:: MATRIX11 semantic presentation begin notation let X be set ; synonym 2Set X for TWOELEMENTSETS X; end; theorem Th1: :: MATRIX11:1 for X being set for n being Nat holds ( X in 2Set (Seg n) iff ex i, j being Nat st ( i in Seg n & j in Seg n & i < j & X = {i,j} ) ) proof let X be set ; ::_thesis: for n being Nat holds ( X in 2Set (Seg n) iff ex i, j being Nat st ( i in Seg n & j in Seg n & i < j & X = {i,j} ) ) let n be Nat; ::_thesis: ( X in 2Set (Seg n) iff ex i, j being Nat st ( i in Seg n & j in Seg n & i < j & X = {i,j} ) ) thus ( X in 2Set (Seg n) implies ex i, j being Nat st ( i in Seg n & j in Seg n & i < j & X = {i,j} ) ) ::_thesis: ( ex i, j being Nat st ( i in Seg n & j in Seg n & i < j & X = {i,j} ) implies X in 2Set (Seg n) ) proof assume X in 2Set (Seg n) ; ::_thesis: ex i, j being Nat st ( i in Seg n & j in Seg n & i < j & X = {i,j} ) then consider x, y being set such that A1: x in Seg n and A2: y in Seg n and A3: x <> y and A4: X = {x,y} by SGRAPH1:8; reconsider x = x, y = y as Element of NAT by A1, A2; ( x > y or y > x ) by A3, XXREAL_0:1; hence ex i, j being Nat st ( i in Seg n & j in Seg n & i < j & X = {i,j} ) by A1, A2, A4; ::_thesis: verum end; assume ex i, j being Nat st ( i in Seg n & j in Seg n & i < j & X = {i,j} ) ; ::_thesis: X in 2Set (Seg n) then consider i, j being Nat such that A5: i in Seg n and A6: j in Seg n and A7: i < j and A8: X = {i,j} ; {i,j} c= Seg n by A5, A6, ZFMISC_1:32; hence X in 2Set (Seg n) by A5, A6, A7, A8, SGRAPH1:8; ::_thesis: verum end; theorem Th2: :: MATRIX11:2 ( 2Set (Seg 0) = {} & 2Set (Seg 1) = {} ) proof thus 2Set (Seg 0) = {} ::_thesis: 2Set (Seg 1) = {} proof assume 2Set (Seg 0) <> {} ; ::_thesis: contradiction then consider x being set such that A1: x in 2Set (Seg 0) by XBOOLE_0:def_1; ex i, j being Nat st ( i in Seg 0 & j in Seg 0 & i < j & x = {i,j} ) by A1, Th1; hence contradiction ; ::_thesis: verum end; thus 2Set (Seg 1) = {} ::_thesis: verum proof assume 2Set (Seg 1) <> {} ; ::_thesis: contradiction then consider x being set such that A2: x in 2Set (Seg 1) by XBOOLE_0:def_1; consider i, j being Nat such that A3: i in Seg 1 and A4: j in Seg 1 and A5: i < j and x = {i,j} by A2, Th1; i = 1 by A3, FINSEQ_1:2, TARSKI:def_1; hence contradiction by A4, A5, FINSEQ_1:2, TARSKI:def_1; ::_thesis: verum end; end; theorem Th3: :: MATRIX11:3 for n being Nat st n >= 2 holds {1,2} in 2Set (Seg n) proof let n be Nat; ::_thesis: ( n >= 2 implies {1,2} in 2Set (Seg n) ) assume A1: n >= 2 ; ::_thesis: {1,2} in 2Set (Seg n) 1 <= n by A1, XXREAL_0:2; then A2: 1 in Seg n ; 2 in Seg n by A1; hence {1,2} in 2Set (Seg n) by A2, Th1; ::_thesis: verum end; registration let n be Nat; cluster 2Set (Seg (n + 2)) -> non empty finite ; coherence ( not 2Set (Seg (n + 2)) is empty & 2Set (Seg (n + 2)) is finite ) proof n + 2 >= 0 + 2 by XREAL_1:6; hence ( not 2Set (Seg (n + 2)) is empty & 2Set (Seg (n + 2)) is finite ) by Th3, SGRAPH1:13; ::_thesis: verum end; end; registration let n be Nat; let x be set ; let perm be Element of Permutations n; clusterperm . x -> natural ; coherence perm . x is natural proof percases ( x in dom perm or not x in dom perm ) ; supposeA1: x in dom perm ; ::_thesis: perm . x is natural perm is Permutation of (Seg n) by MATRIX_2:def_9; then A2: rng perm = Seg n by FUNCT_2:def_3; perm . x in rng perm by A1, FUNCT_1:def_3; hence perm . x is natural by A2; ::_thesis: verum end; suppose not x in dom perm ; ::_thesis: perm . x is natural hence perm . x is natural by FUNCT_1:def_2; ::_thesis: verum end; end; end; end; registration let K be Field; cluster the multF of K -> having_a_unity ; coherence the multF of K is having_a_unity ; cluster the multF of K -> associative ; coherence the multF of K is associative ; end; definition let n be Nat; let K be Field; let perm2 be Element of Permutations (n + 2); func Part_sgn (perm2,K) -> Function of (2Set (Seg (n + 2))), the carrier of K means :Def1: :: MATRIX11:def 1 for i, j being Element of NAT st i in Seg (n + 2) & j in Seg (n + 2) & i < j holds ( ( perm2 . i < perm2 . j implies it . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies it . {i,j} = - (1_ K) ) ); existence ex b1 being Function of (2Set (Seg (n + 2))), the carrier of K st for i, j being Element of NAT st i in Seg (n + 2) & j in Seg (n + 2) & i < j holds ( ( perm2 . i < perm2 . j implies b1 . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies b1 . {i,j} = - (1_ K) ) ) proof set n9 = n + 2; defpred S1[ set , set ] means for i, j being Element of NAT st i in Seg (n + 2) & j in Seg (n + 2) & i < j & $1 = {i,j} holds ( ( perm2 . i < perm2 . j implies $2 = 1_ K ) & ( perm2 . i > perm2 . j implies $2 = - (1_ K) ) ); A1: for x being set st x in 2Set (Seg (n + 2)) holds ex y being set st ( y in the carrier of K & S1[x,y] ) proof let x be set ; ::_thesis: ( x in 2Set (Seg (n + 2)) implies ex y being set st ( y in the carrier of K & S1[x,y] ) ) assume x in 2Set (Seg (n + 2)) ; ::_thesis: ex y being set st ( y in the carrier of K & S1[x,y] ) then consider i, j being Nat such that A2: i in Seg (n + 2) and A3: j in Seg (n + 2) and A4: i < j and A5: x = {i,j} by Th1; perm2 is Permutation of (Seg (n + 2)) by MATRIX_2:def_9; then A6: perm2 . i <> perm2 . j by A2, A3, A4, FUNCT_2:19; now__::_thesis:_(_(_perm2_._i_<_perm2_._j_&_ex_y_being_set_st_ (_y_in_the_carrier_of_K_&_S1[x,y]_)_)_or_(_perm2_._i_>_perm2_._j_&_ex_y_being_set_st_ (_y_in_the_carrier_of_K_&_S1[x,y]_)_)_) percases ( perm2 . i < perm2 . j or perm2 . i > perm2 . j ) by A6, XXREAL_0:1; caseA7: perm2 . i < perm2 . j ; ::_thesis: ex y being set st ( y in the carrier of K & S1[x,y] ) S1[x, 1_ K] proof let i9, j9 be Element of NAT ; ::_thesis: ( i9 in Seg (n + 2) & j9 in Seg (n + 2) & i9 < j9 & x = {i9,j9} implies ( ( perm2 . i9 < perm2 . j9 implies 1_ K = 1_ K ) & ( perm2 . i9 > perm2 . j9 implies 1_ K = - (1_ K) ) ) ) assume that i9 in Seg (n + 2) and j9 in Seg (n + 2) and A8: i9 < j9 and A9: x = {i9,j9} ; ::_thesis: ( ( perm2 . i9 < perm2 . j9 implies 1_ K = 1_ K ) & ( perm2 . i9 > perm2 . j9 implies 1_ K = - (1_ K) ) ) ( ( i = i9 & j = j9 ) or ( i = j9 & j = i9 ) ) by A5, A9, ZFMISC_1:22; hence ( ( perm2 . i9 < perm2 . j9 implies 1_ K = 1_ K ) & ( perm2 . i9 > perm2 . j9 implies 1_ K = - (1_ K) ) ) by A4, A7, A8; ::_thesis: verum end; hence ex y being set st ( y in the carrier of K & S1[x,y] ) ; ::_thesis: verum end; caseA10: perm2 . i > perm2 . j ; ::_thesis: ex y being set st ( y in the carrier of K & S1[x,y] ) S1[x, - (1_ K)] proof let i9, j9 be Element of NAT ; ::_thesis: ( i9 in Seg (n + 2) & j9 in Seg (n + 2) & i9 < j9 & x = {i9,j9} implies ( ( perm2 . i9 < perm2 . j9 implies - (1_ K) = 1_ K ) & ( perm2 . i9 > perm2 . j9 implies - (1_ K) = - (1_ K) ) ) ) assume that i9 in Seg (n + 2) and j9 in Seg (n + 2) and A11: i9 < j9 and A12: x = {i9,j9} ; ::_thesis: ( ( perm2 . i9 < perm2 . j9 implies - (1_ K) = 1_ K ) & ( perm2 . i9 > perm2 . j9 implies - (1_ K) = - (1_ K) ) ) ( ( i = i9 & j = j9 ) or ( i = j9 & j = i9 ) ) by A5, A12, ZFMISC_1:22; hence ( ( perm2 . i9 < perm2 . j9 implies - (1_ K) = 1_ K ) & ( perm2 . i9 > perm2 . j9 implies - (1_ K) = - (1_ K) ) ) by A4, A10, A11; ::_thesis: verum end; hence ex y being set st ( y in the carrier of K & S1[x,y] ) ; ::_thesis: verum end; end; end; hence ex y being set st ( y in the carrier of K & S1[x,y] ) ; ::_thesis: verum end; consider Path being Function of (2Set (Seg (n + 2))), the carrier of K such that A13: for x being set st x in 2Set (Seg (n + 2)) holds S1[x,Path . x] from FUNCT_2:sch_1(A1); take Path ; ::_thesis: for i, j being Element of NAT st i in Seg (n + 2) & j in Seg (n + 2) & i < j holds ( ( perm2 . i < perm2 . j implies Path . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies Path . {i,j} = - (1_ K) ) ) let i, j be Element of NAT ; ::_thesis: ( i in Seg (n + 2) & j in Seg (n + 2) & i < j implies ( ( perm2 . i < perm2 . j implies Path . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies Path . {i,j} = - (1_ K) ) ) ) assume that A14: i in Seg (n + 2) and A15: j in Seg (n + 2) and A16: i < j ; ::_thesis: ( ( perm2 . i < perm2 . j implies Path . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies Path . {i,j} = - (1_ K) ) ) {i,j} in 2Set (Seg (n + 2)) by A14, A15, A16, Th1; hence ( ( perm2 . i < perm2 . j implies Path . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies Path . {i,j} = - (1_ K) ) ) by A13, A14, A15, A16; ::_thesis: verum end; uniqueness for b1, b2 being Function of (2Set (Seg (n + 2))), the carrier of K st ( for i, j being Element of NAT st i in Seg (n + 2) & j in Seg (n + 2) & i < j holds ( ( perm2 . i < perm2 . j implies b1 . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies b1 . {i,j} = - (1_ K) ) ) ) & ( for i, j being Element of NAT st i in Seg (n + 2) & j in Seg (n + 2) & i < j holds ( ( perm2 . i < perm2 . j implies b2 . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies b2 . {i,j} = - (1_ K) ) ) ) holds b1 = b2 proof set n9 = n + 2; let P1, P2 be Function of (2Set (Seg (n + 2))), the carrier of K; ::_thesis: ( ( for i, j being Element of NAT st i in Seg (n + 2) & j in Seg (n + 2) & i < j holds ( ( perm2 . i < perm2 . j implies P1 . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies P1 . {i,j} = - (1_ K) ) ) ) & ( for i, j being Element of NAT st i in Seg (n + 2) & j in Seg (n + 2) & i < j holds ( ( perm2 . i < perm2 . j implies P2 . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies P2 . {i,j} = - (1_ K) ) ) ) implies P1 = P2 ) assume that A17: for i, j being Element of NAT st i in Seg (n + 2) & j in Seg (n + 2) & i < j holds ( ( perm2 . i < perm2 . j implies P1 . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies P1 . {i,j} = - (1_ K) ) ) and A18: for i, j being Element of NAT st i in Seg (n + 2) & j in Seg (n + 2) & i < j holds ( ( perm2 . i < perm2 . j implies P2 . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies P2 . {i,j} = - (1_ K) ) ) ; ::_thesis: P1 = P2 for x being set st x in 2Set (Seg (n + 2)) holds P1 . x = P2 . x proof let x be set ; ::_thesis: ( x in 2Set (Seg (n + 2)) implies P1 . x = P2 . x ) assume x in 2Set (Seg (n + 2)) ; ::_thesis: P1 . x = P2 . x then consider i, j being Nat such that A19: i in Seg (n + 2) and A20: j in Seg (n + 2) and A21: i < j and A22: x = {i,j} by Th1; perm2 is Permutation of (Seg (n + 2)) by MATRIX_2:def_9; then A23: perm2 . i <> perm2 . j by A19, A20, A21, FUNCT_2:19; now__::_thesis:_(_(_perm2_._i_<_perm2_._j_&_P1_._x_=_P2_._x_)_or_(_perm2_._i_>_perm2_._j_&_P1_._x_=_P2_._x_)_) percases ( perm2 . i < perm2 . j or perm2 . i > perm2 . j ) by A23, XXREAL_0:1; caseA24: perm2 . i < perm2 . j ; ::_thesis: P1 . x = P2 . x then P1 . {i,j} = 1_ K by A17, A19, A20, A21; hence P1 . x = P2 . x by A18, A19, A20, A21, A22, A24; ::_thesis: verum end; caseA25: perm2 . i > perm2 . j ; ::_thesis: P1 . x = P2 . x then P1 . {i,j} = - (1_ K) by A17, A19, A20, A21; hence P1 . x = P2 . x by A18, A19, A20, A21, A22, A25; ::_thesis: verum end; end; end; hence P1 . x = P2 . x ; ::_thesis: verum end; hence P1 = P2 by FUNCT_2:12; ::_thesis: verum end; end; :: deftheorem Def1 defines Part_sgn MATRIX11:def_1_:_ for n being Nat for K being Field for perm2 being Element of Permutations (n + 2) for b4 being Function of (2Set (Seg (n + 2))), the carrier of K holds ( b4 = Part_sgn (perm2,K) iff for i, j being Element of NAT st i in Seg (n + 2) & j in Seg (n + 2) & i < j holds ( ( perm2 . i < perm2 . j implies b4 . {i,j} = 1_ K ) & ( perm2 . i > perm2 . j implies b4 . {i,j} = - (1_ K) ) ) ); theorem Th4: :: MATRIX11:4 for n being Nat for K being Field for p2 being Element of Permutations (n + 2) for X being Element of Fin (2Set (Seg (n + 2))) st ( for x being set st x in X holds (Part_sgn (p2,K)) . x = 1_ K ) holds the multF of K $$ (X,(Part_sgn (p2,K))) = 1_ K proof let n be Nat; ::_thesis: for K being Field for p2 being Element of Permutations (n + 2) for X being Element of Fin (2Set (Seg (n + 2))) st ( for x being set st x in X holds (Part_sgn (p2,K)) . x = 1_ K ) holds the multF of K $$ (X,(Part_sgn (p2,K))) = 1_ K let K be Field; ::_thesis: for p2 being Element of Permutations (n + 2) for X being Element of Fin (2Set (Seg (n + 2))) st ( for x being set st x in X holds (Part_sgn (p2,K)) . x = 1_ K ) holds the multF of K $$ (X,(Part_sgn (p2,K))) = 1_ K let p2 be Element of Permutations (n + 2); ::_thesis: for X being Element of Fin (2Set (Seg (n + 2))) st ( for x being set st x in X holds (Part_sgn (p2,K)) . x = 1_ K ) holds the multF of K $$ (X,(Part_sgn (p2,K))) = 1_ K let X be Element of Fin (2Set (Seg (n + 2))); ::_thesis: ( ( for x being set st x in X holds (Part_sgn (p2,K)) . x = 1_ K ) implies the multF of K $$ (X,(Part_sgn (p2,K))) = 1_ K ) assume A1: for x being set st x in X holds (Part_sgn (p2,K)) . x = 1_ K ; ::_thesis: the multF of K $$ (X,(Part_sgn (p2,K))) = 1_ K set Path = Part_sgn (p2,K); set 2S = 2Set (Seg (n + 2)); set KK = the carrier of K; set mm = the multF of K; consider G being Function of (Fin (2Set (Seg (n + 2)))), the carrier of K such that A2: the multF of K $$ (X,(Part_sgn (p2,K))) = G . X and A3: for e being Element of the carrier of K st e is_a_unity_wrt the multF of K holds G . {} = e and A4: for x being Element of 2Set (Seg (n + 2)) holds G . {x} = (Part_sgn (p2,K)) . x and A5: for B being Element of Fin (2Set (Seg (n + 2))) st B c= X & B <> {} holds for x being Element of 2Set (Seg (n + 2)) st x in X \ B holds G . (B \/ {x}) = the multF of K . ((G . B),((Part_sgn (p2,K)) . x)) by SETWISEO:def_3; defpred S1[ Nat] means for B being Element of Fin (2Set (Seg (n + 2))) st card B = $1 & B c= X holds G . B = 1_ K; A6: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A7: S1[k] ; ::_thesis: S1[k + 1] set k1 = k + 1; let B be Element of Fin (2Set (Seg (n + 2))); ::_thesis: ( card B = k + 1 & B c= X implies G . B = 1_ K ) assume that A8: card B = k + 1 and A9: B c= X ; ::_thesis: G . B = 1_ K now__::_thesis:_(_(_k_=_0_&_G_._B_=_1__K_)_or_(_k_>_0_&_G_._B_=_1__K_)_) percases ( k = 0 or k > 0 ) ; case k = 0 ; ::_thesis: G . B = 1_ K then consider x being set such that A10: B = {x} by A8, CARD_2:42; A11: x in B by A10, TARSKI:def_1; B c= 2Set (Seg (n + 2)) by FINSUB_1:def_5; then reconsider x = x as Element of 2Set (Seg (n + 2)) by A11; G . B = (Part_sgn (p2,K)) . x by A4, A10; hence G . B = 1_ K by A1, A9, A11; ::_thesis: verum end; caseA12: k > 0 ; ::_thesis: G . B = 1_ K consider x being set such that A13: x in B by A8, CARD_1:27, XBOOLE_0:def_1; B c= 2Set (Seg (n + 2)) by FINSUB_1:def_5; then reconsider x = x as Element of 2Set (Seg (n + 2)) by A13; A14: (Part_sgn (p2,K)) . x = 1_ K by A1, A9, A13; B c= 2Set (Seg (n + 2)) by FINSUB_1:def_5; then B \ {x} c= 2Set (Seg (n + 2)) by XBOOLE_1:1; then reconsider B9 = B \ {x} as Element of Fin (2Set (Seg (n + 2))) by FINSUB_1:def_5; A15: not x in B9 by ZFMISC_1:56; then A16: x in X \ B9 by A9, A13, XBOOLE_0:def_5; A17: {x} \/ B9 = B by A13, ZFMISC_1:116; then A18: k + 1 = (card B9) + 1 by A8, A15, CARD_2:41; then G . B9 = 1_ K by A7, A9, XBOOLE_1:1; then G . B = (1_ K) * (1_ K) by A5, A9, A12, A17, A18, A16, A14, CARD_1:27, XBOOLE_1:1; hence G . B = 1_ K by VECTSP_1:def_4; ::_thesis: verum end; end; end; hence G . B = 1_ K ; ::_thesis: verum end; A19: S1[ 0 ] proof let B be Element of Fin (2Set (Seg (n + 2))); ::_thesis: ( card B = 0 & B c= X implies G . B = 1_ K ) assume that A20: card B = 0 and B c= X ; ::_thesis: G . B = 1_ K B = {} by A20; hence G . B = 1_ K by A3, FVSUM_1:4; ::_thesis: verum end; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A19, A6); then S1[ card X] ; hence the multF of K $$ (X,(Part_sgn (p2,K))) = 1_ K by A2; ::_thesis: verum end; theorem Th5: :: MATRIX11:5 for n being Nat for K being Field for p2 being Element of Permutations (n + 2) for s being Element of 2Set (Seg (n + 2)) holds ( (Part_sgn (p2,K)) . s = 1_ K or (Part_sgn (p2,K)) . s = - (1_ K) ) proof let n be Nat; ::_thesis: for K being Field for p2 being Element of Permutations (n + 2) for s being Element of 2Set (Seg (n + 2)) holds ( (Part_sgn (p2,K)) . s = 1_ K or (Part_sgn (p2,K)) . s = - (1_ K) ) let K be Field; ::_thesis: for p2 being Element of Permutations (n + 2) for s being Element of 2Set (Seg (n + 2)) holds ( (Part_sgn (p2,K)) . s = 1_ K or (Part_sgn (p2,K)) . s = - (1_ K) ) let p2 be Element of Permutations (n + 2); ::_thesis: for s being Element of 2Set (Seg (n + 2)) holds ( (Part_sgn (p2,K)) . s = 1_ K or (Part_sgn (p2,K)) . s = - (1_ K) ) let s be Element of 2Set (Seg (n + 2)); ::_thesis: ( (Part_sgn (p2,K)) . s = 1_ K or (Part_sgn (p2,K)) . s = - (1_ K) ) consider i, j being Nat such that A1: i in Seg (n + 2) and A2: j in Seg (n + 2) and A3: i < j and A4: s = {i,j} by Th1; p2 is Permutation of (Seg (n + 2)) by MATRIX_2:def_9; then p2 . i <> p2 . j by A1, A2, A3, FUNCT_2:19; then ( p2 . i > p2 . j or p2 . i < p2 . j ) by XXREAL_0:1; hence ( (Part_sgn (p2,K)) . s = 1_ K or (Part_sgn (p2,K)) . s = - (1_ K) ) by A1, A2, A3, A4, Def1; ::_thesis: verum end; theorem Th6: :: MATRIX11:6 for n being Nat for K being Field for p2, q2 being Element of Permutations (n + 2) for i, j being Nat st i in Seg (n + 2) & j in Seg (n + 2) & i < j & p2 . i = q2 . i & p2 . j = q2 . j holds (Part_sgn (p2,K)) . {i,j} = (Part_sgn (q2,K)) . {i,j} proof let n be Nat; ::_thesis: for K being Field for p2, q2 being Element of Permutations (n + 2) for i, j being Nat st i in Seg (n + 2) & j in Seg (n + 2) & i < j & p2 . i = q2 . i & p2 . j = q2 . j holds (Part_sgn (p2,K)) . {i,j} = (Part_sgn (q2,K)) . {i,j} let K be Field; ::_thesis: for p2, q2 being Element of Permutations (n + 2) for i, j being Nat st i in Seg (n + 2) & j in Seg (n + 2) & i < j & p2 . i = q2 . i & p2 . j = q2 . j holds (Part_sgn (p2,K)) . {i,j} = (Part_sgn (q2,K)) . {i,j} let p2, q2 be Element of Permutations (n + 2); ::_thesis: for i, j being Nat st i in Seg (n + 2) & j in Seg (n + 2) & i < j & p2 . i = q2 . i & p2 . j = q2 . j holds (Part_sgn (p2,K)) . {i,j} = (Part_sgn (q2,K)) . {i,j} set n2 = n + 2; let i, j be Nat; ::_thesis: ( i in Seg (n + 2) & j in Seg (n + 2) & i < j & p2 . i = q2 . i & p2 . j = q2 . j implies (Part_sgn (p2,K)) . {i,j} = (Part_sgn (q2,K)) . {i,j} ) assume that A1: i in Seg (n + 2) and A2: j in Seg (n + 2) and A3: i < j and A4: p2 . i = q2 . i and A5: p2 . j = q2 . j ; ::_thesis: (Part_sgn (p2,K)) . {i,j} = (Part_sgn (q2,K)) . {i,j} reconsider p29 = p2 as Permutation of (Seg (n + 2)) by MATRIX_2:def_9; A6: p29 . i <> p29 . j by A1, A2, A3, FUNCT_2:19; now__::_thesis:_(Part_sgn_(p2,K))_._{i,j}_=_(Part_sgn_(q2,K))_._{i,j} percases ( p2 . i < p2 . j or p2 . i > p2 . j ) by A6, XXREAL_0:1; supposeA7: p2 . i < p2 . j ; ::_thesis: (Part_sgn (p2,K)) . {i,j} = (Part_sgn (q2,K)) . {i,j} then (Part_sgn (p2,K)) . {i,j} = 1_ K by A1, A2, A3, Def1; hence (Part_sgn (p2,K)) . {i,j} = (Part_sgn (q2,K)) . {i,j} by A1, A2, A3, A4, A5, A7, Def1; ::_thesis: verum end; supposeA8: p2 . i > p2 . j ; ::_thesis: (Part_sgn (p2,K)) . {i,j} = (Part_sgn (q2,K)) . {i,j} then (Part_sgn (p2,K)) . {i,j} = - (1_ K) by A1, A2, A3, Def1; hence (Part_sgn (p2,K)) . {i,j} = (Part_sgn (q2,K)) . {i,j} by A1, A2, A3, A4, A5, A8, Def1; ::_thesis: verum end; end; end; hence (Part_sgn (p2,K)) . {i,j} = (Part_sgn (q2,K)) . {i,j} ; ::_thesis: verum end; theorem Th7: :: MATRIX11:7 for n being Nat for K being Field for X being Element of Fin (2Set (Seg (n + 2))) for p2, q2 being Element of Permutations (n + 2) for F being finite set st F = { s where s is Element of 2Set (Seg (n + 2)) : ( s in X & (Part_sgn (p2,K)) . s <> (Part_sgn (q2,K)) . s ) } holds ( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) ) proof let n be Nat; ::_thesis: for K being Field for X being Element of Fin (2Set (Seg (n + 2))) for p2, q2 being Element of Permutations (n + 2) for F being finite set st F = { s where s is Element of 2Set (Seg (n + 2)) : ( s in X & (Part_sgn (p2,K)) . s <> (Part_sgn (q2,K)) . s ) } holds ( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) ) let K be Field; ::_thesis: for X being Element of Fin (2Set (Seg (n + 2))) for p2, q2 being Element of Permutations (n + 2) for F being finite set st F = { s where s is Element of 2Set (Seg (n + 2)) : ( s in X & (Part_sgn (p2,K)) . s <> (Part_sgn (q2,K)) . s ) } holds ( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) ) let X be Element of Fin (2Set (Seg (n + 2))); ::_thesis: for p2, q2 being Element of Permutations (n + 2) for F being finite set st F = { s where s is Element of 2Set (Seg (n + 2)) : ( s in X & (Part_sgn (p2,K)) . s <> (Part_sgn (q2,K)) . s ) } holds ( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) ) let p2, q2 be Element of Permutations (n + 2); ::_thesis: for F being finite set st F = { s where s is Element of 2Set (Seg (n + 2)) : ( s in X & (Part_sgn (p2,K)) . s <> (Part_sgn (q2,K)) . s ) } holds ( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) ) let F be finite set ; ::_thesis: ( F = { s where s is Element of 2Set (Seg (n + 2)) : ( s in X & (Part_sgn (p2,K)) . s <> (Part_sgn (q2,K)) . s ) } implies ( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) ) ) assume A1: F = { s where s is Element of 2Set (Seg (n + 2)) : ( s in X & (Part_sgn (p2,K)) . s <> (Part_sgn (q2,K)) . s ) } ; ::_thesis: ( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) ) set Pq = Part_sgn (q2,K); set Pp = Part_sgn (p2,K); set 2S = 2Set (Seg (n + 2)); X c= 2Set (Seg (n + 2)) by FINSUB_1:def_5; then X \ F c= 2Set (Seg (n + 2)) by XBOOLE_1:1; then reconsider Y = X \ F as Element of Fin (2Set (Seg (n + 2))) by FINSUB_1:def_5; A2: F c= X proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F or x in X ) assume x in F ; ::_thesis: x in X then ex s being Element of 2Set (Seg (n + 2)) st ( x = s & s in X & (Part_sgn (p2,K)) . s <> (Part_sgn (q2,K)) . s ) by A1; hence x in X ; ::_thesis: verum end; then A3: F \/ Y = X by XBOOLE_1:45; X c= 2Set (Seg (n + 2)) by FINSUB_1:def_5; then F c= 2Set (Seg (n + 2)) by A2, XBOOLE_1:1; then reconsider F9 = F as Element of Fin (2Set (Seg (n + 2))) by FINSUB_1:def_5; set KK = the carrier of K; set mm = the multF of K; consider Gp being Function of (Fin (2Set (Seg (n + 2)))), the carrier of K such that A4: the multF of K $$ (F9,(Part_sgn (p2,K))) = Gp . F and A5: for e being Element of the carrier of K st e is_a_unity_wrt the multF of K holds Gp . {} = e and A6: for x being Element of 2Set (Seg (n + 2)) holds Gp . {x} = (Part_sgn (p2,K)) . x and A7: for B being Element of Fin (2Set (Seg (n + 2))) st B c= F & B <> {} holds for x being Element of 2Set (Seg (n + 2)) st x in F9 \ B holds Gp . (B \/ {x}) = the multF of K . ((Gp . B),((Part_sgn (p2,K)) . x)) by SETWISEO:def_3; A8: Y c= 2Set (Seg (n + 2)) by FINSUB_1:def_5; consider Gq being Function of (Fin (2Set (Seg (n + 2)))), the carrier of K such that A9: the multF of K $$ (F9,(Part_sgn (q2,K))) = Gq . F and A10: for e being Element of the carrier of K st e is_a_unity_wrt the multF of K holds Gq . {} = e and A11: for x being Element of 2Set (Seg (n + 2)) holds Gq . {x} = (Part_sgn (q2,K)) . x and A12: for B being Element of Fin (2Set (Seg (n + 2))) st B c= F & B <> {} holds for x being Element of 2Set (Seg (n + 2)) st x in F \ B holds Gq . (B \/ {x}) = the multF of K . ((Gq . B),((Part_sgn (q2,K)) . x)) by SETWISEO:def_3; defpred S1[ Nat] means for B being Element of Fin (2Set (Seg (n + 2))) st card B = $1 & B c= F holds ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ); A13: for s being Element of 2Set (Seg (n + 2)) st s in F holds (Part_sgn (p2,K)) . s = - ((Part_sgn (q2,K)) . s) proof let s be Element of 2Set (Seg (n + 2)); ::_thesis: ( s in F implies (Part_sgn (p2,K)) . s = - ((Part_sgn (q2,K)) . s) ) assume s in F ; ::_thesis: (Part_sgn (p2,K)) . s = - ((Part_sgn (q2,K)) . s) then A14: ex s9 being Element of 2Set (Seg (n + 2)) st ( s9 = s & s9 in X & (Part_sgn (p2,K)) . s9 <> (Part_sgn (q2,K)) . s9 ) by A1; A15: ( (Part_sgn (q2,K)) . s = 1_ K or (Part_sgn (q2,K)) . s = - (1_ K) ) by Th5; ( (Part_sgn (p2,K)) . s = 1_ K or (Part_sgn (p2,K)) . s = - (1_ K) ) by Th5; then ((Part_sgn (p2,K)) . s) + ((Part_sgn (q2,K)) . s) = 0. K by A14, A15, RLVECT_1:def_10; hence (Part_sgn (p2,K)) . s = - ((Part_sgn (q2,K)) . s) by VECTSP_1:16; ::_thesis: verum end; A16: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A17: S1[k] ; ::_thesis: S1[k + 1] set k1 = k + 1; let B be Element of Fin (2Set (Seg (n + 2))); ::_thesis: ( card B = k + 1 & B c= F implies ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ) ) assume that A18: card B = k + 1 and A19: B c= F ; ::_thesis: ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ) now__::_thesis:_(_(_k_=_0_&_(_(card_B)_mod_2_=_0_implies_Gp_._B_=_Gq_._B_)_&_(_(card_B)_mod_2_=_1_implies_Gp_._B_=_-_(Gq_._B)_)_)_or_(_k_>_0_&_(_(card_B)_mod_2_=_0_implies_Gp_._B_=_Gq_._B_)_&_(_(card_B)_mod_2_=_1_implies_Gp_._B_=_-_(Gq_._B)_)_)_) percases ( k = 0 or k > 0 ) ; caseA20: k = 0 ; ::_thesis: ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ) then consider x being set such that A21: B = {x} by A18, CARD_2:42; A22: x in B by A21, TARSKI:def_1; B c= 2Set (Seg (n + 2)) by FINSUB_1:def_5; then reconsider x = x as Element of 2Set (Seg (n + 2)) by A22; A23: Gq . B = (Part_sgn (q2,K)) . x by A11, A21; Gp . B = (Part_sgn (p2,K)) . x by A6, A21; hence ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ) by A13, A18, A19, A20, A22, A23, NAT_D:14; ::_thesis: verum end; caseA24: k > 0 ; ::_thesis: ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ) consider x being set such that A25: x in B by A18, CARD_1:27, XBOOLE_0:def_1; B c= 2Set (Seg (n + 2)) by FINSUB_1:def_5; then reconsider x = x as Element of 2Set (Seg (n + 2)) by A25; B c= 2Set (Seg (n + 2)) by FINSUB_1:def_5; then B \ {x} c= 2Set (Seg (n + 2)) by XBOOLE_1:1; then reconsider B9 = B \ {x} as Element of Fin (2Set (Seg (n + 2))) by FINSUB_1:def_5; A26: not x in B9 by ZFMISC_1:56; then A27: x in F \ B9 by A19, A25, XBOOLE_0:def_5; A28: B9 c= F by A19, XBOOLE_1:1; A29: {x} \/ B9 = B by A25, ZFMISC_1:116; then A30: k + 1 = (card B9) + 1 by A18, A26, CARD_2:41; then A31: Gq . B = the multF of K . ((Gq . B9),((Part_sgn (q2,K)) . x)) by A12, A19, A24, A29, A27, CARD_1:27, XBOOLE_1:1; A32: Gp . B = the multF of K . ((Gp . B9),((Part_sgn (p2,K)) . x)) by A7, A19, A24, A29, A30, A27, CARD_1:27, XBOOLE_1:1; now__::_thesis:_(_(_k_mod_2_=_0_&_(_(card_B)_mod_2_=_0_implies_Gp_._B_=_Gq_._B_)_&_(_(card_B)_mod_2_=_1_implies_Gp_._B_=_-_(Gq_._B)_)_)_or_(_k_mod_2_=_1_&_(_(card_B)_mod_2_=_0_implies_Gp_._B_=_Gq_._B_)_&_(_(card_B)_mod_2_=_1_implies_Gp_._B_=_-_(Gq_._B)_)_)_) percases ( k mod 2 = 0 or k mod 2 = 1 ) by NAT_D:12; caseA33: k mod 2 = 0 ; ::_thesis: ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ) 0 < 2 - 1 ; then A34: (k + 1) mod 2 = 0 + 1 by A33, NAT_D:70; A35: Gp . B = (Gp . B9) * (- ((Part_sgn (q2,K)) . x)) by A13, A19, A25, A32; Gq . B = (Gp . B9) * ((Part_sgn (q2,K)) . x) by A17, A30, A28, A31, A33; hence ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ) by A18, A35, A34, VECTSP_1:8; ::_thesis: verum end; caseA36: k mod 2 = 1 ; ::_thesis: ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ) A37: (Part_sgn (p2,K)) . x = - ((Part_sgn (q2,K)) . x) by A13, A19, A25; Gp . B9 = - (Gq . B9) by A17, A30, A28, A36; then A38: Gp . B = (- (Gq . B9)) * (- ((Part_sgn (q2,K)) . x)) by A7, A19, A24, A29, A30, A27, A37, CARD_1:27, XBOOLE_1:1; A39: 2 - 1 = 1 ; Gq . B = (Gq . B9) * ((Part_sgn (q2,K)) . x) by A12, A19, A24, A29, A30, A27, CARD_1:27, XBOOLE_1:1; hence ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ) by A18, A36, A38, A39, NAT_D:69, VECTSP_1:10; ::_thesis: verum end; end; end; hence ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ) ; ::_thesis: verum end; end; end; hence ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ) ; ::_thesis: verum end; A40: S1[ 0 ] proof let B be Element of Fin (2Set (Seg (n + 2))); ::_thesis: ( card B = 0 & B c= F implies ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ) ) assume that A41: card B = 0 and B c= F ; ::_thesis: ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ) A42: 0 = 0 mod 2 by NAT_D:26; A43: B = {} by A41; then Gp . B = 1_ K by A5, FVSUM_1:4; hence ( ( (card B) mod 2 = 0 implies Gp . B = Gq . B ) & ( (card B) mod 2 = 1 implies Gp . B = - (Gq . B) ) ) by A10, A43, A42, FVSUM_1:4; ::_thesis: verum end; A44: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A40, A16); A45: Y misses F by XBOOLE_1:79; then A46: the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K . (( the multF of K $$ (Y,(Part_sgn (p2,K)))),( the multF of K $$ (F9,(Part_sgn (p2,K))))) by A3, SETWOP_2:4; A47: the multF of K $$ (X,(Part_sgn (q2,K))) = the multF of K . (( the multF of K $$ (Y,(Part_sgn (q2,K)))),( the multF of K $$ (F9,(Part_sgn (q2,K))))) by A45, A3, SETWOP_2:4; A48: dom (Part_sgn (p2,K)) = 2Set (Seg (n + 2)) by FUNCT_2:def_1; then A49: dom ((Part_sgn (p2,K)) | Y) = Y by A8, RELAT_1:62; dom (Part_sgn (q2,K)) = 2Set (Seg (n + 2)) by FUNCT_2:def_1; then A50: dom ((Part_sgn (q2,K)) | Y) = Y by A8, RELAT_1:62; for x being set st x in dom ((Part_sgn (p2,K)) | Y) holds ((Part_sgn (p2,K)) | Y) . x = ((Part_sgn (q2,K)) | Y) . x proof let x be set ; ::_thesis: ( x in dom ((Part_sgn (p2,K)) | Y) implies ((Part_sgn (p2,K)) | Y) . x = ((Part_sgn (q2,K)) | Y) . x ) assume A51: x in dom ((Part_sgn (p2,K)) | Y) ; ::_thesis: ((Part_sgn (p2,K)) | Y) . x = ((Part_sgn (q2,K)) | Y) . x Y c= 2Set (Seg (n + 2)) by FINSUB_1:def_5; then reconsider x9 = x as Element of 2Set (Seg (n + 2)) by A49, A51; A52: ((Part_sgn (p2,K)) | Y) . x9 = (Part_sgn (p2,K)) . x9 by A51, FUNCT_1:47; A53: not x9 in F by A49, A51, XBOOLE_0:def_5; assume A54: ((Part_sgn (p2,K)) | Y) . x <> ((Part_sgn (q2,K)) | Y) . x ; ::_thesis: contradiction ((Part_sgn (q2,K)) | Y) . x9 = (Part_sgn (q2,K)) . x9 by A49, A50, A51, FUNCT_1:47; hence contradiction by A1, A49, A51, A54, A52, A53; ::_thesis: verum end; then A55: (Part_sgn (p2,K)) | Y = (Part_sgn (q2,K)) | Y by A48, A8, A50, FUNCT_1:2, RELAT_1:62; then A56: the multF of K $$ (Y,(Part_sgn (p2,K))) = the multF of K $$ (Y,(Part_sgn (q2,K))) by SETWOP_2:7; now__::_thesis:_(_(_(card_F)_mod_2_=_0_&_(_(card_F)_mod_2_=_0_implies_the_multF_of_K_$$_(X,(Part_sgn_(p2,K)))_=_the_multF_of_K_$$_(X,(Part_sgn_(q2,K)))_)_&_(_(card_F)_mod_2_=_1_implies_the_multF_of_K_$$_(X,(Part_sgn_(p2,K)))_=_-_(_the_multF_of_K_$$_(X,(Part_sgn_(q2,K))))_)_)_or_(_(card_F)_mod_2_=_1_&_(_(card_F)_mod_2_=_0_implies_the_multF_of_K_$$_(X,(Part_sgn_(p2,K)))_=_the_multF_of_K_$$_(X,(Part_sgn_(q2,K)))_)_&_(_(card_F)_mod_2_=_1_implies_the_multF_of_K_$$_(X,(Part_sgn_(p2,K)))_=_-_(_the_multF_of_K_$$_(X,(Part_sgn_(q2,K))))_)_)_) percases ( (card F) mod 2 = 0 or (card F) mod 2 = 1 ) by NAT_D:12; case (card F) mod 2 = 0 ; ::_thesis: ( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) ) hence ( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) ) by A4, A9, A44, A56, A46, A47; ::_thesis: verum end; caseA57: (card F) mod 2 = 1 ; ::_thesis: ( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) ) A58: the multF of K $$ (X,(Part_sgn (q2,K))) = ( the multF of K $$ (Y,(Part_sgn (p2,K)))) * ( the multF of K $$ (F9,(Part_sgn (q2,K)))) by A55, A47, SETWOP_2:7; the multF of K $$ (X,(Part_sgn (p2,K))) = ( the multF of K $$ (Y,(Part_sgn (p2,K)))) * (- ( the multF of K $$ (F9,(Part_sgn (q2,K))))) by A4, A9, A44, A46, A57; hence ( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) ) by A57, A58, VECTSP_1:8; ::_thesis: verum end; end; end; hence ( ( (card F) mod 2 = 0 implies the multF of K $$ (X,(Part_sgn (p2,K))) = the multF of K $$ (X,(Part_sgn (q2,K))) ) & ( (card F) mod 2 = 1 implies the multF of K $$ (X,(Part_sgn (p2,K))) = - ( the multF of K $$ (X,(Part_sgn (q2,K)))) ) ) ; ::_thesis: verum end; theorem Th8: :: MATRIX11:8 for n being Nat for P being Permutation of (Seg n) st P is being_transposition holds for i, j being Nat st i < j holds ( P . i = j iff ( i in dom P & j in dom P & P . i = j & P . j = i & ( for k being Nat st k <> i & k <> j & k in dom P holds P . k = k ) ) ) proof let n be Nat; ::_thesis: for P being Permutation of (Seg n) st P is being_transposition holds for i, j being Nat st i < j holds ( P . i = j iff ( i in dom P & j in dom P & P . i = j & P . j = i & ( for k being Nat st k <> i & k <> j & k in dom P holds P . k = k ) ) ) let P be Permutation of (Seg n); ::_thesis: ( P is being_transposition implies for i, j being Nat st i < j holds ( P . i = j iff ( i in dom P & j in dom P & P . i = j & P . j = i & ( for k being Nat st k <> i & k <> j & k in dom P holds P . k = k ) ) ) ) assume P is being_transposition ; ::_thesis: for i, j being Nat st i < j holds ( P . i = j iff ( i in dom P & j in dom P & P . i = j & P . j = i & ( for k being Nat st k <> i & k <> j & k in dom P holds P . k = k ) ) ) then consider i9, j9 being Nat such that i9 in dom P and j9 in dom P and i9 <> j9 and A1: P . i9 = j9 and A2: P . j9 = i9 and A3: for k being Nat st k <> i9 & k <> j9 & k in dom P holds P . k = k by MATRIX_2:def_11; let i, j be Nat; ::_thesis: ( i < j implies ( P . i = j iff ( i in dom P & j in dom P & P . i = j & P . j = i & ( for k being Nat st k <> i & k <> j & k in dom P holds P . k = k ) ) ) ) assume A4: i < j ; ::_thesis: ( P . i = j iff ( i in dom P & j in dom P & P . i = j & P . j = i & ( for k being Nat st k <> i & k <> j & k in dom P holds P . k = k ) ) ) thus ( P . i = j implies ( i in dom P & j in dom P & P . i = j & P . j = i & ( for k being Nat st k <> i & k <> j & k in dom P holds P . k = k ) ) ) ::_thesis: ( i in dom P & j in dom P & P . i = j & P . j = i & ( for k being Nat st k <> i & k <> j & k in dom P holds P . k = k ) implies P . i = j ) proof A5: dom P = Seg n by FUNCT_2:52; A6: rng P = Seg n by FUNCT_2:def_3; assume A7: P . i = j ; ::_thesis: ( i in dom P & j in dom P & P . i = j & P . j = i & ( for k being Nat st k <> i & k <> j & k in dom P holds P . k = k ) ) then A8: i in dom P by A4, FUNCT_1:def_2; then ( i = j9 or i = i9 ) by A4, A3, A7; hence ( i in dom P & j in dom P & P . i = j & P . j = i & ( for k being Nat st k <> i & k <> j & k in dom P holds P . k = k ) ) by A1, A2, A3, A7, A8, A6, A5, FUNCT_1:def_3; ::_thesis: verum end; thus ( i in dom P & j in dom P & P . i = j & P . j = i & ( for k being Nat st k <> i & k <> j & k in dom P holds P . k = k ) implies P . i = j ) ; ::_thesis: verum end; theorem Th9: :: MATRIX11:9 for n being Nat for K being Field for p2, q2, pq2 being Element of Permutations (n + 2) for i, j being Nat st pq2 = p2 * q2 & q2 is being_transposition & q2 . i = j & i < j holds for s being Element of 2Set (Seg (n + 2)) holds ( not (Part_sgn (p2,K)) . s <> (Part_sgn (pq2,K)) . s or i in s or j in s ) proof let n be Nat; ::_thesis: for K being Field for p2, q2, pq2 being Element of Permutations (n + 2) for i, j being Nat st pq2 = p2 * q2 & q2 is being_transposition & q2 . i = j & i < j holds for s being Element of 2Set (Seg (n + 2)) holds ( not (Part_sgn (p2,K)) . s <> (Part_sgn (pq2,K)) . s or i in s or j in s ) let K be Field; ::_thesis: for p2, q2, pq2 being Element of Permutations (n + 2) for i, j being Nat st pq2 = p2 * q2 & q2 is being_transposition & q2 . i = j & i < j holds for s being Element of 2Set (Seg (n + 2)) holds ( not (Part_sgn (p2,K)) . s <> (Part_sgn (pq2,K)) . s or i in s or j in s ) set n2 = n + 2; let p, q, pq be Element of Permutations (n + 2); ::_thesis: for i, j being Nat st pq = p * q & q is being_transposition & q . i = j & i < j holds for s being Element of 2Set (Seg (n + 2)) holds ( not (Part_sgn (p,K)) . s <> (Part_sgn (pq,K)) . s or i in s or j in s ) let i, j be Nat; ::_thesis: ( pq = p * q & q is being_transposition & q . i = j & i < j implies for s being Element of 2Set (Seg (n + 2)) holds ( not (Part_sgn (p,K)) . s <> (Part_sgn (pq,K)) . s or i in s or j in s ) ) assume that A1: pq = p * q and A2: q is being_transposition and A3: q . i = j and A4: i < j ; ::_thesis: for s being Element of 2Set (Seg (n + 2)) holds ( not (Part_sgn (p,K)) . s <> (Part_sgn (pq,K)) . s or i in s or j in s ) reconsider q9 = q, pq9 = pq as Permutation of (Seg (n + 2)) by MATRIX_2:def_9; let s be Element of 2Set (Seg (n + 2)); ::_thesis: ( not (Part_sgn (p,K)) . s <> (Part_sgn (pq,K)) . s or i in s or j in s ) assume A5: (Part_sgn (p,K)) . s <> (Part_sgn (pq,K)) . s ; ::_thesis: ( i in s or j in s ) A6: dom q9 = Seg (n + 2) by FUNCT_2:52; A7: dom pq9 = Seg (n + 2) by FUNCT_2:52; assume that A8: not i in s and A9: not j in s ; ::_thesis: contradiction consider i9, j9 being Nat such that A10: i9 in Seg (n + 2) and A11: j9 in Seg (n + 2) and A12: i9 < j9 and A13: s = {i9,j9} by Th1; A14: j <> j9 by A13, A9, TARSKI:def_2; A15: j <> i9 by A13, A9, TARSKI:def_2; i <> j9 by A13, A8, TARSKI:def_2; then q . j9 = j9 by A2, A3, A4, A11, A14, A6, Th8; then A16: pq . j9 = p . j9 by A1, A11, A7, FUNCT_1:12; i <> i9 by A13, A8, TARSKI:def_2; then q . i9 = i9 by A2, A3, A4, A10, A15, A6, Th8; then pq . i9 = p . i9 by A1, A10, A7, FUNCT_1:12; hence contradiction by A5, A10, A11, A12, A13, A16, Th6; ::_thesis: verum end; Lm1: for n being Nat for K being Field for p2 being Element of Permutations (n + 2) for i, j being Nat st i in Seg (n + 2) & j in Seg (n + 2) & i < j & 1_ K <> - (1_ K) holds ( ( (Part_sgn (p2,K)) . {i,j} = 1_ K implies p2 . i < p2 . j ) & ( (Part_sgn (p2,K)) . {i,j} = - (1_ K) implies p2 . i > p2 . j ) ) proof let n be Nat; ::_thesis: for K being Field for p2 being Element of Permutations (n + 2) for i, j being Nat st i in Seg (n + 2) & j in Seg (n + 2) & i < j & 1_ K <> - (1_ K) holds ( ( (Part_sgn (p2,K)) . {i,j} = 1_ K implies p2 . i < p2 . j ) & ( (Part_sgn (p2,K)) . {i,j} = - (1_ K) implies p2 . i > p2 . j ) ) let K be Field; ::_thesis: for p2 being Element of Permutations (n + 2) for i, j being Nat st i in Seg (n + 2) & j in Seg (n + 2) & i < j & 1_ K <> - (1_ K) holds ( ( (Part_sgn (p2,K)) . {i,j} = 1_ K implies p2 . i < p2 . j ) & ( (Part_sgn (p2,K)) . {i,j} = - (1_ K) implies p2 . i > p2 . j ) ) let p2 be Element of Permutations (n + 2); ::_thesis: for i, j being Nat st i in Seg (n + 2) & j in Seg (n + 2) & i < j & 1_ K <> - (1_ K) holds ( ( (Part_sgn (p2,K)) . {i,j} = 1_ K implies p2 . i < p2 . j ) & ( (Part_sgn (p2,K)) . {i,j} = - (1_ K) implies p2 . i > p2 . j ) ) set n2 = n + 2; let i, j be Nat; ::_thesis: ( i in Seg (n + 2) & j in Seg (n + 2) & i < j & 1_ K <> - (1_ K) implies ( ( (Part_sgn (p2,K)) . {i,j} = 1_ K implies p2 . i < p2 . j ) & ( (Part_sgn (p2,K)) . {i,j} = - (1_ K) implies p2 . i > p2 . j ) ) ) assume that A1: i in Seg (n + 2) and A2: j in Seg (n + 2) and A3: i < j and A4: 1_ K <> - (1_ K) ; ::_thesis: ( ( (Part_sgn (p2,K)) . {i,j} = 1_ K implies p2 . i < p2 . j ) & ( (Part_sgn (p2,K)) . {i,j} = - (1_ K) implies p2 . i > p2 . j ) ) reconsider p9 = p2 as Permutation of (Seg (n + 2)) by MATRIX_2:def_9; p9 . i <> p9 . j by A1, A2, A3, FUNCT_2:19; then A5: ( p2 . i < p2 . j or p2 . i > p2 . j ) by XXREAL_0:1; thus ( (Part_sgn (p2,K)) . {i,j} = 1_ K implies p2 . i < p2 . j ) ::_thesis: ( (Part_sgn (p2,K)) . {i,j} = - (1_ K) implies p2 . i > p2 . j ) proof p9 . i <> p9 . j by A1, A2, A3, FUNCT_2:19; then A6: ( p2 . i < p2 . j or p2 . i > p2 . j ) by XXREAL_0:1; assume (Part_sgn (p2,K)) . {i,j} = 1_ K ; ::_thesis: p2 . i < p2 . j hence p2 . i < p2 . j by A1, A2, A3, A4, A6, Def1; ::_thesis: verum end; assume (Part_sgn (p2,K)) . {i,j} = - (1_ K) ; ::_thesis: p2 . i > p2 . j hence p2 . i > p2 . j by A1, A2, A3, A4, A5, Def1; ::_thesis: verum end; theorem Th10: :: MATRIX11:10 for n being Nat for p2, q2, pq2 being Element of Permutations (n + 2) for i, j being Nat for K being Field st pq2 = p2 * q2 & q2 is being_transposition & q2 . i = j & i < j & 1_ K <> - (1_ K) holds ( (Part_sgn (p2,K)) . {i,j} <> (Part_sgn (pq2,K)) . {i,j} & ( for k being Nat st k in Seg (n + 2) & i <> k & j <> k holds ( (Part_sgn (p2,K)) . {i,k} <> (Part_sgn (pq2,K)) . {i,k} iff (Part_sgn (p2,K)) . {j,k} <> (Part_sgn (pq2,K)) . {j,k} ) ) ) proof let n be Nat; ::_thesis: for p2, q2, pq2 being Element of Permutations (n + 2) for i, j being Nat for K being Field st pq2 = p2 * q2 & q2 is being_transposition & q2 . i = j & i < j & 1_ K <> - (1_ K) holds ( (Part_sgn (p2,K)) . {i,j} <> (Part_sgn (pq2,K)) . {i,j} & ( for k being Nat st k in Seg (n + 2) & i <> k & j <> k holds ( (Part_sgn (p2,K)) . {i,k} <> (Part_sgn (pq2,K)) . {i,k} iff (Part_sgn (p2,K)) . {j,k} <> (Part_sgn (pq2,K)) . {j,k} ) ) ) set n2 = n + 2; let p, q, pq be Element of Permutations (n + 2); ::_thesis: for i, j being Nat for K being Field st pq = p * q & q is being_transposition & q . i = j & i < j & 1_ K <> - (1_ K) holds ( (Part_sgn (p,K)) . {i,j} <> (Part_sgn (pq,K)) . {i,j} & ( for k being Nat st k in Seg (n + 2) & i <> k & j <> k holds ( (Part_sgn (p,K)) . {i,k} <> (Part_sgn (pq,K)) . {i,k} iff (Part_sgn (p,K)) . {j,k} <> (Part_sgn (pq,K)) . {j,k} ) ) ) let i, j be Nat; ::_thesis: for K being Field st pq = p * q & q is being_transposition & q . i = j & i < j & 1_ K <> - (1_ K) holds ( (Part_sgn (p,K)) . {i,j} <> (Part_sgn (pq,K)) . {i,j} & ( for k being Nat st k in Seg (n + 2) & i <> k & j <> k holds ( (Part_sgn (p,K)) . {i,k} <> (Part_sgn (pq,K)) . {i,k} iff (Part_sgn (p,K)) . {j,k} <> (Part_sgn (pq,K)) . {j,k} ) ) ) let K be Field; ::_thesis: ( pq = p * q & q is being_transposition & q . i = j & i < j & 1_ K <> - (1_ K) implies ( (Part_sgn (p,K)) . {i,j} <> (Part_sgn (pq,K)) . {i,j} & ( for k being Nat st k in Seg (n + 2) & i <> k & j <> k holds ( (Part_sgn (p,K)) . {i,k} <> (Part_sgn (pq,K)) . {i,k} iff (Part_sgn (p,K)) . {j,k} <> (Part_sgn (pq,K)) . {j,k} ) ) ) ) assume that A1: pq = p * q and A2: q is being_transposition and A3: q . i = j and A4: i < j and A5: 1_ K <> - (1_ K) ; ::_thesis: ( (Part_sgn (p,K)) . {i,j} <> (Part_sgn (pq,K)) . {i,j} & ( for k being Nat st k in Seg (n + 2) & i <> k & j <> k holds ( (Part_sgn (p,K)) . {i,k} <> (Part_sgn (pq,K)) . {i,k} iff (Part_sgn (p,K)) . {j,k} <> (Part_sgn (pq,K)) . {j,k} ) ) ) A6: i in dom q by A2, A3, A4, Th8; set P2 = Part_sgn (pq,K); set P1 = Part_sgn (p,K); reconsider p9 = p, q9 = q, pq9 = pq as Permutation of (Seg (n + 2)) by MATRIX_2:def_9; A7: dom q9 = Seg (n + 2) by FUNCT_2:52; A8: j in dom q by A2, A3, A4, Th8; A9: dom pq9 = Seg (n + 2) by FUNCT_2:52; then A10: pq . i = p . j by A1, A3, A6, A7, FUNCT_1:12; q . j = i by A2, A3, A4, Th8; then A11: pq . j = p . i by A1, A8, A9, A7, FUNCT_1:12; dom p9 = Seg (n + 2) by FUNCT_2:52; then A12: p9 . i <> p9 . j by A4, A6, A8, A7, FUNCT_1:def_4; now__::_thesis:_(Part_sgn_(p,K))_._{i,j}_<>_(Part_sgn_(pq,K))_._{i,j} percases ( p . i < p . j or p . i > p . j ) by A12, XXREAL_0:1; supposeA13: p . i < p . j ; ::_thesis: (Part_sgn (p,K)) . {i,j} <> (Part_sgn (pq,K)) . {i,j} then (Part_sgn (p,K)) . {i,j} = 1_ K by A4, A6, A8, A7, Def1; hence (Part_sgn (p,K)) . {i,j} <> (Part_sgn (pq,K)) . {i,j} by A4, A5, A6, A8, A7, A10, A11, A13, Def1; ::_thesis: verum end; supposeA14: p . i > p . j ; ::_thesis: (Part_sgn (p,K)) . {i,j} <> (Part_sgn (pq,K)) . {i,j} then (Part_sgn (p,K)) . {i,j} = - (1_ K) by A4, A6, A8, A7, Def1; hence (Part_sgn (p,K)) . {i,j} <> (Part_sgn (pq,K)) . {i,j} by A4, A5, A6, A8, A7, A10, A11, A14, Def1; ::_thesis: verum end; end; end; hence (Part_sgn (p,K)) . {i,j} <> (Part_sgn (pq,K)) . {i,j} ; ::_thesis: for k being Nat st k in Seg (n + 2) & i <> k & j <> k holds ( (Part_sgn (p,K)) . {i,k} <> (Part_sgn (pq,K)) . {i,k} iff (Part_sgn (p,K)) . {j,k} <> (Part_sgn (pq,K)) . {j,k} ) let k be Nat; ::_thesis: ( k in Seg (n + 2) & i <> k & j <> k implies ( (Part_sgn (p,K)) . {i,k} <> (Part_sgn (pq,K)) . {i,k} iff (Part_sgn (p,K)) . {j,k} <> (Part_sgn (pq,K)) . {j,k} ) ) assume that A15: k in Seg (n + 2) and A16: i <> k and A17: j <> k ; ::_thesis: ( (Part_sgn (p,K)) . {i,k} <> (Part_sgn (pq,K)) . {i,k} iff (Part_sgn (p,K)) . {j,k} <> (Part_sgn (pq,K)) . {j,k} ) A18: q . k = k by A2, A3, A4, A7, A15, A16, A17, Th8; A19: pq . k = p . (q . k) by A1, A9, A15, FUNCT_1:12; ( i < k or k < i ) by A16, XXREAL_0:1; then A20: {i,k} in 2Set (Seg (n + 2)) by A6, A7, A15, Th1; A21: ( (Part_sgn (p,K)) . {i,k} = (Part_sgn (pq,K)) . {i,k} implies (Part_sgn (p,K)) . {j,k} = (Part_sgn (pq,K)) . {j,k} ) proof A22: ( j < k or k < j ) by A17, XXREAL_0:1; A23: ( i < k or i > k ) by A16, XXREAL_0:1; assume A24: (Part_sgn (p,K)) . {i,k} = (Part_sgn (pq,K)) . {i,k} ; ::_thesis: (Part_sgn (p,K)) . {j,k} = (Part_sgn (pq,K)) . {j,k} ( (Part_sgn (p,K)) . {k,i} = 1_ K or (Part_sgn (p,K)) . {k,i} = - (1_ K) ) by A20, Th5; then ( ( pq . j < pq . k & p . j < p . k ) or ( pq . j > pq . k & p . j > p . k ) ) by A5, A6, A7, A10, A11, A15, A18, A19, A24, A23, Lm1; then ( ( (Part_sgn (pq,K)) . {j,k} = 1_ K & (Part_sgn (p,K)) . {j,k} = 1_ K ) or ( (Part_sgn (pq,K)) . {j,k} = - (1_ K) & (Part_sgn (p,K)) . {j,k} = - (1_ K) ) ) by A8, A7, A15, A22, Def1; hence (Part_sgn (p,K)) . {j,k} = (Part_sgn (pq,K)) . {j,k} ; ::_thesis: verum end; ( j < k or k < j ) by A17, XXREAL_0:1; then A25: {j,k} in 2Set (Seg (n + 2)) by A8, A7, A15, Th1; ( (Part_sgn (p,K)) . {j,k} = (Part_sgn (pq,K)) . {j,k} implies (Part_sgn (p,K)) . {i,k} = (Part_sgn (pq,K)) . {i,k} ) proof A26: ( i < k or k < i ) by A16, XXREAL_0:1; A27: ( j < k or j > k ) by A17, XXREAL_0:1; assume A28: (Part_sgn (p,K)) . {j,k} = (Part_sgn (pq,K)) . {j,k} ; ::_thesis: (Part_sgn (p,K)) . {i,k} = (Part_sgn (pq,K)) . {i,k} ( (Part_sgn (p,K)) . {k,j} = 1_ K or (Part_sgn (p,K)) . {k,j} = - (1_ K) ) by A25, Th5; then ( ( pq . i < pq . k & p . i < p . k ) or ( pq . i > pq . k & p . i > p . k ) ) by A5, A8, A7, A10, A11, A15, A18, A19, A28, A27, Lm1; then ( ( (Part_sgn (pq,K)) . {i,k} = 1_ K & (Part_sgn (p,K)) . {i,k} = 1_ K ) or ( (Part_sgn (pq,K)) . {i,k} = - (1_ K) & (Part_sgn (p,K)) . {i,k} = - (1_ K) ) ) by A6, A7, A15, A26, Def1; hence (Part_sgn (p,K)) . {i,k} = (Part_sgn (pq,K)) . {i,k} ; ::_thesis: verum end; hence ( (Part_sgn (p,K)) . {i,k} <> (Part_sgn (pq,K)) . {i,k} iff (Part_sgn (p,K)) . {j,k} <> (Part_sgn (pq,K)) . {j,k} ) by A21; ::_thesis: verum end; definition let n be Nat; let K be Field; let perm2 be Element of Permutations (n + 2); func sgn (perm2,K) -> Element of K equals :: MATRIX11:def 2 the multF of K $$ ((FinOmega (2Set (Seg (n + 2)))),(Part_sgn (perm2,K))); coherence the multF of K $$ ((FinOmega (2Set (Seg (n + 2)))),(Part_sgn (perm2,K))) is Element of K ; end; :: deftheorem defines sgn MATRIX11:def_2_:_ for n being Nat for K being Field for perm2 being Element of Permutations (n + 2) holds sgn (perm2,K) = the multF of K $$ ((FinOmega (2Set (Seg (n + 2)))),(Part_sgn (perm2,K))); theorem Th11: :: MATRIX11:11 for n being Nat for K being Field for p2 being Element of Permutations (n + 2) holds ( sgn (p2,K) = 1_ K or sgn (p2,K) = - (1_ K) ) proof let n be Nat; ::_thesis: for K being Field for p2 being Element of Permutations (n + 2) holds ( sgn (p2,K) = 1_ K or sgn (p2,K) = - (1_ K) ) let K be Field; ::_thesis: for p2 being Element of Permutations (n + 2) holds ( sgn (p2,K) = 1_ K or sgn (p2,K) = - (1_ K) ) let p2 be Element of Permutations (n + 2); ::_thesis: ( sgn (p2,K) = 1_ K or sgn (p2,K) = - (1_ K) ) set KK = the carrier of K; set n2 = n + 2; set 2S = 2Set (Seg (n + 2)); set mm = the multF of K; set Path = Part_sgn (p2,K); A1: FinOmega (2Set (Seg (n + 2))) = 2Set (Seg (n + 2)) by MATRIX_2:def_14; then reconsider 2S9 = 2Set (Seg (n + 2)) as Element of Fin (2Set (Seg (n + 2))) ; consider G being Function of (Fin (2Set (Seg (n + 2)))), the carrier of K such that A2: the multF of K $$ (2S9,(Part_sgn (p2,K))) = G . 2S9 and A3: for e being Element of the carrier of K st e is_a_unity_wrt the multF of K holds G . {} = e and A4: for s being Element of 2Set (Seg (n + 2)) holds G . {s} = (Part_sgn (p2,K)) . s and A5: for B being Element of Fin (2Set (Seg (n + 2))) st B c= 2S9 & B <> {} holds for s being Element of 2Set (Seg (n + 2)) st s in 2S9 \ B holds G . (B \/ {s}) = the multF of K . ((G . B),((Part_sgn (p2,K)) . s)) by SETWISEO:def_3; defpred S1[ Nat] means for B being Element of Fin (2Set (Seg (n + 2))) st card B = $1 & B c= 2Set (Seg (n + 2)) & not G . B = 1_ K holds G . B = - (1_ K); A6: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A7: S1[k] ; ::_thesis: S1[k + 1] set k1 = k + 1; let B be Element of Fin (2Set (Seg (n + 2))); ::_thesis: ( card B = k + 1 & B c= 2Set (Seg (n + 2)) & not G . B = 1_ K implies G . B = - (1_ K) ) assume that A8: card B = k + 1 and A9: B c= 2Set (Seg (n + 2)) ; ::_thesis: ( G . B = 1_ K or G . B = - (1_ K) ) now__::_thesis:_(_(_k_=_0_&_(_G_._B_=_1__K_or_G_._B_=_-_(1__K)_)_)_or_(_k_>_0_&_(_G_._B_=_1__K_or_G_._B_=_-_(1__K)_)_)_) percases ( k = 0 or k > 0 ) ; case k = 0 ; ::_thesis: ( G . B = 1_ K or G . B = - (1_ K) ) then consider x being set such that A10: B = {x} by A8, CARD_2:42; x in B by A10, TARSKI:def_1; then reconsider x = x as Element of 2Set (Seg (n + 2)) by A9; G . B = (Part_sgn (p2,K)) . x by A4, A10; hence ( G . B = 1_ K or G . B = - (1_ K) ) by Th5; ::_thesis: verum end; caseA11: k > 0 ; ::_thesis: ( G . B = 1_ K or G . B = - (1_ K) ) consider x being set such that A12: x in B by A8, CARD_1:27, XBOOLE_0:def_1; reconsider x = x as Element of 2Set (Seg (n + 2)) by A9, A12; B \ {x} c= 2Set (Seg (n + 2)) by A9, XBOOLE_1:1; then reconsider B9 = B \ {x} as Element of Fin (2Set (Seg (n + 2))) by FINSUB_1:def_5; A13: not x in B9 by ZFMISC_1:56; A14: {x} \/ B9 = B by A12, ZFMISC_1:116; then A15: k + 1 = (card B9) + 1 by A8, A13, CARD_2:41; then A16: ( G . B9 = 1_ K or G . B9 = - (1_ K) ) by A7, A9, XBOOLE_1:1; x in (2Set (Seg (n + 2))) \ B9 by A13, XBOOLE_0:def_5; then G . B = the multF of K . ((G . B9),((Part_sgn (p2,K)) . x)) by A5, A9, A11, A14, A15, CARD_1:27, XBOOLE_1:1; then ( G . B = (1_ K) * (1_ K) or G . B = (1_ K) * (- (1_ K)) or G . B = (- (1_ K)) * (1_ K) or G . B = (- (1_ K)) * (- (1_ K)) ) by A16, Th5; then ( G . B = (1_ K) * (1_ K) or G . B = (1_ K) * (- (1_ K)) ) by VECTSP_1:10; hence ( G . B = 1_ K or G . B = - (1_ K) ) by VECTSP_1:def_4; ::_thesis: verum end; end; end; hence ( G . B = 1_ K or G . B = - (1_ K) ) ; ::_thesis: verum end; A17: S1[ 0 ] proof let B be Element of Fin (2Set (Seg (n + 2))); ::_thesis: ( card B = 0 & B c= 2Set (Seg (n + 2)) & not G . B = 1_ K implies G . B = - (1_ K) ) assume that A18: card B = 0 and B c= 2Set (Seg (n + 2)) ; ::_thesis: ( G . B = 1_ K or G . B = - (1_ K) ) B = {} by A18; hence ( G . B = 1_ K or G . B = - (1_ K) ) by A3, FVSUM_1:4; ::_thesis: verum end; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A17, A6); then S1[ card 2S9] ; hence ( sgn (p2,K) = 1_ K or sgn (p2,K) = - (1_ K) ) by A1, A2; ::_thesis: verum end; theorem Th12: :: MATRIX11:12 for n being Nat for K being Field for Id being Element of Permutations (n + 2) st Id = idseq (n + 2) holds sgn (Id,K) = 1_ K proof let n be Nat; ::_thesis: for K being Field for Id being Element of Permutations (n + 2) st Id = idseq (n + 2) holds sgn (Id,K) = 1_ K let K be Field; ::_thesis: for Id being Element of Permutations (n + 2) st Id = idseq (n + 2) holds sgn (Id,K) = 1_ K set n2 = n + 2; let Id be Element of Permutations (n + 2); ::_thesis: ( Id = idseq (n + 2) implies sgn (Id,K) = 1_ K ) assume A1: Id = idseq (n + 2) ; ::_thesis: sgn (Id,K) = 1_ K set Path = Part_sgn (Id,K); set 2S = 2Set (Seg (n + 2)); A2: FinOmega (2Set (Seg (n + 2))) = 2Set (Seg (n + 2)) by MATRIX_2:def_14; then reconsider 2S9 = 2Set (Seg (n + 2)) as Element of Fin (2Set (Seg (n + 2))) ; now__::_thesis:_for_x_being_set_st_x_in_2S9_holds_ (Part_sgn_(Id,K))_._x_=_1__K let x be set ; ::_thesis: ( x in 2S9 implies (Part_sgn (Id,K)) . x = 1_ K ) assume x in 2S9 ; ::_thesis: (Part_sgn (Id,K)) . x = 1_ K then consider i, j being Nat such that A3: i in Seg (n + 2) and A4: j in Seg (n + 2) and A5: i < j and A6: x = {i,j} by Th1; A7: Id . j = j by A1, A4, FUNCT_1:18; Id . i = i by A1, A3, FUNCT_1:18; hence (Part_sgn (Id,K)) . x = 1_ K by A3, A4, A5, A6, A7, Def1; ::_thesis: verum end; hence sgn (Id,K) = 1_ K by A2, Th4; ::_thesis: verum end; Lm2: for X being set for n, i being Nat st X in 2Set (Seg n) & i in X holds ( i in Seg n & ex j being Nat st ( j in Seg n & i <> j & X = {i,j} ) ) proof let X be set ; ::_thesis: for n, i being Nat st X in 2Set (Seg n) & i in X holds ( i in Seg n & ex j being Nat st ( j in Seg n & i <> j & X = {i,j} ) ) let n, i be Nat; ::_thesis: ( X in 2Set (Seg n) & i in X implies ( i in Seg n & ex j being Nat st ( j in Seg n & i <> j & X = {i,j} ) ) ) assume that A1: X in 2Set (Seg n) and A2: i in X ; ::_thesis: ( i in Seg n & ex j being Nat st ( j in Seg n & i <> j & X = {i,j} ) ) consider i9, j9 being Nat such that A3: i9 in Seg n and A4: j9 in Seg n and A5: i9 < j9 and A6: X = {i9,j9} by A1, Th1; now__::_thesis:_(_(_i_=_i9_&_i_in_Seg_n_&_ex_j_being_Nat_st_ (_j_in_Seg_n_&_i_<>_j_&_X_=_{i,j}_)_)_or_(_i_=_j9_&_i_in_Seg_n_&_ex_j_being_Nat_st_ (_j_in_Seg_n_&_i_<>_j_&_X_=_{i,j}_)_)_) percases ( i = i9 or i = j9 ) by A2, A6, TARSKI:def_2; case i = i9 ; ::_thesis: ( i in Seg n & ex j being Nat st ( j in Seg n & i <> j & X = {i,j} ) ) hence ( i in Seg n & ex j being Nat st ( j in Seg n & i <> j & X = {i,j} ) ) by A3, A4, A5, A6; ::_thesis: verum end; case i = j9 ; ::_thesis: ( i in Seg n & ex j being Nat st ( j in Seg n & i <> j & X = {i,j} ) ) hence ( i in Seg n & ex j being Nat st ( j in Seg n & i <> j & X = {i,j} ) ) by A3, A4, A5, A6; ::_thesis: verum end; end; end; hence ( i in Seg n & ex j being Nat st ( j in Seg n & i <> j & X = {i,j} ) ) ; ::_thesis: verum end; theorem Th13: :: MATRIX11:13 for n being Nat for K being Field for p2, q2, pq2 being Element of Permutations (n + 2) st pq2 = p2 * q2 & q2 is being_transposition holds sgn (pq2,K) = - (sgn (p2,K)) proof let n be Nat; ::_thesis: for K being Field for p2, q2, pq2 being Element of Permutations (n + 2) st pq2 = p2 * q2 & q2 is being_transposition holds sgn (pq2,K) = - (sgn (p2,K)) let K be Field; ::_thesis: for p2, q2, pq2 being Element of Permutations (n + 2) st pq2 = p2 * q2 & q2 is being_transposition holds sgn (pq2,K) = - (sgn (p2,K)) set n2 = n + 2; set 2SS = 2Set (Seg (n + 2)); let p, q, pq be Element of Permutations (n + 2); ::_thesis: ( pq = p * q & q is being_transposition implies sgn (pq,K) = - (sgn (p,K)) ) assume that A1: pq = p * q and A2: q is being_transposition ; ::_thesis: sgn (pq,K) = - (sgn (p,K)) A3: FinOmega (2Set (Seg (n + 2))) = 2Set (Seg (n + 2)) by MATRIX_2:def_14; then reconsider 2S = 2Set (Seg (n + 2)) as Element of Fin (2Set (Seg (n + 2))) ; A4: for i, j being Nat st i < j & q . i = j holds sgn (pq,K) = - (sgn (p,K)) proof let i, j be Nat; ::_thesis: ( i < j & q . i = j implies sgn (pq,K) = - (sgn (p,K)) ) assume that A5: i < j and A6: q . i = j ; ::_thesis: sgn (pq,K) = - (sgn (p,K)) now__::_thesis:_sgn_(pq,K)_=_-_(sgn_(p,K)) percases ( 1_ K = - (1_ K) or 1_ K <> - (1_ K) ) ; supposeA7: 1_ K = - (1_ K) ; ::_thesis: sgn (pq,K) = - (sgn (p,K)) then sgn (pq,K) = - (1_ K) by Th11; hence sgn (pq,K) = - (sgn (p,K)) by A7, Th11; ::_thesis: verum end; supposeA8: 1_ K <> - (1_ K) ; ::_thesis: sgn (pq,K) = - (sgn (p,K)) set P2 = Part_sgn (p,K); set P1 = Part_sgn (pq,K); A9: (Part_sgn (pq,K)) . {i,j} <> (Part_sgn (p,K)) . {i,j} by A1, A2, A5, A6, A8, Th10; defpred S1[ set , set ] means for k being Nat st k in $1 & k <> i holds ( ( k <> j implies $2 = {j,k} ) & ( k = j implies $2 = {i,j} ) ); set D = { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s ) } ; { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s ) } c= 2S proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s ) } or x in 2S ) assume x in { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s ) } ; ::_thesis: x in 2S then ex s being Element of 2Set (Seg (n + 2)) st ( x = s & s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s ) ; hence x in 2S ; ::_thesis: verum end; then reconsider D = { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s ) } as finite set ; set D1 = { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & i in s ) } ; set D2 = { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & j in s ) } ; A10: { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & i in s ) } c= D proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & i in s ) } or x in D ) assume x in { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & i in s ) } ; ::_thesis: x in D then ex s being Element of 2Set (Seg (n + 2)) st ( x = s & s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & i in s ) ; hence x in D ; ::_thesis: verum end; A11: { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & j in s ) } c= D proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & j in s ) } or x in D ) assume x in { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & j in s ) } ; ::_thesis: x in D then ex s being Element of 2Set (Seg (n + 2)) st ( x = s & s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & j in s ) ; hence x in D ; ::_thesis: verum end; then reconsider D1 = { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & i in s ) } , D2 = { s where s is Element of 2Set (Seg (n + 2)) : ( s in 2S & (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s & j in s ) } as finite set by A10; A12: j in dom q by A2, A5, A6, Th8; A13: D c= D1 \/ D2 proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in D or x in D1 \/ D2 ) assume x in D ; ::_thesis: x in D1 \/ D2 then consider s being Element of 2Set (Seg (n + 2)) such that A14: x = s and s in 2S and A15: (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s ; ( i in s or j in s ) by A1, A2, A5, A6, A15, Th9; then ( x in D1 or x in D2 ) by A14, A15; hence x in D1 \/ D2 by XBOOLE_0:def_3; ::_thesis: verum end; D1 \/ D2 c= D by A10, A11, XBOOLE_1:8; then A16: D1 \/ D2 = D by A13, XBOOLE_0:def_10; A17: D1 /\ D2 c= {{i,j}} proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in D1 /\ D2 or x in {{i,j}} ) assume A18: x in D1 /\ D2 ; ::_thesis: x in {{i,j}} then x in D1 by XBOOLE_0:def_4; then A19: ex s1 being Element of 2Set (Seg (n + 2)) st ( x = s1 & s1 in 2S & (Part_sgn (pq,K)) . s1 <> (Part_sgn (p,K)) . s1 & i in s1 ) ; then consider i9, j9 being Nat such that i9 in Seg (n + 2) and j9 in Seg (n + 2) and i9 < j9 and A20: {i9,j9} = x by Th1; x in D2 by A18, XBOOLE_0:def_4; then ex s2 being Element of 2Set (Seg (n + 2)) st ( x = s2 & s2 in 2S & (Part_sgn (pq,K)) . s2 <> (Part_sgn (p,K)) . s2 & j in s2 ) ; then A21: ( j = i9 or j = j9 ) by A20, TARSKI:def_2; ( i = i9 or i = j9 ) by A19, A20, TARSKI:def_2; hence x in {{i,j}} by A5, A20, A21, TARSKI:def_1; ::_thesis: verum end; q is Permutation of (Seg (n + 2)) by MATRIX_2:def_9; then A22: dom q = Seg (n + 2) by FUNCT_2:52; A23: i in dom q by A2, A5, A6, Th8; then A24: {i,j} in 2S by A5, A12, A22, Th1; A25: i in {i,j} by TARSKI:def_2; then {i,j} in D1 by A24, A9; then card D1 > 0 ; then reconsider c1 = (card D1) - 1 as Nat by NAT_1:20; A26: j in {i,j} by TARSKI:def_2; then A27: {i,j} in D2 by A24, A9; A28: for x being set st x in D1 holds ex y being set st ( y in D2 & S1[x,y] ) proof let x be set ; ::_thesis: ( x in D1 implies ex y being set st ( y in D2 & S1[x,y] ) ) assume x in D1 ; ::_thesis: ex y being set st ( y in D2 & S1[x,y] ) then consider s being Element of 2Set (Seg (n + 2)) such that A29: x = s and s in 2S and A30: (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s and A31: i in s ; consider j9 being Nat such that A32: j9 in Seg (n + 2) and A33: j9 <> i and A34: s = {i,j9} by A31, Lm2; now__::_thesis:_ex_X_being_set_st_ (_X_in_D2_&_(_for_k_being_Nat_st_k_in_x_&_k_<>_i_holds_ (_(_k_<>_j_implies_X_=_{j,k}_)_&_(_k_=_j_implies_X_=_{i,j}_)_)_)_) percases ( j9 = j or j9 <> j ) ; supposeA35: j9 = j ; ::_thesis: ex X being set st ( X in D2 & ( for k being Nat st k in x & k <> i holds ( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) ) ) ) take X = {i,j}; ::_thesis: ( X in D2 & ( for k being Nat st k in x & k <> i holds ( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) ) ) ) thus X in D2 by A26, A24, A9; ::_thesis: for k being Nat st k in x & k <> i holds ( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) ) let k be Nat; ::_thesis: ( k in x & k <> i implies ( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) ) ) assume that A36: k in x and k <> i ; ::_thesis: ( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) ) thus ( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) ) by A29, A34, A35, A36, TARSKI:def_2; ::_thesis: verum end; supposeA37: j9 <> j ; ::_thesis: ex X being set st ( X in D2 & ( for k being Nat st k in x & k <> i holds ( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) ) ) ) take X = {j,j9}; ::_thesis: ( X in D2 & ( for k being Nat st k in x & k <> i holds ( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) ) ) ) ( j < j9 or j > j9 ) by A37, XXREAL_0:1; then A38: X in 2Set (Seg (n + 2)) by A12, A22, A32, Th1; A39: j in X by TARSKI:def_2; (Part_sgn (pq,K)) . X <> (Part_sgn (p,K)) . X by A1, A2, A5, A6, A8, A30, A32, A33, A34, A37, Th10; hence X in D2 by A39, A38; ::_thesis: for k being Nat st k in x & k <> i holds ( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) ) let k be Nat; ::_thesis: ( k in x & k <> i implies ( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) ) ) assume that A40: k in x and A41: k <> i ; ::_thesis: ( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) ) thus ( ( k <> j implies X = {j,k} ) & ( k = j implies X = {i,j} ) ) by A29, A34, A37, A40, A41, TARSKI:def_2; ::_thesis: verum end; end; end; hence ex y being set st ( y in D2 & S1[x,y] ) ; ::_thesis: verum end; consider f being Function of D1,D2 such that A42: for x being set st x in D1 holds S1[x,f . x] from FUNCT_2:sch_1(A28); A43: {i,j} in D2 by A26, A24, A9; then A44: dom f = D1 by FUNCT_2:def_1; for y being set st y in D2 holds ex x being set st ( x in D1 & y = f . x ) proof let y be set ; ::_thesis: ( y in D2 implies ex x being set st ( x in D1 & y = f . x ) ) assume y in D2 ; ::_thesis: ex x being set st ( x in D1 & y = f . x ) then consider s being Element of 2Set (Seg (n + 2)) such that A45: s = y and s in 2S and A46: (Part_sgn (pq,K)) . s <> (Part_sgn (p,K)) . s and A47: j in s ; consider i1 being Nat such that A48: i1 in Seg (n + 2) and A49: i1 <> j and A50: s = {j,i1} by A47, Lm2; now__::_thesis:_ex_x_being_set_st_ (_x_in_D1_&_y_=_f_._x_) percases ( i1 = i or i1 <> i ) ; supposeA51: i1 = i ; ::_thesis: ex x being set st ( x in D1 & y = f . x ) A52: {i,j} in D1 by A25, A24, A9; then f . s = y by A5, A26, A42, A45, A50, A51; hence ex x being set st ( x in D1 & y = f . x ) by A50, A51, A52; ::_thesis: verum end; supposeA53: i1 <> i ; ::_thesis: ex x being set st ( x in D1 & y = f . x ) then ( i < i1 or i > i1 ) by XXREAL_0:1; then A54: {i,i1} in 2Set (Seg (n + 2)) by A23, A22, A48, Th1; A55: i in {i,i1} by TARSKI:def_2; (Part_sgn (pq,K)) . {i,i1} <> (Part_sgn (p,K)) . {i,i1} by A1, A2, A5, A6, A8, A46, A48, A49, A50, A53, Th10; then A56: {i,i1} in D1 by A54, A55; i1 in {i,i1} by TARSKI:def_2; then f . {i,i1} = {j,i1} by A42, A49, A53, A56; hence ex x being set st ( x in D1 & y = f . x ) by A45, A50, A56; ::_thesis: verum end; end; end; hence ex x being set st ( x in D1 & y = f . x ) ; ::_thesis: verum end; then A57: rng f = D2 by FUNCT_2:10; for x1, x2 being set st x1 in D1 & x2 in D1 & f . x1 = f . x2 holds x1 = x2 proof let x1, x2 be set ; ::_thesis: ( x1 in D1 & x2 in D1 & f . x1 = f . x2 implies x1 = x2 ) assume that A58: x1 in D1 and A59: x2 in D1 and A60: f . x1 = f . x2 ; ::_thesis: x1 = x2 consider s1 being Element of 2Set (Seg (n + 2)) such that A61: x1 = s1 and s1 in 2S and (Part_sgn (pq,K)) . s1 <> (Part_sgn (p,K)) . s1 and A62: i in s1 by A58; consider j1 being Nat such that j1 in Seg (n + 2) and A63: i <> j1 and A64: {i,j1} = s1 by A62, Lm2; consider s2 being Element of 2Set (Seg (n + 2)) such that A65: x2 = s2 and s2 in 2S and (Part_sgn (pq,K)) . s2 <> (Part_sgn (p,K)) . s2 and A66: i in s2 by A59; consider j2 being Nat such that j2 in Seg (n + 2) and A67: i <> j2 and A68: {i,j2} = s2 by A66, Lm2; A69: j2 in s2 by A68, TARSKI:def_2; A70: j1 in s1 by A64, TARSKI:def_2; now__::_thesis:_(_(_j_=_j1_&_j_=_j2_&_x1_=_x2_)_or_(_j_<>_j1_&_j_=_j2_&_x1_=_x2_)_or_(_j_=_j1_&_j_<>_j2_&_x1_=_x2_)_or_(_j_<>_j1_&_j_<>_j2_&_x1_=_x2_)_) percases ( ( j = j1 & j = j2 ) or ( j <> j1 & j = j2 ) or ( j = j1 & j <> j2 ) or ( j <> j1 & j <> j2 ) ) ; case ( j = j1 & j = j2 ) ; ::_thesis: x1 = x2 hence x1 = x2 by A61, A64, A65, A68; ::_thesis: verum end; caseA71: ( j <> j1 & j = j2 ) ; ::_thesis: x1 = x2 then A72: f . x2 = {i,j} by A42, A59, A65, A67, A69; f . x1 = {j,j1} by A42, A58, A61, A63, A70, A71; hence x1 = x2 by A5, A60, A63, A66, A68, A71, A72, TARSKI:def_2; ::_thesis: verum end; caseA73: ( j = j1 & j <> j2 ) ; ::_thesis: x1 = x2 then A74: f . x2 = {j,j2} by A42, A59, A65, A67, A69; f . x1 = {i,j} by A42, A58, A61, A63, A70, A73; hence x1 = x2 by A5, A60, A62, A64, A67, A73, A74, TARSKI:def_2; ::_thesis: verum end; caseA75: ( j <> j1 & j <> j2 ) ; ::_thesis: x1 = x2 then A76: f . x2 = {j,j2} by A42, A59, A65, A67, A69; A77: j1 in {j,j1} by TARSKI:def_2; f . x1 = {j,j1} by A42, A58, A61, A63, A70, A75; hence x1 = x2 by A60, A61, A64, A65, A68, A75, A76, A77, TARSKI:def_2; ::_thesis: verum end; end; end; hence x1 = x2 ; ::_thesis: verum end; then f is one-to-one by A43, FUNCT_2:19; then D1,D2 are_equipotent by A57, A44, WELLORD2:def_4; then A78: card D1 = card D2 by CARD_1:5; {i,j} in D1 by A25, A24, A9; then {i,j} in D1 /\ D2 by A27, XBOOLE_0:def_4; then {{i,j}} c= D1 /\ D2 by ZFMISC_1:31; then D1 /\ D2 = {{i,j}} by A17, XBOOLE_0:def_10; then card D = ((card D1) + (card D1)) - (card {{i,j}}) by A78, A16, CARD_2:45 .= ((c1 + 1) + (c1 + 1)) - 1 by CARD_1:30 .