n) st b1 is homogeneous & b1 is additive & b1 is rotation holds
b1 is being_homeomorphism
proof
set TR = TOP-REAL n;
set cTR = the carrier of (TOP-REAL n);
let f be Function of (TOP-REAL n),(TOP-REAL n); ::_thesis: ( f is homogeneous & f is additive & f is rotation implies f is being_homeomorphism )
assume A1: ( f is homogeneous & f is additive & f is rotation ) ; ::_thesis: f is being_homeomorphism
then reconsider F = f as additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) ;
set M = AutMt F;
percases ( Det (AutMt F) = 1. F_Real or ( Det (AutMt F) = - (1. F_Real) & not f is base_rotation ) ) by A1, Th37, Th38;
suppose Det (AutMt F) = 1. F_Real ; ::_thesis: f is being_homeomorphism
then f is base_rotation by A1, Th37;
hence f is being_homeomorphism ; ::_thesis: verum
end;
supposeA2: ( Det (AutMt F) = - (1. F_Real) & not f is base_rotation ) ; ::_thesis: f is being_homeomorphism
n <> 0
proof
A3: ( dom f = the carrier of (TOP-REAL n) & dom (id the carrier of (TOP-REAL n)) = the carrier of (TOP-REAL n) ) by FUNCT_2:52;
assume n = 0 ; ::_thesis: contradiction
then A4: the carrier of (TOP-REAL n) = {(0. (TOP-REAL n))} by EUCLID:22, EUCLID:77;
rng f c= the carrier of (TOP-REAL n) by RELAT_1:def_19;
then ( rng (id the carrier of (TOP-REAL n)) = the carrier of (TOP-REAL n) & rng f = the carrier of (TOP-REAL n) ) by A4, ZFMISC_1:33;
then f = id (TOP-REAL n) by A3, A4, FUNCT_1:7;
hence contradiction by A2; ::_thesis: verum
end;
then A5: n in Seg n by FINSEQ_1:3;
A6: dom f = the carrier of (TOP-REAL n) by FUNCT_2:52;
set A = AxialSymmetry (n,n);
set MA = Mx2Tran (AxialSymmetry (n,n));
A7: (Mx2Tran (AxialSymmetry (n,n))) " is being_homeomorphism by TOPS_2:56;
A8: ( Mx2Tran (AxialSymmetry (n,n)) is one-to-one & rng (Mx2Tran (AxialSymmetry (n,n))) = [#] (TOP-REAL n) ) by TOPS_2:def_5;
AutMt (Mx2Tran (AxialSymmetry (n,n))) = AxialSymmetry (n,n) by Def6;
then A9: f * (Mx2Tran (AxialSymmetry (n,n))) is base_rotation by A1, A2, A5, Th39;
(f * (Mx2Tran (AxialSymmetry (n,n)))) * ((Mx2Tran (AxialSymmetry (n,n))) ") = f * ((Mx2Tran (AxialSymmetry (n,n))) * ((Mx2Tran (AxialSymmetry (n,n))) ")) by RELAT_1:36
.= f * (id the carrier of (TOP-REAL n)) by A8, TOPS_2:52
.= f by A6, RELAT_1:51 ;
hence f is being_homeomorphism by A9, A7, TOPS_2:57; ::_thesis: verum
end;
end;
end;
end;
begin
theorem :: MATRTOP3:40
for n being Nat
for p, q being Point of (TOP-REAL n) st n = 1 & |.p.| = |.q.| holds
ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is rotation & f . p = q & ( AutMt f = AxialSymmetry (n,n) or AutMt f = 1. (F_Real,n) ) )
proof
let n be Nat; ::_thesis: for p, q being Point of (TOP-REAL n) st n = 1 & |.p.| = |.q.| holds
ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is rotation & f . p = q & ( AutMt f = AxialSymmetry (n,n) or AutMt f = 1. (F_Real,n) ) )
let p, q be Point of (TOP-REAL n); ::_thesis: ( n = 1 & |.p.| = |.q.| implies ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is rotation & f . p = q & ( AutMt f = AxialSymmetry (n,n) or AutMt f = 1. (F_Real,n) ) ) )
set TR = TOP-REAL n;
assume that
A1: n = 1 and
A2: |.p.| = |.q.| ; ::_thesis: ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is rotation & f . p = q & ( AutMt f = AxialSymmetry (n,n) or AutMt f = 1. (F_Real,n) ) )
percases ( p = q or p <> q ) ;
supposeA3: p = q ; ::_thesis: ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is rotation & f . p = q & ( AutMt f = AxialSymmetry (n,n) or AutMt f = 1. (F_Real,n) ) )
take I = id (TOP-REAL n); ::_thesis: ( I is rotation & I . p = q & ( AutMt I = AxialSymmetry (n,n) or AutMt I = 1. (F_Real,n) ) )
id (TOP-REAL n) = Mx2Tran (1. (F_Real,1)) by A1, MATRTOP1:33;
hence ( I is rotation & I . p = q & ( AutMt I = AxialSymmetry (n,n) or AutMt I = 1. (F_Real,n) ) ) by A1, A3, Def6, FUNCT_1:17; ::_thesis: verum
end;
supposeA4: p <> q ; ::_thesis: ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is rotation & f . p = q & ( AutMt f = AxialSymmetry (n,n) or AutMt f = 1. (F_Real,n) ) )
A5: len p = 1 by A1, CARD_1:def_7;
then A6: p = <*(p . 1)*> by FINSEQ_1:40;
A7: 1 in Seg 1 ;
then reconsider f = Mx2Tran (AxialSymmetry (1,1)) as additive homogeneous rotation Function of (TOP-REAL n),(TOP-REAL n) by A1, Th27;
take f ; ::_thesis: ( f is rotation & f . p = q & ( AutMt f = AxialSymmetry (n,n) or AutMt f = 1. (F_Real,n) ) )
A8: ( (q . 1) ^2 >= 0 & (p . 1) ^2 >= 0 ) by XREAL_1:63;
A9: |.p.| = sqrt (Sum (sqr <*(p . 1)*>)) by A5, FINSEQ_1:40
.= sqrt (Sum <*((p . 1) ^2)*>) by RVSUM_1:55
.= sqrt ((p . 1) ^2) by RVSUM_1:73 ;
A10: len q = 1 by A1, CARD_1:def_7;
then A11: q = <*(q . 1)*> by FINSEQ_1:40;
|.q.| = sqrt (Sum (sqr <*(q . 1)*>)) by A10, FINSEQ_1:40
.= sqrt (Sum <*((q . 1) ^2)*>) by RVSUM_1:55
.= sqrt ((q . 1) ^2) by RVSUM_1:73 ;
then A12: (q . 1) ^2 = (p . 1) ^2 by A2, A8, A9, SQUARE_1:28;
len (f . p) = 1 by A1, CARD_1:def_7;
then f . p = <*((f . p) . 1)*> by FINSEQ_1:40
.= <*(- (p . 1))*> by A1, A7, Th9
.= q by A4, A6, A11, A12, SQUARE_1:40 ;
hence ( f is rotation & f . p = q & ( AutMt f = AxialSymmetry (n,n) or AutMt f = 1. (F_Real,n) ) ) by A1, Def6; ::_thesis: verum
end;
end;
end;
theorem :: MATRTOP3:41
for n being Nat
for p, q being Point of (TOP-REAL n) st n <> 1 & |.p.| = |.q.| holds
ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is base_rotation & f . p = q )
proof
let n be Nat; ::_thesis: for p, q being Point of (TOP-REAL n) st n <> 1 & |.p.| = |.q.| holds
ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is base_rotation & f . p = q )
let p, q be Point of (TOP-REAL n); ::_thesis: ( n <> 1 & |.p.| = |.q.| implies ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is base_rotation & f . p = q ) )
set TR = TOP-REAL n;
assume A1: n <> 1 ; ::_thesis: ( not |.p.| = |.q.| or ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is base_rotation & f . p = q ) )