= (2 * c1) + 1 ; then (card D) mod 2 = 1 mod 2 by NAT_D:21; hence sgn (pq,K) = - (sgn (p,K)) by A3, Th7, NAT_D:14; ::_thesis: verum end; end; end; hence sgn (pq,K) = - (sgn (p,K)) ; ::_thesis: verum end; consider i, j being Nat such that i in dom q and j in dom q and A79: i <> j and A80: q . i = j and A81: q . j = i and for k being Nat st k <> i & k <> j & k in dom q holds q . k = k by A2, MATRIX_2:def_11; ( i < j or j < i ) by A79, XXREAL_0:1; hence sgn (pq,K) = - (sgn (p,K)) by A4, A80, A81; ::_thesis: verum end; theorem Th14: :: MATRIX11:14 for n being Nat for K being Field for tr being Element of Permutations (n + 2) st tr is being_transposition holds sgn (tr,K) = - (1_ K) proof let n be Nat; ::_thesis: for K being Field for tr being Element of Permutations (n + 2) st tr is being_transposition holds sgn (tr,K) = - (1_ K) let K be Field; ::_thesis: for tr being Element of Permutations (n + 2) st tr is being_transposition holds sgn (tr,K) = - (1_ K) set n2 = n + 2; set S = Seg (n + 2); let tr be Element of Permutations (n + 2); ::_thesis: ( tr is being_transposition implies sgn (tr,K) = - (1_ K) ) assume A1: tr is being_transposition ; ::_thesis: sgn (tr,K) = - (1_ K) reconsider Tr = tr as Permutation of (Seg (n + 2)) by MATRIX_2:def_9; reconsider Id = idseq (n + 2), IdTr = (id (Seg (n + 2))) * Tr as Element of Permutations (n + 2) by MATRIX_2:def_9; rng Tr = Seg (n + 2) by FUNCT_2:def_3; then IdTr = Tr by RELAT_1:54; then sgn (tr,K) = - (sgn (Id,K)) by A1, Th13; hence sgn (tr,K) = - (1_ K) by Th12; ::_thesis: verum end; theorem Th15: :: MATRIX11:15 for n being Nat for K being Field for P being FinSequence of (Group_of_Perm (n + 2)) for p2 being Element of Permutations (n + 2) st p2 = Product P & ( for i being Nat st i in dom P holds ex trans being Element of Permutations (n + 2) st ( P . i = trans & trans is being_transposition ) ) holds ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) proof let n be Nat; ::_thesis: for K being Field for P being FinSequence of (Group_of_Perm (n + 2)) for p2 being Element of Permutations (n + 2) st p2 = Product P & ( for i being Nat st i in dom P holds ex trans being Element of Permutations (n + 2) st ( P . i = trans & trans is being_transposition ) ) holds ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) let K be Field; ::_thesis: for P being FinSequence of (Group_of_Perm (n + 2)) for p2 being Element of Permutations (n + 2) st p2 = Product P & ( for i being Nat st i in dom P holds ex trans being Element of Permutations (n + 2) st ( P . i = trans & trans is being_transposition ) ) holds ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) set n2 = n + 2; set G = Group_of_Perm (n + 2); defpred S1[ Nat] means for P being FinSequence of (Group_of_Perm (n + 2)) for p2 being Element of Permutations (n + 2) st p2 = Product P & len P = $1 & ( for i being Nat st i in dom P holds ex trans being Element of Permutations (n + 2) st ( P . i = trans & trans is being_transposition ) ) holds ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ); A1: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A2: S1[k] ; ::_thesis: S1[k + 1] set k1 = k + 1; let P be FinSequence of (Group_of_Perm (n + 2)); ::_thesis: for p2 being Element of Permutations (n + 2) st p2 = Product P & len P = k + 1 & ( for i being Nat st i in dom P holds ex trans being Element of Permutations (n + 2) st ( P . i = trans & trans is being_transposition ) ) holds ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) let p2 be Element of Permutations (n + 2); ::_thesis: ( p2 = Product P & len P = k + 1 & ( for i being Nat st i in dom P holds ex trans being Element of Permutations (n + 2) st ( P . i = trans & trans is being_transposition ) ) implies ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) ) assume that A3: p2 = Product P and A4: len P = k + 1 and A5: for i being Nat st i in dom P holds ex trans being Element of Permutations (n + 2) st ( P . i = trans & trans is being_transposition ) ; ::_thesis: ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) Seg (len P) = dom P by FINSEQ_1:def_3; then consider x being set , Q being FinSequence such that A6: P = <*x*> ^ Q and A7: len P = (len Q) + 1 by A4, RELAT_1:38, REWRITE1:5; reconsider X = <*x*>, Q = Q as FinSequence of (Group_of_Perm (n + 2)) by A6, FINSEQ_1:36; A8: for i being Nat st i in dom Q holds ex trans being Element of Permutations (n + 2) st ( Q . i = trans & trans is being_transposition ) proof let i be Nat; ::_thesis: ( i in dom Q implies ex trans being Element of Permutations (n + 2) st ( Q . i = trans & trans is being_transposition ) ) assume A9: i in dom Q ; ::_thesis: ex trans being Element of Permutations (n + 2) st ( Q . i = trans & trans is being_transposition ) Q . i = P . ((len X) + i) by A6, A9, FINSEQ_1:def_7; hence ex trans being Element of Permutations (n + 2) st ( Q . i = trans & trans is being_transposition ) by A5, A6, A9, FINSEQ_1:28; ::_thesis: verum end; 1 + 0 <= k + 1 by XREAL_1:7; then 1 in Seg (k + 1) ; then A10: 1 in dom P by A4, FINSEQ_1:def_3; P . 1 = x by A6, FINSEQ_1:41; then consider tr being Element of Permutations (n + 2) such that A11: x = tr and A12: tr is being_transposition by A5, A10; reconsider PQ = Product Q as Element of Permutations (n + 2) by MATRIX_2:def_10; reconsider Tr = tr as Element of (Group_of_Perm (n + 2)) by MATRIX_2:def_10; A13: p2 = Tr * (Product Q) by A3, A6, A11, GROUP_4:7 .= PQ * tr by MATRIX_2:def_10 ; then A14: sgn (p2,K) = - (sgn (PQ,K)) by A12, Th13; now__::_thesis:_(_(_(len_P)_mod_2_=_0_implies_sgn_(p2,K)_=_1__K_)_&_(_(len_P)_mod_2_=_1_implies_sgn_(p2,K)_=_-_(1__K)_)_) percases ( (len Q) mod 2 = 0 or (len Q) mod 2 = 1 ) by NAT_D:12; supposeA15: (len Q) mod 2 = 0 ; ::_thesis: ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) 0 < 2 - 1 ; then A16: (len P) mod 2 = 0 + 1 by A7, A15, NAT_D:70; sgn (PQ,K) = 1_ K by A2, A4, A7, A8, A15; hence ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) by A12, A13, A16, Th13; ::_thesis: verum end; supposeA17: (len Q) mod 2 = 1 ; ::_thesis: ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) A18: 2 - 1 = 1 ; sgn (PQ,K) = - (1_ K) by A2, A4, A7, A8, A17; hence ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) by A7, A14, A17, A18, NAT_D:69, RLVECT_1:17; ::_thesis: verum end; end; end; hence ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) ; ::_thesis: verum end; A19: S1[ 0 ] proof let P be FinSequence of (Group_of_Perm (n + 2)); ::_thesis: for p2 being Element of Permutations (n + 2) st p2 = Product P & len P = 0 & ( for i being Nat st i in dom P holds ex trans being Element of Permutations (n + 2) st ( P . i = trans & trans is being_transposition ) ) holds ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) let p2 be Element of Permutations (n + 2); ::_thesis: ( p2 = Product P & len P = 0 & ( for i being Nat st i in dom P holds ex trans being Element of Permutations (n + 2) st ( P . i = trans & trans is being_transposition ) ) implies ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) ) assume that A20: p2 = Product P and A21: len P = 0 and for i being Nat st i in dom P holds ex trans being Element of Permutations (n + 2) st ( P . i = trans & trans is being_transposition ) ; ::_thesis: ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) P = <*> the carrier of (Group_of_Perm (n + 2)) by A21; then Product P = 1_ (Group_of_Perm (n + 2)) by GROUP_4:8; then Product P = idseq (n + 2) by MATRIX_2:24; hence ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) by A20, A21, Th12, NAT_D:26; ::_thesis: verum end; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A19, A1); hence for P being FinSequence of (Group_of_Perm (n + 2)) for p2 being Element of Permutations (n + 2) st p2 = Product P & ( for i being Nat st i in dom P holds ex trans being Element of Permutations (n + 2) st ( P . i = trans & trans is being_transposition ) ) holds ( ( (len P) mod 2 = 0 implies sgn (p2,K) = 1_ K ) & ( (len P) mod 2 = 1 implies sgn (p2,K) = - (1_ K) ) ) ; ::_thesis: verum end; theorem Th16: :: MATRIX11:16 for i, j, n being Nat st i < j & i in Seg n & j in Seg n holds ex tr being Element of Permutations n st ( tr is being_transposition & tr . i = j ) proof let i, j, n be Nat; ::_thesis: ( i < j & i in Seg n & j in Seg n implies ex tr being Element of Permutations n st ( tr is being_transposition & tr . i = j ) ) assume that A1: i < j and A2: i in Seg n and A3: j in Seg n ; ::_thesis: ex tr being Element of Permutations n st ( tr is being_transposition & tr . i = j ) defpred S1[ set , set ] means for k being Nat st k in Seg n & k = $1 holds ( ( k = i implies $2 = j ) & ( k = j implies $2 = i ) & ( k <> i & k <> j implies $2 = k ) ); A4: for x being set st x in Seg n holds ex y being set st ( y in Seg n & S1[x,y] ) proof let x be set ; ::_thesis: ( x in Seg n implies ex y being set st ( y in Seg n & S1[x,y] ) ) assume A5: x in Seg n ; ::_thesis: ex y being set st ( y in Seg n & S1[x,y] ) reconsider m = x as Nat by A5; now__::_thesis:_ex_y_being_set_st_ (_y_in_Seg_n_&_S1[x,y]_) percases ( m = i or m = j or ( m <> i & m <> j ) ) ; suppose m = i ; ::_thesis: ex y being set st ( y in Seg n & S1[x,y] ) then S1[x,j] ; hence ex y being set st ( y in Seg n & S1[x,y] ) by A3; ::_thesis: verum end; suppose m = j ; ::_thesis: ex y being set st ( y in Seg n & S1[x,y] ) then S1[x,i] ; hence ex y being set st ( y in Seg n & S1[x,y] ) by A2; ::_thesis: verum end; suppose ( m <> i & m <> j ) ; ::_thesis: ex y being set st ( y in Seg n & S1[x,y] ) then S1[x,x] ; hence ex y being set st ( y in Seg n & S1[x,y] ) by A5; ::_thesis: verum end; end; end; hence ex y being set st ( y in Seg n & S1[x,y] ) ; ::_thesis: verum end; consider f being Function of (Seg n),(Seg n) such that A6: for x being set st x in Seg n holds S1[x,f . x] from FUNCT_2:sch_1(A4); for x1, x2 being set st x1 in Seg n & x2 in Seg n & f . x1 = f . x2 holds x1 = x2 proof let x1, x2 be set ; ::_thesis: ( x1 in Seg n & x2 in Seg n & f . x1 = f . x2 implies x1 = x2 ) assume that A7: x1 in Seg n and A8: x2 in Seg n and A9: f . x1 = f . x2 ; ::_thesis: x1 = x2 reconsider k1 = x1 as Nat by A7; ( x1 = i or x1 = j or ( x1 <> i & x1 <> j ) ) ; then A10: ( ( x1 = i & f . x1 = j ) or ( x1 = j & f . x1 = i ) or ( x1 <> i & x1 <> j & f . x1 = k1 ) ) by A6, A7; ( x2 = i or x2 = j or ( x2 <> i & x2 <> j ) ) ; hence x1 = x2 by A6, A8, A9, A10; ::_thesis: verum end; then A11: f is one-to-one by A2, FUNCT_2:19; for y being set st y in Seg n holds ex x being set st ( x in Seg n & y = f . x ) proof let y be set ; ::_thesis: ( y in Seg n implies ex x being set st ( x in Seg n & y = f . x ) ) assume A12: y in Seg n ; ::_thesis: ex x being set st ( x in Seg n & y = f . x ) reconsider k = y as Nat by A12; ( ( k = i & f . j = i ) or ( k = j & f . i = j ) or ( k <> i & k <> j & f . k = k ) ) by A2, A3, A6, A12; hence ex x being set st ( x in Seg n & y = f . x ) by A2, A3, A12; ::_thesis: verum end; then rng f = Seg n by FUNCT_2:10; then f is onto by FUNCT_2:def_3; then reconsider P = f as Element of Permutations n by A11, MATRIX_2:def_9; A13: P . j = i by A3, A6; dom P = Seg n by A2, FUNCT_2:def_1; then A14: for k being Nat st k <> i & k <> j & k in dom P holds P . k = k by A6; take P ; ::_thesis: ( P is being_transposition & P . i = j ) A15: i in dom P by A2, FUNCT_2:def_1; A16: j in dom P by A3, FUNCT_2:def_1; P . i = j by A2, A6; hence ( P is being_transposition & P . i = j ) by A1, A15, A16, A13, A14, MATRIX_2:def_11; ::_thesis: verum end; theorem Th17: :: MATRIX11:17 for k being Nat for p being Element of Permutations (k + 1) st p . (k + 1) <> k + 1 holds ex tr being Element of Permutations (k + 1) st ( tr is being_transposition & tr . (p . (k + 1)) = k + 1 & (tr * p) . (k + 1) = k + 1 ) proof let k be Nat; ::_thesis: for p being Element of Permutations (k + 1) st p . (k + 1) <> k + 1 holds ex tr being Element of Permutations (k + 1) st ( tr is being_transposition & tr . (p . (k + 1)) = k + 1 & (tr * p) . (k + 1) = k + 1 ) set k1 = k + 1; let p be Element of Permutations (k + 1); ::_thesis: ( p . (k + 1) <> k + 1 implies ex tr being Element of Permutations (k + 1) st ( tr is being_transposition & tr . (p . (k + 1)) = k + 1 & (tr * p) . (k + 1) = k + 1 ) ) assume A1: p . (k + 1) <> k + 1 ; ::_thesis: ex tr being Element of Permutations (k + 1) st ( tr is being_transposition & tr . (p . (k + 1)) = k + 1 & (tr * p) . (k + 1) = k + 1 ) reconsider p9 = p as Permutation of (Seg (k + 1)) by MATRIX_2:def_9; A2: dom p9 = Seg (k + 1) by FUNCT_2:52; A3: rng p9 = Seg (k + 1) by FUNCT_2:def_3; A4: k + 1 in Seg (k + 1) by FINSEQ_1:3; then A5: p . (k + 1) in Seg (k + 1) by A2, A3, FUNCT_1:def_3; then p . (k + 1) <= k + 1 by FINSEQ_1:1; then p . (k + 1) < k + 1 by A1, XXREAL_0:1; then consider tr being Element of Permutations (k + 1) such that A6: tr is being_transposition and A7: tr . (p . (k + 1)) = k + 1 by A4, A5, Th16; reconsider tr9 = tr as Permutation of (Seg (k + 1)) by MATRIX_2:def_9; dom tr9 = Seg (k + 1) by FUNCT_2:52; then dom (tr * p) = Seg (k + 1) by A2, A3, RELAT_1:27; then (tr * p) . (k + 1) = tr . (p . (k + 1)) by FINSEQ_1:3, FUNCT_1:12; hence ex tr being Element of Permutations (k + 1) st ( tr is being_transposition & tr . (p . (k + 1)) = k + 1 & (tr * p) . (k + 1) = k + 1 ) by A6, A7; ::_thesis: verum end; theorem Th18: :: MATRIX11:18 for X, x being set st not x in X holds for p1 being Permutation of (X \/ {x}) st p1 . x = x holds ex p being Permutation of X st p1 | X = p proof let X, x be set ; ::_thesis: ( not x in X implies for p1 being Permutation of (X \/ {x}) st p1 . x = x holds ex p being Permutation of X st p1 | X = p ) assume A1: not x in X ; ::_thesis: for p1 being Permutation of (X \/ {x}) st p1 . x = x holds ex p being Permutation of X st p1 | X = p let p1 be Permutation of (X \/ {x}); ::_thesis: ( p1 . x = x implies ex p being Permutation of X st p1 | X = p ) assume A2: p1 . x = x ; ::_thesis: ex p being Permutation of X st p1 | X = p A3: X c= X \/ {x} by XBOOLE_1:7; set pX = p1 | X; A4: dom p1 = X \/ {x} by FUNCT_2:52; then A5: dom (p1 | X) = X by RELAT_1:62, XBOOLE_1:7; A6: rng p1 = X \/ {x} by FUNCT_2:def_3; then A7: rng (p1 | X) c= X \/ {x} by RELAT_1:70; A8: rng (p1 | X) c= X proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng (p1 | X) or y in X ) assume A9: y in rng (p1 | X) ; ::_thesis: y in X consider x9 being set such that A10: x9 in dom (p1 | X) and A11: (p1 | X) . x9 = y by A9, FUNCT_1:def_3; assume A12: not y in X ; ::_thesis: contradiction y in rng (p1 | X) by A10, A11, FUNCT_1:def_3; then y in {x} by A7, A12, XBOOLE_0:def_3; then A13: y = x by TARSKI:def_1; (p1 | X) . x9 = p1 . x9 by A10, FUNCT_1:47; hence contradiction by A1, A2, A3, A4, A5, A7, A9, A10, A11, A13, FUNCT_1:def_4; ::_thesis: verum end; X c= rng (p1 | X) proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in X or y in rng (p1 | X) ) assume A14: y in X ; ::_thesis: y in rng (p1 | X) consider x9 being set such that A15: x9 in dom p1 and A16: p1 . x9 = y by A3, A6, A14, FUNCT_1:def_3; A17: x9 in X proof assume not x9 in X ; ::_thesis: contradiction then x9 in {x} by A4, A15, XBOOLE_0:def_3; hence contradiction by A1, A2, A14, A16, TARSKI:def_1; ::_thesis: verum end; then (p1 | X) . x9 = p1 . x9 by A5, FUNCT_1:47; hence y in rng (p1 | X) by A5, A16, A17, FUNCT_1:def_3; ::_thesis: verum end; then A18: rng (p1 | X) = X by A8, XBOOLE_0:def_10; A19: p1 | X is one-to-one by FUNCT_1:52; reconsider pX = p1 | X as Function of X,X by A5, A18, FUNCT_2:1; pX is onto by A18, FUNCT_2:def_3; hence ex p being Permutation of X st p1 | X = p by A19; ::_thesis: verum end; theorem Th19: :: MATRIX11:19 for X, x being set for p, q being Permutation of X for p1, q1 being Permutation of (X \/ {x}) st p1 | X = p & q1 | X = q & p1 . x = x & q1 . x = x holds ( (p1 * q1) | X = p * q & (p1 * q1) . x = x ) proof let X, x be set ; ::_thesis: for p, q being Permutation of X for p1, q1 being Permutation of (X \/ {x}) st p1 | X = p & q1 | X = q & p1 . x = x & q1 . x = x holds ( (p1 * q1) | X = p * q & (p1 * q1) . x = x ) let p, q be Permutation of X; ::_thesis: for p1, q1 being Permutation of (X \/ {x}) st p1 | X = p & q1 | X = q & p1 . x = x & q1 . x = x holds ( (p1 * q1) | X = p * q & (p1 * q1) . x = x ) let p1, q1 be Permutation of (X \/ {x}); ::_thesis: ( p1 | X = p & q1 | X = q & p1 . x = x & q1 . x = x implies ( (p1 * q1) | X = p * q & (p1 * q1) . x = x ) ) assume that A1: p1 | X = p and A2: q1 | X = q and A3: p1 . x = x and A4: q1 . x = x ; ::_thesis: ( (p1 * q1) | X = p * q & (p1 * q1) . x = x ) set pq = p * q; set pq1 = p1 * q1; set X1 = X \/ {x}; A5: X c= X \/ {x} by XBOOLE_1:7; A6: rng q = X by FUNCT_2:def_3; A7: dom q = X by FUNCT_2:52; dom (p1 * q1) = X \/ {x} by FUNCT_2:52; then A8: dom ((p1 * q1) | X) = X by RELAT_1:62, XBOOLE_1:7; A9: dom (p * q) = X by FUNCT_2:52; A10: dom p = X by FUNCT_2:52; for y being set st y in dom (p * q) holds (p * q) . y = ((p1 * q1) | X) . y proof let y be set ; ::_thesis: ( y in dom (p * q) implies (p * q) . y = ((p1 * q1) | X) . y ) assume A11: y in dom (p * q) ; ::_thesis: (p * q) . y = ((p1 * q1) | X) . y A12: (p * q) . y = p . (q . y) by A9, A11, FUNCT_2:15; A13: (p1 * q1) . y = ((p1 * q1) | X) . y by A9, A8, A11, FUNCT_1:47; A14: q . y in rng q by A7, A9, A11, FUNCT_1:def_3; A15: (p1 * q1) . y = p1 . (q1 . y) by A5, A9, A11, FUNCT_2:15; q1 . y = q . y by A2, A7, A9, A11, FUNCT_1:47; hence (p * q) . y = ((p1 * q1) | X) . y by A1, A10, A6, A14, A13, A12, A15, FUNCT_1:47; ::_thesis: verum end; hence (p1 * q1) | X = p * q by A8, FUNCT_1:2, FUNCT_2:52; ::_thesis: (p1 * q1) . x = x x in {x} by TARSKI:def_1; then x in X \/ {x} by XBOOLE_0:def_3; hence (p1 * q1) . x = x by A3, A4, FUNCT_2:15; ::_thesis: verum end; theorem Th20: :: MATRIX11:20 for k being Nat for tr being Element of Permutations k st tr is being_transposition holds ( tr * tr = idseq k & tr = tr " ) proof let k be Nat; ::_thesis: for tr being Element of Permutations k st tr is being_transposition holds ( tr * tr = idseq k & tr = tr " ) set I = idseq k; let tr be Element of Permutations k; ::_thesis: ( tr is being_transposition implies ( tr * tr = idseq k & tr = tr " ) ) assume tr is being_transposition ; ::_thesis: ( tr * tr = idseq k & tr = tr " ) then consider i, j being Nat such that i in dom tr and j in dom tr and i <> j and A1: tr . i = j and A2: tr . j = i and A3: for m being Nat st m <> i & m <> j & m in dom tr holds tr . m = m by MATRIX_2:def_11; reconsider TR = tr as Permutation of (Seg k) by MATRIX_2:def_9; set TT = TR * TR; A4: dom (TR * TR) = Seg k by FUNCT_2:52; A5: dom TR = Seg k by FUNCT_2:52; A6: for x being set st x in dom (TR * TR) holds (TR * TR) . x = (idseq k) . x proof let x be set ; ::_thesis: ( x in dom (TR * TR) implies (TR * TR) . x = (idseq k) . x ) assume A7: x in dom (TR * TR) ; ::_thesis: (TR * TR) . x = (idseq k) . x reconsider m = x as Nat by A4, A7; now__::_thesis:_(TR_*_TR)_._m_=_m percases ( m = i or m = j or ( m <> i & m <> j ) ) ; suppose ( m = i or m = j ) ; ::_thesis: (TR * TR) . m = m hence (TR * TR) . m = m by A1, A2, A7, FUNCT_1:12; ::_thesis: verum end; suppose ( m <> i & m <> j ) ; ::_thesis: (TR * TR) . m = m then tr . m = m by A3, A4, A5, A7; hence (TR * TR) . m = m by A7, FUNCT_1:12; ::_thesis: verum end; end; end; hence (TR * TR) . x = (idseq k) . x by A4, A7, FUNCT_1:18; ::_thesis: verum end; A8: dom (idseq k) = Seg k by FUNCT_2:52; hence tr * tr = idseq k by A6, FUNCT_1:2, FUNCT_2:52; ::_thesis: tr = tr " rng TR = Seg k by FUNCT_2:def_3; hence tr = tr " by A4, A8, A5, A6, FUNCT_1:2, FUNCT_1:42; ::_thesis: verum end; theorem Th21: :: MATRIX11:21 for n being Nat for perm being Element of Permutations n ex P being FinSequence of (Group_of_Perm n) st ( perm = Product P & ( for i being Nat st i in dom P holds ex trans being Element of Permutations n st ( P . i = trans & trans is being_transposition ) ) ) proof let n be Nat; ::_thesis: for perm being Element of Permutations n ex P being FinSequence of (Group_of_Perm n) st ( perm = Product P & ( for i being Nat st i in dom P holds ex trans being Element of Permutations n st ( P . i = trans & trans is being_transposition ) ) ) defpred S1[ Nat] means for perm being Element of Permutations $1 ex P being FinSequence of (Group_of_Perm $1) st ( perm = Product P & ( for i being Nat st i in dom P holds ex trans being Element of Permutations $1 st ( P . i = trans & trans is being_transposition ) ) ); let perm be Element of Permutations n; ::_thesis: ex P being FinSequence of (Group_of_Perm n) st ( perm = Product P & ( for i being Nat st i in dom P holds ex trans being Element of Permutations n st ( P . i = trans & trans is being_transposition ) ) ) A1: n is Element of NAT by ORDINAL1:def_12; A2: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A3: S1[k] ; ::_thesis: S1[k + 1] set k1 = k + 1; let p be Element of Permutations (k + 1); ::_thesis: ex P being FinSequence of (Group_of_Perm (k + 1)) st ( p = Product P & ( for i being Nat st i in dom P holds ex trans being Element of Permutations (k + 1) st ( P . i = trans & trans is being_transposition ) ) ) reconsider p9 = p as Permutation of (Seg (k + 1)) by MATRIX_2:def_9; set Gk1 = Group_of_Perm (k + 1); A4: for p being Element of Permutations (k + 1) st p . (k + 1) = k + 1 holds ex P being FinSequence of (Group_of_Perm (k + 1)) st ( p = Product P & ( for i being Nat st i in dom P holds ex trans being Element of Permutations (k + 1) st ( P . i = trans & trans is being_transposition ) ) ) proof set Ik = idseq k; set Ik1 = idseq (k + 1); set Gk1 = Group_of_Perm (k + 1); set Gk = Group_of_Perm k; let p be Element of Permutations (k + 1); ::_thesis: ( p . (k + 1) = k + 1 implies ex P being FinSequence of (Group_of_Perm (k + 1)) st ( p = Product P & ( for i being Nat st i in dom P holds ex trans being Element of Permutations (k + 1) st ( P . i = trans & trans is being_transposition ) ) ) ) assume A5: p . (k + 1) = k + 1 ; ::_thesis: ex P being FinSequence of (Group_of_Perm (k + 1)) st ( p = Product P & ( for i being Nat st i in dom P holds ex trans being Element of Permutations (k + 1) st ( P . i = trans & trans is being_transposition ) ) ) set mG1 = the multF of (Group_of_Perm (k + 1)); set mG = the multF of (Group_of_Perm k); reconsider p9 = p as Permutation of (Seg (k + 1)) by MATRIX_2:def_9; A6: Seg (k + 1) = (Seg k) \/ {(k + 1)} by FINSEQ_1:9; then consider pk being Permutation of (Seg k) such that A7: p9 | (Seg k) = pk by A5, Th18, FINSEQ_3:8; reconsider pk9 = pk as Element of Permutations k by MATRIX_2:def_9; consider P being FinSequence of (Group_of_Perm k) such that A8: pk9 = Product P and A9: for i being Nat st i in dom P holds ex trans being Element of Permutations k st ( P . i = trans & trans is being_transposition ) by A3; A10: pk9 = the multF of (Group_of_Perm k) "**" P by A8, GROUP_4:def_2; defpred S2[ set , set ] means for i being Nat for tr being Element of Permutations k st i in dom P & P . i = tr & i = $1 holds ex newtr being Element of Permutations (k + 1) st ( newtr = $2 & newtr is being_transposition & newtr . (k + 1) = k + 1 & tr = newtr | (Seg k) ); A11: not k + 1 in Seg k by FINSEQ_3:8; A12: for m being Nat st m in Seg (len P) holds ex x being Element of (Group_of_Perm (k + 1)) st S2[m,x] proof let m be Nat; ::_thesis: ( m in Seg (len P) implies ex x being Element of (Group_of_Perm (k + 1)) st S2[m,x] ) assume m in Seg (len P) ; ::_thesis: ex x being Element of (Group_of_Perm (k + 1)) st S2[m,x] then m in dom P by FINSEQ_1:def_3; then consider tr being Element of Permutations k such that A13: P . m = tr and A14: tr is being_transposition by A9; consider i9, j9 being Nat such that A15: i9 in dom tr and A16: j9 in dom tr and A17: i9 <> j9 and A18: tr . i9 = j9 and A19: tr . j9 = i9 and A20: for k being Nat st k <> i9 & k <> j9 & k in dom tr holds tr . k = k by A14, MATRIX_2:def_11; reconsider tr9 = tr as Permutation of (Seg k) by MATRIX_2:def_9; consider newt being Function of (Seg (k + 1)),(Seg (k + 1)) such that A21: newt | (Seg k) = tr9 and A22: newt . (k + 1) = k + 1 by A6, A11, STIRL2_1:57; A23: newt . j9 = tr . j9 by A21, A16, FUNCT_1:47; A24: ( Seg k is empty implies Seg k is empty ) ; then A25: newt is onto by A6, A21, A22, STIRL2_1:58; newt is one-to-one by A6, A11, A24, A21, A22, STIRL2_1:58; then reconsider NT = newt as Element of Permutations (k + 1) by A25, MATRIX_2:def_9; reconsider NT9 = NT as Element of (Group_of_Perm (k + 1)) by MATRIX_2:def_10; take NT9 ; ::_thesis: S2[m,NT9] let I be Nat; ::_thesis: for tr being Element of Permutations k st I in dom P & P . I = tr & I = m holds ex newtr being Element of Permutations (k + 1) st ( newtr = NT9 & newtr is being_transposition & newtr . (k + 1) = k + 1 & tr = newtr | (Seg k) ) let TR be Element of Permutations k; ::_thesis: ( I in dom P & P . I = TR & I = m implies ex newtr being Element of Permutations (k + 1) st ( newtr = NT9 & newtr is being_transposition & newtr . (k + 1) = k + 1 & TR = newtr | (Seg k) ) ) assume that I in dom P and A26: P . I = TR and A27: I = m ; ::_thesis: ex newtr being Element of Permutations (k + 1) st ( newtr = NT9 & newtr is being_transposition & newtr . (k + 1) = k + 1 & TR = newtr | (Seg k) ) take NT ; ::_thesis: ( NT = NT9 & NT is being_transposition & NT . (k + 1) = k + 1 & TR = NT | (Seg k) ) A28: dom tr c= dom newt by A21, RELAT_1:60; A29: for m being Nat st m <> i9 & m <> j9 & m in dom newt holds newt . m = m proof A30: dom tr9 = Seg k by FUNCT_2:52; let m be Nat; ::_thesis: ( m <> i9 & m <> j9 & m in dom newt implies newt . m = m ) assume that A31: m <> i9 and A32: m <> j9 and A33: m in dom newt ; ::_thesis: newt . m = m dom newt = Seg (k + 1) by FUNCT_2:52; then ( m in Seg k or m in {(k + 1)} ) by A6, A33, XBOOLE_0:def_3; then ( m in dom tr or m = k + 1 ) by A30, TARSKI:def_1; then ( ( tr . m = newt . m & tr . m = m ) or newt . m = m ) by A21, A22, A20, A31, A32, FUNCT_1:47; hence newt . m = m ; ::_thesis: verum end; newt . i9 = tr . i9 by A21, A15, FUNCT_1:47; hence ( NT = NT9 & NT is being_transposition & NT . (k + 1) = k + 1 & TR = NT | (Seg k) ) by A13, A21, A22, A26, A27, A15, A16, A17, A18, A19, A28, A23, A29, MATRIX_2:def_11; ::_thesis: verum end; consider Pr being FinSequence of (Group_of_Perm (k + 1)) such that A34: dom Pr = Seg (len P) and A35: for m being Nat st m in Seg (len P) holds S2[m,Pr . m] from FINSEQ_1:sch_5(A12); take Pr ; ::_thesis: ( p = Product Pr & ( for i being Nat st i in dom Pr holds ex trans being Element of Permutations (k + 1) st ( Pr . i = trans & trans is being_transposition ) ) ) A36: Product Pr = the multF of (Group_of_Perm (k + 1)) "**" Pr by GROUP_4:def_2; now__::_thesis:_(_p_=_Product_Pr_&_(_for_i_being_Nat_st_i_in_dom_Pr_holds_ ex_trans_being_Element_of_Permutations_(k_+_1)_st_ (_Pr_._i_=_trans_&_trans_is_being_transposition_)_)_) percases ( len Pr = 0 or len Pr > 0 ) ; supposeA37: len Pr = 0 ; ::_thesis: ( p = Product Pr & ( for i being Nat st i in dom Pr holds ex trans being Element of Permutations (k + 1) st ( Pr . i = trans & trans is being_transposition ) ) ) then A38: Seg (len Pr) = 0 ; A39: Product Pr = the_unity_wrt the multF of (Group_of_Perm (k + 1)) by A36, A37, FINSOP_1:def_1; the_unity_wrt the multF of (Group_of_Perm (k + 1)) = 1_ (Group_of_Perm (k + 1)) by GROUP_1:22; then A40: Product Pr = idseq (k + 1) by A39, MATRIX_2:24; len P = 0 by A34, A37, FINSEQ_1:def_3; then A41: pk9 = the_unity_wrt the multF of (Group_of_Perm k) by A10, FINSOP_1:def_1; A42: dom p9 = Seg (k + 1) by FUNCT_2:52; A43: the_unity_wrt the multF of (Group_of_Perm k) = 1_ (Group_of_Perm k) by GROUP_1:22; A44: for y being set st y in dom p holds p . y = (idseq (k + 1)) . y proof let y be set ; ::_thesis: ( y in dom p implies p . y = (idseq (k + 1)) . y ) assume A45: y in dom p ; ::_thesis: p . y = (idseq (k + 1)) . y reconsider y9 = y as Nat by A42, A45; A46: (idseq (k + 1)) . y9 = y9 by A42, A45, FUNCT_1:18; A47: dom pk = Seg k by FUNCT_2:52; ( y in Seg k or y in {(k + 1)} ) by A6, A42, A45, XBOOLE_0:def_3; then ( ( pk . y = p . y & (idseq k) . y9 = y9 ) or ( p . (k + 1) = k + 1 & y = k + 1 ) ) by A5, A7, A47, FUNCT_1:18, FUNCT_1:47, TARSKI:def_1; hence p . y = (idseq (k + 1)) . y by A41, A43, A46, MATRIX_2:24; ::_thesis: verum end; dom (idseq (k + 1)) = Seg (k + 1) by FUNCT_2:52; hence ( p = Product Pr & ( for i being Nat st i in dom Pr holds ex trans being Element of Permutations (k + 1) st ( Pr . i = trans & trans is being_transposition ) ) ) by A40, A42, A44, A38, FINSEQ_1:def_3, FUNCT_1:2; ::_thesis: verum end; supposeA48: len Pr > 0 ; ::_thesis: ( p = Product Pr & ( for i being Nat st i in dom Pr holds ex trans being Element of Permutations (k + 1) st ( Pr . i = trans & trans is being_transposition ) ) ) consider fPr being Function of NAT,(Group_of_Perm (k + 1)) such that A49: fPr . 1 = Pr . 1 and A50: for n being Element of NAT st 0 <> n & n < len Pr holds fPr . (n + 1) = the multF of (Group_of_Perm (k + 1)) . ((fPr . n),(Pr . (n + 1))) and A51: Product Pr = fPr . (len Pr) by A36, A48, FINSOP_1:def_1; len P > 0 by A34, A48, FINSEQ_1:def_3; then consider fP being Function of NAT,(Group_of_Perm k) such that A52: fP . 1 = P . 