assume A2: |.p.| = |.q.| ; ::_thesis: ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is base_rotation & f . p = q )
percases ( p = q or p <> q ) ;
supposeA3: p = q ; ::_thesis: ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is base_rotation & f . p = q )
take I = id (TOP-REAL n); ::_thesis: ( I is base_rotation & I . p = q )
thus ( I is base_rotation & I . p = q ) by A3, FUNCT_1:17; ::_thesis: verum
end;
supposeA4: p <> q ; ::_thesis: ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is base_rotation & f . p = q )
A5: n <> 0
proof
assume A6: n = 0 ; ::_thesis: contradiction
then p = 0. (TOP-REAL n) by EUCLID:77;
hence contradiction by A4, A6, EUCLID:77; ::_thesis: verum
end;
then A7: n >= 1 by NAT_1:14;
defpred S1[ Nat] means ( $1 + 1 <= n implies ex f being base_rotation Function of (TOP-REAL n),(TOP-REAL n) ex X being set st
( card X = $1 & X c= Seg n & ( for k being Nat st k in X holds
(f . p) . k = q . k ) ) );
A8: Sum (sqr q) >= 0 by RVSUM_1:86;
A9: card (Seg n) = n by FINSEQ_1:57;
A10: for m being Nat st S1[m] holds
S1[m + 1]
proof
let m be Nat; ::_thesis: ( S1[m] implies S1[m + 1] )
assume A11: S1[m] ; ::_thesis: S1[m + 1]
set m1 = m + 1;
assume A12: (m + 1) + 1 <= n ; ::_thesis: ex f being base_rotation Function of (TOP-REAL n),(TOP-REAL n) ex X being set st
( card X = m + 1 & X c= Seg n & ( for k being Nat st k in X holds
(f . p) . k = q . k ) )
then consider f being base_rotation Function of (TOP-REAL n),(TOP-REAL n), Xm being set such that
A13: card Xm = m and
A14: Xm c= Seg n and
A15: for k being Nat st k in Xm holds
(f . p) . k = q . k by A11, NAT_1:13;
set fp = f . p;
set sfp = sqr (f . p);
set sq = sqr q;
A16: m + 1 < n by A12, NAT_1:13;
percases ( ex k being Nat st
( k in (Seg n) \ Xm & (sqr (f . p)) . k >= (sqr q) . k ) or for k being Nat st k in (Seg n) \ Xm holds
(sqr (f . p)) . k < (sqr q) . k ) ;
suppose ex k being Nat st
( k in (Seg n) \ Xm & (sqr (f . p)) . k >= (sqr q) . k ) ; ::_thesis: ex f being base_rotation Function of (TOP-REAL n),(TOP-REAL n) ex X being set st
( card X = m + 1 & X c= Seg n & ( for k being Nat st k in X holds
(f . p) . k = q . k ) )
then consider k being Nat such that
A17: k in (Seg n) \ Xm and
A18: (sqr (f . p)) . k >= (sqr q) . k ;
A19: k in Seg n by A17, XBOOLE_0:def_5;
then A20: 1 <= k by FINSEQ_1:1;
set Xmk = Xm \/ {k};
A21: ( (sqr (f . p)) . k = ((f . p) . k) ^2 & (sqr q) . k = (q . k) ^2 ) by VALUED_1:11;
A22: {k} c= Seg n by A19, ZFMISC_1:31;
then A23: Xm \/ {k} c= Seg n by A14, XBOOLE_1:8;
A24: not k in Xm by A17, XBOOLE_0:def_5;
then card (Xm \/ {k}) = m + 1 by A13, A14, CARD_2:41;
then Xm \/ {k} c< Seg n by A9, A16, A23, XBOOLE_0:def_8;
then consider z being set such that
A25: z in Seg n and
A26: not z in Xm \/ {k} by XBOOLE_0:6;
reconsider z = z as Nat by A25;