1 and A53: for n being Element of NAT st 0 <> n & n < len P holds fP . (n + 1) = the multF of (Group_of_Perm k) . ((fP . n),(P . (n + 1))) and A54: pk = fP . (len P) by A10, FINSOP_1:def_1; A55: len P = len Pr by A34, FINSEQ_1:def_3; defpred S3[ Nat] means ( $1 > 0 & $1 <= len P implies ex Prod1 being Element of Permutations (k + 1) ex Prod being Element of Permutations k st ( Prod1 = fPr . $1 & fP . $1 = Prod & Prod1 | (Seg k) = Prod & Prod1 . (k + 1) = k + 1 ) ); A56: for m being Element of NAT st S3[m] holds S3[m + 1] proof let m be Element of NAT ; ::_thesis: ( S3[m] implies S3[m + 1] ) assume A57: S3[m] ; ::_thesis: S3[m + 1] set m1 = m + 1; assume that m + 1 > 0 and A58: m + 1 <= len P ; ::_thesis: ex Prod1 being Element of Permutations (k + 1) ex Prod being Element of Permutations k st ( Prod1 = fPr . (m + 1) & fP . (m + 1) = Prod & Prod1 | (Seg k) = Prod & Prod1 . (k + 1) = k + 1 ) (m + 1) + 0 > 0 ; then m + 1 >= 1 by NAT_1:19; then A59: m + 1 in Seg (len P) by A58; A60: dom P = Seg (len P) by FINSEQ_1:def_3; then consider tr being Element of Permutations k such that A61: P . (m + 1) = tr and tr is being_transposition by A9, A59; consider tr1 being Element of Permutations (k + 1) such that A62: tr1 = Pr . (m + 1) and tr1 is being_transposition and A63: tr1 . (k + 1) = k + 1 and A64: tr = tr1 | (Seg k) by A35, A59, A60, A61; now__::_thesis:_ex_Prod1_being_Element_of_Permutations_(k_+_1)_ex_Prod_being_Element_of_Permutations_k_st_ (_Prod1_=_fPr_._(m_+_1)_&_fP_._(m_+_1)_=_Prod_&_Prod1_|_(Seg_k)_=_Prod_&_Prod1_._(k_+_1)_=_k_+_1_) percases ( m = 0 or m > 0 ) ; suppose m = 0 ; ::_thesis: ex Prod1 being Element of Permutations (k + 1) ex Prod being Element of Permutations k st ( Prod1 = fPr . (m + 1) & fP . (m + 1) = Prod & Prod1 | (Seg k) = Prod & Prod1 . (k + 1) = k + 1 ) hence ex Prod1 being Element of Permutations (k + 1) ex Prod being Element of Permutations k st ( Prod1 = fPr . (m + 1) & fP . (m + 1) = Prod & Prod1 | (Seg k) = Prod & Prod1 . (k + 1) = k + 1 ) by A52, A49, A61, A62, A63, A64; ::_thesis: verum end; supposeA65: m > 0 ; ::_thesis: ex Prod1 being Element of Permutations (k + 1) ex Prod being Element of Permutations k st ( Prod1 = fPr . (m + 1) & fP . (m + 1) = Prod & Prod1 | (Seg k) = Prod & Prod1 . (k + 1) = k + 1 ) A66: m + 0 < m + 1 by XREAL_1:6; then consider Q1 being Element of Permutations (k + 1), Q being Element of Permutations k such that A67: Q1 = fPr . m and A68: fP . m = Q and A69: Q1 | (Seg k) = Q and A70: Q1 . (k + 1) = k + 1 by A57, A58, A65, XXREAL_0:2; reconsider Q = Q, tr = tr as Permutation of (Seg k) by MATRIX_2:def_9; reconsider trQ = tr * Q as Element of Permutations k by MATRIX_2:def_9; A71: m < len P by A58, A66, XXREAL_0:2; then A72: fP . (m + 1) = the multF of (Group_of_Perm k) . (Q,tr) by A53, A61, A65, A68; then A73: fP . (m + 1) = trQ by MATRIX_2:def_10; reconsider Q1 = Q1, tr1 = tr1 as Permutation of (Seg (k + 1)) by MATRIX_2:def_9; reconsider trQ1 = tr1 * Q1 as Element of Permutations (k + 1) by MATRIX_2:def_9; A74: trQ1 | (Seg k) = trQ by A6, A63, A64, A69, A70, Th19; len P = len Pr by A34, FINSEQ_1:def_3; then fPr . (m + 1) = the multF of (Group_of_Perm (k + 1)) . (Q1,tr1) by A50, A62, A65, A67, A71; then A75: fPr . (m + 1) = trQ1 by MATRIX_2:def_10; trQ1 . (k + 1) = k + 1 by A6, A63, A64, A69, A70, A72, Th19; hence ex Prod1 being Element of Permutations (k + 1) ex Prod being Element of Permutations k st ( Prod1 = fPr . (m + 1) & fP . (m + 1) = Prod & Prod1 | (Seg k) = Prod & Prod1 . (k + 1) = k + 1 ) by A73, A75, A74; ::_thesis: verum end; end; end; hence ex Prod1 being Element of Permutations (k + 1) ex Prod being Element of Permutations k st ( Prod1 = fPr . (m + 1) & fP . (m + 1) = Prod & Prod1 | (Seg k) = Prod & Prod1 . (k + 1) = k + 1 ) ; ::_thesis: verum end; A76: S3[ 0 ] ; for m being Element of NAT holds S3[m] from NAT_1:sch_1(A76, A56); then consider Prod1 being Element of Permutations (k + 1), Prod being Element of Permutations k such that A77: Prod1 = fPr . (len P) and A78: fP . (len P) = Prod and A79: Prod1 | (Seg k) = Prod and A80: Prod1 . (k + 1) = k + 1 by A48, A55; reconsider Prod1 = Prod1 as Permutation of (Seg (k + 1)) by MATRIX_2:def_9; A81: dom p9 = Seg (k + 1) by FUNCT_2:52; A82: for y being set st y in dom p holds p . y = Prod1 . y proof let y be set ; ::_thesis: ( y in dom p implies p . y = Prod1 . y ) assume y in dom p ; ::_thesis: p . y = Prod1 . y then A83: ( y in Seg k or y in {(k + 1)} ) by A6, A81, XBOOLE_0:def_3; dom pk = Seg k by FUNCT_2:52; then ( ( Prod . y = p . y & Prod . y = Prod1 . y ) or ( y = k + 1 & p . (k + 1) = Prod1 . (k + 1) ) ) by A5, A7, A54, A78, A79, A80, A83, FUNCT_1:47, TARSKI:def_1; hence p . y = Prod1 . y ; ::_thesis: verum end; dom Prod1 = Seg (k + 1) by FUNCT_2:52; hence p = Product Pr by A51, A55, A77, A81, A82, FUNCT_1:2; ::_thesis: for i being Nat st i in dom Pr holds ex trans being Element of Permutations (k + 1) st ( Pr . i = trans & trans is being_transposition ) thus for i being Nat st i in dom Pr holds ex trans being Element of Permutations (k + 1) st ( Pr . i = trans & trans is being_transposition ) ::_thesis: verum proof A84: Seg (len P) = dom P by FINSEQ_1:def_3; let m be Nat; ::_thesis: ( m in dom Pr implies ex trans being Element of Permutations (k + 1) st ( Pr . m = trans & trans is being_transposition ) ) assume A85: m in dom Pr ; ::_thesis: ex trans being Element of Permutations (k + 1) st ( Pr . m = trans & trans is being_transposition ) consider t being Element of Permutations k such that A86: P . m = t and t is being_transposition by A9, A34, A85, A84; reconsider m9 = m as Element of NAT by ORDINAL1:def_12; ex T being Element of Permutations (k + 1) st ( T = Pr . m9 & T is being_transposition & T . (k + 1) = k + 1 & t = T | (Seg k) ) by A34, A35, A85, A84, A86; hence ex trans being Element of Permutations (k + 1) st ( Pr . m = trans & trans is being_transposition ) ; ::_thesis: verum end; end; end; end; hence ( p = Product Pr & ( for i being Nat st i in dom Pr holds ex trans being Element of Permutations (k + 1) st ( Pr . i = trans & trans is being_transposition ) ) ) ; ::_thesis: verum end; now__::_thesis:_ex_P_being_FinSequence_of_(Group_of_Perm_(k_+_1))_st_ (_p_=_Product_P_&_(_for_i_being_Nat_st_i_in_dom_P_holds_ ex_trans_being_Element_of_Permutations_(k_+_1)_st_ (_P_._i_=_trans_&_trans_is_being_transposition_)_)_) percases ( p . (k + 1) = k + 1 or p . (k + 1) <> k + 1 ) ; suppose p . (k + 1) = k + 1 ; ::_thesis: ex P being FinSequence of (Group_of_Perm (k + 1)) st ( p = Product P & ( for i being Nat st i in dom P holds ex trans being Element of Permutations (k + 1) st ( P . i = trans & trans is being_transposition ) ) ) hence ex P being FinSequence of (Group_of_Perm (k + 1)) st ( p = Product P & ( for i being Nat st i in dom P holds ex trans being Element of Permutations (k + 1) st ( P . i = trans & trans is being_transposition ) ) ) by A4; ::_thesis: verum end; supposeA87: p . (k + 1) <> k + 1 ; ::_thesis: ex PT being FinSequence of the carrier of (Group_of_Perm (k + 1)) st ( Product PT = p & ( for m being Nat st m in dom PT holds ex trans being Element of Permutations (k + 1) st ( PT . m = trans & trans is being_transposition ) ) ) A88: rng p9 = Seg (k + 1) by FUNCT_2:def_3; consider tr being Element of Permutations (k + 1) such that A89: tr is being_transposition and tr . (p . (k + 1)) = k + 1 and A90: (tr * p) . (k + 1) = k + 1 by A87, Th17; reconsider tr9 = tr as Permutation of (Seg (k + 1)) by MATRIX_2:def_9; reconsider trp = tr9 * p9 as Element of Permutations (k + 1) by MATRIX_2:def_9; consider P being FinSequence of (Group_of_Perm (k + 1)) such that A91: trp = Product P and A92: for i being Nat st i in dom P holds ex trans being Element of Permutations (k + 1) st ( P . i = trans & trans is being_transposition ) by A4, A90; reconsider TRP = trp as Element of (Group_of_Perm (k + 1)) by MATRIX_2:def_10; reconsider T = tr as Element of (Group_of_Perm (k + 1)) by MATRIX_2:def_10; take PT = P ^ <*T*>; ::_thesis: ( Product PT = p & ( for m being Nat st m in dom PT holds ex trans being Element of Permutations (k + 1) st ( PT . m = trans & trans is being_transposition ) ) ) Product PT = TRP * T by A91, GROUP_4:6; hence Product PT = tr * (tr * p) by MATRIX_2:def_10 .= (tr * tr) * p by RELAT_1:36 .= (idseq (k + 1)) * p by A89, Th20 .= p by A88, RELAT_1:54 ; ::_thesis: for m being Nat st m in dom PT holds ex trans being Element of Permutations (k + 1) st ( PT . m = trans & trans is being_transposition ) thus for m being Nat st m in dom PT holds ex trans being Element of Permutations (k + 1) st ( PT . m = trans & trans is being_transposition ) ::_thesis: verum proof set L = len P; set L1 = (len P) + 1; A93: Seg ((len P) + 1) = (Seg (len P)) \/ {((len P) + 1)} by FINSEQ_1:9; len PT = (len P) + 1 by FINSEQ_2:16; then A94: dom PT = Seg ((len P) + 1) by FINSEQ_1:def_3; let m be Nat; ::_thesis: ( m in dom PT implies ex trans being Element of Permutations (k + 1) st ( PT . m = trans & trans is being_transposition ) ) assume A95: m in dom PT ; ::_thesis: ex trans being Element of Permutations (k + 1) st ( PT . m = trans & trans is being_transposition ) now__::_thesis:_ex_trans_being_Element_of_Permutations_(k_+_1)_st_ (_PT_._m_=_trans_&_trans_is_being_transposition_) percases ( m in Seg (len P) or m in {((len P) + 1)} ) by A95, A94, A93, XBOOLE_0:def_3; suppose m in Seg (len P) ; ::_thesis: ex trans being Element of Permutations (k + 1) st ( PT . m = trans & trans is being_transposition ) then A96: m in dom P by FINSEQ_1:def_3; then PT . m = P . m by FINSEQ_1:def_7; hence ex trans being Element of Permutations (k + 1) st ( PT . m = trans & trans is being_transposition ) by A92, A96; ::_thesis: verum end; suppose m in {((len P) + 1)} ; ::_thesis: ex trans being Element of Permutations (k + 1) st ( PT . m = trans & trans is being_transposition ) then m = (len P) + 1 by TARSKI:def_1; hence ex trans being Element of Permutations (k + 1) st ( PT . m = trans & trans is being_transposition ) by A89, FINSEQ_1:42; ::_thesis: verum end; end; end; hence ex trans being Element of Permutations (k + 1) st ( PT . m = trans & trans is being_transposition ) ; ::_thesis: verum end; end; end; end; hence ex P being FinSequence of (Group_of_Perm (k + 1)) st ( p = Product P & ( for i being Nat st i in dom P holds ex trans being Element of Permutations (k + 1) st ( P . i = trans & trans is being_transposition ) ) ) ; ::_thesis: verum end; A97: S1[ 0 ] proof let perm be Element of Permutations 0; ::_thesis: ex P being FinSequence of (Group_of_Perm 0) st ( perm = Product P & ( for i being Nat st i in dom P holds ex trans being Element of Permutations 0 st ( P . i = trans & trans is being_transposition ) ) ) take <*> the carrier of (Group_of_Perm 0) ; ::_thesis: ( perm = Product (<*> the carrier of (Group_of_Perm 0)) & ( for i being Nat st i in dom (<*> the carrier of (Group_of_Perm 0)) holds ex trans being Element of Permutations 0 st ( (<*> the carrier of (Group_of_Perm 0)) . i = trans & trans is being_transposition ) ) ) perm is Permutation of (Seg 0) by MATRIX_2:def_9; then perm = idseq 0 ; then perm = 1_ (Group_of_Perm 0) by MATRIX_2:24; hence ( perm = Product (<*> the carrier of (Group_of_Perm 0)) & ( for i being Nat st i in dom (<*> the carrier of (Group_of_Perm 0)) holds ex trans being Element of Permutations 0 st ( (<*> the carrier of (Group_of_Perm 0)) . i = trans & trans is being_transposition ) ) ) by GROUP_4:8; ::_thesis: verum end; for n being Element of NAT holds S1[n] from NAT_1:sch_1(A97, A2); hence ex P being FinSequence of (Group_of_Perm n) st ( perm = Product P & ( for i being Nat st i in dom P holds ex trans being Element of Permutations n st ( P . i = trans & trans is being_transposition ) ) ) by A1; ::_thesis: verum end; theorem Th22: :: MATRIX11:22 for K being Field holds ( K is Fanoian iff 1_ K <> - (1_ K) ) proof let K be Field; ::_thesis: ( K is Fanoian iff 1_ K <> - (1_ K) ) thus ( K is Fanoian implies 1_ K <> - (1_ K) ) ::_thesis: ( 1_ K <> - (1_ K) implies K is Fanoian ) proof assume A1: K is Fanoian ; ::_thesis: 1_ K <> - (1_ K) assume 1_ K = - (1_ K) ; ::_thesis: contradiction then (1_ K) + (1_ K) = 0. K by RLVECT_1:def_10; hence contradiction by A1, VECTSP_1:def_18; ::_thesis: verum end; assume A2: 1_ K <> - (1_ K) ; ::_thesis: K is Fanoian assume not K is Fanoian ; ::_thesis: contradiction then consider a being Element of K such that A3: a + a = 0. K and A4: a <> 0. K by VECTSP_1:def_18; a = a * (1_ K) by VECTSP_1:def_4; then 0. K = a * ((1_ K) + (1_ K)) by A3, VECTSP_1:def_7; then 0. K = (1_ K) + (1_ K) by A4, VECTSP_1:12; hence contradiction by A2, VECTSP_1:16; ::_thesis: verum end; theorem Th23: :: MATRIX11:23 for n being Nat for perm2 being Element of Permutations (n + 2) for K being Fanoian Field holds ( ( perm2 is even implies sgn (perm2,K) = 1_ K ) & ( sgn (perm2,K) = 1_ K implies perm2 is even ) & ( perm2 is odd implies sgn (perm2,K) = - (1_ K) ) & ( sgn (perm2,K) = - (1_ K) implies perm2 is odd ) ) proof let n be Nat; ::_thesis: for perm2 being Element of Permutations (n + 2) for K being Fanoian Field holds ( ( perm2 is even implies sgn (perm2,K) = 1_ K ) & ( sgn (perm2,K) = 1_ K implies perm2 is even ) & ( perm2 is odd implies sgn (perm2,K) = - (1_ K) ) & ( sgn (perm2,K) = - (1_ K) implies perm2 is odd ) ) let perm2 be Element of Permutations (n + 2); ::_thesis: for K being Fanoian Field holds ( ( perm2 is even implies sgn (perm2,K) = 1_ K ) & ( sgn (perm2,K) = 1_ K implies perm2 is even ) & ( perm2 is odd implies sgn (perm2,K) = - (1_ K) ) & ( sgn (perm2,K) = - (1_ K) implies perm2 is odd ) ) set n2 = n + 2; let K be Fanoian Field; ::_thesis: ( ( perm2 is even implies sgn (perm2,K) = 1_ K ) & ( sgn (perm2,K) = 1_ K implies perm2 is even ) & ( perm2 is odd implies sgn (perm2,K) = - (1_ K) ) & ( sgn (perm2,K) = - (1_ K) implies perm2 is odd ) ) A1: len (Permutations (n + 2)) = n + 2 by MATRIX_2:18; thus A2: ( perm2 is even implies sgn (perm2,K) = 1_ K ) ::_thesis: ( ( sgn (perm2,K) = 1_ K implies perm2 is even ) & ( perm2 is odd implies sgn (perm2,K) = - (1_ K) ) & ( sgn (perm2,K) = - (1_ K) implies perm2 is odd ) ) proof assume perm2 is even ; ::_thesis: sgn (perm2,K) = 1_ K then ex L being FinSequence of (Group_of_Perm (n + 2)) st ( (len L) mod 2 = 0 & perm2 = Product L & ( for i being Nat st i in dom L holds ex q2 being Element of Permutations (n + 2) st ( L . i = q2 & q2 is being_transposition ) ) ) by A1, MATRIX_2:def_12; hence sgn (perm2,K) = 1_ K by Th15; ::_thesis: verum end; thus ( sgn (perm2,K) = 1_ K implies perm2 is even ) ::_thesis: ( perm2 is odd iff sgn (perm2,K) = - (1_ K) ) proof assume A3: sgn (perm2,K) = 1_ K ; ::_thesis: perm2 is even consider P being FinSequence of (Group_of_Perm (n + 2)) such that A4: perm2 = Product P and A5: for i being Nat st i in dom P holds ex trans being Element of Permutations (n + 2) st ( P . i = trans & trans is being_transposition ) by Th21; assume perm2 is odd ; ::_thesis: contradiction then (len P) mod 2 <> 0 by A1, A4, A5, MATRIX_2:def_12; then (len P) mod 2 = 1 by NAT_D:12; then sgn (perm2,K) = - (1_ K) by A4, A5, Th15; hence contradiction by A3, Th22; ::_thesis: verum end; hence ( perm2 is odd iff sgn (perm2,K) = - (1_ K) ) by A2, Th11, Th22; ::_thesis: verum end; theorem Th24: :: MATRIX11:24 for n being Nat for K being Field for p2, q2, pq2 being Element of Permutations (n + 2) st pq2 = p2 * q2 holds sgn (pq2,K) = (sgn (p2,K)) * (sgn (q2,K)) proof let n be Nat; ::_thesis: for K being Field for p2, q2, pq2 being Element of Permutations (n + 2) st pq2 = p2 * q2 holds sgn (pq2,K) = (sgn (p2,K)) * (sgn (q2,K)) let K be Field; ::_thesis: for p2, q2, pq2 being Element of Permutations (n + 2) st pq2 = p2 * q2 holds sgn (pq2,K) = (sgn (p2,K)) * (sgn (q2,K)) set n2 = n + 2; let p, q, pq be Element of Permutations (n + 2); ::_thesis: ( pq = p * q implies sgn (pq,K) = (sgn (p,K)) * (sgn (q,K)) ) assume A1: pq = p * q ; ::_thesis: sgn (pq,K) = (sgn (p,K)) * (sgn (q,K)) consider P2 being FinSequence of (Group_of_Perm (n + 2)) such that A2: q = Product P2 and A3: for i being Nat st i in dom P2 holds ex trans being Element of Permutations (n + 2) st ( P2 . i = trans & trans is being_transposition ) by Th21; consider P1 being FinSequence of (Group_of_Perm (n + 2)) such that A4: p = Product P1 and A5: for i being Nat st i in dom P1 holds ex trans being Element of Permutations (n + 2) st ( P1 . i = trans & trans is being_transposition ) by Th21; set PP = P2 ^ P1; A6: for i being Nat st i in dom (P2 ^ P1) holds ex trans being Element of Permutations (n + 2) st ( (P2 ^ P1) . i = trans & trans is being_transposition ) proof let i be Nat; ::_thesis: ( i in dom (P2 ^ P1) implies ex trans being Element of Permutations (n + 2) st ( (P2 ^ P1) . i = trans & trans is being_transposition ) ) assume A7: i in dom (P2 ^ P1) ; ::_thesis: ex trans being Element of Permutations (n + 2) st ( (P2 ^ P1) . i = trans & trans is being_transposition ) now__::_thesis:_ex_trans_being_Element_of_Permutations_(n_+_2)_st_ (_(P2_^_P1)_._i_=_trans_&_trans_is_being_transposition_) percases ( i in dom P2 or ex k being Nat st ( k in dom P1 & i = (len P2) + k ) ) by A7, FINSEQ_1:25; supposeA8: i in dom P2 ; ::_thesis: ex trans being Element of Permutations (n + 2) st ( (P2 ^ P1) . i = trans & trans is being_transposition ) then P2 . i = (P2 ^ P1) . i by FINSEQ_1:def_7; hence ex trans being Element of Permutations (n + 2) st ( (P2 ^ P1) . i = trans & trans is being_transposition ) by A3, A8; ::_thesis: verum end; suppose ex k being Nat st ( k in dom P1 & i = (len P2) + k ) ; ::_thesis: ex trans being Element of Permutations (n + 2) st ( (P2 ^ P1) . i = trans & trans is being_transposition ) then consider k being Nat such that A9: k in dom P1 and A10: i = (len P2) + k ; P1 . k = (P2 ^ P1) . i by A9, A10, FINSEQ_1:def_7; hence ex trans being Element of Permutations (n + 2) st ( (P2 ^ P1) . i = trans & trans is being_transposition ) by A5, A9; ::_thesis: verum end; end; end; hence ex trans being Element of Permutations (n + 2) st ( (P2 ^ P1) . i = trans & trans is being_transposition ) ; ::_thesis: verum end; A11: Product (P2 ^ P1) = (Product P2) * (Product P1) by GROUP_4:5 .= pq by A1, A4, A2, MATRIX_2:def_10 ; now__::_thesis:_sgn_(pq,K)_=_(sgn_(p,K))_*_(sgn_(q,K)) percases ( ( (len P1) mod 2 = 0 & (len P2) mod 2 = 0 ) or ( (len P1) mod 2 = 1 & (len P2) mod 2 = 0 ) or ( (len P1) mod 2 = 0 & (len P2) mod 2 = 1 ) or ( (len P1) mod 2 = 1 & (len P2) mod 2 = 1 ) ) by NAT_D:12; supposeA12: ( (len P1) mod 2 = 0 & (len P2) mod 2 = 0 ) ; ::_thesis: sgn (pq,K) = (sgn (p,K)) * (sgn (q,K)) (len (P2 ^ P1)) mod 2 = ((len P2) + (len P1)) mod 2 by FINSEQ_1:22 .= ((0 + (len P1)) + 0) mod 2 by A12, NAT_D:22 .= 0 by A12 ; then A13: sgn (pq,K) = 1_ K by A11, A6, Th15; A14: sgn (q,K) = 1_ K by A2, A3, A12, Th15; sgn (p,K) = 1_ K by A4, A5, A12, Th15; hence sgn (pq,K) = (sgn (p,K)) * (sgn (q,K)) by A13, A14, VECTSP_1:def_4; ::_thesis: verum end; supposeA15: ( (len P1) mod 2 = 1 & (len P2) mod 2 = 0 ) ; ::_thesis: sgn (pq,K) = (sgn (p,K)) * (sgn (q,K)) (len (P2 ^ P1)) mod 2 = ((len P2) + (len P1)) mod 2 by FINSEQ_1:22 .= ((0 + (len P1)) + 0) mod 2 by A15, NAT_D:22 .= 1 by A15 ; then A16: sgn (pq,K) = - (1_ K) by A11, A6, Th15; A17: sgn (q,K) = 1_ K by A2, A3, A15, Th15; sgn (p,K) = - (1_ K) by A4, A5, A15, Th15; hence sgn (pq,K) = (sgn (p,K)) * (sgn (q,K)) by A16, A17, VECTSP_1:def_4; ::_thesis: verum end; supposeA18: ( (len P1) mod 2 = 0 & (len P2) mod 2 = 1 ) ; ::_thesis: sgn (pq,K) = (sgn (p,K)) * (sgn (q,K)) (len (P2 ^ P1)) mod 2 = ((len P2) + (len P1)) mod 2 by FINSEQ_1:22 .= (1 + (len P1)) mod 2 by A18, NAT_D:22 .= (1 + 0) mod 2 by A18, NAT_D:22 .= 1 by NAT_D:14 ; then A19: sgn (pq,K) = - (1_ K) by A11, A6, Th15; A20: sgn (q,K) = - (1_ K) by A2, A3, A18, Th15; sgn (p,K) = 1_ K by A4, A5, A18, Th15; hence sgn (pq,K) = (sgn (p,K)) * (sgn (q,K)) by A19, A20, VECTSP_1:def_4; ::_thesis: verum end; supposeA21: ( (len P1) mod 2 = 1 & (len P2) mod 2 = 1 ) ; ::_thesis: sgn (pq,K) = (sgn (p,K)) * (sgn (q,K)) (len (P2 ^ P1)) mod 2 = ((len P2) + (len P1)) mod 2 by FINSEQ_1:22 .= (1 + (len P1)) mod 2 by A21, NAT_D:22 .= (1 + 1) mod 2 by A21, NAT_D:22 .= 0 by NAT_D:25 ; then A22: sgn (pq,K) = 1_ K by A11, A6, Th15; A23: (1_ K) * (1_ K) = 1_ K by VECTSP_1:def_4; A24: sgn (q,K) = - (1_ K) by A2, A3, A21, Th15; sgn (p,K) = - (1_ K) by A4, A5, A21, Th15; hence sgn (pq,K) = (sgn (p,K)) * (sgn (q,K)) by A22, A24, A23, VECTSP_1:10; ::_thesis: verum end; end; end; hence sgn (pq,K) = (sgn (p,K)) * (sgn (q,K)) ; ::_thesis: verum end; Lm3: for n being Nat for p being Element of Permutations n st n < 2 holds ( p is even & p = idseq n ) proof let n be Nat; ::_thesis: for p being Element of Permutations n st n < 2 holds ( p is even & p = idseq n ) let p be Element of Permutations n; ::_thesis: ( n < 2 implies ( p is even & p = idseq n ) ) reconsider P = p as Permutation of (Seg n) by MATRIX_2:def_9; assume A1: n < 2 ; ::_thesis: ( p is even & p = idseq n ) now__::_thesis:_(_p_is_even_&_p_=_idseq_n_) percases ( n = 0 or n = 1 ) by A1, NAT_1:23; supposeA2: n = 0 ; ::_thesis: ( p is even & p = idseq n ) then A3: Seg n = {} ; A4: len (Permutations n) = n by MATRIX_2:18; P = {} by A2; hence ( p is even & p = idseq n ) by A4, A3, MATRIX_2:25, RELAT_1:55; ::_thesis: verum end; supposeA5: n = 1 ; ::_thesis: ( p is even & p = idseq n ) A6: len (Permutations n) = n by MATRIX_2:18; P = id (Seg n) by A5, MATRIX_2:19, TARSKI:def_1; hence ( p is even & p = idseq n ) by A6, MATRIX_2:25; ::_thesis: verum end; end; end; hence ( p is even & p = idseq n ) ; ::_thesis: verum end; theorem Th25: :: MATRIX11:25 for n being Nat for p, q being Element of Permutations n holds ( ( ( p is even & q is even ) or ( p is odd & q is odd ) ) iff p * q is even ) proof let n be Nat; ::_thesis: for p, q being Element of Permutations n holds ( ( ( p is even & q is even ) or ( p is odd & q is odd ) ) iff p * q is even ) let p, q be Element of Permutations n; ::_thesis: ( ( ( p is even & q is even ) or ( p is odd & q is odd ) ) iff p * q is even ) reconsider pq = p * q as Element of Permutations n by MATRIX_9:39; now__::_thesis:_(_(_(_p_is_even_&_q_is_even_)_or_(_p_is_odd_&_q_is_odd_)_)_iff_p_*_q_is_even_) percases ( n < 2 or n >= 2 ) ; supposeA1: n < 2 ; ::_thesis: ( ( ( p is even & q is even ) or ( p is odd & q is odd ) ) iff p * q is even ) then pq is even by Lm3; hence ( ( ( p is even & q is even ) or ( p is odd & q is odd ) ) iff p * q is even ) by A1, Lm3; ::_thesis: verum end; suppose n >= 2 ; ::_thesis: ( ( ( ( p is even & q is even ) or ( p is odd & q is odd ) ) implies p * q is even ) & ( not p * q is even or ( p is even & q is even ) or ( p is odd & q is odd ) ) ) then reconsider n2 = n - 2 as Nat by NAT_1:21; set K = the Fanoian Field; reconsider p9 = p, q9 = q, pq = pq as Element of Permutations (n2 + 2) ; thus ( ( ( p is even & q is even ) or ( p is odd & q is odd ) ) implies p * q is even ) ::_thesis: ( not p * q is even or ( p is even & q is even ) or ( p is odd & q is odd ) ) proof assume ( ( p is even & q is even ) or ( p is odd & q is odd ) ) ; ::_thesis: p * q is even then ( ( sgn (p9, the Fanoian Field) = 1_ the Fanoian Field & sgn (q9, the Fanoian Field) = 1_ the Fanoian Field ) or ( sgn (p9, the Fanoian Field) = - (1_ the Fanoian Field) & sgn (q9, the Fanoian Field) = - (1_ the Fanoian Field) ) ) by Th23; then A2: (sgn (p9, the Fanoian Field)) * (sgn (q9, the Fanoian Field)) = (1_ the Fanoian Field) * (1_ the Fanoian Field) by VECTSP_1:10; (1_ the Fanoian Field) * (1_ the Fanoian Field) = 1_ the Fanoian Field by VECTSP_1:def_4; then sgn (pq, the Fanoian Field) = 1_ the Fanoian Field by A2, Th24; hence p * q is even by Th23; ::_thesis: verum end; thus ( not p * q is even or ( p is even & q is even ) or ( p is odd & q is odd ) ) ::_thesis: verum proof assume p * q is even ; ::_thesis: ( ( p is even & q is even ) or ( p is odd & q is odd ) ) then sgn (pq, the Fanoian Field) = 1_ the Fanoian Field by Th23; then A3: (sgn (p9, the Fanoian Field)) * (sgn (q9, the Fanoian Field)) = 1_ the Fanoian Field by Th24; assume A4: ( not ( p is even & q is even ) & not ( p is odd & q is odd ) ) ; ::_thesis: contradiction now__::_thesis:_contradiction percases ( ( p is even & q is odd ) or ( p is odd & q is even ) ) by A4; supposeA5: ( p is even & q is odd ) ; ::_thesis: contradiction then A6: sgn (q9, the Fanoian Field) = - (1_ the Fanoian Field) by Th23; sgn (p9, the Fanoian Field) = 1_ the Fanoian Field by A5, Th23; then (sgn (p9, the Fanoian Field)) * (sgn (q9, the Fanoian Field)) = - (1_ the Fanoian Field) by A6, VECTSP_1:def_4; hence contradiction by A3, Th22; ::_thesis: verum end; supposeA7: ( p is odd & q is even ) ; ::_thesis: contradiction then A8: sgn (q9, the Fanoian Field) = 1_ the Fanoian Field by Th23; sgn (p9, the Fanoian Field) = - (1_ the Fanoian Field) by A7, Th23; then (sgn (p9, the Fanoian Field)) * (sgn (q9, the Fanoian Field)) = - (1_ the Fanoian Field) by A8, VECTSP_1:def_4; hence contradiction by A3, Th22; ::_thesis: verum end; end; end; hence contradiction ; ::_thesis: verum end; end; end; end; hence ( ( ( p is even & q is even ) or ( p is odd & q is odd ) ) iff p * q is even ) ; ::_thesis: verum end; theorem Th26: :: MATRIX11:26 for n being Nat for K being Field for a being Element of K for perm2 being Element of Permutations (n + 2) holds - (a,perm2) = (sgn (perm2,K)) * a proof let n be Nat; ::_thesis: for K being Field for a being Element of K for perm2 being Element of Permutations (n + 2) holds - (a,perm2) = (sgn (perm2,K)) * a let K be Field; ::_thesis: for a being Element of K for perm2 being Element of Permutations (n + 2) holds - (a,perm2) = (sgn (perm2,K)) * a let a be Element of K; ::_thesis: for perm2 being Element of Permutations (n + 2) holds - (a,perm2) = (sgn (perm2,K)) * a let perm2 be Element of Permutations (n + 2); ::_thesis: - (a,perm2) = (sgn (perm2,K)) * a percases ( ( perm2 is even & K is Fanoian ) or ( perm2 is odd & K is Fanoian ) or ( perm2 is even & not K is Fanoian ) or ( perm2 is odd & not K is Fanoian ) ) ; supposeA1: ( perm2 is even & K is Fanoian ) ; ::_thesis: - (a,perm2) = (sgn (perm2,K)) * a then A2: - (a,perm2) = a by MATRIX_2:def_13; sgn (perm2,K) = 1_ K by A1, Th23; hence - (a,perm2) = (sgn (perm2,K)) * a by A2, VECTSP_1:def_4; ::_thesis: verum end; supposeA3: ( perm2 is odd & K is Fanoian ) ; ::_thesis: - (a,perm2) = (sgn (perm2,K)) * a then A4: - (a,perm2) = - a by MATRIX_2:def_13; A5: (- (1_ K)) * a = - ((1_ K) * a) by VECTSP_1:8; sgn (perm2,K) = - (1_ K) by A3, Th23; hence - (a,perm2) = (sgn (perm2,K)) * a by A4, A5, VECTSP_1:def_4; ::_thesis: verum end; supposeA6: ( perm2 is even & not K is Fanoian ) ; ::_thesis: - (a,perm2) = (sgn (perm2,K)) * a then A7: - (a,perm2) = a by MATRIX_2:def_13; A8: ( sgn (perm2,K) = 1_ K or sgn (perm2,K) = - (1_ K) ) by Th11; 1_ K = - (1_ K) by A6, Th22; hence - (a,perm2) = (sgn (perm2,K)) * a by A7, A8, VECTSP_1:def_4; ::_thesis: verum end; supposeA9: ( perm2 is odd & not K is Fanoian ) ; ::_thesis: - (a,perm2) = (sgn (perm2,K)) * a then A10: - (a,perm2) = - a by MATRIX_2:def_13; A11: (- (1_ K)) * a = - ((1_ K) * a) by VECTSP_1:8; A12: ( sgn (perm2,K) = 1_ K or sgn (perm2,K) = - (1_ K) ) by Th11; 1_ K = - (1_ K) by A9, Th22; hence - (a,perm2) = (sgn (perm2,K)) * a by A10, A11, A12, VECTSP_1:def_4; ::_thesis: verum end; end; end; theorem Th27: :: MATRIX11:27 for n being Nat for tr being Element of Permutations (n + 2) st tr is being_transposition holds tr is odd proof let n be Nat; ::_thesis: for tr being Element of Permutations (n + 2) st tr is being_transposition holds tr is odd set K = the Fanoian Field; let tr be Element of Permutations (n + 2); ::_thesis: ( tr is being_transposition implies tr is odd ) assume tr is being_transposition ; ::_thesis: tr is odd then sgn (tr, the Fanoian Field) = - (1_ the Fanoian Field) by Th14; hence tr is odd by Th23; ::_thesis: verum end; registration let n be Nat; cluster Relation-like Seg (n + 2) -defined Seg (n + 2) -valued Function-like one-to-one non empty total quasi_total onto bijective finite odd for Element of bool [:(Seg (n + 2)),(Seg (n + 2)):]; existence ex b1 being Permutation of (Seg (n + 2)) st b1 is odd proof set n2 = n + 2; A1: len (Permutations (n + 2)) = n + 2 by MATRIX_2:18; n + 2 >= 2 + 0 by XREAL_1:6; then {1,2} in 2Set (Seg (n + 2)) by Th3; then consider i, j being Nat such that A2: i in Seg (n + 2) and A3: j in Seg (n + 2) and A4: i < j and {1,2} = {i,j} by Th1; consider tr being Element of Permutations (n + 2) such that A5: tr is being_transposition and tr . i = j by A2, A3, A4, Th16; tr is odd by A5, Th27; hence ex b1 being Permutation of (Seg (n + 2)) st b1 is odd by A1; ::_thesis: verum end; end; begin definition let l, n, m be Nat; let D be non empty set ; let M be Matrix of n,m,D; let pD be FinSequence of D; func ReplaceLine (M,l,pD) -> Matrix of n,m,D means :Def3: :: MATRIX11:def 3 ( len it = len M & width it = width M & ( for i, j being Nat st [i,j] in Indices M holds ( ( i <> l implies it * (i,j) = M * (i,j) ) & ( i = l implies it * (l,j) = pD . j ) ) ) ) if len pD = width M otherwise it = M; consistency for b1 being Matrix of n,m,D holds verum ; existence ( ( len pD = width M implies ex b1 being Matrix of n,m,D st ( len b1 = len M & width b1 = width M & ( for i, j being Nat st [i,j] in Indices M holds ( ( i <> l implies b1 * (i,j) = M * (i,j) ) & ( i = l implies b1 * (l,j) = pD . j ) ) ) ) ) & ( not len pD = width M implies ex b1 being Matrix of n,m,D st b1 = M ) ) proof thus ( len pD = width M implies ex M1 being Matrix of n,m,D st ( len M1 = len M & width M1 = width M & ( for i, j being Nat st [i,j] in Indices M holds ( ( i <> l implies M1 * (i,j) = M * (i,j) ) & ( i = l implies M1 * (l,j) = pD . j ) ) ) ) ) ::_thesis: ( not len pD = width M implies ex b1 being Matrix of n,m,D st b1 = M ) proof reconsider M9 = M as Matrix of len M, width M,D by MATRIX_2:7; reconsider n1 = n, m1 = m as Element of NAT by ORDINAL1:def_12; defpred S1[ set , set , set ] means for i, j being Nat st i = $1 & j = $2 holds ( ( i <> l implies $3 = M * (i,j) ) & ( i = l implies $3 = pD . j ) ); assume A1: len pD = width M ; ::_thesis: ex M1 being Matrix of n,m,D st ( len M1 = len M & width M1 = width M & ( for i, j being Nat st [i,j] in Indices M holds ( ( i <> l implies M1 * (i,j) = M * (i,j) ) & ( i = l implies M1 * (l,j) = pD . j ) ) ) ) A2: for i, j being Nat st [i,j] in [:(Seg n1),(Seg m1):] holds ex x being Element of D st S1[i,j,x] proof let i, j be Nat; ::_thesis: ( [i,j] in [:(Seg n1),(Seg m1):] implies ex x being Element of D st S1[i,j,x] ) assume A3: [i,j] in [:(Seg n1),(Seg m1):] ; ::_thesis: ex x being Element of D st S1[i,j,x] now__::_thesis:_(_(_i_=_l_&_ex_x_being_Element_of_D_st_S1[i,j,x]_)_or_(_i_<>_l_&_ex_x_being_Element_of_D_st_S1[i,j,x]_)_) percases ( i = l or i <> l ) ; caseA4: i = l ; ::_thesis: ex x being Element of D st S1[i,j,x] A5: rng pD c= D by FINSEQ_1:def_4; n1 <> 0 by A3, ZFMISC_1:87; then len pD = m by A1, MATRIX_1:23; then j in Seg (len pD) by A3, ZFMISC_1:87; then j in dom pD by FINSEQ_1:def_3; then A6: pD . j in rng pD by FUNCT_1:def_3; S1[i,j,pD . j] by A4; hence ex x being Element of D st S1[i,j,x] by A6, A5; ::_thesis: verum end; case i <> l ; ::_thesis: ex x being Element of D st S1[i,j,x] then S1[i,j,M * (i,j)] ; hence ex x being Element of D st S1[i,j,x] ; ::_thesis: verum end; end; end; hence ex x being Element of D st S1[i,j,x] ; ::_thesis: verum end; consider M1 being Matrix of n1,m1,D such that A7: for i, j being Nat st [i,j] in Indices M1 holds S1[i,j,M1 * (i,j)] from MATRIX_1:sch_2(A2); reconsider M1 = M1 as Matrix of n,m,D ; take M1 ; ::_thesis: ( len M1 = len M & width M1 = width M & ( for i, j being Nat st [i,j] in Indices M holds ( ( i <> l implies M1 * (i,j) = M * (i,j) ) & ( i = l implies M1 * (l,j) = pD . j ) ) ) ) A8: now__::_thesis:_(_len_M_=_len_M1_&_width_M1_=_width_M_) percases ( n = 0 or n > 0 ) ; supposeA9: n = 0 ; ::_thesis: ( len M = len M1 & width M1 = width M ) then len M1 = 0 by MATRIX_1:def_2; then A10: width M1 = 0 by MATRIX_1:def_3; len M = 0 by A9, MATRIX_1:def_2; hence ( len M = len M1 & width M1 = width M ) by A9, A10, MATRIX_1:def_2, MATRIX_1:def_3; ::_thesis: verum end; supposeA11: n > 0 ; ::_thesis: ( len M = len M1 & width M = width M1 ) then A12: width M = m by MATRIX_1:23; len M = n by A11, MATRIX_1:23; hence ( len M = len M1 & width M = width M1 ) by A11, A12, MATRIX_1:23; ::_thesis: verum end; end; end; Indices M9 = Indices M1 by MATRIX_1:26; hence ( len M1 = len M & width M1 = width M & ( for i, j being Nat st [i,j] in Indices M holds ( ( i <> l implies M1 * (i,j) = M * (i,j) ) & ( i = l implies M1 * (l,j) = pD . j ) ) ) ) by A7, A8; ::_thesis: verum end; thus ( not len pD = width M implies ex b1 being Matrix of n,m,D st b1 = M ) ; ::_thesis: verum end; uniqueness for b1, b2 being Matrix of n,m,D holds ( ( len pD = width M & len b1 = len M & width b1 = width M & ( for i, j being Nat st [i,j] in Indices M holds ( ( i <> l implies b1 * (i,j) = M * (i,j) ) & ( i = l implies b1 * (l,j) = pD . j ) ) ) & len b2 = len M & width b2 = width M & ( for i, j being Nat st [i,j] in Indices M holds ( ( i <> l implies b2 * (i,j) = M * (i,j) ) & ( i = l implies b2 * (l,j) = pD . j ) ) ) implies b1 = b2 ) & ( not len pD = width M & b1 = M & b2 = M implies b1 = b2 ) ) proof let M1, M2 be Matrix of n,m,D; ::_thesis: ( ( len pD = width M & len M1 = len M & width M1 = width M & ( for i, j being Nat st [i,j] in Indices M holds ( ( i <> l implies M1 * (i,j) = M * (i,j) ) & ( i = l implies M1 * (l,j) = pD . j ) ) ) & len M2 = len M & width M2 = width M & ( for i, j being Nat st [i,j] in Indices M holds ( ( i <> l implies M2 * (i,j) = M * (i,j) ) & ( i = l implies M2 * (l,j) = pD . j ) ) ) implies M1 = M2 ) & ( not len pD = width M & M1 = M & M2 = M implies M1 = M2 ) ) thus ( len pD = width M & len M1 = len M & width M1 = width M & ( for i, j being Nat st [i,j] in Indices M holds ( ( i <> l implies M1 * (i,j) = M * (i,j) ) & ( i = l implies M1 * (l,j) = pD . j ) ) ) & len M2 = len M & width M2 = width M & ( for i, j being Nat st [i,j] in Indices M holds ( ( i <> l implies M2 * (i,j) = M * (i,j) ) & ( i = l implies M2 * (l,j) = pD . j ) ) ) implies M1 = M2 ) ::_thesis: ( not len pD = width M & M1 = M & M2 = M implies M1 = M2 ) proof assume len pD = width M ; ::_thesis: ( not len M1 = len M or not width M1 = width M or ex i, j being Nat st ( [i,j] in Indices M & not ( ( i <> l implies M1 * (i,j) = M * (i,j) ) & ( i = l implies M1 * (l,j) = pD . j ) ) ) or not len M2 = len M or not width M2 = width M or ex i, j being Nat st ( [i,j] in Indices M & not ( ( i <> l implies M2 * (i,j) = M * (i,j) ) & ( i = l implies M2 * (l,j) = pD . j ) ) ) or M1 = M2 ) assume that A13: len M1 = len M and A14: width M1 = width M and A15: for i, j being Nat st [i,j] in Indices M holds ( ( i <> l implies M1 * (i,j) = M * (i,j) ) & ( i = l implies M1 * (l,j) = pD . j ) ) ; ::_thesis: ( not len M2 = len M or not width M2 = width M or ex i, j being Nat st ( [i,j] in Indices M & not ( ( i <> l implies M2 * (i,j) = M * (i,j) ) & ( i = l implies M2 * (l,j) = pD . j ) ) ) or M1 = M2 ) assume that len M2 = len M and width M2 = width M and A16: for i, j being Nat st [i,j] in Indices M holds ( ( i <> l implies M2 * (i,j) = M * (i,j) ) & ( i = l implies M2 * (l,j) = pD . j ) ) ; ::_thesis: M1 = M2 for i, j being Nat st [i,j] in Indices M1 holds M1 * (i,j) = M2 * (i,j) proof let i, j be Nat; ::_thesis: ( [i,j] in Indices M1 implies M1 * (i,j) = M2 * (i,j) ) assume [i,j] in Indices M1 ; ::_thesis: M1 * (i,j) = M2 * (i,j) then A17: [i,j] in Indices M by A13, A14, MATRIX_4:55; then A18: ( i = l implies M1 * (l,j) = pD . j ) by A15; A19: ( i <> l implies M2 * (i,j) = M * (i,j) ) by A16, A17; ( i <> l implies M1 * (i,j) = M * (i,j) ) by A15, A17; hence M1 * (i,j) = M2 * (i,j) by A16, A17, A18, A19; ::_thesis: verum end; hence M1 = M2 by MATRIX_1:27; ::_thesis: verum end; thus ( not len pD = width M & M1 = M & M2 = M implies M1 = M2 ) ; ::_thesis: verum end; end; :: deftheorem Def3 defines ReplaceLine MATRIX11:def_3_:_ for l, n, m being Nat for D being non empty set for M being Matrix of n,m,D for pD being FinSequence of D for b7 being Matrix of n,m,D holds ( ( len pD = width M implies ( b7 = ReplaceLine (M,l,pD) iff ( len b7 = len M & width b7 = width M & ( for i, j being Nat st [i,j] in Indices M holds ( ( i <> l implies b7 * (i,j) = M * (i,j) ) & ( i = l implies b7 * (l,j) = pD . j ) ) ) ) ) ) & ( not len pD = width M implies ( b7 = ReplaceLine (M,l,pD) iff b7 = M ) ) ); notation let l, n, m be Nat; let D be non empty set ; let M be Matrix of n,m,D; let pD be FinSequence of D; synonym RLine (M,l,pD) for ReplaceLine (M,l,pD); end; Lm4: for m, n being Nat for D being non empty set for l being Nat for M being Matrix of n,m,D for pD being FinSequence of D holds ( Indices M = Indices (RLine (M,l,pD)) & len M = len (RLine (M,l,pD)) & width M = width (RLine (M,l,pD)) ) proof let m, n be Nat; ::_thesis: for D being non empty set for l being Nat for M being Matrix of n,m,D for pD being FinSequence of D holds ( Indices M = Indices (RLine (M,l,pD)) & len M = len (RLine (M,l,pD)) & width M = width (RLine (M,l,pD)) ) let D be non empty set ; ::_thesis: for l being Nat for M being Matrix of n,m,D for pD being FinSequence of D holds ( Indices M = Indices (RLine (M,l,pD)) & len M = len (RLine (M,l,pD)) & width M = width (RLine (M,l,pD)) ) let l be Nat; ::_thesis: for M being Matrix of n,m,D for pD being FinSequence of D holds ( Indices M = Indices (RLine (M,l,pD)) & len M = len (RLine (M,l,pD)) & width M = width (RLine (M,l,pD)) ) let M be Matrix of n,m,D; ::_thesis: for pD being FinSequence of D holds ( Indices M = Indices (RLine (M,l,pD)) & len M = len (RLine (M,l,pD)) & width M = width (RLine (M,l,pD)) ) let pD be FinSequence of D; ::_thesis: ( Indices M = Indices (RLine (M,l,pD)) & len M = len (RLine (M,l,pD)) & width M = width (RLine (M,l,pD)) ) now__::_thesis:_(_(_len_pD_=_width_M_&_Indices_M_=_Indices_(RLine_(M,l,pD))_&_len_M_=_len_(RLine_(M,l,pD))_&_width_M_=_width_(RLine_(M,l,pD))_)_or_(_len_pD_<>_width_M_&_Indices_M_=_Indices_(RLine_(M,l,pD))_&_len_M_=_len_(RLine_(M,l,pD))_&_width_M_=_width_(RLine_(M,l,pD))_)_) percases ( len pD = width M or len pD <> width M ) ; caseA1: len pD = width M ; ::_thesis: ( Indices M = Indices (RLine (M,l,pD)) & len M = len (RLine (M,l,pD)) & width M = width (RLine (M,l,pD)) ) then A2: width M = width (RLine (M,l,pD)) by Def3; len M = len (ReplaceLine (M,l,pD)) by A1, Def3; hence ( Indices M = Indices (RLine (M,l,pD)) & len M = len (RLine (M,l,pD)) & width M = width (RLine (M,l,pD)) ) by A2, MATRIX_4:55; ::_thesis: verum end; case len pD <> width M ; ::_thesis: ( Indices M = Indices (RLine (M,l,pD)) & len M = len (RLine (M,l,pD)) & width M = width (RLine (M,l,pD)) ) hence ( Indices M = Indices (RLine (M,l,pD)) & len M = len (RLine (M,l,pD)) & width M = width (RLine (M,l,pD)) ) by Def3; ::_thesis: verum end; end; end; hence ( Indices M = Indices (RLine (M,l,pD)) & len M = len (RLine (M,l,pD)) & width M = width (RLine (M,l,pD)) ) ; ::_thesis: verum end; theorem Th28: :: MATRIX11:28 for m, n being Nat for D being non empty set for l being Nat for M being Matrix of n,m,D for pD being FinSequence of D for i being Nat st i in Seg n holds ( ( i = l & len pD = width M implies Line ((RLine (M,l,pD)),i) = pD ) & ( i <> l implies Line ((RLine (M,l,pD)),i) = Line (M,i) ) ) proof let m, n be Nat; ::_thesis: for D being non empty set for l being Nat for M being Matrix of n,m,D for pD being FinSequence of D for i being Nat st i in Seg n holds ( ( i = l & len pD = width M implies Line ((RLine (M,l,pD)),i) = pD ) & ( i <> l implies Line ((RLine (M,l,pD)),i) = Line (M,i) ) ) let D be non empty set ; ::_thesis: for l being Nat for M being Matrix of n,m,D for pD being FinSequence of D for i being Nat st i in Seg n holds ( ( i = l & len pD = width M implies Line ((RLine (M,l,pD)),i) = pD ) & ( i <> l implies Line ((RLine (M,l,pD)),i) = Line (M,i) ) ) let l be Nat; ::_thesis: for M being Matrix of n,m,D for pD being FinSequence of D for i being Nat st i in Seg n holds ( ( i = l & len pD = width M implies Line ((RLine (M,l,pD)),i) = pD ) & ( i <> l implies Line ((RLine (M,l,pD)),i) = Line (M,i) ) ) let M be Matrix of n,m,D; ::_thesis: for pD being FinSequence of D for i being Nat st i in Seg n holds ( ( i = l & len pD = width M implies Line ((RLine (M,l,pD)),i) = pD ) & ( i <> l implies Line ((RLine (M,l,pD)),i) = Line (M,i) ) ) let pD be FinSequence of D; ::_thesis: for i being Nat st i in Seg n holds ( ( i = l & len pD = width M implies Line ((RLine (M,l,pD)),i) = pD ) & ( i <> l implies Line ((RLine (M,l,pD)),i) = Line (M,i) ) ) let i be Nat; ::_thesis: ( i in Seg n implies ( ( i = l & len pD = width M implies Line ((RLine (M,l,pD)),i) = pD ) & ( i <> l implies Line ((RLine (M,l,pD)),i) = Line (M,i) ) ) ) assume A1: i in Seg n ; ::_thesis: ( ( i = l & len pD = width M implies Line ((RLine (M,l,pD)),i) = pD ) & ( i <> l implies Line ((RLine (M,l,pD)),i) = Line (M,i) ) ) set R = RLine (M,l,pD); set LR = Line ((RLine (M,l,pD)),i); thus ( i = l & len pD = width M implies Line ((RLine (M,l,pD)),i) = pD ) ::_thesis: ( i <> l implies Line ((RLine (M,l,pD)),i) = Line (M,i) ) proof assume that A2: i = l and A3: len pD = width M ; ::_thesis: Line ((RLine (M,l,pD)),i) = pD A4: width (RLine (M,l,pD)) = len pD by A3, Def3; A5: now__::_thesis:_for_j_being_Nat_st_1_<=_j_&_j_<=_len_pD_holds_ (Line_((RLine_(M,l,pD)),i))_._j_=_pD_._j let j be Nat; ::_thesis: ( 1 <= j & j <= len pD implies (Line ((RLine (M,l,pD)),i)) . j = pD . j ) assume that A6: 1 <= j and A7: j <= len pD ; ::_thesis: (Line ((RLine (M,l,pD)),i)) . j = pD . j j in NAT by ORDINAL1:def_12; then A8: j in Seg (width (RLine (M,l,pD))) by A4, A6, A7; n = len (RLine (M,l,pD)) by MATRIX_1:def_2; then i in dom (RLine (M,l,pD)) by A1, FINSEQ_1:def_3; then A9: [i,j] in Indices (RLine (M,l,pD)) by A8, ZFMISC_1:87; A10: Indices (RLine (M,l,pD)) = Indices M by Lm4; (Line ((RLine (M,l,pD)),i)) . j = (RLine (M,l,pD)) * (i,j) by A8, MATRIX_1:def_7; hence (Line ((RLine (M,l,pD)),i)) . j = pD . j by A2, A3, A9, A10, Def3; ::_thesis: verum end; len (Line ((RLine (M,l,pD)),i)) = len pD by A4, MATRIX_1:def_7; hence Line ((RLine (M,l,pD)),i) = pD by A5, FINSEQ_1:14; ::_thesis: verum end; set LM = Line (M,i); A11: width M = len (Line (M,i)) by MATRIX_1:def_7; A12: width M = width (RLine (M,l,pD)) by Lm4; assume A13: i <> l ; ::_thesis: Line ((RLine (M,l,pD)),i) = Line (M,i) A14: now__::_thesis:_for_j_being_Nat_st_1_<=_j_&_j_<=_len_(Line_(M,i))_holds_ (Line_(M,i))_._j_=_(Line_((RLine_(M,l,pD)),i))_._j let j be Nat; ::_thesis: ( 1 <= j & j <= len (Line (M,i)) implies (Line (M,i)) . j = (Line ((RLine (M,l,pD)),i)) . j ) assume that A15: 1 <= j and A16: j <= len (Line (M,i)) ; ::_thesis: (Line (M,i)) . j = (Line ((RLine (M,l,pD)),i)) . j j in NAT by ORDINAL1:def_12; then A17: j in Seg (len (Line (M,i))) by A15, A16; then A18: (Line (M,i)) . j = M * (i,j) by A11, MATRIX_1:def_7; i in Seg (len M) by A1, MATRIX_1:def_2; then i in dom M by FINSEQ_1:def_3; then A19: [i,j] in Indices M by A11, A17, ZFMISC_1:87; A20: (Line ((RLine (M,l,pD)),i)) . j = (RLine (M,l,pD)) * (i,j) by A12, A11, A17, MATRIX_1:def_7; now__::_thesis:_(_(_len_pD_=_width_M_&_(Line_(M,i))_._j_=_(Line_((RLine_(M,l,pD)),i))_._j_)_or_(_len_pD_<>_width_M_&_(Line_(M,i))_._j_=_(Line_((RLine_(M,l,pD)),i))_._j_)_) percases ( len pD = width M or len pD <> width M ) ; case len pD = width M ; ::_thesis: (Line (M,i)) . j = (Line ((RLine (M,l,pD)),i)) . j hence (Line (M,i)) . j = (Line ((RLine (M,l,pD)),i)) . j by A13, A18, A20, A19, Def3; ::_thesis: verum end; case len pD <> width M ; ::_thesis: (Line (M,i)) . j = (Line ((RLine (M,l,pD)),i)) . j hence (Line (M,i)) . j = (Line ((RLine (M,l,pD)),i)) . j by Def3; ::_thesis: verum end; end; end; hence (Line (M,i)) . j = (Line ((RLine (M,l,pD)),i)) . j ; ::_thesis: verum end; len (Line ((RLine (M,l,pD)),i)) = width (RLine (M,l,pD)) by MATRIX_1:def_7; hence Line ((RLine (M,l,pD)),i) = Line (M,i) by A12, A11, A14, FINSEQ_1:14; ::_thesis: verum end; theorem :: MATRIX11:29 for m, n, l being Nat for D being non empty set for M being Matrix of n,m,D for pD being FinSequence of D st len pD = width M holds for p9 being Element of D * st pD = p9 holds RLine (M,l,pD) = Replace (M,l,p9) proof let m, n, l be Nat; ::_thesis: for D being non empty set for M being Matrix of n,m,D for pD being FinSequence of D st len pD = width M holds for p9 being Element of D * st pD = p9 holds RLine (M,l,pD) = Replace (M,l,p9) let D be non empty set ; ::_thesis: for M being Matrix of n,m,D for pD being FinSequence of D st len pD = width M holds for p9 being Element of D * st pD = p9 holds RLine (M,l,pD) = Replace (M,l,p9) let M be Matrix of n,m,D; ::_thesis: for pD being FinSequence of D st len pD = width M holds for p9 being Element of D * st pD = p9 holds RLine (M,l,pD) = Replace (M,l,p9) let pD be FinSequence of D; ::_thesis: ( len pD = width M implies for p9 being Element of D * st pD = p9 holds RLine (M,l,pD) = Replace (M,l,p9) ) assume A1: len pD = width M ; ::_thesis: for p9 being Element of D * st pD = p9 holds RLine (M,l,pD) = Replace (M,l,p9) set RL = RLine (M,l,pD); let p9 be Element of D * ; ::_thesis: ( pD = p9 implies RLine (M,l,pD) = Replace (M,l,p9) ) assume A2: pD = p9 ; ::_thesis: RLine (M,l,pD) = Replace (M,l,p9) set R = Replace (M,l,p9); A3: len (Replace (M,l,p9)) = len M by FINSEQ_7:5; A4: now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_len_(Replace_(M,l,p9))_holds_ (Replace_(M,l,p9))_._i_=_(RLine_(M,l,pD))_._i let i be Nat; ::_thesis: ( 1 <= i & i <= len (Replace (M,l,p9)) implies (Replace (M,l,p9)) . i = (RLine (M,l,pD)) . i ) assume that A5: 1 <= i and A6: i <= len (Replace (M,l,p9)) ; ::_thesis: (Replace (M,l,p9)) . i = (RLine (M,l,pD)) . i i in NAT by ORDINAL1:def_12; then A7: i in Seg (len (Replace (M,l,p9))) by A5, A6; then A8: i in dom (Replace (M,l,p9)) by FINSEQ_1:def_3; A9: i in Seg n by A3, A7, MATRIX_1:def_2; A10: i in dom M by A3, A7, FINSEQ_1:def_3; now__::_thesis:_(_(_i_=_l_&_(Replace_(M,l,p9))_._i_=_(RLine_(M,l,pD))_._i_)_or_(_i_<>_l_&_(Replace_(M,l,p9))_._i_=_(RLine_(M,l,pD))_._i_)_) percases ( i = l or i <> l ) ; caseA11: i = l ; ::_thesis: (Replace (M,l,p9)) . i = (RLine (M,l,pD)) . i then A12: Line ((RLine (M,l,pD)),i) = pD by A1, A9, Th28; A13: (Replace (M,l,p9)) /. i = (Replace (M,l,p9)) . i by A8, PARTFUN1:def_6; (Replace (M,l,p9)) /. i = p9 by A3, A5, A6, A11, FINSEQ_7:8; hence (Replace (M,l,p9)) . i = (RLine (M,l,pD)) . i by A2, A9, A13, A12, MATRIX_2:8; ::_thesis: verum end; caseA14: i <> l ; ::_thesis: (Replace (M,l,p9)) . i = (RLine (M,l,pD)) . i then A15: Line (M,i) = Line ((RLine (M,l,pD)),i) by A9, Th28; A16: (Replace (M,l,p9)) . i = (Replace (M,l,p9)) /. i by A8, PARTFUN1:def_6; A17: M . i = Line (M,i) by A9, MATRIX_2:8; A18: M /. i = M . i by A10, PARTFUN1:def_6; (Replace (M,l,p9)) /. i = M /. i by A3, A5, A6, A14, FINSEQ_7:10; hence (Replace (M,l,p9)) . i = (RLine (M,l,pD)) . i by A9, A16, A18, A17, A15, MATRIX_2:8; ::_thesis: verum end; end; end; hence (Replace (M,l,p9)) . i = (RLine (M,l,pD)) . i ; ::_thesis: verum end; len M = len (RLine (M,l,pD)) by Lm4; hence RLine (M,l,pD) = Replace (M,l,p9) by A4, FINSEQ_1:14, FINSEQ_7:5; ::_thesis: verum end; theorem Th30: :: MATRIX11:30 for n, m, l being Nat for D being non empty set for M being Matrix of n,m,D holds M = RLine (M,l,(Line (M,l))) proof let n, m, l be Nat; ::_thesis: for D being non empty set for M being Matrix of n,m,D holds M = RLine (M,l,(Line (M,l))) let D be non empty set ; ::_thesis: for M being Matrix of n,m,D holds M = RLine (M,l,(Line (M,l))) let M be Matrix of n,m,D; ::_thesis: M = RLine (M,l,(Line (M,l))) set L = Line (M,l); set RL = RLine (M,l,(Line (M,l))); A1: width M = len (Line (M,l)) by MATRIX_1:def_7; A2: now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_len_M_holds_ (RLine_(M,l,(Line_(M,l))))_._i_=_M_._i let i be Nat; ::_thesis: ( 1 <= i & i <= len M implies (RLine (M,l,(Line (M,l)))) . i = M . i ) assume that A3: 1 <= i and A4: i <= len M ; ::_thesis: (RLine (M,l,(Line (M,l)))) . i = M . i i in NAT by ORDINAL1:def_12; then A5: i in Seg (len M) by A3, A4; A6: n = len M by MATRIX_1:def_2; then A7: (RLine (M,l,(Line (M,l)))) . i = Line ((RLine (M,l,(Line (M,l)))),i) by A5, MATRIX_2:8; A8: Line (M,i) = M . i by A5, A6, MATRIX_2:8; now__::_thesis:_(_(_i_=_l_&_(RLine_(M,l,(Line_(M,l))))_._i_=_M_._i_)_or_(_i_<>_l_&_(RLine_(M,l,(Line_(M,l))))_._i_=_M_._i_)_) percases ( i = l or i <> l ) ; case i = l ; ::_thesis: (RLine (M,l,(Line (M,l)))) . i = M . i hence (RLine (M,l,(Line (M,l)))) . i = M . i by A1, A5, A6, A8, A7, Th28; ::_thesis: verum end; case i <> l ; ::_thesis: (RLine (M,l,(Line (M,l)))) . i = M . i hence (RLine (M,l,(Line (M,l)))) . i = M . i by A5, A6, A8, A7, Th28; ::_thesis: verum end; end; end; hence (RLine (M,l,(Line (M,l)))) . i = M . i ; ::_thesis: verum end; len M = len (RLine (M,l,(Line (M,l)))) by Lm4; hence M = RLine (M,l,(Line (M,l))) by A2, FINSEQ_1:14; ::_thesis: verum end; Lm5: for K being Field for pK being FinSequence of K for a being Element of K holds len pK = len (a * pK) proof let K be Field; ::_thesis: for pK being FinSequence of K for a being Element of K holds len pK = len (a * pK) let pK be FinSequence of K; ::_thesis: for a being Element of K holds len pK = len (a * pK) let a be Element of K; ::_thesis: len pK = len (a * pK) pK is Element of (len pK) -tuples_on the carrier of K by FINSEQ_2:92; then a * pK is Element of (len pK) -tuples_on the carrier of K by FINSEQ_2:113; hence len pK = len (a * pK) by CARD_1:def_7; ::_thesis: verum end; Lm6: for K being Field for pK, qK being FinSequence of K st len pK = len qK holds len pK = len (pK + qK) proof let K be Field; ::_thesis: for pK, qK being FinSequence of K st len pK = len qK holds len pK = len (pK + qK) let pK, qK be FinSequence of K; ::_thesis: ( len pK = len qK implies len pK = len (pK + qK) ) assume len pK = len qK ; ::_thesis: len pK = len (pK + qK) then A1: qK is Element of (len pK) -tuples_on the carrier of K by FINSEQ_2:92; pK is Element of (len pK) -tuples_on the carrier of K by FINSEQ_2:92; then pK + qK is Element of (len pK) -tuples_on the carrier of K by A1, FINSEQ_2:120; hence len pK = len (pK + qK) by CARD_1:def_7; ::_thesis: verum end; theorem Th31: :: MATRIX11:31 for n being Nat for K being Field for a, b being Element of K for l being Nat for pK, qK being FinSequence of K for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds for M being Matrix of n,K holds the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK)))))) proof let n be Nat; ::_thesis: for K being Field for a, b being Element of K for l being Nat for pK, qK being FinSequence of K for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds for M being Matrix of n,K holds the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK)))))) let K be Field; ::_thesis: for a, b being Element of K for l being Nat for pK, qK being FinSequence of K for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds for M being Matrix of n,K holds the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK)))))) let a, b be Element of K; ::_thesis: for l being Nat for pK, qK being FinSequence of K for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds for M being Matrix of n,K holds the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK)))))) let l be Nat; ::_thesis: for pK, qK being FinSequence of K for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds for M being Matrix of n,K holds the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK)))))) let pK, qK be FinSequence of K; ::_thesis: for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds for M being Matrix of n,K holds the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK)))))) let perm be Element of Permutations n; ::_thesis: ( l in Seg n & len pK = n & len qK = n implies for M being Matrix of n,K holds the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK)))))) ) assume that A1: l in Seg n and A2: len pK = n and A3: len qK = n ; ::_thesis: for M being Matrix of n,K holds the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK)))))) Seg n <> {} by A1; then A4: n <> 0 ; reconsider L = l as Element of NAT by ORDINAL1:def_12; set mm = the multF of K; let M be Matrix of n,K; ::_thesis: the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK)))))) set Rpq = RLine (M,l,((a * pK) + (b * qK))); set Ppq = Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK))))); A5: len (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = n by MATRIX_3:def_7; then consider fpq being Function of NAT, the carrier of K such that A6: fpq . 1 = (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . 1 and A7: for k being Element of NAT st 0 <> k & k < len (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) holds fpq . (k + 1) = the multF of K . ((fpq . k),((Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . (k + 1))) and A8: the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = fpq . (len (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK))))))) by A4, FINSOP_1:def_1; set Rq = RLine (M,l,qK); set Pq = Path_matrix (perm,(RLine (M,l,qK))); A9: len (Path_matrix (perm,(RLine (M,l,qK)))) = n by MATRIX_3:def_7; then consider fq being Function of NAT, the carrier of K such that A10: fq . 1 = (Path_matrix (perm,(RLine (M,l,qK)))) . 1 and A11: for k being Element of NAT st 0 <> k & k < len (Path_matrix (perm,(RLine (M,l,qK)))) holds fq . (k + 1) = the multF of K . ((fq . k),((Path_matrix (perm,(RLine (M,l,qK)))) . (k + 1))) and A12: the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK)))) = fq . (len (Path_matrix (perm,(RLine (M,l,qK))))) by A4, FINSOP_1:def_1; set Rp = RLine (M,l,pK); set Pp = Path_matrix (perm,(RLine (M,l,pK))); A13: len (Path_matrix (perm,(RLine (M,l,pK)))) = n by MATRIX_3:def_7; then consider fp being Function of NAT, the carrier of K such that A14: fp . 1 = (Path_matrix (perm,(RLine (M,l,pK)))) . 1 and A15: for k being Element of NAT st 0 <> k & k < len (Path_matrix (perm,(RLine (M,l,pK)))) holds fp . (k + 1) = the multF of K . ((fp . k),((Path_matrix (perm,(RLine (M,l,pK)))) . (k + 1))) and A16: the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))) = fp . (len (Path_matrix (perm,(RLine (M,l,pK))))) by A4, FINSOP_1:def_1; A17: n >= 1 by A4, NAT_1:14; defpred S1[ Nat] means ( 1 <= $1 & $1 < L implies ( fp . $1 = fq . $1 & fpq . $1 = fp . $1 ) ); A18: for k being Element of NAT st k in Seg n holds ( ( k <> L implies ( (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (Path_matrix (perm,(RLine (M,l,pK)))) . k & (Path_matrix (perm,(RLine (M,l,pK)))) . k = (Path_matrix (perm,(RLine (M,l,qK)))) . k ) ) & ( k = L implies ex Ppk, Pqk being Element of K st ( Ppk = (Path_matrix (perm,(RLine (M,l,pK)))) . k & Pqk = (Path_matrix (perm,(RLine (M,l,qK)))) . k & (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (a * Ppk) + (b * Pqk) ) ) ) proof let k be Element of NAT ; ::_thesis: ( k in Seg n implies ( ( k <> L implies ( (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (Path_matrix (perm,(RLine (M,l,pK)))) . k & (Path_matrix (perm,(RLine (M,l,pK)))) . k = (Path_matrix (perm,(RLine (M,l,qK)))) . k ) ) & ( k = L implies ex Ppk, Pqk being Element of K st ( Ppk = (Path_matrix (perm,(RLine (M,l,pK)))) . k & Pqk = (Path_matrix (perm,(RLine (M,l,qK)))) . k & (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (a * Ppk) + (b * Pqk) ) ) ) ) assume A19: k in Seg n ; ::_thesis: ( ( k <> L implies ( (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (Path_matrix (perm,(RLine (M,l,pK)))) . k & (Path_matrix (perm,(RLine (M,l,pK)))) . k = (Path_matrix (perm,(RLine (M,l,qK)))) . k ) ) & ( k = L implies ex Ppk, Pqk being Element of K st ( Ppk = (Path_matrix (perm,(RLine (M,l,pK)))) . k & Pqk = (Path_matrix (perm,(RLine (M,l,qK)))) . k & (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (a * Ppk) + (b * Pqk) ) ) ) A20: perm . k in Seg n by A5, A19, MATRIX_7:14; then reconsider pk = perm . k as Element of NAT ; A21: k in dom (Path_matrix (perm,(RLine (M,l,pK)))) by A13, A19, FINSEQ_1:def_3; A22: k in dom (Path_matrix (perm,(RLine (M,l,qK)))) by A9, A19, FINSEQ_1:def_3; [k,pk] in [:(Seg n),(Seg n):] by A19, A20, ZFMISC_1:87; then A23: [k,pk] in Indices M by MATRIX_1:24; dom qK = Seg n by A3, FINSEQ_1:def_3; then A24: qK /. pk = qK . pk by A5, A19, MATRIX_7:14, PARTFUN1:def_6; dom pK = Seg n by A2, FINSEQ_1:def_3; then pK /. pk = pK . pk by A5, A19, MATRIX_7:14, PARTFUN1:def_6; then reconsider ppk = pK . pk, qpk = qK . pk as Element of K by A24; A25: len (b * qK) = n by A3, Lm5; then dom (b * qK) = Seg n by FINSEQ_1:def_3; then A26: (b * qK) . pk = b * qpk by A5, A19, FVSUM_1:50, MATRIX_7:14; A27: len (a * pK) = n by A2, Lm5; then A28: len ((a * pK) + (b * qK)) = n by A25, Lm6; then A29: dom ((a * pK) + (b * qK)) = Seg n by FINSEQ_1:def_3; dom (a * pK) = Seg n by A27, FINSEQ_1:def_3; then A30: (a * pK) . pk = a * ppk by A5, A19, FVSUM_1:50, MATRIX_7:14; A31: width M = n by MATRIX_1:24; A32: k in dom (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) by A5, A19, FINSEQ_1:def_3; thus ( k <> L implies ( (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (Path_matrix (perm,(RLine (M,l,pK)))) . k & (Path_matrix (perm,(RLine (M,l,pK)))) . k = (Path_matrix (perm,(RLine (M,l,qK)))) . k ) ) ::_thesis: ( k = L implies ex Ppk, Pqk being Element of K st ( Ppk = (Path_matrix (perm,(RLine (M,l,pK)))) . k & Pqk = (Path_matrix (perm,(RLine (M,l,qK)))) . k & (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (a * Ppk) + (b * Pqk) ) ) proof assume A33: k <> L ; ::_thesis: ( (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (Path_matrix (perm,(RLine (M,l,pK)))) . k & (Path_matrix (perm,(RLine (M,l,pK)))) . k = (Path_matrix (perm,(RLine (M,l,qK)))) . k ) then A34: (RLine (M,l,qK)) * (k,pk) = M * (k,pk) by A3, A23, A31, Def3; (RLine (M,l,pK)) * (k,pk) = M * (k,pk) by A2, A23, A31, A33, Def3; then A35: (Path_matrix (perm,(RLine (M,l,pK)))) . k = M * (k,pk) by A21, MATRIX_3:def_7; (RLine (M,l,((a * pK) + (b * qK)))) * (k,pk) = M * (k,pk) by A28, A23, A31, A33, Def3; hence ( (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (Path_matrix (perm,(RLine (M,l,pK)))) . k & (Path_matrix (perm,(RLine (M,l,pK)))) . k = (Path_matrix (perm,(RLine (M,l,qK)))) . k ) by A32, A22, A34, A35, MATRIX_3:def_7; ::_thesis: verum end; assume A36: k = L ; ::_thesis: ex Ppk, Pqk being Element of K st ( Ppk = (Path_matrix (perm,(RLine (M,l,pK)))) . k & Pqk = (Path_matrix (perm,(RLine (M,l,qK)))) . k & (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (a * Ppk) + (b * Pqk) ) then A37: (RLine (M,l,pK)) * (k,pk) = pK . pk by A2, A23, A31, Def3; A38: (RLine (M,l,qK)) * (k,pk) = qK . pk by A3, A23, A31, A36, Def3; take ppk ; ::_thesis: ex Pqk being Element of K st ( ppk = (Path_matrix (perm,(RLine (M,l,pK)))) . k & Pqk = (Path_matrix (perm,(RLine (M,l,qK)))) . k & (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (a * ppk) + (b * Pqk) ) take qpk ; ::_thesis: ( ppk = (Path_matrix (perm,(RLine (M,l,pK)))) . k & qpk = (Path_matrix (perm,(RLine (M,l,qK)))) . k & (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (a * ppk) + (b * qpk) ) (RLine (M,l,((a * pK) + (b * qK)))) * (k,pk) = ((a * pK) + (b * qK)) . pk by A28, A23, A31, A36, Def3; then (RLine (M,l,((a * pK) + (b * qK)))) * (k,pk) = (a * ppk) + (b * qpk) by A5, A19, A29, A30, A26, FVSUM_1:17, MATRIX_7:14; hence ( ppk = (Path_matrix (perm,(RLine (M,l,pK)))) . k & qpk = (Path_matrix (perm,(RLine (M,l,qK)))) . k & (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . k = (a * ppk) + (b * qpk) ) by A32, A21, A22, A37, A38, MATRIX_3:def_7; ::_thesis: verum end; A39: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A40: S1[k] ; ::_thesis: S1[k + 1] set k1 = k + 1; assume that A41: 1 <= k + 1 and A42: k + 1 < L ; ::_thesis: ( fp . (k + 1) = fq . (k + 1) & fpq . (k + 1) = fp . (k + 1) ) L <= n by A1, FINSEQ_1:1; then A43: k + 1 <= n by A42, XXREAL_0:2; then A44: k < n by NAT_1:13; A45: k + 1 in Seg n by A41, A43; now__::_thesis:_(_(_k_=_0_&_fp_._(k_+_1)_=_fq_._(k_+_1)_&_fpq_._(k_+_1)_=_fp_._(k_+_1)_)_or_(_k_>_0_&_fp_._(k_+_1)_=_fq_._(k_+_1)_&_fpq_._(k_+_1)_=_fp_._(k_+_1)_)_) percases ( k = 0 or k > 0 ) ; case k = 0 ; ::_thesis: ( fp . (k + 1) = fq . (k + 1) & fpq . (k + 1) = fp . (k + 1) ) hence ( fp . (k + 1) = fq . (k + 1) & fpq . (k + 1) = fp . (k + 1) ) by A6, A14, A10, A18, A42, A45; ::_thesis: verum end; caseA46: k > 0 ; ::_thesis: ( fp . (k + 1) = fq . (k + 1) & fpq . (k + 1) = fp . (k + 1) ) then A47: fp . (k + 1) = the multF of K . ((fp . k),((Path_matrix (perm,(RLine (M,l,pK)))) . (k + 1))) by A13, A15, A44; A48: 0 < k + 0 by A46; A49: fq . (k + 1) = the multF of K . ((fq . k),((Path_matrix (perm,(RLine (M,l,qK)))) . (k + 1))) by A9, A11, A44, A46; fpq . (k + 1) = the multF of K . ((fpq . k),((Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . (k + 1))) by A5, A7, A44, A46; hence ( fp . (k + 1) = fq . (k + 1) & fpq . (k + 1) = fp . (k + 1) ) by A18, A40, A42, A45, A48, A47, A49, NAT_1:13, NAT_1:19; ::_thesis: verum end; end; end; hence ( fp . (k + 1) = fq . (k + 1) & fpq . (k + 1) = fp . (k + 1) ) ; ::_thesis: verum end; defpred S2[ Nat] means ( 1 <= $1 & L <= $1 & $1 <= n implies for k being Element of NAT st $1 = k holds fpq . k = (a * (fp . k)) + (b * (fq . k)) ); A50: S1[ 0 ] ; A51: fpq . L = (a * (fp . L)) + (b * (fq . L)) proof consider PpL, PqL being Element of K such that A52: PpL = (Path_matrix (perm,(RLine (M,l,pK)))) . L and A53: PqL = (Path_matrix (perm,(RLine (M,l,qK)))) . L and A54: (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . L = (a * PpL) + (b * PqL) by A1, A18; A55: L >= 1 by A1, FINSEQ_1:1; now__::_thesis:_(_(_L_>_1_&_fpq_._L_=_(a_*_(fp_._L))_+_(b_*_(fq_._L))_)_or_(_L_=_1_&_fpq_._L_=_(a_*_(fp_._L))_+_(b_*_(fq_._L))_)_) percases ( L > 1 or L = 1 ) by A55, XXREAL_0:1; caseA56: L > 1 ; ::_thesis: fpq . L = (a * (fp . L)) + (b * (fq . L)) then reconsider L1 = L - 1 as Element of NAT by NAT_1:20; A57: L1 + 1 > 1 + 0 by A56; A58: L1 < L1 + 1 by NAT_1:19; L <= n by A1, FINSEQ_1:1; then A59: L1 < n by A58, XXREAL_0:2; then fp . L = (fp . L1) * PpL by A13, A15, A52, A57; then A60: (fp . L1) * (a * PpL) = a * (fp . L) by GROUP_1:def_3; A61: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A50, A39); A62: 1 <= L1 by A57, NAT_1:19; then fp . L1 = fq . L1 by A61, A58; then fq . L = (fp . L1) * PqL by A9, A11, A53, A57, A59; then A63: (fp . L1) * (b * PqL) = b * (fq . L) by GROUP_1:def_3; fpq . L1 = fp . L1 by A61, A58, A62; then fpq . L = (fp . L1) * ((a * PpL) + (b * PqL)) by A5, A7, A54, A57, A59; hence fpq . L = (a * (fp . L)) + (b * (fq . L)) by A60, A63, VECTSP_1:def_7; ::_thesis: verum end; case L = 1 ; ::_thesis: fpq . L = (a * (fp . L)) + (b * (fq . L)) hence fpq . L = (a * (fp . L)) + (b * (fq . L)) by A6, A14, A10, A52, A53, A54; ::_thesis: verum end; end; end; hence fpq . L = (a * (fp . L)) + (b * (fq . L)) ; ::_thesis: verum end; A64: for k being Element of NAT st S2[k] holds S2[k + 1] proof let k be Element of NAT ; ::_thesis: ( S2[k] implies S2[k + 1] ) assume A65: S2[k] ; ::_thesis: S2[k + 1] set k1 = k + 1; assume that A66: 1 <= k + 1 and A67: L <= k + 1 and A68: k + 1 <= n ; ::_thesis: for k being Element of NAT st k + 1 = k holds fpq . k = (a * (fp . k)) + (b * (fq . k)) let k9 be Element of NAT ; ::_thesis: ( k + 1 = k9 implies fpq . k9 = (a * (fp . k9)) + (b * (fq . k9)) ) assume A69: k9 = k + 1 ; ::_thesis: fpq . k9 = (a * (fp . k9)) + (b * (fq . k9)) now__::_thesis:_(_(_k_+_1_=_L_&_fpq_._k9_=_(a_*_(fp_._k9))_+_(b_*_(fq_._k9))_)_or_(_k_+_1_>_L_&_fpq_._k9_=_(a_*_(fp_._k9))_+_(b_*_(fq_._k9))_)_) percases ( k + 1 = L or k + 1 > L ) by A67, XXREAL_0:1; case k + 1 = L ; ::_thesis: fpq . k9 = (a * (fp . k9)) + (b * (fq . k9)) hence fpq . k9 = (a * (fp . k9)) + (b * (fq . k9)) by A51, A69; ::_thesis: verum end; caseA70: k + 1 > L ; ::_thesis: fpq . k9 = (a * (fp . k9)) + (b * (fq . k9)) A71: k + 1 in Seg n by A66, A68; then A72: (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) . (k + 1) = (Path_matrix (perm,(RLine (M,l,pK)))) . (k + 1) by A18, A70; k + 1 in dom (Path_matrix (perm,(RLine (M,l,pK)))) by A13, A71, FINSEQ_1:def_3; then (Path_matrix (perm,(RLine (M,l,pK)))) /. (k + 1) = (Path_matrix (perm,(RLine (M,l,pK)))) . (k + 1) by PARTFUN1:def_6; then reconsider Ppk1 = (Path_matrix (perm,(RLine (M,l,pK)))) . (k + 1) as Element of K ; A73: k < n by A68, NAT_1:13; A74: (b * (fq . k)) * Ppk1 = b * ((fq . k) * Ppk1) by GROUP_1:def_3; A75: (a * (fp . k)) * Ppk1 = a * ((fp . k) * Ppk1) by GROUP_1:def_3; A76: 1 <= L by A1, FINSEQ_1:1; A77: k >= L by A70, NAT_1:13; then fpq . k = (a * (fp . k)) + (b * (fq . k)) by A65, A68, A76, NAT_1:13, XXREAL_0:2; then A78: fpq . (k + 1) = ((a * (fp . k)) + (b * (fq . k))) * Ppk1 by A5, A7, A77, A73, A76, A72; (Path_matrix (perm,(RLine (M,l,pK)))) . (k + 1) = (Path_matrix (perm,(RLine (M,l,qK)))) . (k + 1) by A18, A70, A71; then A79: fq . (k + 1) = (fq . k) * Ppk1 by A9, A11, A77, A73, A76; fp . (k + 1) = (fp . k) * Ppk1 by A13, A15, A77, A73, A76; hence fpq . k9 = (a * (fp . k9)) + (b * (fq . k9)) by A69, A78, A79, A75, A74, VECTSP_1:def_7; ::_thesis: verum end; end; end; hence fpq . k9 = (a * (fp . k9)) + (b * (fq . k9)) ; ::_thesis: verum end; A80: L <= n by A1, FINSEQ_1:1; A81: S2[ 0 ] ; for k being Element of NAT holds S2[k] from NAT_1:sch_1(A81, A64); hence the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK)))))) by A17, A5, A13, A9, A8, A16, A12, A80; ::_thesis: verum end; theorem Th32: :: MATRIX11:32 for n being Nat for K being Field for a, b being Element of K for l being Nat for pK, qK being FinSequence of K for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds for M being Matrix of n,K holds (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm)) proof let n be Nat; ::_thesis: for K being Field for a, b being Element of K for l being Nat for pK, qK being FinSequence of K for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds for M being Matrix of n,K holds (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm)) let K be Field; ::_thesis: for a, b being Element of K for l being Nat for pK, qK being FinSequence of K for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds for M being Matrix of n,K holds (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm)) let a, b be Element of K; ::_thesis: for l being Nat for pK, qK being FinSequence of K for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds for M being Matrix of n,K holds (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm)) let l be Nat; ::_thesis: for pK, qK being FinSequence of K for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds for M being Matrix of n,K holds (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm)) let pK, qK be FinSequence of K; ::_thesis: for perm being Element of Permutations n st l in Seg n & len pK = n & len qK = n holds for M being Matrix of n,K holds (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm)) let perm be Element of Permutations n; ::_thesis: ( l in Seg n & len pK = n & len qK = n implies for M being Matrix of n,K holds (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm)) ) assume that A1: l in Seg n and A2: len pK = n and A3: len qK = n ; ::_thesis: for M being Matrix of n,K holds (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm)) set mm = the multF of K; let M be Matrix of n,K; ::_thesis: (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm)) set Rpq = RLine (M,l,((a * pK) + (b * qK))); set Rp = RLine (M,l,pK); set Rq = RLine (M,l,qK); set Ppq = Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK))))); set Pathpq = Path_product (RLine (M,l,((a * pK) + (b * qK)))); set Pp = Path_matrix (perm,(RLine (M,l,pK))); set Pathp = Path_product (RLine (M,l,pK)); set Pq = Path_matrix (perm,(RLine (M,l,qK))); set Pathq = Path_product (RLine (M,l,qK)); now__::_thesis:_(_(_perm_is_even_&_(Path_product_(RLine_(M,l,((a_*_pK)_+_(b_*_qK)))))_._perm_=_(a_*_((Path_product_(RLine_(M,l,pK)))_._perm))_+_(b_*_((Path_product_(RLine_(M,l,qK)))_._perm))_)_or_(_perm_is_odd_&_(Path_product_(RLine_(M,l,((a_*_pK)_+_(b_*_qK)))))_._perm_=_(a_*_((Path_product_(RLine_(M,l,pK)))_._perm))_+_(b_*_((Path_product_(RLine_(M,l,qK)))_._perm))_)_) percases ( perm is even or perm is odd ) ; caseA4: perm is even ; ::_thesis: (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm)) then the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = - (( the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK))))))),perm) by MATRIX_2:def_13; then A5: (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) by MATRIX_3:def_8; the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK)))) = - (( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK))))),perm) by A4, MATRIX_2:def_13; then A6: (Path_product (RLine (M,l,qK))) . perm = the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK)))) by MATRIX_3:def_8; the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))) = - (( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK))))),perm) by A4, MATRIX_2:def_13; then (Path_product (RLine (M,l,pK))) . perm = the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))) by MATRIX_3:def_8; hence (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm)) by A1, A2, A3, A6, A5, Th31; ::_thesis: verum end; caseA7: perm is odd ; ::_thesis: (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm)) then - ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK))))))) = - (( the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK))))))),perm) by MATRIX_2:def_13; then A8: (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = - ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK))))))) by MATRIX_3:def_8; - ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK))))) = - (( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK))))),perm) by A7, MATRIX_2:def_13; then A9: (Path_product (RLine (M,l,pK))) . perm = - ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK))))) by MATRIX_3:def_8; A10: - (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) = a * (- ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) by VECTSP_1:8; - ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK))))) = - (( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK))))),perm) by A7, MATRIX_2:def_13; then A11: (Path_product (RLine (M,l,qK))) . perm = - ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK))))) by MATRIX_3:def_8; A12: - ((a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK))))))) = (- (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK))))))) - (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK)))))) by VECTSP_1:17; the multF of K $$ (Path_matrix (perm,(RLine (M,l,((a * pK) + (b * qK)))))) = (a * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,pK)))))) + (b * ( the multF of K $$ (Path_matrix (perm,(RLine (M,l,qK)))))) by A1, A2, A3, Th31; hence (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm)) by A9, A11, A8, A10, A12, VECTSP_1:8; ::_thesis: verum end; end; end; hence (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . perm = (a * ((Path_product (RLine (M,l,pK))) . perm)) + (b * ((Path_product (RLine (M,l,qK))) . perm)) ; ::_thesis: verum end; theorem Th33: :: MATRIX11:33 for n being Nat for K being Field for a, b being Element of K for l being Nat for pK, qK being FinSequence of K st l in Seg n & len pK = n & len qK = n holds for M being Matrix of n,K holds Det (RLine (M,l,((a * pK) + (b * qK)))) = (a * (Det (RLine (M,l,pK)))) + (b * (Det (RLine (M,l,qK)))) proof let n be Nat; ::_thesis: for K being Field for a, b being Element of K for l being Nat for pK, qK being FinSequence of K st l in Seg n & len pK = n & len qK = n holds for M being Matrix of n,K holds Det (RLine (M,l,((a * pK) + (b * qK)))) = (a * (Det (RLine (M,l,pK)))) + (b * (Det (RLine (M,l,qK)))) let K be Field; ::_thesis: for a, b being Element of K for l being Nat for pK, qK being FinSequence of K st l in Seg n & len pK = n & len qK = n holds for M being Matrix of n,K holds Det (RLine (M,l,((a * pK) + (b * qK)))) = (a * (Det (RLine (M,l,pK)))) + (b * (Det (RLine (M,l,qK)))) let a, b be Element of K; ::_thesis: for l being Nat for pK, qK being FinSequence of K st l in Seg n & len pK = n & len qK = n holds for M being Matrix of n,K holds Det (RLine (M,l,((a * pK) + (b * qK)))) = (a * (Det (RLine (M,l,pK)))) + (b * (Det (RLine (M,l,qK)))) let l be Nat; ::_thesis: for pK, qK being FinSequence of K st l in Seg n & len pK = n & len qK = n holds for M being Matrix of n,K holds Det (RLine (M,l,((a * pK) + (b * qK)))) = (a * (Det (RLine (M,l,pK)))) + (b * (Det (RLine (M,l,qK)))) let pK, qK be FinSequence of K; ::_thesis: ( l in Seg n & len pK = n & len qK = n implies for M being Matrix of n,K holds Det (RLine (M,l,((a * pK) + (b * qK)))) = (a * (Det (RLine (M,l,pK)))) + (b * (Det (RLine (M,l,qK)))) ) assume that A1: l in Seg n and A2: len pK = n and A3: len qK = n ; ::_thesis: for M being Matrix of n,K holds Det (RLine (M,l,((a * pK) + (b * qK)))) = (a * (Det (RLine (M,l,pK)))) + (b * (Det (RLine (M,l,qK)))) set P = Permutations n; set KK = the carrier of K; set aa = the addF of K; let M be Matrix of n,K; ::_thesis: Det (RLine (M,l,((a * pK) + (b * qK)))) = (a * (Det (RLine (M,l,pK)))) + (b * (Det (RLine (M,l,qK)))) set Rpq = RLine (M,l,((a * pK) + (b * qK))); set Rp = RLine (M,l,pK); set Rq = RLine (M,l,qK); set Pathpq = Path_product (RLine (M,l,((a * pK) + (b * qK)))); set Pathp = Path_product (RLine (M,l,pK)); set Pathq = Path_product (RLine (M,l,qK)); set F = FinOmega (Permutations n); A4: FinOmega (Permutations n) = Permutations n by MATRIX_2:26, MATRIX_2:def_14; then consider Gpq being Function of (Fin (Permutations n)), the carrier of K such that A5: Det (RLine (M,l,((a * pK) + (b * qK)))) = Gpq . (FinOmega (Permutations n)) and A6: for e being Element of the carrier of K st e is_a_unity_wrt the addF of K holds Gpq . {} = e and A7: for x being Element of Permutations n holds Gpq . {x} = (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . x and A8: for B9 being Element of Fin (Permutations n) st B9 c= FinOmega (Permutations n) & B9 <> {} holds for x being Element of Permutations n st x in (FinOmega (Permutations n)) \ B9 holds Gpq . (B9 \/ {x}) = the addF of K . ((Gpq . B9),((Path_product (RLine (M,l,((a * pK) + (b * qK))))) . x)) by SETWISEO:def_3; consider Gq being Function of (Fin (Permutations n)), the carrier of K such that A9: Det (RLine (M,l,qK)) = Gq . (FinOmega (Permutations n)) and A10: for e being Element of the carrier of K st e is_a_unity_wrt the addF of K holds Gq . {} = e and A11: for x being Element of Permutations n holds Gq . {x} = (Path_product (RLine (M,l,qK))) . x and A12: for B9 being Element of Fin (Permutations n) st B9 c= FinOmega (Permutations n) & B9 <> {} holds for x being Element of Permutations n st x in (FinOmega (Permutations n)) \ B9 holds Gq . (B9 \/ {x}) = the addF of K . ((Gq . B9),((Path_product (RLine (M,l,qK))) . x)) by A4, SETWISEO:def_3; consider Gp being Function of (Fin (Permutations n)), the carrier of K such that A13: Det (RLine (M,l,pK)) = Gp . (FinOmega (Permutations n)) and A14: for e being Element of the carrier of K st e is_a_unity_wrt the addF of K holds Gp . {} = e and A15: for x being Element of Permutations n holds Gp . {x} = (Path_product (RLine (M,l,pK))) . x and A16: for B9 being Element of Fin (Permutations n) st B9 c= FinOmega (Permutations n) & B9 <> {} holds for x being Element of Permutations n st x in (FinOmega (Permutations n)) \ B9 holds Gp . (B9 \/ {x}) = the addF of K . ((Gp . B9),((Path_product (RLine (M,l,pK))) . x)) by A4, SETWISEO:def_3; defpred S1[ Nat] means for B being Element of Fin (Permutations n) st card B = $1 holds Gpq . B = (a * (Gp . B)) + (b * (Gq . B)); A17: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A18: S1[k] ; ::_thesis: S1[k + 1] let B be Element of Fin (Permutations n); ::_thesis: ( card B = k + 1 implies Gpq . B = (a * (Gp . B)) + (b * (Gq . B)) ) assume A19: card B = k + 1 ; ::_thesis: Gpq . B = (a * (Gp . B)) + (b * (Gq . B)) now__::_thesis:_(_(_k_=_0_&_Gpq_._B_=_(a_*_(Gp_._B))_+_(b_*_(Gq_._B))_)_or_(_k_>_0_&_Gpq_._B_=_(a_*_(Gp_._B))_+_(b_*_(Gq_._B))_)_) percases ( k = 0 or k > 0 ) ; case k = 0 ; ::_thesis: Gpq . B = (a * (Gp . B)) + (b * (Gq . B)) then consider x being set such that A20: B = {x} by A19, CARD_2:42; A21: x in B by A20, TARSKI:def_1; B c= Permutations n by FINSUB_1:def_5; then reconsider x = x as Element of Permutations n by A21; A22: Gp . B = (Path_product (RLine (M,l,pK))) . x by A15, A20; A23: Gq . B = (Path_product (RLine (M,l,qK))) . x by A11, A20; Gpq . B = (Path_product (RLine (M,l,((a * pK) + (b * qK))))) . x by A7, A20; hence Gpq . B = (a * (Gp . B)) + (b * (Gq . B)) by A1, A2, A3, A22, A23, Th32; ::_thesis: verum end; caseA24: k > 0 ; ::_thesis: Gpq . B = (a * (Gp . B)) + (b * (Gq . B)) consider x being set such that A25: x in B by A19, CARD_1:27, XBOOLE_0:def_1; B c= Permutations n by FINSUB_1:def_5; then reconsider x = x as Element of Permutations n by A25; B c= Permutations n by FINSUB_1:def_5; then B \ {x} c= Permutations n by XBOOLE_1:1; then reconsider B9 = B \ {x} as Element of Fin (Permutations n) by FINSUB_1:def_5; A26: not x in B9 by ZFMISC_1:56; then A27: x in (FinOmega (Permutations n)) \ B9 by A4, XBOOLE_0:def_5; A28: {x} \/ B9 = B by A25, ZFMISC_1:116; then A29: k + 1 = (card B9) + 1 by A19, A26, CARD_2:41; then A30: Gpq . B9 = (a * (Gp . B9)) + (b * (Gq . B9)) by A18; A31: B9 c= FinOmega (Permutations n) by A4, FINSUB_1:def_5; then Gpq . B = the addF of K . ((Gpq . B9),((Path_product (RLine (M,l,((a * pK) + (b * qK))))) . x)) by A8, A24, A28, A29, A27, CARD_1:27; then A32: Gpq . B = ((a * (Gp . B9)) + (b * (Gq . B9))) + ((a * ((Path_product (RLine (M,l,pK))) . x)) + (b * ((Path_product (RLine (M,l,qK))) . x))) by A1, A2, A3, A30, Th32 .= (a * (Gp . B9)) + ((b * (Gq . B9)) + ((a * ((Path_product (RLine (M,l,pK))) . x)) + (b * ((Path_product (RLine (M,l,qK))) . x)))) by RLVECT_1:def_3 .= (a * (Gp . B9)) + ((a * ((Path_product (RLine (M,l,pK))) . x)) + ((b * (Gq . B9)) + (b * ((Path_product (RLine (M,l,qK))) . x)))) by RLVECT_1:def_3 .= ((a * (Gp . B9)) + (a * ((Path_product (RLine (M,l,pK))) . x))) + ((b * (Gq . B9)) + (b * ((Path_product (RLine (M,l,qK))) . x))) by RLVECT_1:def_3 .= (a * ((Gp . B9) + ((Path_product (RLine (M,l,pK))) . x))) + ((b * (Gq . B9)) + (b * ((Path_product (RLine (M,l,qK))) . x))) by VECTSP_1:def_7 .= (a * ( the addF of K . ((Gp . B9),((Path_product (RLine (M,l,pK))) . x)))) + (b * ((Gq . B9) + ((Path_product (RLine (M,l,qK))) . x))) by VECTSP_1:def_7 .= (a * ( the addF of K . ((Gp . B9),((Path_product (RLine (M,l,pK))) . x)))) + (b * ( the addF of K . ((Gq . B9),((Path_product (RLine (M,l,qK))) . x)))) ; Gp . B = the addF of K . ((Gp . B9),((Path_product (RLine (M,l,pK))) . x)) by A16, A24, A28, A29, A27, A31, CARD_1:27; hence Gpq . B = (a * (Gp . B)) + (b * (Gq . B)) by A12, A24, A28, A29, A27, A31, A32, CARD_1:27; ::_thesis: verum end; end; end; hence Gpq . B = (a * (Gp . B)) + (b * (Gq . B)) ; ::_thesis: verum end; A33: S1[ 0 ] proof A34: b * (0. K) = 0. K by VECTSP_1:6; A35: a * (0. K) = 0. K by VECTSP_1:6; let B be Element of Fin (Permutations n); ::_thesis: ( card B = 0 implies Gpq . B = (a * (Gp . B)) + (b * (Gq . B)) ) assume card B = 0 ; ::_thesis: Gpq . B = (a * (Gp . B)) + (b * (Gq . B)) then A36: B = {} ; then A37: Gp . B = 0. K by A14, FVSUM_1:6; A38: Gq . B = 0. K by A10, A36, FVSUM_1:6; Gpq . B = 0. K by A6, A36, FVSUM_1:6; hence Gpq . B = (a * (Gp . B)) + (b * (Gq . B)) by A37, A38, A35, A34, RLVECT_1:4; ::_thesis: verum end; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A33, A17); then S1[ card (FinOmega (Permutations n))] ; hence Det (RLine (M,l,((a * pK) + (b * qK)))) = (a * (Det (RLine (M,l,pK)))) + (b * (Det (RLine (M,l,qK)))) by A5, A13, A9; ::_thesis: verum end; theorem Th34: :: MATRIX11:34 for l, n being Nat for K being Field for a being Element of K for pK being FinSequence of K for A being Matrix of n,K st l in Seg n & len pK = n holds Det (RLine (A,l,(a * pK))) = a * (Det (RLine (A,l,pK))) proof let l, n be Nat; ::_thesis: for K being Field for a being Element of K for pK being FinSequence of K for A being Matrix of n,K st l in Seg n & len pK = n holds Det (RLine (A,l,(a * pK))) = a * (Det (RLine (A,l,pK))) let K be Field; ::_thesis: for a being Element of K for pK being FinSequence of K for A being Matrix of n,K st l in Seg n & len pK = n holds Det (RLine (A,l,(a * pK))) = a * (Det (RLine (A,l,pK))) let a be Element of K; ::_thesis: for pK being FinSequence of K for A being Matrix of n,K st l in Seg n & len pK = n holds Det (RLine (A,l,(a * pK))) = a * (Det (RLine (A,l,pK))) let pK be FinSequence of K; ::_thesis: for A being Matrix of n,K st l in Seg n & len pK = n holds Det (RLine (A,l,(a * pK))) = a * (Det (RLine (A,l,pK))) let A be Matrix of n,K; ::_thesis: ( l in Seg n & len pK = n implies Det (RLine (A,l,(a * pK))) = a * (Det (RLine (A,l,pK))) ) assume that A1: l in Seg n and A2: len pK = n ; ::_thesis: Det (RLine (A,l,(a * pK))) = a * (Det (RLine (A,l,pK))) pK is Element of (len pK) -tuples_on the carrier of K by FINSEQ_2:92; then A3: (a * pK) + ((0. K) * pK) = (a + (0. K)) * pK by FVSUM_1:55; a + (0. K) = a by RLVECT_1:4; hence Det (RLine (A,l,(a * pK))) = (a * (Det (RLine (A,l,pK)))) + ((0. K) * (Det (RLine (A,l,pK)))) by A1, A2, A3, Th33 .= (a * (Det (RLine (A,l,pK)))) + (0. K) by VECTSP_1:6 .= a * (Det (RLine (A,l,pK))) by RLVECT_1:4 ; ::_thesis: verum end; theorem :: MATRIX11:35 for l, n being Nat for K being Field for a being Element of K for A being Matrix of n,K st l in Seg n holds Det (RLine (A,l,(a * (Line (A,l))))) = a * (Det A) proof let l, n be Nat; ::_thesis: for K being Field for a being Element of K for A being Matrix of n,K st l in Seg n holds Det (RLine (A,l,(a * (Line (A,l))))) = a * (Det A) let K be Field; ::_thesis: for a being Element of K for A being Matrix of n,K st l in Seg n holds Det (RLine (A,l,(a * (Line (A,l))))) = a * (Det A) let a be Element of K; ::_thesis: for A being Matrix of n,K st l in Seg n holds Det (RLine (A,l,(a * (Line (A,l))))) = a * (Det A) let A be Matrix of n,K; ::_thesis: ( l in Seg n implies Det (RLine (A,l,(a * (Line (A,l))))) = a * (Det A) ) A1: len (Line (A,l)) = width A by MATRIX_1:def_7; assume l in Seg n ; ::_thesis: Det (RLine (A,l,(a * (Line (A,l))))) = a * (Det A) then Det (RLine (A,l,(a * (Line (A,l))))) = a * (Det (RLine (A,l,(Line (A,l))))) by A1, Th34, MATRIX_1:24; hence Det (RLine (A,l,(a * (Line (A,l))))) = a * (Det A) by Th30; ::_thesis: verum end; theorem Th36: :: MATRIX11:36 for l, n being Nat for K being Field for pK, qK being FinSequence of K for A being Matrix of n,K st l in Seg n & len pK = n & len qK = n holds Det (RLine (A,l,(pK + qK))) = (Det (RLine (A,l,pK))) + (Det (RLine (A,l,qK))) proof let l, n be Nat; ::_thesis: for K being Field for pK, qK being FinSequence of K for A being Matrix of n,K st l in Seg n & len pK = n & len qK = n holds Det (RLine (A,l,(pK + qK))) = (Det (RLine (A,l,pK))) + (Det (RLine (A,l,qK))) let K be Field; ::_thesis: for pK, qK being FinSequence of K for A being Matrix of n,K st l in Seg n & len pK = n & len qK = n holds Det (RLine (A,l,(pK + qK))) = (Det (RLine (A,l,pK))) + (Det (RLine (A,l,qK))) let pK, qK be FinSequence of K; ::_thesis: for A being Matrix of n,K st l in Seg n & len pK = n & len qK = n holds Det (RLine (A,l,(pK + qK))) = (Det (RLine (A,l,pK))) + (Det (RLine (A,l,qK))) let A be Matrix of n,K; ::_thesis: ( l in Seg n & len pK = n & len qK = n implies Det (RLine (A,l,(pK + qK))) = (Det (RLine (A,l,pK))) + (Det (RLine (A,l,qK))) ) assume that A1: l in Seg n and A2: len pK = n and A3: len qK = n ; ::_thesis: Det (RLine (A,l,(pK + qK))) = (Det (RLine (A,l,pK))) + (Det (RLine (A,l,qK))) pK is Element of (len pK) -tuples_on the carrier of K by FINSEQ_2:92; then A4: (1_ K) * pK = pK by FVSUM_1:57; qK is Element of (len pK) -tuples_on the carrier of K by A2, A3, FINSEQ_2:92; then (1_ K) * qK = qK by FVSUM_1:57; hence Det (RLine (A,l,(pK + qK))) = ((1_ K) * (Det (RLine (A,l,pK)))) + ((1_ K) * (Det (RLine (A,l,qK)))) by A1, A2, A3, A4, Th33 .= (Det (RLine (A,l,pK))) + ((1_ K) * (Det (RLine (A,l,qK)))) by VECTSP_1:def_4 .= (Det (RLine (A,l,pK))) + (Det (RLine (A,l,qK))) by VECTSP_1:def_4 ; ::_thesis: verum end; Lm7: for n, m being Nat for D being non empty set for F being Function of (Seg n),(Seg n) for M being Matrix of n,m,D holds M * F is Matrix of n,m,D proof let n, m be Nat; ::_thesis: for D being non empty set for F being Function of (Seg n),(Seg n) for M being Matrix of n,m,D holds M * F is Matrix of n,m,D let D be non empty set ; ::_thesis: for F being Function of (Seg n),(Seg n) for M being Matrix of n,m,D holds M * F is Matrix of n,m,D let F be Function of (Seg n),(Seg n); ::_thesis: for M being Matrix of n,m,D holds M * F is Matrix of n,m,D let M be Matrix of n,m,D; ::_thesis: M * F is Matrix of n,m,D A1: rng F c= Seg n by RELAT_1:def_19; len M = n by MATRIX_1:def_2; then A2: dom M = Seg n by FINSEQ_1:def_3; dom F = Seg n by FUNCT_2:52; then A3: dom (M * F) = Seg n by A1, A2, RELAT_1:27; then reconsider Mp = M * F as FinSequence by FINSEQ_1:def_2; A4: for x being set st x in rng Mp holds ex p being FinSequence of D st ( x = p & len p = m ) proof A5: rng M c= D * by FINSEQ_1:def_4; let x be set ; ::_thesis: ( x in rng Mp implies ex p being FinSequence of D st ( x = p & len p = m ) ) assume A6: x in rng Mp ; ::_thesis: ex p being FinSequence of D st ( x = p & len p = m ) rng Mp c= rng M by RELAT_1:26; then x in D * by A6, A5, TARSKI:def_3; then reconsider p = x as FinSequence of D by FINSEQ_1:def_11; take p ; ::_thesis: ( x = p & len p = m ) p in rng M by A6, FUNCT_1:14; hence ( x = p & len p = m ) by MATRIX_1:def_2; ::_thesis: verum end; then reconsider Mp = Mp as Matrix of D by MATRIX_1:9; A7: n is Element of NAT by ORDINAL1:def_12; ( len Mp = n & ( for p being FinSequence of D st p in rng Mp holds len p = m ) ) proof thus len Mp = n by A3, A7, FINSEQ_1:def_3; ::_thesis: for p being FinSequence of D st p in rng Mp holds len p = m let p be FinSequence of D; ::_thesis: ( p in rng Mp implies len p = m ) assume p in rng Mp ; ::_thesis: len p = m then ex q being FinSequence of D st ( p = q & len q = m ) by A4; hence len p = m ; ::_thesis: verum end; hence M * F is Matrix of n,m,D by MATRIX_1:def_2; ::_thesis: verum end; begin definition let n, m be Nat; let D be non empty set ; let F be Function of (Seg n),(Seg n); let M be Matrix of n,m,D; :: original: * redefine funcM * F -> Matrix of n,m,D means :Def4: :: MATRIX11:def 4 ( len it = len M & width it = width M & ( for i, j, k being Nat st [i,j] in Indices M & F . i = k holds it * (i,j) = M * (k,j) ) ); compatibility for b1 being Matrix of n,m,D holds ( b1 = F * M iff ( len b1 = len M & width b1 = width M & ( for i, j, k being Nat st [i,j] in Indices M & F . i = k holds b1 * (i,j) = M * (k,j) ) ) ) proof reconsider Mf = M * F as Matrix of n,m,D by Lm7; let Mp be Matrix of n,m,D; ::_thesis: ( Mp = F * M iff ( len Mp = len M & width Mp = width M & ( for i, j, k being Nat st [i,j] in Indices M & F . i = k holds Mp * (i,j) = M * (k,j) ) ) ) thus ( Mp = M * F implies ( len Mp = len M & width Mp = width M & ( for i, j, k being Nat st [i,j] in Indices M & F . i = k holds Mp * (i,j) = M * (k,j) ) ) ) ::_thesis: ( len Mp = len M & width Mp = width M & ( for i, j, k being Nat st [i,j] in Indices M & F . i = k holds Mp * (i,j) = M * (k,j) ) implies Mp = F * M ) proof A1: rng F c= Seg n by RELAT_1:def_19; assume A2: Mp = M * F ; ::_thesis: ( len Mp = len M & width Mp = width M & ( for i, j, k being Nat st [i,j] in Indices M & F . i = k holds Mp * (i,j) = M * (k,j) ) ) A3: len M = n by MATRIX_1:def_2; A4: len Mp = n by MATRIX_1:def_2; A5: now__::_thesis:_(_(_n_=_0_&_width_M_=_width_Mp_)_or_(_n_>_0_&_width_M_=_width_Mp_)_) percases ( n = 0 or n > 0 ) ; caseA6: n = 0 ; ::_thesis: width M = width Mp then width M = 0 by A3, MATRIX_1:def_3; hence width M = width Mp by A4, A6, MATRIX_1:def_3; ::_thesis: verum end; caseA7: n > 0 ; ::_thesis: width M = width Mp then width M = m by A3, MATRIX_1:20; hence width M = width Mp by A4, A7, MATRIX_1:20; ::_thesis: verum end; end; end; hence ( len Mp = len M & width Mp = width M ) by A3, MATRIX_1:def_2; ::_thesis: for i, j, k being Nat st [i,j] in Indices M & F . i = k holds Mp * (i,j) = M * (k,j) let i, j, k be Nat; ::_thesis: ( [i,j] in Indices M & F . i = k implies Mp * (i,j) = M * (k,j) ) assume that A8: [i,j] in Indices M and A9: F . i = k ; ::_thesis: Mp * (i,j) = M * (k,j) Indices M = [:(Seg n),(Seg (width M)):] by MATRIX_1:25; then A10: i in Seg n by A8, ZFMISC_1:87; then A11: Line (Mp,i) = Mp . i by MATRIX_2:8; dom F = Seg n by FUNCT_2:52; then A12: F . i in rng F by A10, FUNCT_1:def_3; len Mp = n by MATRIX_1:25; then dom Mp = Seg n by FINSEQ_1:def_3; then Mp . i = M . k by A2, A9, A10, FUNCT_1:12; then A13: Line (Mp,i) = Line (M,k) by A9, A12, A1, A11, MATRIX_2:8; A14: j in Seg (width M) by A8, ZFMISC_1:87; then (Line (M,k)) . j = M * (k,j) by MATRIX_1:def_7; hence Mp * (i,j) = M * (k,j) by A5, A14, A13, MATRIX_1:def_7; ::_thesis: verum end; assume that A15: len Mp = len M and A16: width Mp = width M ; ::_thesis: ( ex i, j, k being Nat st ( [i,j] in Indices M & F . i = k & not Mp * (i,j) = M * (k,j) ) or Mp = F * M ) assume A17: for i, j, k being Nat st [i,j] in Indices M & F . i = k holds Mp * (i,j) = M * (k,j) ; ::_thesis: Mp = F * M for i, j being Nat st [i,j] in Indices Mp holds Mp * (i,j) = Mf * (i,j) proof A18: Indices Mp = Indices M by A15, A16, MATRIX_4:55; let i, j be Nat; ::_thesis: ( [i,j] in Indices Mp implies Mp * (i,j) = Mf * (i,j) ) assume A19: [i,j] in Indices Mp ; ::_thesis: Mp * (i,j) = Mf * (i,j) Indices Mp = [:(Seg n),(Seg (width M)):] by A16, MATRIX_1:25; then A20: i in Seg n by A19, ZFMISC_1:87; then A21: Line (Mf,i) = Mf . i by MATRIX_2:8; A22: rng F c= Seg n by RELAT_1:def_19; dom F = Seg n by FUNCT_2:52; then A23: F . i in rng F by A20, FUNCT_1:def_3; then F . i in Seg n by A22; then reconsider k = F . i as Element of NAT ; len Mf = n by MATRIX_1:25; then dom Mf = Seg n by FINSEQ_1:def_3; then Mf . i = M . k by A20, FUNCT_1:12; then A24: Line (Mf,i) = Line (M,k) by A23, A22, A21, MATRIX_2:8; A25: width M = len (Line (M,k)) by MATRIX_1:def_7; A26: width Mf = len (Line (Mf,i)) by MATRIX_1:def_7; A27: j in Seg (width M) by A16, A19, ZFMISC_1:87; then (Line (M,k)) . j = M * (k,j) by MATRIX_1:def_7; then Mf * (i,j) = M * (k,j) by A27, A24, A25, A26, MATRIX_1:def_7; hence Mp * (i,j) = Mf * (i,j) by A17, A19, A18; ::_thesis: verum end; hence Mp = F * M by MATRIX_1:27; ::_thesis: verum end; correctness coherence F * M is Matrix of n,m,D; by Lm7; end; :: deftheorem Def4 defines * MATRIX11:def_4_:_ for n, m being Nat for D being non empty set for F being Function of (Seg n),(Seg n) for M, b6 being Matrix of n,m,D holds ( b6 = M * F iff ( len b6 = len M & width b6 = width M & ( for i, j, k being Nat st [i,j] in Indices M & F . i = k holds b6 * (i,j) = M * (k,j) ) ) ); theorem Th37: :: MATRIX11:37 for n, m being Nat for D being non empty set for F being Function of (Seg n),(Seg n) for M being Matrix of n,m,D holds ( Indices M = Indices (M * F) & ( for i, j being Nat st [i,j] in Indices M holds ex k being Nat st ( F . i = k & [k,j] in Indices M & (M * F) * (i,j) = M * (k,j) ) ) ) proof let n, m be Nat; ::_thesis: for D being non empty set for F being Function of (Seg n),(Seg n) for M being Matrix of n,m,D holds ( Indices M = Indices (M * F) & ( for i, j being Nat st [i,j] in Indices M holds ex k being Nat st ( F . i = k & [k,j] in Indices M & (M * F) * (i,j) = M * (k,j) ) ) ) let D be non empty set ; ::_thesis: for F being Function of (Seg n),(Seg n) for M being Matrix of n,m,D holds ( Indices M = Indices (M * F) & ( for i, j being Nat st [i,j] in Indices M holds ex k being Nat st ( F . i = k & [k,j] in Indices M & (M * F) * (i,j) = M * (k,j) ) ) ) let F be Function of (Seg n),(Seg n); ::_thesis: for M being Matrix of n,m,D holds ( Indices M = Indices (M * F) & ( for i, j being Nat st [i,j] in Indices M holds ex k being Nat st ( F . i = k & [k,j] in Indices M & (M * F) * (i,j) = M * (k,j) ) ) ) let M be Matrix of n,m,D; ::_thesis: ( Indices M = Indices (M * F) & ( for i, j being Nat st [i,j] in Indices M holds ex k being Nat st ( F . i = k & [k,j] in Indices M & (M * F) * (i,j) = M * (k,j) ) ) ) set Mp = M * F; A1: dom F = Seg n by FUNCT_2:52; A2: width M = width (M * F) by Def4; len M = len (M * F) by Def4; hence Indices M = Indices (M * F) by A2, MATRIX_4:55; ::_thesis: for i, j being Nat st [i,j] in Indices M holds ex k being Nat st ( F . i = k & [k,j] in Indices M & (M * F) * (i,j) = M * (k,j) ) let i, j be Nat; ::_thesis: ( [i,j] in Indices M implies ex k being Nat st ( F . i = k & [k,j] in Indices M & (M * F) * (i,j) = M * (k,j) ) ) assume A3: [i,j] in Indices M ; ::_thesis: ex k being Nat st ( F . i = k & [k,j] in Indices M & (M * F) * (i,j) = M * (k,j) ) Indices M = [:(Seg n),(Seg (width M)):] by MATRIX_1:25; then i in Seg n by A3, ZFMISC_1:87; then A4: F . i in rng F by A1, FUNCT_1:def_3; A5: rng F c= Seg n by RELAT_1:def_19; then F . i in Seg n by A4; then reconsider k = F . i as Element of NAT ; j in Seg (width M) by A3, ZFMISC_1:87; then [k,j] in [:(Seg n),(Seg (width M)):] by A4, A5, ZFMISC_1:87; then A6: [k,j] in Indices M by MATRIX_1:25; (M * F) * (i,j) = M * (k,j) by A3, Def4; hence ex k being Nat st ( F . i = k & [k,j] in Indices M & (M * F) * (i,j) = M * (k,j) ) by A6; ::_thesis: verum end; theorem Th38: :: MATRIX11:38 for n, m being Nat for D being non empty set for M being Matrix of n,m,D for F being Function of (Seg n),(Seg n) for k being Nat st k in Seg n holds Line ((M * F),k) = M . (F . k) proof let n, m be Nat; ::_thesis: for D being non empty set for M being Matrix of n,m,D for F being Function of (Seg n),(Seg n) for k being Nat st k in Seg n holds Line ((M * F),k) = M . (F . k) let D be non empty set ; ::_thesis: for M being Matrix of n,m,D for F being Function of (Seg n),(Seg n) for k being Nat st k in Seg n holds Line ((M * F),k) = M . (F . k) let M be Matrix of n,m,D; ::_thesis: for F being Function of (Seg n),(Seg n) for k being Nat st k in Seg n holds Line ((M * F),k) = M . (F . k) let F be Function of (Seg n),(Seg n); ::_thesis: for k being Nat st k in Seg n holds Line ((M * F),k) = M . (F . k) let k be Nat; ::_thesis: ( k in Seg n implies Line ((M * F),k) = M . (F . k) ) assume A1: k in Seg n ; ::_thesis: Line ((M * F),k) = M . (F . k) len (M * F) = n by MATRIX_1:def_2; then k in dom (M * F) by A1, FINSEQ_1:def_3; then (M * F) . k = M . (F . k) by FUNCT_1:12; hence Line ((M * F),k) = M . (F . k) by A1, MATRIX_2:8; ::_thesis: verum end; theorem Th39: :: MATRIX11:39 for m, n being Nat for D being non empty set for M being Matrix of n,m,D holds M * (idseq n) = M proof let m, n be Nat; ::_thesis: for D being non empty set for M being Matrix of n,m,D holds M * (idseq n) = M let D be non empty set ; ::_thesis: for M being Matrix of n,m,D holds M * (idseq n) = M let M be Matrix of n,m,D; ::_thesis: M * (idseq n) = M reconsider I = idseq n as Permutation of (Seg n) ; A1: width (M * I) = width M by Def4; A2: for i, j being Nat st [i,j] in Indices M holds M * (i,j) = (M * I) * (i,j) proof let i, j be Nat; ::_thesis: ( [i,j] in Indices M implies M * (i,j) = (M * I) * (i,j) ) assume A3: [i,j] in Indices M ; ::_thesis: M * (i,j) = (M * I) * (i,j) [i,j] in [:(Seg n),(Seg (width M)):] by A3, MATRIX_1:25; then A4: i in Seg n by ZFMISC_1:87; ex k being Nat st ( I . i = k & [k,j] in Indices M & (M * I) * (i,j) = M * (k,j) ) by A3, Th37; hence M * (i,j) = (M * I) * (i,j) by A4, FUNCT_1:17; ::_thesis: verum end; len (M * I) = len M by Def4; hence M * (idseq n) = M by A1, A2, MATRIX_1:21; ::_thesis: verum end; theorem Th40: :: MATRIX11:40 for n being Nat for K being Field for A being Matrix of n,K for p being Element of Permutations n for Perm being Permutation of (Seg n) for q being Element of Permutations n st q = p * (Perm ") holds Path_matrix (p,(A * Perm)) = (Path_matrix (q,A)) * Perm proof let n be Nat; ::_thesis: for K being Field for A being Matrix of n,K for p being Element of Permutations n for Perm being Permutation of (Seg n) for q being Element of Permutations n st q = p * (Perm ") holds Path_matrix (p,(A * Perm)) = (Path_matrix (q,A)) * Perm let K be Field; ::_thesis: for A being Matrix of n,K for p being Element of Permutations n for Perm being Permutation of (Seg n) for q being Element of Permutations n st q = p * (Perm ") holds Path_matrix (p,(A * Perm)) = (Path_matrix (q,A)) * Perm let A be Matrix of n,K; ::_thesis: for p being Element of Permutations n for Perm being Permutation of (Seg n) for q being Element of Permutations n st q = p * (Perm ") holds Path_matrix (p,(A * Perm)) = (Path_matrix (q,A)) * Perm let p be Element of Permutations n; ::_thesis: for Perm being Permutation of (Seg n) for q being Element of Permutations n st q = p * (Perm ") holds Path_matrix (p,(A * Perm)) = (Path_matrix (q,A)) * Perm let Perm be Permutation of (Seg n); ::_thesis: for q being Element of Permutations n st q = p * (Perm ") holds Path_matrix (p,(A * Perm)) = (Path_matrix (q,A)) * Perm let q be Element of Permutations n; ::_thesis: ( q = p * (Perm ") implies Path_matrix (p,(A * Perm)) = (Path_matrix (q,A)) * Perm ) assume A1: q = p * (Perm ") ; ::_thesis: Path_matrix (p,(A * Perm)) = (Path_matrix (q,A)) * Perm reconsider perm = Perm as Element of Permutations n by MATRIX_2:def_9; set Ap = A * Perm; set P2 = Path_matrix (q,A); set P1 = Path_matrix (p,(A * Perm)); A2: dom perm = Seg n by FUNCT_2:52; A3: p is Permutation of (Seg n) by MATRIX_2:def_9; then A4: dom p = Seg n by FUNCT_2:52; A5: rng p = Seg n by A3, FUNCT_2:def_3; A6: q is Permutation of (Seg n) by MATRIX_2:def_9; then A7: dom q = Seg n by FUNCT_2:52; len (Path_matrix (q,A)) = n by MATRIX_3:def_7; then A8: dom (Path_matrix (q,A)) = Seg n by FINSEQ_1:def_3; A9: rng perm = Seg n by FUNCT_2:def_3; then A10: dom ((Path_matrix (q,A)) * perm) = Seg n by A2, A8, RELAT_1:27; then reconsider P2p = (Path_matrix (q,A)) * perm as FinSequence by FINSEQ_1:def_2; A11: len (Path_matrix (p,(A * Perm))) = n by MATRIX_3:def_7; A12: rng q = Seg n by A6, FUNCT_2:def_3; A13: now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_k_<=_len_(Path_matrix_(p,(A_*_Perm)))_holds_ P2p_._k_=_(Path_matrix_(p,(A_*_Perm)))_._k let k be Nat; ::_thesis: ( 1 <= k & k <= len (Path_matrix (p,(A * Perm))) implies P2p . k = (Path_matrix (p,(A * Perm))) . k ) assume that A14: 1 <= k and A15: k <= len (Path_matrix (p,(A * Perm))) ; ::_thesis: P2p . k = (Path_matrix (p,(A * Perm))) . k k in NAT by ORDINAL1:def_12; then A16: k in Seg n by A11, A14, A15; then A17: p . k in Seg n by A4, A5, FUNCT_1:3; then reconsider pk = p . k as Element of NAT ; A18: k = (perm ") . (perm . k) by A2, A16, FUNCT_1:34; [k,pk] in [:(Seg n),(Seg n):] by A16, A17, ZFMISC_1:87; then [k,pk] in Indices A by MATRIX_1:24; then consider permk being Nat such that A19: perm . k = permk and A20: [permk,pk] in Indices A and A21: (A * Perm) * (k,pk) = A * (permk,pk) by Th37; dom P2p = Seg n by A2, A9, A8, RELAT_1:27; then A22: P2p . k = (Path_matrix (q,A)) . permk by A16, A19, FUNCT_1:12; Indices A = [:(Seg n),(Seg n):] by MATRIX_1:24; then A23: permk in Seg n by A20, ZFMISC_1:87; then q . permk in Seg n by A7, A12, FUNCT_1:3; then reconsider qpermk = q . permk as Element of NAT ; A24: (Path_matrix (q,A)) . permk = A * (permk,qpermk) by A8, A23, MATRIX_3:def_7; A25: dom (Path_matrix (p,(A * Perm))) = Seg n by A11, FINSEQ_1:def_3; q . permk = p . ((perm ") . (perm . k)) by A1, A7, A19, A23, FUNCT_1:12; hence P2p . k = (Path_matrix (p,(A * Perm))) . k by A16, A21, A24, A22, A18, A25, MATRIX_3:def_7; ::_thesis: verum end; n is Element of NAT by ORDINAL1:def_12; then len P2p = n by A10, FINSEQ_1:def_3; hence Path_matrix (p,(A * Perm)) = (Path_matrix (q,A)) * Perm by A11, A13, FINSEQ_1:14; ::_thesis: verum end; theorem Th41: :: MATRIX11:41 for n being Nat for K being Field for A being Matrix of n,K for p being Element of Permutations n for Perm being Permutation of (Seg n) for q being Element of Permutations n st q = p * (Perm ") holds the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A)) proof let n be Nat; ::_thesis: for K being Field for A being Matrix of n,K for p being Element of Permutations n for Perm being Permutation of (Seg n) for q being Element of Permutations n st q = p * (Perm ") holds the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A)) let K be Field; ::_thesis: for A being Matrix of n,K for p being Element of Permutations n for Perm being Permutation of (Seg n) for q being Element of Permutations n st q = p * (Perm ") holds the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A)) let A be Matrix of n,K; ::_thesis: for p being Element of Permutations n for Perm being Permutation of (Seg n) for q being Element of Permutations n st q = p * (Perm ") holds the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A)) let p be Element of Permutations n; ::_thesis: for Perm being Permutation of (Seg n) for q being Element of Permutations n st q = p * (Perm ") holds the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A)) let Perm be Permutation of (Seg n); ::_thesis: for q being Element of Permutations n st q = p * (Perm ") holds the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A)) let q be Element of Permutations n; ::_thesis: ( q = p * (Perm ") implies the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A)) ) assume A1: q = p * (Perm ") ; ::_thesis: the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A)) set mm = the multF of K; set P2 = Path_matrix (q,A); set P1 = Path_matrix (p,(A * Perm)); now__::_thesis:_(_(_n_=_0_&_the_multF_of_K_$$_(Path_matrix_(p,(A_*_Perm)))_=_the_multF_of_K_$$_(Path_matrix_(q,A))_)_or_(_n_+_0_>_0_&_the_multF_of_K_$$_(Path_matrix_(p,(A_*_Perm)))_=_the_multF_of_K_$$_(Path_matrix_(q,A))_)_) percases ( n = 0 or n + 0 > 0 ) ; caseA2: n = 0 ; ::_thesis: the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A)) then len (Path_matrix (p,(A * Perm))) = 0 by MATRIX_3:def_7; then A3: the multF of K $$ (Path_matrix (p,(A * Perm))) = the_unity_wrt the multF of K by FINSOP_1:def_1; len (Path_matrix (q,A)) = 0 by A2, MATRIX_3:def_7; hence the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A)) by A3, FINSOP_1:def_1; ::_thesis: verum end; case n + 0 > 0 ; ::_thesis: the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A)) then A4: n >= 1 by NAT_1:19; A5: len (Path_matrix (q,A)) = n by MATRIX_3:def_7; A6: Perm is Element of Permutations n by MATRIX_2:def_9; Path_matrix (p,(A * Perm)) = (Path_matrix (q,A)) * Perm by A1, Th40; hence the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A)) by A4, A5, A6, MATRIX_7:33; ::_thesis: verum end; end; end; hence the multF of K $$ (Path_matrix (p,(A * Perm))) = the multF of K $$ (Path_matrix (q,A)) ; ::_thesis: verum end; theorem Th42: :: MATRIX11:42 for n being Nat for K being Field for p2, q2 being Element of Permutations (n + 2) st q2 = p2 " holds sgn (p2,K) = sgn (q2,K) proof let n be Nat; ::_thesis: for K being Field for p2, q2 being Element of Permutations (n + 2) st q2 = p2 " holds sgn (p2,K) = sgn (q2,K) let K be Field; ::_thesis: for p2, q2 being Element of Permutations (n + 2) st q2 = p2 " holds sgn (p2,K) = sgn (q2,K) A1: (n + 1) + 1 >= 0 + 1 by XREAL_1:6; let p2, q2 be Element of Permutations (n + 2); ::_thesis: ( q2 = p2 " implies sgn (p2,K) = sgn (q2,K) ) assume q2 = p2 " ; ::_thesis: sgn (p2,K) = sgn (q2,K) then A2: - ((1_ K),p2) = - ((1_ K),q2) by A1, MATRIX_7:29; A3: - ((1_ K),q2) = (sgn (q2,K)) * (1_ K) by Th26; - ((1_ K),p2) = (sgn (p2,K)) * (1_ K) by Th26; then (sgn (p2,K)) * (1_ K) = sgn (q2,K) by A2, A3, VECTSP_1:def_4; hence sgn (p2,K) = sgn (q2,K) by VECTSP_1:def_4; ::_thesis: verum end; theorem Th43: :: MATRIX11:43 for n being Nat for K being Field for M being Matrix of n + 2,K for perm2 being Element of Permutations (n + 2) for Perm2 being Permutation of (Seg (n + 2)) st perm2 = Perm2 holds for p2, q2 being Element of Permutations (n + 2) st q2 = p2 * (Perm2 ") holds (Path_product M) . q2 = (sgn (perm2,K)) * ((Path_product (M * Perm2)) . p2) proof let n be Nat; ::_thesis: for K being Field for M being Matrix of n + 2,K for perm2 being Element of Permutations (n + 2) for Perm2 being Permutation of (Seg (n + 2)) st perm2 = Perm2 holds for p2, q2 being Element of Permutations (n + 2) st q2 = p2 * (Perm2 ") holds (Path_product M) . q2 = (sgn (perm2,K)) * ((Path_product (M * Perm2)) . p2) let K be Field; ::_thesis: for M being Matrix of n + 2,K for perm2 being Element of Permutations (n + 2) for Perm2 being Permutation of (Seg (n + 2)) st perm2 = Perm2 holds for p2, q2 being Element of Permutations (n + 2) st q2 = p2 * (Perm2 ") holds (Path_product M) . q2 = (sgn (perm2,K)) * ((Path_product (M * Perm2)) . p2) let M be Matrix of n + 2,K; ::_thesis: for perm2 being Element of Permutations (n + 2) for Perm2 being Permutation of (Seg (n + 2)) st perm2 = Perm2 holds for p2, q2 being Element of Permutations (n + 2) st q2 = p2 * (Perm2 ") holds (Path_product M) . q2 = (sgn (perm2,K)) * ((Path_product (M * Perm2)) . p2) let perm2 be Element of Permutations (n + 2); ::_thesis: for Perm2 being Permutation of (Seg (n + 2)) st perm2 = Perm2 holds for p2, q2 being Element of Permutations (n + 2) st q2 = p2 * (Perm2 ") holds (Path_product M) . q2 = (sgn (perm2,K)) * ((Path_product (M * Perm2)) . p2) let Perm2 be Permutation of (Seg (n + 2)); ::_thesis: ( perm2 = Perm2 implies for p2, q2 being Element of Permutations (n + 2) st q2 = p2 * (Perm2 ") holds (Path_product M) . q2 = (sgn (perm2,K)) * ((Path_product (M * Perm2)) . p2) ) assume A1: perm2 = Perm2 ; ::_thesis: for p2, q2 being Element of Permutations (n + 2) st q2 = p2 * (Perm2 ") holds (Path_product M) . q2 = (sgn (perm2,K)) * ((Path_product (M * Perm2)) . p2) set P = Permutations (n + 2); set mm = the multF of K; let p2, q2 be Element of Permutations (n + 2); ::_thesis: ( q2 = p2 * (Perm2 ") implies (Path_product M) . q2 = (sgn (perm2,K)) * ((Path_product (M * Perm2)) . p2) ) assume A2: q2 = p2 * (Perm2 ") ; ::_thesis: (Path_product M) . q2 = (sgn (perm2,K)) * ((Path_product (M * Perm2)) . p2) reconsider perm29 = perm2 " as Element of Permutations (n + 2) by MATRIX_7:18; set PM = the multF of K $$ (Path_matrix (q2,M)); set PMp = the multF of K $$ (Path_matrix (p2,(M * Perm2))); sgn (q2,K) = (sgn (p2,K)) * (sgn (perm29,K)) by A1, A2, Th24 .