A27: 1 <= z by A25, FINSEQ_1:1;
((f . p) . z) ^2 >= 0 by XREAL_1:63;
then A28: 0 + ((q . k) ^2) <= (((f . p) . z) ^2) + (((f . p) . k) ^2) by A18, A21, XREAL_1:7;
A29: z <= n by A25, FINSEQ_1:1;
A30: k <= n by A19, FINSEQ_1:1;
not z in {k} by A26, XBOOLE_0:def_3;
then A31: z <> k by TARSKI:def_1;
now__::_thesis:_ex_g_being_base_rotation_Function_of_(TOP-REAL_n),(TOP-REAL_n)_st_
(_g_is_{k,z}_-support-yielding_&_(g_._(f_._p))_._k_=_q_._k_)
percases ( z < k or z > k ) by A31, XXREAL_0:1;
supposeA32: z < k ; ::_thesis: ex g being base_rotation Function of (TOP-REAL n),(TOP-REAL n) st
( g is {k,z} -support-yielding & (g . (f . p)) . k = q . k )
then consider r being real number such that
A33: ((Mx2Tran (Rotation (z,k,n,r))) . (f . p)) . k = q . k by A27, A28, A30, Th25;
( Mx2Tran (Rotation (z,k,n,r)) is {k,z} -support-yielding & Mx2Tran (Rotation (z,k,n,r)) is base_rotation ) by A27, A30, A32, Th28, Th26;
hence ex g being base_rotation Function of (TOP-REAL n),(TOP-REAL n) st
( g is {k,z} -support-yielding & (g . (f . p)) . k = q . k ) by A33; ::_thesis: verum
end;
supposeA34: z > k ; ::_thesis: ex g being base_rotation Function of (TOP-REAL n),(TOP-REAL n) st
( g is {k,z} -support-yielding & (g . (f . p)) . k = q . k )
then consider r being real number such that
A35: ((Mx2Tran (Rotation (k,z,n,r))) . (f . p)) . k = q . k by A20, A28, A29, Th24;
( Mx2Tran (Rotation (k,z,n,r)) is {z,k} -support-yielding & Mx2Tran (Rotation (k,z,n,r)) is base_rotation ) by A20, A29, A34, Th28, Th26;
hence ex g being base_rotation Function of (TOP-REAL n),(TOP-REAL n) st
( g is {k,z} -support-yielding & (g . (f . p)) . k = q . k ) by A35; ::_thesis: verum
end;
end;
end;
then consider g being base_rotation Function of (TOP-REAL n),(TOP-REAL n) such that
A36: g is {k,z} -support-yielding and
A37: (g . (f . p)) . k = q . k ;
take gf = g * f; ::_thesis: ex X being set st
( card X = m + 1 & X c= Seg n & ( for k being Nat st k in X holds
(gf . p) . k = q . k ) )
take Xm \/ {k} ; ::_thesis: ( card (Xm \/ {k}) = m + 1 & Xm \/ {k} c= Seg n & ( for k being Nat st k in Xm \/ {k} holds
(gf . p) . k = q . k ) )
thus ( card (Xm \/ {k}) = m + 1 & Xm \/ {k} c= Seg n ) by A13, A14, A22, A24, CARD_2:41, XBOOLE_1:8; ::_thesis: for k being Nat st k in Xm \/ {k} holds
(gf . p) . k = q . k
let m be Nat; ::_thesis: ( m in Xm \/ {k} implies (gf . p) . m = q . m )
A38: dom gf = the carrier of (TOP-REAL n) by FUNCT_2:52;
A39: dom g = the carrier of (TOP-REAL n) by FUNCT_2:52;
assume A40: m in Xm \/ {k} ; ::_thesis: (gf . p) . m = q . m
then A41: ( m in Xm or m in {k} ) by XBOOLE_0:def_3;
percases ( m in Xm or m = k ) by A41, TARSKI:def_1;
supposeA42: m in Xm ; ::_thesis: (gf . p) . m = q . m
then m <> k by A17, XBOOLE_0:def_5;
then not m in {k,z} by A26, A40, TARSKI:def_2;
then (g . (f . p)) . m = (f . p) . m by A36, A39, Def1;
hence (gf . p) . m = (f . p) . m by A38, FUNCT_1:12