= (sgn (p2,K)) * (sgn (perm2,K)) by Th42 ; then - (( the multF of K $$ (Path_matrix (q2,M))),q2) = ((sgn (perm2,K)) * (sgn (p2,K))) * ( the multF of K $$ (Path_matrix (q2,M))) by Th26 .= (sgn (perm2,K)) * ((sgn (p2,K)) * ( the multF of K $$ (Path_matrix (q2,M)))) by GROUP_1:def_3 .= (sgn (perm2,K)) * ((sgn (p2,K)) * ( the multF of K $$ (Path_matrix (p2,(M * Perm2))))) by A2, Th41 .= (sgn (perm2,K)) * (- (( the multF of K $$ (Path_matrix (p2,(M * Perm2)))),p2)) by Th26 .= (sgn (perm2,K)) * ((Path_product (M * Perm2)) . p2) by MATRIX_3:def_8 ; hence (Path_product M) . q2 = (sgn (perm2,K)) * ((Path_product (M * Perm2)) . p2) by MATRIX_3:def_8; ::_thesis: verum end; theorem Th44: :: MATRIX11:44 for n being Nat for perm being Element of Permutations n ex P being Permutation of (Permutations n) st for p being Element of Permutations n holds P . p = p * perm proof let n be Nat; ::_thesis: for perm being Element of Permutations n ex P being Permutation of (Permutations n) st for p being Element of Permutations n holds P . p = p * perm let perm be Element of Permutations n; ::_thesis: ex P being Permutation of (Permutations n) st for p being Element of Permutations n holds P . p = p * perm set P = Permutations n; defpred S1[ set , set ] means for p being Element of Permutations n st $1 = p holds $2 = p * perm; A1: card (Permutations n) = card (Permutations n) ; A2: for x being set st x in Permutations n holds ex y being set st ( y in Permutations n & S1[x,y] ) proof let x be set ; ::_thesis: ( x in Permutations n implies ex y being set st ( y in Permutations n & S1[x,y] ) ) assume x in Permutations n ; ::_thesis: ex y being set st ( y in Permutations n & S1[x,y] ) then reconsider p = x as Element of Permutations n ; reconsider pp = p * perm as Element of Permutations n by MATRIX_9:39; take pp ; ::_thesis: ( pp in Permutations n & S1[x,pp] ) thus ( pp in Permutations n & S1[x,pp] ) ; ::_thesis: verum end; consider G being Function of (Permutations n),(Permutations n) such that A3: for x being set st x in Permutations n holds S1[x,G . x] from FUNCT_2:sch_1(A2); for x1, x2 being set st x1 in Permutations n & x2 in Permutations n & G . x1 = G . x2 holds x1 = x2 proof let x1, x2 be set ; ::_thesis: ( x1 in Permutations n & x2 in Permutations n & G . x1 = G . x2 implies x1 = x2 ) assume that A4: x1 in Permutations n and A5: x2 in Permutations n and A6: G . x1 = G . x2 ; ::_thesis: x1 = x2 reconsider p1 = x1, p2 = x2 as Element of Permutations n by A4, A5; p2 is Permutation of (Seg n) by MATRIX_2:def_9; then A7: dom p2 = Seg n by FUNCT_2:52; A8: G . p2 = p2 * perm by A3; A9: G . p1 = p1 * perm by A3; perm is Permutation of (Seg n) by MATRIX_2:def_9; then A10: rng perm = Seg n by FUNCT_2:def_3; p1 is Permutation of (Seg n) by MATRIX_2:def_9; then dom p1 = Seg n by FUNCT_2:52; then p1 = p1 * (id (rng perm)) by A10, RELAT_1:52 .= p1 * (perm * (perm ")) by FUNCT_1:39 .= (p2 * perm) * (perm ") by A6, A9, A8, RELAT_1:36 .= p2 * (perm * (perm ")) by RELAT_1:36 .= p2 * (id (rng perm)) by FUNCT_1:39 .= p2 by A10, A7, RELAT_1:52 ; hence x1 = x2 ; ::_thesis: verum end; then A11: G is one-to-one by FUNCT_2:19; Permutations n is finite set by MATRIX_2:26; then G is onto by A11, A1, STIRL2_1:60; then reconsider G = G as Permutation of (Permutations n) by A11; take G ; ::_thesis: for p being Element of Permutations n holds G . p = p * perm thus for p being Element of Permutations n holds G . p = p * perm by A3; ::_thesis: verum end; theorem Th45: :: MATRIX11:45 for n being Nat for K being Field for M being Matrix of n + 2,n + 2,K for perm2 being Element of Permutations (n + 2) for Perm2 being Permutation of (Seg (n + 2)) st perm2 = Perm2 holds Det (M * Perm2) = (sgn (perm2,K)) * (Det M) proof let n be Nat; ::_thesis: for K being Field for M being Matrix of n + 2,n + 2,K for perm2 being Element of Permutations (n + 2) for Perm2 being Permutation of (Seg (n + 2)) st perm2 = Perm2 holds Det (M * Perm2) = (sgn (perm2,K)) * (Det M) let K be Field; ::_thesis: for M being Matrix of n + 2,n + 2,K for perm2 being Element of Permutations (n + 2) for Perm2 being Permutation of (Seg (n + 2)) st perm2 = Perm2 holds Det (M * Perm2) = (sgn (perm2,K)) * (Det M) set n2 = n + 2; let M be Matrix of n + 2,n + 2,K; ::_thesis: for perm2 being Element of Permutations (n + 2) for Perm2 being Permutation of (Seg (n + 2)) st perm2 = Perm2 holds Det (M * Perm2) = (sgn (perm2,K)) * (Det M) let perm2 be Element of Permutations (n + 2); ::_thesis: for Perm2 being Permutation of (Seg (n + 2)) st perm2 = Perm2 holds Det (M * Perm2) = (sgn (perm2,K)) * (Det M) let Perm2 be Permutation of (Seg (n + 2)); ::_thesis: ( perm2 = Perm2 implies Det (M * Perm2) = (sgn (perm2,K)) * (Det M) ) assume A1: perm2 = Perm2 ; ::_thesis: Det (M * Perm2) = (sgn (perm2,K)) * (Det M) set PathM = Path_product M; set Mperm = M * Perm2; set P = Permutations (n + 2); set KK = the carrier of K; set aa = the addF of K; set PathMp = Path_product (M * Perm2); set F = FinOmega (Permutations (n + 2)); reconsider perm29 = perm2 " as Element of Permutations (n + 2) by MATRIX_7:18; A2: FinOmega (Permutations (n + 2)) = Permutations (n + 2) by MATRIX_2:26, MATRIX_2:def_14; then consider GM being Function of (Fin (Permutations (n + 2))), the carrier of K such that A3: Det M = GM . (FinOmega (Permutations (n + 2))) and for e being Element of the carrier of K st e is_a_unity_wrt the addF of K holds GM . {} = e and A4: for x being Element of Permutations (n + 2) holds GM . {x} = (Path_product M) . x and A5: for B9 being Element of Fin (Permutations (n + 2)) st B9 c= FinOmega (Permutations (n + 2)) & B9 <> {} holds for x being Element of Permutations (n + 2) st x in (FinOmega (Permutations (n + 2))) \ B9 holds GM . (B9 \/ {x}) = the addF of K . ((GM . B9),((Path_product M) . x)) by SETWISEO:def_3; consider PERM being Permutation of (Permutations (n + 2)) such that A6: for p being Element of Permutations (n + 2) holds PERM . p = p * perm29 by Th44; consider GMp being Function of (Fin (Permutations (n + 2))), the carrier of K such that A7: Det (M * Perm2) = GMp . (FinOmega (Permutations (n + 2))) and for e being Element of the carrier of K st e is_a_unity_wrt the addF of K holds GMp . {} = e and A8: for x being Element of Permutations (n + 2) holds GMp . {x} = (Path_product (M * Perm2)) . x and A9: for B9 being Element of Fin (Permutations (n + 2)) st B9 c= FinOmega (Permutations (n + 2)) & B9 <> {} holds for x being Element of Permutations (n + 2) st x in (FinOmega (Permutations (n + 2))) \ B9 holds GMp . (B9 \/ {x}) = the addF of K . ((GMp . B9),((Path_product (M * Perm2)) . x)) by A2, SETWISEO:def_3; defpred S1[ Nat] means ( $1 <> 0 implies for B being Element of Fin (Permutations (n + 2)) st card B = $1 holds (sgn (perm2,K)) * (GMp . B) = GM . (PERM .: B) ); A10: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A11: S1[k] ; ::_thesis: S1[k + 1] set k1 = k + 1; assume k + 1 <> 0 ; ::_thesis: for B being Element of Fin (Permutations (n + 2)) st card B = k + 1 holds (sgn (perm2,K)) * (GMp . B) = GM . (PERM .: B) let B be Element of Fin (Permutations (n + 2)); ::_thesis: ( card B = k + 1 implies (sgn (perm2,K)) * (GMp . B) = GM . (PERM .: B) ) assume A12: card B = k + 1 ; ::_thesis: (sgn (perm2,K)) * (GMp . B) = GM . (PERM .: B) percases ( k = 0 or k > 0 ) ; suppose k = 0 ; ::_thesis: (sgn (perm2,K)) * (GMp . B) = GM . (PERM .: B) then consider x being set such that A13: B = {x} by A12, CARD_2:42; A14: x in B by A13, TARSKI:def_1; B c= Permutations (n + 2) by FINSUB_1:def_5; then reconsider x = x as Element of Permutations (n + 2) by A14; A15: GM . {(PERM . x)} = (Path_product M) . (PERM . x) by A4; A16: PERM . x = x * perm29 by A6; A17: Permutations (n + 2) = dom PERM by FUNCT_2:52; GMp . {x} = (Path_product (M * Perm2)) . x by A8; then (sgn (perm2,K)) * (GMp . B) = GM . {(PERM . x)} by A1, A13, A15, A16, Th43; then (sgn (perm2,K)) * (GMp . B) = GM . (Im (PERM,x)) by A17, FUNCT_1:59; hence (sgn (perm2,K)) * (GMp . B) = GM . (PERM .: B) by A13; ::_thesis: verum end; supposeA18: k > 0 ; ::_thesis: (sgn (perm2,K)) * (GMp . B) = GM . (PERM .: B) consider x being set such that A19: x in B by A12, CARD_1:27, XBOOLE_0:def_1; B c= Permutations (n + 2) by FINSUB_1:def_5; then reconsider x = x as Element of Permutations (n + 2) by A19; PERM .: (B \ {x}) c= rng PERM by RELAT_1:111; then A20: PERM .: (B \ {x}) c= Permutations (n + 2) by FUNCT_2:def_3; reconsider Px = PERM . x as Element of Permutations (n + 2) ; A21: Px in {Px} by TARSKI:def_1; dom PERM = Permutations (n + 2) by FUNCT_2:52; then A22: Im (PERM,x) = {Px} by FUNCT_1:59; A23: B c= Permutations (n + 2) by FINSUB_1:def_5; then B \ {x} c= Permutations (n + 2) by XBOOLE_1:1; then reconsider B9 = B \ {x}, PeBx = PERM .: (B \ {x}), PeB = PERM .: B as Element of Fin (Permutations (n + 2)) by A20, FINSUB_1:def_5; A24: {x} \/ B9 = B by A19, ZFMISC_1:116; then A25: PERM .: B = (Im (PERM,x)) \/ PeBx by RELAT_1:120; PERM . x = x * perm29 by A6; then A26: (sgn (perm2,K)) * ((Path_product (M * Perm2)) . x) = (Path_product M) . Px by A1, Th43; A27: dom PERM = Permutations (n + 2) by FUNCT_2:52; B9 misses {x} by XBOOLE_1:79; then B9 /\ {x} = {} by XBOOLE_0:def_7; then PERM .: {} = {Px} /\ PeBx by A22, FUNCT_1:62; then not Px in PeBx by A21, XBOOLE_0:def_4; then A28: Px in (FinOmega (Permutations (n + 2))) \ PeBx by A2, XBOOLE_0:def_5; A29: B9 c= Permutations (n + 2) by FINSUB_1:def_5; A30: not x in B9 by ZFMISC_1:56; then A31: x in (FinOmega (Permutations (n + 2))) \ B9 by A2, XBOOLE_0:def_5; A32: k + 1 = (card B9) + 1 by A12, A24, A30, CARD_2:41; then ex y being set st y in B9 by A18, CARD_1:27, XBOOLE_0:def_1; then GM . PeB = the addF of K . ((GM . PeBx),((Path_product M) . Px)) by A2, A5, A20, A25, A22, A28, A29, A27; then GM . PeB = ((sgn (perm2,K)) * (GMp . B9)) + ((sgn (perm2,K)) * ((Path_product (M * Perm2)) . x)) by A11, A18, A32, A26 .= (sgn (perm2,K)) * ((GMp . B9) + ((Path_product (M * Perm2)) . x)) by VECTSP_1:def_7 .= (sgn (perm2,K)) * (GMp . B) by A2, A9, A18, A23, A24, A32, A31, CARD_1:27, XBOOLE_1:1 ; hence (sgn (perm2,K)) * (GMp . B) = GM . (PERM .: B) ; ::_thesis: verum end; end; end; A33: S1[ 0 ] ; A34: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A33, A10); A35: rng PERM = Permutations (n + 2) by FUNCT_2:def_3; A36: dom PERM = Permutations (n + 2) by FUNCT_2:52; A37: PERM .: (dom PERM) = rng PERM by RELAT_1:113; A38: (1_ K) * (1_ K) = (- (1_ K)) * (- (1_ K)) by VECTSP_1:10; A39: ( sgn (perm2,K) = 1_ K or sgn (perm2,K) = - (1_ K) ) by Th11; A40: (1_ K) * (1_ K) = 1_ K by VECTSP_1:def_4; card (FinOmega (Permutations (n + 2))) <> 0 by A2; then (sgn (perm2,K)) * (Det (M * Perm2)) = Det M by A2, A3, A7, A34, A37, A36, A35; hence (sgn (perm2,K)) * (Det M) = (1_ K) * (Det (M * Perm2)) by A39, A40, A38, GROUP_1:def_3 .= Det (M * Perm2) by VECTSP_1:def_4 ; ::_thesis: verum end; theorem Th46: :: MATRIX11:46 for n being Nat for K being Field for M being Matrix of n,K for perm being Element of Permutations n for Perm being Permutation of (Seg n) st perm = Perm holds Det (M * Perm) = - ((Det M),perm) proof let n be Nat; ::_thesis: for K being Field for M being Matrix of n,K for perm being Element of Permutations n for Perm being Permutation of (Seg n) st perm = Perm holds Det (M * Perm) = - ((Det M),perm) let K be Field; ::_thesis: for M being Matrix of n,K for perm being Element of Permutations n for Perm being Permutation of (Seg n) st perm = Perm holds Det (M * Perm) = - ((Det M),perm) let M be Matrix of n,K; ::_thesis: for perm being Element of Permutations n for Perm being Permutation of (Seg n) st perm = Perm holds Det (M * Perm) = - ((Det M),perm) let perm be Element of Permutations n; ::_thesis: for Perm being Permutation of (Seg n) st perm = Perm holds Det (M * Perm) = - ((Det M),perm) let Perm be Permutation of (Seg n); ::_thesis: ( perm = Perm implies Det (M * Perm) = - ((Det M),perm) ) assume A1: Perm = perm ; ::_thesis: Det (M * Perm) = - ((Det M),perm) percases ( n < 2 or n >= 2 ) ; supposeA2: n < 2 ; ::_thesis: Det (M * Perm) = - ((Det M),perm) then perm = idseq n by Lm3; then A3: M * perm = M by Th39; perm is even by A2, Lm3; hence Det (M * Perm) = - ((Det M),perm) by A1, A3, MATRIX_2:def_13; ::_thesis: verum end; suppose n >= 2 ; ::_thesis: Det (M * Perm) = - ((Det M),perm) then reconsider n2 = n - 2 as Nat by NAT_1:21; reconsider M9 = M as Matrix of n2 + 2,K ; reconsider Perm2 = Perm as Permutation of (Seg (n2 + 2)) ; reconsider perm2 = perm as Element of Permutations (n2 + 2) ; Det (M9 * Perm2) = (sgn (perm2,K)) * (Det M9) by A1, Th45; hence Det (M * Perm) = - ((Det M),perm) by Th26; ::_thesis: verum end; end; end; theorem Th47: :: MATRIX11:47 for n being Nat for PERM being Permutation of (Permutations n) for perm being Element of Permutations n st perm is odd & ( for p being Element of Permutations n holds PERM . p = p * perm ) holds PERM .: { p where p is Element of Permutations n : p is even } = { q where q is Element of Permutations n : q is odd } proof let n be Nat; ::_thesis: for PERM being Permutation of (Permutations n) for perm being Element of Permutations n st perm is odd & ( for p being Element of Permutations n holds PERM . p = p * perm ) holds PERM .: { p where p is Element of Permutations n : p is even } = { q where q is Element of Permutations n : q is odd } set P = Permutations n; let PERM be Permutation of (Permutations n); ::_thesis: for perm being Element of Permutations n st perm is odd & ( for p being Element of Permutations n holds PERM . p = p * perm ) holds PERM .: { p where p is Element of Permutations n : p is even } = { q where q is Element of Permutations n : q is odd } let perm be Element of Permutations n; ::_thesis: ( perm is odd & ( for p being Element of Permutations n holds PERM . p = p * perm ) implies PERM .: { p where p is Element of Permutations n : p is even } = { q where q is Element of Permutations n : q is odd } ) assume that A1: perm is odd and A2: for p being Element of Permutations n holds PERM . p = p * perm ; ::_thesis: PERM .: { p where p is Element of Permutations n : p is even } = { q where q is Element of Permutations n : q is odd } set E = { p where p is Element of Permutations n : p is even } ; set OD = { q where q is Element of Permutations n : q is odd } ; for y being set holds ( y in { q where q is Element of Permutations n : q is odd } iff ex x being set st ( x in dom PERM & x in { p where p is Element of Permutations n : p is even } & y = PERM . x ) ) proof let y be set ; ::_thesis: ( y in { q where q is Element of Permutations n : q is odd } iff ex x being set st ( x in dom PERM & x in { p where p is Element of Permutations n : p is even } & y = PERM . x ) ) thus ( y in { q where q is Element of Permutations n : q is odd } implies ex x being set st ( x in dom PERM & x in { p where p is Element of Permutations n : p is even } & y = PERM . x ) ) ::_thesis: ( ex x being set st ( x in dom PERM & x in { p where p is Element of Permutations n : p is even } & y = PERM . x ) implies y in { q where q is Element of Permutations n : q is odd } ) proof reconsider perm9 = perm " as Element of Permutations n by MATRIX_7:18; A3: dom PERM = Permutations n by FUNCT_2:52; n >= 2 by A1, Lm3; then A4: n >= 1 by XXREAL_0:2; assume y in { q where q is Element of Permutations n : q is odd } ; ::_thesis: ex x being set st ( x in dom PERM & x in { p where p is Element of Permutations n : p is even } & y = PERM . x ) then consider q being Element of Permutations n such that A5: y = q and A6: q is odd ; A7: q * (idseq n) = q by MATRIX_2:21; n is Element of NAT by ORDINAL1:def_12; then perm9 is odd by A1, A4, MATRIX_7:28; then A8: q * perm9 is even by A6, Th25; reconsider qp9 = q * perm9 as Element of Permutations n by MATRIX_9:39; take qp9 ; ::_thesis: ( qp9 in dom PERM & qp9 in { p where p is Element of Permutations n : p is even } & y = PERM . qp9 ) A9: perm9 * perm = idseq n by MATRIX_2:22; PERM . qp9 = qp9 * perm by A2; hence ( qp9 in dom PERM & qp9 in { p where p is Element of Permutations n : p is even } & y = PERM . qp9 ) by A5, A3, A9, A7, A8, RELAT_1:36; ::_thesis: verum end; assume ex x being set st ( x in dom PERM & x in { p where p is Element of Permutations n : p is even } & y = PERM . x ) ; ::_thesis: y in { q where q is Element of Permutations n : q is odd } then consider x being set such that x in dom PERM and A10: x in { p where p is Element of Permutations n : p is even } and A11: y = PERM . x ; consider p being Element of Permutations n such that A12: p = x and A13: p is even by A10; reconsider pp = p * perm as Element of Permutations n by MATRIX_9:39; A14: PERM . x = p * perm by A2, A12; pp is odd by A1, A13, Th25; hence y in { q where q is Element of Permutations n : q is odd } by A11, A14; ::_thesis: verum end; hence PERM .: { p where p is Element of Permutations n : p is even } = { q where q is Element of Permutations n : q is odd } by FUNCT_1:def_6; ::_thesis: verum end; Lm8: for n, i, j being Nat st i in Seg n & j in Seg n & i < j holds ex ODD, EVEN being finite set st ( EVEN = { p where p is Element of Permutations n : p is even } & ODD = { q where q is Element of Permutations n : q is odd } & EVEN /\ ODD = {} & EVEN \/ ODD = Permutations n & ex PERM being Function of EVEN,ODD ex perm being Element of Permutations n st ( perm is being_transposition & perm . i = j & dom PERM = EVEN & PERM is bijective & ( for p being Element of Permutations n st p in EVEN holds PERM . p = p * perm ) ) ) proof let n, i, j be Nat; ::_thesis: ( i in Seg n & j in Seg n & i < j implies ex ODD, EVEN being finite set st ( EVEN = { p where p is Element of Permutations n : p is even } & ODD = { q where q is Element of Permutations n : q is odd } & EVEN /\ ODD = {} & EVEN \/ ODD = Permutations n & ex PERM being Function of EVEN,ODD ex perm being Element of Permutations n st ( perm is being_transposition & perm . i = j & dom PERM = EVEN & PERM is bijective & ( for p being Element of Permutations n st p in EVEN holds PERM . p = p * perm ) ) ) ) assume that A1: i in Seg n and A2: j in Seg n and A3: i < j ; ::_thesis: ex ODD, EVEN being finite set st ( EVEN = { p where p is Element of Permutations n : p is even } & ODD = { q where q is Element of Permutations n : q is odd } & EVEN /\ ODD = {} & EVEN \/ ODD = Permutations n & ex PERM being Function of EVEN,ODD ex perm being Element of Permutations n st ( perm is being_transposition & perm . i = j & dom PERM = EVEN & PERM is bijective & ( for p being Element of Permutations n st p in EVEN holds PERM . p = p * perm ) ) ) set P = Permutations n; consider tr being Element of Permutations n such that A4: tr is being_transposition and A5: tr . i = j by A1, A2, A3, Th16; {i,j} in 2Set (Seg n) by A1, A2, A3, Th1; then reconsider n2 = n - 2 as Nat by Th2, NAT_1:21, NAT_1:23; set ODD = { q where q is Element of Permutations n : q is odd } ; set EVEN = { p where p is Element of Permutations n : p is even } ; A6: { p where p is Element of Permutations n : p is even } c= Permutations n proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Element of Permutations n : p is even } or x in Permutations n ) assume x in { p where p is Element of Permutations n : p is even } ; ::_thesis: x in Permutations n then ex p being Element of Permutations n st ( p = x & p is even ) ; hence x in Permutations n ; ::_thesis: verum end; A7: { q where q is Element of Permutations n : q is odd } c= Permutations n proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { q where q is Element of Permutations n : q is odd } or x in Permutations n ) assume x in { q where q is Element of Permutations n : q is odd } ; ::_thesis: x in Permutations n then ex q being Element of Permutations n st ( q = x & q is odd ) ; hence x in Permutations n ; ::_thesis: verum end; then reconsider O = { q where q is Element of Permutations n : q is odd } , E = { p where p is Element of Permutations n : p is even } as finite set by A6, FINSET_1:1, MATRIX_2:26; take O ; ::_thesis: ex EVEN being finite set st ( EVEN = { p where p is Element of Permutations n : p is even } & O = { q where q is Element of Permutations n : q is odd } & EVEN /\ O = {} & EVEN \/ O = Permutations n & ex PERM being Function of EVEN,O ex perm being Element of Permutations n st ( perm is being_transposition & perm . i = j & dom PERM = EVEN & PERM is bijective & ( for p being Element of Permutations n st p in EVEN holds PERM . p = p * perm ) ) ) take E ; ::_thesis: ( E = { p where p is Element of Permutations n : p is even } & O = { q where q is Element of Permutations n : q is odd } & E /\ O = {} & E \/ O = Permutations n & ex PERM being Function of E,O ex perm being Element of Permutations n st ( perm is being_transposition & perm . i = j & dom PERM = E & PERM is bijective & ( for p being Element of Permutations n st p in E holds PERM . p = p * perm ) ) ) thus ( E = { p where p is Element of Permutations n : p is even } & O = { q where q is Element of Permutations n : q is odd } ) ; ::_thesis: ( E /\ O = {} & E \/ O = Permutations n & ex PERM being Function of E,O ex perm being Element of Permutations n st ( perm is being_transposition & perm . i = j & dom PERM = E & PERM is bijective & ( for p being Element of Permutations n st p in E holds PERM . p = p * perm ) ) ) thus E /\ O = {} ::_thesis: ( E \/ O = Permutations n & ex PERM being Function of E,O ex perm being Element of Permutations n st ( perm is being_transposition & perm . i = j & dom PERM = E & PERM is bijective & ( for p being Element of Permutations n st p in E holds PERM . p = p * perm ) ) ) proof assume E /\ O <> {} ; ::_thesis: contradiction then consider x being set such that A8: x in E /\ O by XBOOLE_0:def_1; x in O by A8, XBOOLE_0:def_4; then A9: ex q being Element of Permutations n st ( q = x & q is odd ) ; x in E by A8, XBOOLE_0:def_4; then ex p being Element of Permutations n st ( p = x & p is even ) ; hence contradiction by A9; ::_thesis: verum end; thus E \/ O = Permutations n ::_thesis: ex PERM being Function of E,O ex perm being Element of Permutations n st ( perm is being_transposition & perm . i = j & dom PERM = E & PERM is bijective & ( for p being Element of Permutations n st p in E holds PERM . p = p * perm ) ) proof thus E \/ O c= Permutations n by A6, A7, XBOOLE_1:8; :: according to XBOOLE_0:def_10 ::_thesis: Permutations n c= E \/ O let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Permutations n or x in E \/ O ) assume x in Permutations n ; ::_thesis: x in E \/ O then reconsider p = x as Element of Permutations n ; ( p is even or p is odd ) ; then ( p in E or p in O ) ; hence x in E \/ O by XBOOLE_0:def_3; ::_thesis: verum end; consider PE being Permutation of (Permutations n) such that A10: for p being Element of Permutations n holds PE . p = p * tr by Th44; set PERM = PE | E; tr is Element of Permutations (n2 + 2) ; then PE .: E = O by A4, A10, Th27, Th47; then A11: rng (PE | E) = O by RELAT_1:115; A12: dom PE = Permutations n by FUNCT_2:52; then A13: dom (PE | E) = E by A6, RELAT_1:62; then reconsider PERM = PE | E as Function of E,O by A11, FUNCT_2:1; take PERM ; ::_thesis: ex perm being Element of Permutations n st ( perm is being_transposition & perm . i = j & dom PERM = E & PERM is bijective & ( for p being Element of Permutations n st p in E holds PERM . p = p * perm ) ) take tr ; ::_thesis: ( tr is being_transposition & tr . i = j & dom PERM = E & PERM is bijective & ( for p being Element of Permutations n st p in E holds PERM . p = p * tr ) ) ( PERM is one-to-one & PERM is onto ) by A11, FUNCT_1:52, FUNCT_2:def_3; hence ( tr is being_transposition & tr . i = j & dom PERM = E & PERM is bijective ) by A4, A5, A6, A12, RELAT_1:62; ::_thesis: for p being Element of Permutations n st p in E holds PERM . p = p * tr let p be Element of Permutations n; ::_thesis: ( p in E implies PERM . p = p * tr ) assume p in E ; ::_thesis: PERM . p = p * tr then PERM . p = PE . p by A13, FUNCT_1:47; hence PERM . p = p * tr by A10; ::_thesis: verum end; theorem :: MATRIX11:48 for n being Nat st n >= 2 holds ex ODD, EVEN being finite set st ( EVEN = { p where p is Element of Permutations n : p is even } & ODD = { q where q is Element of Permutations n : q is odd } & EVEN /\ ODD = {} & EVEN \/ ODD = Permutations n & card EVEN = card ODD ) proof let n be Nat; ::_thesis: ( n >= 2 implies ex ODD, EVEN being finite set st ( EVEN = { p where p is Element of Permutations n : p is even } & ODD = { q where q is Element of Permutations n : q is odd } & EVEN /\ ODD = {} & EVEN \/ ODD = Permutations n & card EVEN = card ODD ) ) assume A1: n >= 2 ; ::_thesis: ex ODD, EVEN being finite set st ( EVEN = { p where p is Element of Permutations n : p is even } & ODD = { q where q is Element of Permutations n : q is odd } & EVEN /\ ODD = {} & EVEN \/ ODD = Permutations n & card EVEN = card ODD ) 1 <= n by A1, XXREAL_0:2; then A2: 1 in Seg n ; 2 in Seg n by A1; then consider O, E being finite set such that A3: ( E = { p where p is Element of Permutations n : p is even } & O = { q where q is Element of Permutations n : q is odd } ) and A4: ( E /\ O = {} & E \/ O = Permutations n ) and A5: ex P being Function of E,O ex perm being Element of Permutations n st ( perm is being_transposition & perm . 1 = 2 & dom P = E & P is bijective & ( for p being Element of Permutations n st p in E holds P . p = p * perm ) ) by A2, Lm8; consider P being Function of E,O, perm being Element of Permutations n such that perm is being_transposition and perm . 1 = 2 and A6: dom P = E and A7: P is bijective and for p being Element of Permutations n st p in E holds P . p = p * perm by A5; rng P = O by A7, FUNCT_2:def_3; then E,O are_equipotent by A6, A7, WELLORD2:def_4; then card E = card O by CARD_1:5; hence ex ODD, EVEN being finite set st ( EVEN = { p where p is Element of Permutations n : p is even } & ODD = { q where q is Element of Permutations n : q is odd } & EVEN /\ ODD = {} & EVEN \/ ODD = Permutations n & card EVEN = card ODD ) by A3, A4; ::_thesis: verum end; theorem Th49: :: MATRIX11:49 for n being Nat for K being Field for i, j being Nat st i in Seg n & j in Seg n & i < j holds for M being Matrix of n,K st Line (M,i) = Line (M,j) holds for p, q, tr being Element of Permutations n st q = p * tr & tr is being_transposition & tr . i = j holds (Path_product M) . q = - ((Path_product M) . p) proof let n be Nat; ::_thesis: for K being Field for i, j being Nat st i in Seg n & j in Seg n & i < j holds for M being Matrix of n,K st Line (M,i) = Line (M,j) holds for p, q, tr being Element of Permutations n st q = p * tr & tr is being_transposition & tr . i = j holds (Path_product M) . q = - ((Path_product M) . p) let K be Field; ::_thesis: for i, j being Nat st i in Seg n & j in Seg n & i < j holds for M being Matrix of n,K st Line (M,i) = Line (M,j) holds for p, q, tr being Element of Permutations n st q = p * tr & tr is being_transposition & tr . i = j holds (Path_product M) . q = - ((Path_product M) . p) let i, j be Nat; ::_thesis: ( i in Seg n & j in Seg n & i < j implies for M being Matrix of n,K st Line (M,i) = Line (M,j) holds for p, q, tr being Element of Permutations n st q = p * tr & tr is being_transposition & tr . i = j holds (Path_product M) . q = - ((Path_product M) . p) ) assume that A1: i in Seg n and A2: j in Seg n and A3: i < j ; ::_thesis: for M being Matrix of n,K st Line (M,i) = Line (M,j) holds for p, q, tr being Element of Permutations n st q = p * tr & tr is being_transposition & tr . i = j holds (Path_product M) . q = - ((Path_product M) . p) {i,j} in 2Set (Seg n) by A1, A2, A3, Th1; then reconsider n2 = n - 2 as Nat by Th2, NAT_1:21, NAT_1:23; let M be Matrix of n,K; ::_thesis: ( Line (M,i) = Line (M,j) implies for p, q, tr being Element of Permutations n st q = p * tr & tr is being_transposition & tr . i = j holds (Path_product M) . q = - ((Path_product M) . p) ) assume A4: Line (M,i) = Line (M,j) ; ::_thesis: for p, q, tr being Element of Permutations n st q = p * tr & tr is being_transposition & tr . i = j holds (Path_product M) . q = - ((Path_product M) . p) reconsider M9 = M as Matrix of n2 + 2,K ; let p, q, tr be Element of Permutations n; ::_thesis: ( q = p * tr & tr is being_transposition & tr . i = j implies (Path_product M) . q = - ((Path_product M) . p) ) assume that A5: q = p * tr and A6: tr is being_transposition and A7: tr . i = j ; ::_thesis: (Path_product M) . q = - ((Path_product M) . p) reconsider TR = tr as Permutation of (Seg (n2 + 2)) by MATRIX_2:def_9; set Mt = M9 * TR; A8: for k being Nat st 1 <= k & k <= len M9 holds M9 . k = (M9 * TR) . k proof let k be Nat; ::_thesis: ( 1 <= k & k <= len M9 implies M9 . k = (M9 * TR) . k ) assume that A9: 1 <= k and A10: k <= len M9 ; ::_thesis: M9 . k = (M9 * TR) . k k in NAT by ORDINAL1:def_12; then A11: k in Seg (len M9) by A9, A10; A12: Line (M,j) = M . j by A2, MATRIX_2:8; A13: dom TR = Seg n by FUNCT_2:52; A14: Line (M,i) = M . i by A1, MATRIX_2:8; A15: len M9 = n by MATRIX_1:def_2; then A16: Line ((M9 * TR),k) = M . (tr . k) by A11, Th38; percases ( k = i or k = j or ( k <> i & k <> j ) ) ; suppose k = i ; ::_thesis: M9 . k = (M9 * TR) . k hence M9 . k = (M9 * TR) . k by A1, A4, A7, A16, A14, A12, MATRIX_2:8; ::_thesis: verum end; supposeA17: k = j ; ::_thesis: M9 . k = (M9 * TR) . k then A18: M . k = M . i by A2, A4, A14, MATRIX_2:8; Line ((M9 * TR),k) = M . i by A3, A6, A7, A16, A17, Th8; hence M9 . k = (M9 * TR) . k by A2, A17, A18, MATRIX_2:8; ::_thesis: verum end; suppose ( k <> i & k <> j ) ; ::_thesis: M9 . k = (M9 * TR) . k then Line ((M9 * TR),k) = M . k by A3, A6, A7, A11, A15, A13, A16, Th8; hence M9 . k = (M9 * TR) . k by A11, A15, MATRIX_2:8; ::_thesis: verum end; end; end; len (M9 * TR) = len M9 by Def4; then A19: M9 * TR = M by A8, FINSEQ_1:14; reconsider Tr = tr, p2 = p as Element of Permutations (n2 + 2) ; A20: sgn (Tr,K) = - (1_ K) by A6, Th14; tr = tr " by A6, Th20; hence (Path_product M) . q = (- (1_ K)) * ((Path_product M9) . p2) by A5, A19, A20, Th43 .