.= q . m by A15, A42 ;
::_thesis: verum
end;
suppose m = k ; ::_thesis: (gf . p) . m = q . m
hence (gf . p) . m = q . m by A37, A38, FUNCT_1:12; ::_thesis: verum
end;
end;
end;
supposeA43: for k being Nat st k in (Seg n) \ Xm holds
(sqr (f . p)) . k < (sqr q) . k ; ::_thesis: ex f being base_rotation Function of (TOP-REAL n),(TOP-REAL n) ex X being set st
( card X = m + 1 & X c= Seg n & ( for k being Nat st k in X holds
(f . p) . k = q . k ) )
A44: Sum (sqr (f . p)) >= 0 by RVSUM_1:86;
( Sum (sqr q) >= 0 & |.p.| = |.(f . p).| ) by Def4, RVSUM_1:86;
then A45: Sum (sqr (f . p)) = Sum (sqr q) by A2, A44, SQUARE_1:28;
f . p = @ (@ (f . p)) ;
then A46: len (sqr (f . p)) = len (f . p) by RVSUM_1:143;
q = @ (@ q) ;
then A47: len (sqr q) = len q by RVSUM_1:143;
( len (f . p) = n & len q = n ) by CARD_1:def_7;
then reconsider sfp = sqr (f . p), sq = sqr q as Element of n -tuples_on REAL by A46, A47, FINSEQ_2:92;
m < n by A16, NAT_1:13;
then Xm <> Seg n by A13, FINSEQ_1:57;
then Xm c< Seg n by A14, XBOOLE_0:def_8;
then consider z being set such that
A48: z in Seg n and
A49: not z in Xm by XBOOLE_0:6;
reconsider z = z as Nat by A48;
A50: z in (Seg n) \ Xm by A48, A49, XBOOLE_0:def_5;
for k being Nat st k in Seg n holds
sfp . k <= sq . k
proof
let k be Nat; ::_thesis: ( k in Seg n implies sfp . k <= sq . k )
assume A51: k in Seg n ; ::_thesis: sfp . k <= sq . k
percases ( k in (Seg n) \ Xm or k in Xm ) by A51, XBOOLE_0:def_5;
suppose k in (Seg n) \ Xm ; ::_thesis: sfp . k <= sq . k
hence sfp . k <= sq . k by A43; ::_thesis: verum
end;
suppose k in Xm ; ::_thesis: sfp . k <= sq . k
then (f . p) . k = q . k by A15;
then sfp . k = (q . k) ^2 by VALUED_1:11
.= sq . k by VALUED_1:11 ;
hence sfp . k <= sq . k ; ::_thesis: verum
end;
end;
end;
then sfp . z >= sq . z by A45, A48, RVSUM_1:83;
hence ex f being base_rotation Function of (TOP-REAL n),(TOP-REAL n) ex X being set st
( card X = m + 1 & X c= Seg n & ( for k being Nat st k in X holds
(f . p) . k = q . k ) ) by A43, A50; ::_thesis: verum
end;
end;
end;
reconsider n1 = n - 1 as Nat by A5, NAT_1:14, NAT_1:21;
A52: n1 + 1 = n ;
A53: S1[ 0 ]
proof
assume 0 + 1 <= n ; ::_thesis: ex f being base_rotation Function of (TOP-REAL n),(TOP-REAL n) ex X being set st
( card X = 0 & X c= Seg n & ( for k being Nat st k in X holds
(f . p) . k = q . k ) )
take f = id (TOP-REAL n); ::_thesis: ex X being set st
( card X = 0 & X c= Seg n & ( for k being Nat st k in X holds
(f . p) . k = q . k ) )
take X = {} ; ::_thesis: ( card X = 0 & X c= Seg n & ( for k being Nat st k in X holds
(f . p) . k = q . k ) )
thus ( card X = 0 & X c= Seg n ) by XBOOLE_1:2; ::_thesis: for k being Nat st k in X holds
(f . p) . k = q . k
let k be Nat; ::_thesis: ( k in X implies (f . p) . k = q . k )
assume k in X ; ::_thesis: (f . p) . k = q . k
hence (f . p) . k = q . k ; ::_thesis: verum
end;
for m being Nat holds S1[m] from NAT_1:sch_2(A53, A10);
then consider f being base_rotation Function of (TOP-REAL n),(TOP-REAL n), X being set such that
A54: ( card X = n1 & X c= Seg n ) and
A55: for k being Nat st k in X holds
(f . p) . k = q . k by A52;
card ((Seg n) \ X) = n - n1 by A9, A54, CARD_2:44;
then consider z being set such that
A56: {z} = (Seg n) \ X by CARD_2:42;
set fp = f . p;
( Sum (sqr (f . p)) >= 0 & |.p.| = |.(f . p).| ) by Def4, RVSUM_1:86;
then A57: Sum (sqr q) = Sum (sqr (f . p)) by A2, A8, SQUARE_1:28;
A58: z in {z} by TARSKI:def_1;
then A59: z in Seg n by A56, XBOOLE_0:def_5;
reconsider z = z as Nat by A56, A58;
set fpz = (f . p) +* (z,(q . z));
A60: len (f . p) = n by CARD_1:def_7;
then A61: dom (f . p) = Seg n by FINSEQ_1:def_3;
A62: for k being Nat st 1 <= k & k <= n holds
((f . p) +* (z,(q . z))) . k = q . k
proof
let k be Nat; ::_thesis: ( 1 <= k & k <= n implies ((f . p) +* (z,(q . z))) . k = q . k )
assume ( 1 <= k & k <= n ) ; ::_thesis: ((f . p) +* (z,(q . z))) . k = q . k
then A63: k in Seg n by FINSEQ_1:1;
percases ( k = z or k <> z ) ;
suppose k = z ; ::_thesis: ((f . p) +* (z,(q . z))) . k = q . k
hence ((f . p) +* (z,(q . z))) . k = q . k by A61, A63, FUNCT_7:31; ::_thesis: verum
end;
supposeA64: k <> z ; ::_thesis: ((f . p) +* (z,(q . z))) . k = q . k
then not k in (Seg n) \ X by A56, TARSKI:def_1;
then A65: k in X by A63, XBOOLE_0:def_5;
((f . p) +* (z,(q . z))) . k = (f . p) . k by A64, FUNCT_7:32;
hence ((f . p) +* (z,(q . z))) . k = q . k by A55, A65; ::_thesis: verum
end;
end;
end;
A66: ( len ((f . p) +* (z,(q . z))) = len (f . p) & len q = n ) by CARD_1:def_7, FUNCT_7:97;
then A67: (f . p) +* (z,(q . z)) = q by A60, A62, FINSEQ_1:14;
then A68: Sum (sqr q) = ((Sum (sqr (f . p))) - (((f . p) . z) ^2)) + ((q . z) ^2) by A59, A61, Th3;
percases ( (f . p) . z = q . z or (f . p) . z = - (q . z) ) by A68, A57, SQUARE_1:40;
suppose (f . p) . z = q . z ; ::_thesis: ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is base_rotation & f . p = q )
then f . p = q by A67, FUNCT_7:35;
hence ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is base_rotation & f . p = q ) ; ::_thesis: verum
end;
supposeA69: (f . p) . z = - (q . z) ; ::_thesis: ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is base_rotation & f . p = q )
1 < n by A1, A7, XXREAL_0:1;
then 1 + 1 <= n by NAT_1:13;
then consider h being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) such that
A70: h is base_rotation and
A71: h . (f . p) = (f . p) +* (z,(- ((f . p) . z))) by A59, Th34;
dom (h * f) = the carrier of (TOP-REAL n) by FUNCT_2:52;
then (h * f) . p = (f . p) +* (z,(- ((f . p) . z))) by A71, FUNCT_1:12
.= q by A60, A62, A66, A69, FINSEQ_1:14 ;
hence ex f being additive homogeneous Function of (TOP-REAL n),(TOP-REAL n) st
( f is base_rotation & f . p = q ) by A70; ::_thesis: verum
end;
end;
end;
end;
end;