= - ((1_ K) * ((Path_product M9) . p2)) by VECTSP_1:9 .= - ((Path_product M) . p) by VECTSP_1:def_4 ; ::_thesis: verum end; theorem Th50: :: MATRIX11:50 for n being Nat for K being Field for i, j being Nat st i in Seg n & j in Seg n & i < j holds for M being Matrix of n,K st Line (M,i) = Line (M,j) holds Det M = 0. K proof let n be Nat; ::_thesis: for K being Field for i, j being Nat st i in Seg n & j in Seg n & i < j holds for M being Matrix of n,K st Line (M,i) = Line (M,j) holds Det M = 0. K let K be Field; ::_thesis: for i, j being Nat st i in Seg n & j in Seg n & i < j holds for M being Matrix of n,K st Line (M,i) = Line (M,j) holds Det M = 0. K let i, j be Nat; ::_thesis: ( i in Seg n & j in Seg n & i < j implies for M being Matrix of n,K st Line (M,i) = Line (M,j) holds Det M = 0. K ) assume that A1: i in Seg n and A2: j in Seg n and A3: i < j ; ::_thesis: for M being Matrix of n,K st Line (M,i) = Line (M,j) holds Det M = 0. K set P = Permutations n; consider Q, E being finite set such that ( E = { p where p is Element of Permutations n : p is even } & Q = { q where q is Element of Permutations n : q is odd } ) and A4: ( E /\ Q = {} & E \/ Q = Permutations n ) and A5: ex P being Function of E,Q ex tr being Element of Permutations n st ( tr is being_transposition & tr . i = j & dom P = E & P is bijective & ( for p being Element of Permutations n st p in E holds P . p = p * tr ) ) by A1, A2, A3, Lm8; A6: E c= Permutations n by A4, XBOOLE_1:7; set KK = the carrier of K; set aa = the addF of K; let M be Matrix of n,K; ::_thesis: ( Line (M,i) = Line (M,j) implies Det M = 0. K ) assume A7: Line (M,i) = Line (M,j) ; ::_thesis: Det M = 0. K A8: Q c= Permutations n by A4, XBOOLE_1:7; set PathM = Path_product M; consider PERM being Function of E,Q, tr being Element of Permutations n such that A9: tr is being_transposition and A10: tr . i = j and A11: dom PERM = E and A12: PERM is bijective and A13: for p being Element of Permutations n st p in E holds PERM . p = p * tr by A5; reconsider E = E, Q = Q as Element of Fin (Permutations n) by A6, A8, FINSUB_1:def_5; the addF of K is having_a_unity by FVSUM_1:8; then consider GE being Function of (Fin (Permutations n)), the carrier of K such that A14: the addF of K $$ (E,(Path_product M)) = GE . E and A15: for e being Element of the carrier of K st e is_a_unity_wrt the addF of K holds GE . {} = e and A16: for x being Element of Permutations n holds GE . {x} = (Path_product M) . x and A17: for B9 being Element of Fin (Permutations n) st B9 c= E & B9 <> {} holds for x being Element of Permutations n st x in E \ B9 holds GE . (B9 \/ {x}) = the addF of K . ((GE . B9),((Path_product M) . x)) by SETWISEO:def_3; A18: E misses Q by A4, XBOOLE_0:def_7; the addF of K is having_a_unity by FVSUM_1:8; then consider GQ being Function of (Fin (Permutations n)), the carrier of K such that A19: the addF of K $$ (Q,(Path_product M)) = GQ . Q and A20: for e being Element of the carrier of K st e is_a_unity_wrt the addF of K holds GQ . {} = e and A21: for x being Element of Permutations n holds GQ . {x} = (Path_product M) . x and A22: for B9 being Element of Fin (Permutations n) st B9 c= Q & B9 <> {} holds for x being Element of Permutations n st x in Q \ B9 holds GQ . (B9 \/ {x}) = the addF of K . ((GQ . B9),((Path_product M) . x)) by SETWISEO:def_3; defpred S1[ Nat] means for B, PB being Element of Fin (Permutations n) st card B = $1 & B c= E & PERM .: B = PB holds (GE . B) + (GQ . PB) = 0. K; A23: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A24: S1[k] ; ::_thesis: S1[k + 1] let B, PB be Element of Fin (Permutations n); ::_thesis: ( card B = k + 1 & B c= E & PERM .: B = PB implies (GE . B) + (GQ . PB) = 0. K ) assume that A25: card B = k + 1 and A26: B c= E and A27: PERM .: B = PB ; ::_thesis: (GE . B) + (GQ . PB) = 0. K now__::_thesis:_(_(_k_=_0_&_(GE_._B)_+_(GQ_._PB)_=_0._K_)_or_(_k_>_0_&_(GQ_._PB)_+_(GE_._B)_=_0._K_)_) percases ( k = 0 or k > 0 ) ; case k = 0 ; ::_thesis: (GE . B) + (GQ . PB) = 0. K then consider x being set such that A28: B = {x} by A25, CARD_2:42; A29: x in B by A28, TARSKI:def_1; B c= Permutations n by FINSUB_1:def_5; then reconsider x = x as Element of Permutations n by A29; x * tr is Element of Permutations n by MATRIX_9:39; then reconsider Px = PERM . x as Element of Permutations n by A13, A26, A29; A30: Im (PERM,x) = {Px} by A11, A26, A29, FUNCT_1:59; A31: GE . {x} = (Path_product M) . x by A16; A32: GQ . {(PERM . x)} = (Path_product M) . Px by A21; Px = x * tr by A13, A26, A29; then - (GE . B) = GQ . PB by A1, A2, A3, A7, A9, A10, A27, A28, A31, A32, A30, Th49; hence (GE . B) + (GQ . PB) = 0. K by RLVECT_1:def_10; ::_thesis: verum end; caseA33: k > 0 ; ::_thesis: (GQ . PB) + (GE . B) = 0. K consider x being set such that A34: x in B by A25, CARD_1:27, XBOOLE_0:def_1; B c= Permutations n by FINSUB_1:def_5; then reconsider x = x as Element of Permutations n by A34; x * tr is Element of Permutations n by MATRIX_9:39; then reconsider Px = PERM . x as Element of Permutations n by A13, A26, A34; A35: Im (PERM,x) = {Px} by A11, A26, A34, FUNCT_1:59; Px = x * tr by A13, A26, A34; then A36: - ((Path_product M) . x) = (Path_product M) . Px by A1, A2, A3, A7, A9, A10, Th49; A37: Q c= Permutations n by FINSUB_1:def_5; B c= Permutations n by FINSUB_1:def_5; then A38: B \ {x} c= Permutations n by XBOOLE_1:1; A39: rng PERM = Q by A12, FUNCT_2:def_3; then A40: Px in Q by A11, A26, A34, FUNCT_1:def_3; PERM .: (B \ {x}) c= rng PERM by RELAT_1:111; then PERM .: (B \ {x}) c= Permutations n by A39, A37, XBOOLE_1:1; then reconsider B9 = B \ {x}, PeBx = PERM .: (B \ {x}) as Element of Fin (Permutations n) by A38, FINSUB_1:def_5; A41: Px in {Px} by TARSKI:def_1; A42: {x} \/ B9 = B by A34, ZFMISC_1:116; then A43: PERM .: B = (Im (PERM,x)) \/ PeBx by RELAT_1:120; B9 misses {x} by XBOOLE_1:79; then B9 /\ {x} = {} by XBOOLE_0:def_7; then PERM .: {} = {Px} /\ PeBx by A12, A35, FUNCT_1:62; then not Px in PeBx by A41, XBOOLE_0:def_4; then A44: Px in Q \ PeBx by A40, XBOOLE_0:def_5; A45: not x in B9 by ZFMISC_1:56; then A46: x in E \ B9 by A26, A34, XBOOLE_0:def_5; A47: k + 1 = (card B9) + 1 by A25, A42, A45, CARD_2:41; then consider y being set such that A48: y in B9 by A33, CARD_1:27, XBOOLE_0:def_1; B \ {x} c= E by A26, XBOOLE_1:1; then PERM . y in PeBx by A11, A48, FUNCT_1:def_6; then GQ . PB = the addF of K . ((GQ . PeBx),((Path_product M) . Px)) by A22, A27, A39, A43, A35, A44, RELAT_1:111; hence (GQ . PB) + (GE . B) = ((GQ . PeBx) - ((Path_product M) . x)) + ((GE . B9) + ((Path_product M) . x)) by A17, A26, A33, A42, A47, A46, A36, CARD_1:27, XBOOLE_1:1 .= (GQ . PeBx) + ((- ((Path_product M) . x)) + ((GE . B9) + ((Path_product M) . x))) by RLVECT_1:def_3 .= (GQ . PeBx) + ((GE . B9) + (((Path_product M) . x) - ((Path_product M) . x))) by RLVECT_1:def_3 .= (GQ . PeBx) + ((GE . B9) + (0. K)) by RLVECT_1:def_10 .= ((GQ . PeBx) + (GE . B9)) + (0. K) by RLVECT_1:def_3 .= (0. K) + (0. K) by A24, A26, A47, XBOOLE_1:1 .= 0. K by RLVECT_1:4 ; ::_thesis: verum end; end; end; hence (GE . B) + (GQ . PB) = 0. K ; ::_thesis: verum end; set F = FinOmega (Permutations n); A49: Permutations n = FinOmega (Permutations n) by A4, MATRIX_2:def_14; rng PERM = Q by A12, FUNCT_2:def_3; then A50: PERM .: E = Q by A11, RELAT_1:113; A51: S1[ 0 ] proof let B, PB be Element of Fin (Permutations n); ::_thesis: ( card B = 0 & B c= E & PERM .: B = PB implies (GE . B) + (GQ . PB) = 0. K ) assume that A52: card B = 0 and B c= E and A53: PERM .: B = PB ; ::_thesis: (GE . B) + (GQ . PB) = 0. K A54: B = {} by A52; then A55: GE . B = 0. K by A15, FVSUM_1:6; PERM .: {} = {} ; then GQ . PB = 0. K by A20, A53, A54, FVSUM_1:6; hence (GE . B) + (GQ . PB) = 0. K by A55, RLVECT_1:def_4; ::_thesis: verum end; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A51, A23); then S1[ card E] ; then ( the addF of K $$ (E,(Path_product M))) + ( the addF of K $$ (Q,(Path_product M))) = 0. K by A14, A19, A50; hence Det M = 0. K by A4, A18, A49, FVSUM_1:8, SETWOP_2:4; ::_thesis: verum end; theorem Th51: :: MATRIX11:51 for n being Nat for K being Field for A being Matrix of n,K for i, j being Nat st i in Seg n & j in Seg n & i <> j holds Det (RLine (A,i,(Line (A,j)))) = 0. K proof let n be Nat; ::_thesis: for K being Field for A being Matrix of n,K for i, j being Nat st i in Seg n & j in Seg n & i <> j holds Det (RLine (A,i,(Line (A,j)))) = 0. K let K be Field; ::_thesis: for A being Matrix of n,K for i, j being Nat st i in Seg n & j in Seg n & i <> j holds Det (RLine (A,i,(Line (A,j)))) = 0. K let A be Matrix of n,K; ::_thesis: for i, j being Nat st i in Seg n & j in Seg n & i <> j holds Det (RLine (A,i,(Line (A,j)))) = 0. K let i, j be Nat; ::_thesis: ( i in Seg n & j in Seg n & i <> j implies Det (RLine (A,i,(Line (A,j)))) = 0. K ) assume that A1: i in Seg n and A2: j in Seg n and A3: i <> j ; ::_thesis: Det (RLine (A,i,(Line (A,j)))) = 0. K A4: ( i < j or j < i ) by A3, XXREAL_0:1; len (Line (A,j)) = width A by MATRIX_1:def_7; then A5: Line ((RLine (A,i,(Line (A,j)))),i) = Line (A,j) by A1, Th28; Line ((RLine (A,i,(Line (A,j)))),j) = Line (A,j) by A2, A3, Th28; hence Det (RLine (A,i,(Line (A,j)))) = 0. K by A1, A2, A5, A4, Th50; ::_thesis: verum end; theorem Th52: :: MATRIX11:52 for n being Nat for K being Field for a being Element of K for A being Matrix of n,K for i, j being Nat st i in Seg n & j in Seg n & i <> j holds Det (RLine (A,i,(a * (Line (A,j))))) = 0. K proof let n be Nat; ::_thesis: for K being Field for a being Element of K for A being Matrix of n,K for i, j being Nat st i in Seg n & j in Seg n & i <> j holds Det (RLine (A,i,(a * (Line (A,j))))) = 0. K let K be Field; ::_thesis: for a being Element of K for A being Matrix of n,K for i, j being Nat st i in Seg n & j in Seg n & i <> j holds Det (RLine (A,i,(a * (Line (A,j))))) = 0. K let a be Element of K; ::_thesis: for A being Matrix of n,K for i, j being Nat st i in Seg n & j in Seg n & i <> j holds Det (RLine (A,i,(a * (Line (A,j))))) = 0. K let A be Matrix of n,K; ::_thesis: for i, j being Nat st i in Seg n & j in Seg n & i <> j holds Det (RLine (A,i,(a * (Line (A,j))))) = 0. K let i, j be Nat; ::_thesis: ( i in Seg n & j in Seg n & i <> j implies Det (RLine (A,i,(a * (Line (A,j))))) = 0. K ) assume that A1: i in Seg n and A2: j in Seg n and A3: i <> j ; ::_thesis: Det (RLine (A,i,(a * (Line (A,j))))) = 0. K width A = n by MATRIX_1:24; then len (Line (A,j)) = n by MATRIX_1:def_7; hence Det (RLine (A,i,(a * (Line (A,j))))) = a * (Det (RLine (A,i,(Line (A,j))))) by A1, Th34 .= a * (0. K) by A1, A2, A3, Th51 .= 0. K by VECTSP_1:6 ; ::_thesis: verum end; theorem :: MATRIX11:53 for n being Nat for K being Field for a being Element of K for A being Matrix of n,K for i, j being Nat st i in Seg n & j in Seg n & i <> j holds Det A = Det (RLine (A,i,((Line (A,i)) + (a * (Line (A,j)))))) proof let n be Nat; ::_thesis: for K being Field for a being Element of K for A being Matrix of n,K for i, j being Nat st i in Seg n & j in Seg n & i <> j holds Det A = Det (RLine (A,i,((Line (A,i)) + (a * (Line (A,j)))))) let K be Field; ::_thesis: for a being Element of K for A being Matrix of n,K for i, j being Nat st i in Seg n & j in Seg n & i <> j holds Det A = Det (RLine (A,i,((Line (A,i)) + (a * (Line (A,j)))))) let a be Element of K; ::_thesis: for A being Matrix of n,K for i, j being Nat st i in Seg n & j in Seg n & i <> j holds Det A = Det (RLine (A,i,((Line (A,i)) + (a * (Line (A,j)))))) let A be Matrix of n,K; ::_thesis: for i, j being Nat st i in Seg n & j in Seg n & i <> j holds Det A = Det (RLine (A,i,((Line (A,i)) + (a * (Line (A,j)))))) let i, j be Nat; ::_thesis: ( i in Seg n & j in Seg n & i <> j implies Det A = Det (RLine (A,i,((Line (A,i)) + (a * (Line (A,j)))))) ) assume that A1: i in Seg n and A2: j in Seg n and A3: i <> j ; ::_thesis: Det A = Det (RLine (A,i,((Line (A,i)) + (a * (Line (A,j)))))) A4: width A = n by MATRIX_1:24; then A5: len (Line (A,j)) = n by MATRIX_1:def_7; A6: len (Line (A,j)) = len (a * (Line (A,j))) by Lm5; len (Line (A,i)) = n by A4, MATRIX_1:def_7; hence Det (RLine (A,i,((Line (A,i)) + (a * (Line (A,j)))))) = (Det (RLine (A,i,(Line (A,i))))) + (Det (RLine (A,i,(a * (Line (A,j)))))) by A1, A5, A6, Th36 .= (Det A) + (Det (RLine (A,i,(a * (Line (A,j)))))) by Th30 .= (Det A) + (0. K) by A1, A2, A3, Th52 .= Det A by RLVECT_1:4 ; ::_thesis: verum end; theorem Th54: :: MATRIX11:54 for n being Nat for K being Field for F being Function of (Seg n),(Seg n) for A being Matrix of n,K st not F in Permutations n holds Det (A * F) = 0. K proof let n be Nat; ::_thesis: for K being Field for F being Function of (Seg n),(Seg n) for A being Matrix of n,K st not F in Permutations n holds Det (A * F) = 0. K let K be Field; ::_thesis: for F being Function of (Seg n),(Seg n) for A being Matrix of n,K st not F in Permutations n holds Det (A * F) = 0. K let F be Function of (Seg n),(Seg n); ::_thesis: for A being Matrix of n,K st not F in Permutations n holds Det (A * F) = 0. K let A be Matrix of n,K; ::_thesis: ( not F in Permutations n implies Det (A * F) = 0. K ) assume not F in Permutations n ; ::_thesis: Det (A * F) = 0. K then A1: ( not F is onto or not F is one-to-one ) by MATRIX_2:def_9; card (Seg n) = card (Seg n) ; then not F is one-to-one by A1, STIRL2_1:60; then consider x, y being set such that A2: x in dom F and A3: y in dom F and A4: F . x = F . y and A5: x <> y by FUNCT_1:def_4; A6: dom F = Seg n by FUNCT_2:52; then reconsider x = x, y = y as Nat by A2, A3; Line ((A * F),x) = A . (F . x) by A2, A6, Th38; then A7: Line ((A * F),x) = Line ((A * F),y) by A3, A4, A6, Th38; ( x > y or y > x ) by A5, XXREAL_0:1; hence Det (A * F) = 0. K by A2, A3, A6, A7, Th50; ::_thesis: verum end; begin definition let K be non empty addLoopStr ; func addFinS K -> BinOp of ( the carrier of K *) means :Def5: :: MATRIX11:def 5 for p1, p2 being Element of the carrier of K * holds it . (p1,p2) = p1 + p2; existence ex b1 being BinOp of ( the carrier of K *) st for p1, p2 being Element of the carrier of K * holds b1 . (p1,p2) = p1 + p2 proof set KK = the carrier of K; defpred S1[ set , set , set ] means for p1, p2 being Element of the carrier of K * st $1 = p1 & $2 = p2 holds $3 = p1 + p2; A1: for x, y being Element of the carrier of K * ex z being Element of the carrier of K * st S1[x,y,z] proof let x, y be Element of the carrier of K * ; ::_thesis: ex z being Element of the carrier of K * st S1[x,y,z] reconsider p1 = x, p2 = y as FinSequence of the carrier of K ; reconsider pp = p1 + p2 as Element of the carrier of K * by FINSEQ_1:def_11; take pp ; ::_thesis: S1[x,y,pp] thus S1[x,y,pp] ; ::_thesis: verum end; consider A being Function of [:( the carrier of K *),( the carrier of K *):],( the carrier of K *) such that A2: for x, y being Element of the carrier of K * holds S1[x,y,A . (x,y)] from BINOP_1:sch_3(A1); take A ; ::_thesis: for p1, p2 being Element of the carrier of K * holds A . (p1,p2) = p1 + p2 thus for p1, p2 being Element of the carrier of K * holds A . (p1,p2) = p1 + p2 by A2; ::_thesis: verum end; uniqueness for b1, b2 being BinOp of ( the carrier of K *) st ( for p1, p2 being Element of the carrier of K * holds b1 . (p1,p2) = p1 + p2 ) & ( for p1, p2 being Element of the carrier of K * holds b2 . (p1,p2) = p1 + p2 ) holds b1 = b2 proof set KK = the carrier of K; let f1, f2 be Function of [:( the carrier of K *),( the carrier of K *):],( the carrier of K *); ::_thesis: ( ( for p1, p2 being Element of the carrier of K * holds f1 . (p1,p2) = p1 + p2 ) & ( for p1, p2 being Element of the carrier of K * holds f2 . (p1,p2) = p1 + p2 ) implies f1 = f2 ) assume that A3: for p1, p2 being Element of the carrier of K * holds f1 . (p1,p2) = p1 + p2 and A4: for p1, p2 being Element of the carrier of K * holds f2 . (p1,p2) = p1 + p2 ; ::_thesis: f1 = f2 now__::_thesis:_for_p1,_p2_being_Element_of_the_carrier_of_K_*_holds_f1_._(p1,p2)_=_f2_._(p1,p2) let p1, p2 be Element of the carrier of K * ; ::_thesis: f1 . (p1,p2) = f2 . (p1,p2) f1 . (p1,p2) = p1 + p2 by A3; hence f1 . (p1,p2) = f2 . (p1,p2) by A4; ::_thesis: verum end; hence f1 = f2 by BINOP_1:2; ::_thesis: verum end; end; :: deftheorem Def5 defines addFinS MATRIX11:def_5_:_ for K being non empty addLoopStr for b2 being BinOp of ( the carrier of K *) holds ( b2 = addFinS K iff for p1, p2 being Element of the carrier of K * holds b2 . (p1,p2) = p1 + p2 ); Lm9: for K being non empty addLoopStr for p1, p2 being Element of the carrier of K * holds dom (p1 + p2) = (dom p1) /\ (dom p2) proof let K be non empty addLoopStr ; ::_thesis: for p1, p2 being Element of the carrier of K * holds dom (p1 + p2) = (dom p1) /\ (dom p2) let p1, p2 be Element of the carrier of K * ; ::_thesis: dom (p1 + p2) = (dom p1) /\ (dom p2) A1: rng p2 c= the carrier of K by FINSEQ_1:def_4; rng p1 c= the carrier of K by FINSEQ_1:def_4; then [:(rng p1),(rng p2):] c= [: the carrier of K, the carrier of K:] by A1, ZFMISC_1:96; then [:(rng p1),(rng p2):] c= dom the addF of K by FUNCT_2:def_1; hence dom (p1 + p2) = (dom p1) /\ (dom p2) by FUNCOP_1:69; ::_thesis: verum end; registration let K be non empty Abelian addLoopStr ; cluster addFinS K -> commutative ; coherence addFinS K is commutative proof set KK = the carrier of K; let p1, p2 be Element of the carrier of K * ; :: according to BINOP_1:def_2 ::_thesis: (addFinS K) . (p1,p2) = (addFinS K) . (p2,p1) A1: rng p2 c= the carrier of K by FINSEQ_1:def_4; A2: dom (p1 + p2) = (dom p1) /\ (dom p2) by Lm9; A3: dom (p2 + p1) = (dom p2) /\ (dom p1) by Lm9; A4: rng p1 c= the carrier of K by FINSEQ_1:def_4; now__::_thesis:_for_k_being_Nat_st_k_in_dom_(p1_+_p2)_holds_ (p1_+_p2)_._k_=_(p2_+_p1)_._k let k be Nat; ::_thesis: ( k in dom (p1 + p2) implies (p1 + p2) . k = (p2 + p1) . k ) assume A5: k in dom (p1 + p2) ; ::_thesis: (p1 + p2) . k = (p2 + p1) . k k in dom p2 by A2, A5, XBOOLE_0:def_4; then A6: p2 . k in rng p2 by FUNCT_1:def_3; k in dom p1 by A2, A5, XBOOLE_0:def_4; then p1 . k in rng p1 by FUNCT_1:def_3; then reconsider p1k = p1 . k, p2k = p2 . k as Element of K by A4, A1, A6; (p1 + p2) . k = p1k + p2k by A5, FVSUM_1:17; hence (p1 + p2) . k = (p2 + p1) . k by A2, A3, A5, FVSUM_1:17; ::_thesis: verum end; then A7: p1 + p2 = p2 + p1 by A3, Lm9, FINSEQ_1:13; (addFinS K) . (p1,p2) = p1 + p2 by Def5; hence (addFinS K) . (p1,p2) = (addFinS K) . (p2,p1) by A7, Def5; ::_thesis: verum end; end; registration let K be non empty add-associative addLoopStr ; cluster addFinS K -> associative ; coherence addFinS K is associative proof set aK = addFinS K; set KK = the carrier of K; let p1, p2, p3 be Element of the carrier of K * ; :: according to BINOP_1:def_3 ::_thesis: (addFinS K) . (p1,((addFinS K) . (p2,p3))) = (addFinS K) . (((addFinS K) . (p1,p2)),p3) reconsider p12 = p1 + p2, p23 = p2 + p3 as Element of the carrier of K * by FINSEQ_1:def_11; A1: rng p1 c= the carrier of K by FINSEQ_1:def_4; A2: rng p2 c= the carrier of K by FINSEQ_1:def_4; A3: rng p12 c= the carrier of K by FINSEQ_1:def_4; A4: rng p3 c= the carrier of K by FINSEQ_1:def_4; A5: rng p23 c= the carrier of K by FINSEQ_1:def_4; A6: dom p12 = (dom p1) /\ (dom p2) by Lm9; A7: dom p23 = (dom p2) /\ (dom p3) by Lm9; A8: dom (p12 + p3) = (dom p12) /\ (dom p3) by Lm9; A9: dom (p1 + p23) = (dom p1) /\ (dom p23) by Lm9; then A10: dom (p12 + p3) = dom (p1 + p23) by A6, A8, A7, XBOOLE_1:16; now__::_thesis:_for_k_being_Nat_st_k_in_dom_(p12_+_p3)_holds_ (p1_+_p23)_._k_=_(p12_+_p3)_._k let k be Nat; ::_thesis: ( k in dom (p12 + p3) implies (p1 + p23) . k = (p12 + p3) . k ) assume A11: k in dom (p12 + p3) ; ::_thesis: (p1 + p23) . k = (p12 + p3) . k A12: k in dom p12 by A8, A11, XBOOLE_0:def_4; then A13: p12 . k in rng p12 by FUNCT_1:def_3; k in dom p1 by A6, A12, XBOOLE_0:def_4; then A14: p1 . k in rng p1 by FUNCT_1:def_3; A15: k in dom p3 by A8, A11, XBOOLE_0:def_4; then A16: p3 . k in rng p3 by FUNCT_1:def_3; A17: k in dom p2 by A6, A12, XBOOLE_0:def_4; then A18: p2 . k in rng p2 by FUNCT_1:def_3; A19: k in dom p23 by A7, A15, A17, XBOOLE_0:def_4; then p23 . k in rng p23 by FUNCT_1:def_3; then reconsider p1k = p1 . k, p12k = p12 . k, p2k = p2 . k, p23k = p23 . k, p3k = p3 . k as Element of K by A1, A2, A4, A3, A5, A14, A13, A16, A18; A20: p12 . k = p1k + p2k by A12, FVSUM_1:17; A21: (p12 + p3) . k = p12k + p3k by A11, FVSUM_1:17; A22: p23 . k = p2k + p3k by A19, FVSUM_1:17; (p1 + p23) . k = p1k + p23k by A10, A11, FVSUM_1:17; hence (p1 + p23) . k = (p12 + p3) . k by A20, A22, A21, RLVECT_1:def_3; ::_thesis: verum end; then A23: p1 + p23 = p12 + p3 by A6, A8, A7, A9, FINSEQ_1:13, XBOOLE_1:16; thus (addFinS K) . (p1,((addFinS K) . (p2,p3))) = (addFinS K) . (p1,p23) by Def5 .= p1 + p23 by Def5 .= (addFinS K) . (p12,p3) by A23, Def5 .= (addFinS K) . (((addFinS K) . (p1,p2)),p3) by Def5 ; ::_thesis: verum end; end; theorem Th55: :: MATRIX11:55 for K being Field for A, B being Matrix of K st width A = len B & len B > 0 holds for i being Nat st i in Seg (len A) holds ex P being FinSequence of the carrier of K * st ( len P = len B & Line ((A * B),i) = (addFinS K) "**" P & ( for j being Nat st j in Seg (len B) holds P . j = (A * (i,j)) * (Line (B,j)) ) ) proof let K be Field; ::_thesis: for A, B being Matrix of K st width A = len B & len B > 0 holds for i being Nat st i in Seg (len A) holds ex P being FinSequence of the carrier of K * st ( len P = len B & Line ((A * B),i) = (addFinS K) "**" P & ( for j being Nat st j in Seg (len B) holds P . j = (A * (i,j)) * (Line (B,j)) ) ) let A, B be Matrix of K; ::_thesis: ( width A = len B & len B > 0 implies for i being Nat st i in Seg (len A) holds ex P being FinSequence of the carrier of K * st ( len P = len B & Line ((A * B),i) = (addFinS K) "**" P & ( for j being Nat st j in Seg (len B) holds P . j = (A * (i,j)) * (Line (B,j)) ) ) ) assume that A1: width A = len B and A2: len B > 0 ; ::_thesis: for i being Nat st i in Seg (len A) holds ex P being FinSequence of the carrier of K * st ( len P = len B & Line ((A * B),i) = (addFinS K) "**" P & ( for j being Nat st j in Seg (len B) holds P . j = (A * (i,j)) * (Line (B,j)) ) ) set aa = the addF of K; set mm = the multF of K; set a = addFinS K; set KK = the carrier of K; reconsider m = len B, w = width B as Nat ; let i be Nat; ::_thesis: ( i in Seg (len A) implies ex P being FinSequence of the carrier of K * st ( len P = len B & Line ((A * B),i) = (addFinS K) "**" P & ( for j being Nat st j in Seg (len B) holds P . j = (A * (i,j)) * (Line (B,j)) ) ) ) assume A3: i in Seg (len A) ; ::_thesis: ex P being FinSequence of the carrier of K * st ( len P = len B & Line ((A * B),i) = (addFinS K) "**" P & ( for j being Nat st j in Seg (len B) holds P . j = (A * (i,j)) * (Line (B,j)) ) ) deffunc H1( Nat) -> Element of (width B) -tuples_on the carrier of K = (A * (i,$1)) * (Line (B,$1)); consider P being FinSequence such that A4: len P = len B and A5: for k being Nat st k in dom P holds P . k = H1(k) from FINSEQ_1:sch_2(); A6: dom P = dom B by A4, FINSEQ_3:29 .= Seg (len B) by FINSEQ_1:def_3 ; rng P c= the carrier of K * proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in rng P or y in the carrier of K * ) assume y in rng P ; ::_thesis: y in the carrier of K * then consider x being set such that A7: x in dom P and A8: y = P . x by FUNCT_1:def_3; reconsider x = x as Element of NAT by A7; P . x = (A * (i,x)) * (Line (B,x)) by A5, A7; hence y in the carrier of K * by A8, FINSEQ_1:def_11; ::_thesis: verum end; then reconsider P = P as FinSequence of the carrier of K * by FINSEQ_1:def_4; A9: m >= 1 by A2, NAT_1:14; then consider F being Function of NAT,( the carrier of K *) such that A10: F . 1 = P . 1 and A11: for n being Element of NAT st 0 <> n & n < len P holds F . (n + 1) = (addFinS K) . ((F . n),(P . (n + 1))) and A12: (addFinS K) "**" P = F . (len P) by A4, FINSOP_1:def_1; defpred S1[ Nat] means ( 1 <= $1 & $1 <= m implies for F1 being FinSequence of the carrier of K st F . $1 = F1 holds ( len F1 = w & ( for j being Element of NAT st j in Seg w holds ex LC being FinSequence of the carrier of K st ( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg $1) & the addF of K "**" LC = F1 . j ) ) ) ); A13: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A14: S1[k] ; ::_thesis: S1[k + 1] set k1 = k + 1; assume that A15: 1 <= k + 1 and A16: k + 1 <= m ; ::_thesis: for F1 being FinSequence of the carrier of K st F . (k + 1) = F1 holds ( len F1 = w & ( for j being Element of NAT st j in Seg w holds ex LC being FinSequence of the carrier of K st ( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC = F1 . j ) ) ) A17: k + 1 in Seg m by A15, A16; let Fk1 be FinSequence of the carrier of K; ::_thesis: ( F . (k + 1) = Fk1 implies ( len Fk1 = w & ( for j being Element of NAT st j in Seg w holds ex LC being FinSequence of the carrier of K st ( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC = Fk1 . j ) ) ) ) assume A18: F . (k + 1) = Fk1 ; ::_thesis: ( len Fk1 = w & ( for j being Element of NAT st j in Seg w holds ex LC being FinSequence of the carrier of K st ( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC = Fk1 . j ) ) ) percases ( k = 0 or k > 0 ) ; supposeA19: k = 0 ; ::_thesis: ( len Fk1 = w & ( for j being Element of NAT st j in Seg w holds ex LC being FinSequence of the carrier of K st ( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC = Fk1 . j ) ) ) then A20: P . 1 = (A * (i,1)) * (Line (B,1)) by A5, A6, A17; A21: len (Line (B,1)) = w by MATRIX_1:def_7; hence len Fk1 = w by A10, A18, A19, A20, Lm5; ::_thesis: for j being Element of NAT st j in Seg w holds ex LC being FinSequence of the carrier of K st ( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC = Fk1 . j ) let j be Element of NAT ; ::_thesis: ( j in Seg w implies ex LC being FinSequence of the carrier of K st ( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC = Fk1 . j ) ) assume A22: j in Seg w ; ::_thesis: ex LC being FinSequence of the carrier of K st ( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC = Fk1 . j ) len Fk1 = w by A10, A18, A19, A20, A21, Lm5; then A23: j in dom Fk1 by A22, FINSEQ_1:def_3; (Line (B,1)) . j = B * (1,j) by A22, MATRIX_1:def_7; then A24: Fk1 . j = (A * (i,1)) * (B * (1,j)) by A10, A18, A19, A20, A23, FVSUM_1:50; set C = Col (B,j); set L = Line (A,i); reconsider LC1 = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)), mLC = mlt ((Line (A,i)),(Col (B,j))) as FinSequence of the carrier of K by FINSEQ_1:18; A25: the multF of K .: ((Line (A,i)),(Col (B,j))) is Element of m -tuples_on the carrier of K by A1, FINSEQ_2:120; then len mLC = m by CARD_1:def_7; then A26: dom mLC = Seg m by FINSEQ_1:def_3; Seg 1 = (Seg m) /\ (Seg 1) by A2, FINSEQ_1:7, NAT_1:14; then A27: dom LC1 = Seg 1 by A19, A26, RELAT_1:61; then A28: len LC1 = 1 by FINSEQ_1:def_3; 1 in Seg 1 ; then LC1 . 1 = mLC . 1 by A27, FUNCT_1:47; then A29: LC1 = <*(mLC . 1)*> by A28, FINSEQ_1:40; take LC1 ; ::_thesis: ( LC1 = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC1 = Fk1 . j ) A30: 1 in Seg m by A9; len mLC = m by A25, CARD_1:def_7; then A31: 1 in dom mLC by A30, FINSEQ_1:def_3; Seg m = dom B by FINSEQ_1:def_3; then A32: (Col (B,j)) . 1 = B * (1,j) by A30, MATRIX_1:def_8; (Line (A,i)) . 1 = A * (i,1) by A1, A30, MATRIX_1:def_7; then mLC . 1 = (A * (i,1)) * (B * (1,j)) by A32, A31, FVSUM_1:60; hence ( LC1 = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC1 = Fk1 . j ) by A24, A29, FINSOP_1:11; ::_thesis: verum end; supposeA33: k > 0 ; ::_thesis: ( len Fk1 = w & ( for j being Element of NAT st j in Seg w holds ex LC being FinSequence of the carrier of K st ( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC = Fk1 . j ) ) ) dom P = Seg m by A4, FINSEQ_1:def_3; then A34: P . (k + 1) in rng P by A17, FUNCT_1:def_3; rng P c= the carrier of K * by FINSEQ_1:def_4; then reconsider Pk1 = P . (k + 1), Fk = F . k as FinSequence of the carrier of K by A34, FINSEQ_1:def_11; A35: Pk1 is Element of the carrier of K * by FINSEQ_1:def_11; then A36: (addFinS K) . (Fk,Pk1) = Fk + Pk1 by Def5; A37: k + 0 < k + 1 by XREAL_1:8; then k < m by A16, XXREAL_0:2; then A38: Fk1 = Fk + Pk1 by A4, A11, A18, A33, A36; A39: len (Line (B,(k + 1))) = w by MATRIX_1:def_7; Pk1 = H1(k + 1) by A5, A6, A17; then len Pk1 = w by A39, Lm5; then A40: dom Pk1 = Seg w by FINSEQ_1:def_3; A41: len Fk = w by A14, A16, A33, A37, NAT_1:14, XXREAL_0:2; then dom Fk = Seg w by FINSEQ_1:def_3; then A42: dom (Fk + Pk1) = (Seg w) /\ (Seg w) by A40, A35, Lm9; hence len Fk1 = w by A38, FINSEQ_1:def_3; ::_thesis: for j being Element of NAT st j in Seg w holds ex LC being FinSequence of the carrier of K st ( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC = Fk1 . j ) A43: rng Fk c= the carrier of K by FINSEQ_1:def_4; set L = Line (A,i); A44: Pk1 = H1(k + 1) by A5, A6, A17; let j be Element of NAT ; ::_thesis: ( j in Seg w implies ex LC being FinSequence of the carrier of K st ( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC = Fk1 . j ) ) assume A45: j in Seg w ; ::_thesis: ex LC being FinSequence of the carrier of K st ( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC = Fk1 . j ) A46: (Line (B,(k + 1))) . j = B * ((k + 1),j) by A45, MATRIX_1:def_7; then Pk1 . j = (A * (i,(k + 1))) * (B * ((k + 1),j)) by A40, A45, A44, FVSUM_1:50; then reconsider Pk1j = Pk1 . j as Element of the carrier of K ; set C = Col (B,j); consider LC being FinSequence of the carrier of K such that A47: LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg k) and A48: the addF of K "**" LC = Fk . j by A14, A16, A33, A37, A45, NAT_1:14, XXREAL_0:2; reconsider LC1 = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)), mLC = mlt ((Line (A,i)),(Col (B,j))) as FinSequence of the carrier of K by FINSEQ_1:18; take LC1 ; ::_thesis: ( LC1 = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC1 = Fk1 . j ) the multF of K .: ((Line (A,i)),(Col (B,j))) is Element of m -tuples_on the carrier of K by A1, FINSEQ_2:120; then len mLC = m by CARD_1:def_7; then A49: k + 1 in dom mLC by A17, FINSEQ_1:def_3; A50: k + 1 in Seg m by A15, A16; then A51: (Line (A,i)) . (k + 1) = A * (i,(k + 1)) by A1, MATRIX_1:def_7; Seg m = dom B by FINSEQ_1:def_3; then (Col (B,j)) . (k + 1) = B * ((k + 1),j) by A50, MATRIX_1:def_8; then A52: mLC . (k + 1) = (A * (i,(k + 1))) * (B * ((k + 1),j)) by A51, A49, FVSUM_1:60; LC1 = LC ^ <*(mLC . (k + 1))*> by A47, A49, FINSEQ_5:10; then A53: the addF of K "**" LC1 = the addF of K . ((Fk . j),((A * (i,(k + 1))) * (B * ((k + 1),j)))) by A48, A52, FINSOP_1:4, FVSUM_1:8; j in dom Fk by A41, A45, FINSEQ_1:def_3; then Fk . j in rng Fk by FUNCT_1:def_3; then reconsider Fkj = Fk . j as Element of the carrier of K by A43; Fk1 . j = Fkj + Pk1j by A42, A38, A45, FVSUM_1:17; hence ( LC1 = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg (k + 1)) & the addF of K "**" LC1 = Fk1 . j ) by A40, A45, A46, A44, A53, FVSUM_1:50; ::_thesis: verum end; end; end; set L = Line ((A * B),i); width (A * B) = w by A1, MATRIX_3:def_4; then A54: len (Line ((A * B),i)) = w by MATRIX_1:def_7; (addFinS K) "**" P is Element of the carrier of K * ; then reconsider Fm = F . m as FinSequence of the carrier of K by A4, A12; A55: S1[ 0 ] ; A56: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A55, A13); A57: for j being Nat st 1 <= j & j <= len (Line ((A * B),i)) holds (Line ((A * B),i)) . j = Fm . j proof set AB = A * B; set LA = Line (A,i); let j be Nat; ::_thesis: ( 1 <= j & j <= len (Line ((A * B),i)) implies (Line ((A * B),i)) . j = Fm . j ) assume that A58: 1 <= j and A59: j <= len (Line ((A * B),i)) ; ::_thesis: (Line ((A * B),i)) . j = Fm . j set CB = Col (B,j); the multF of K .: ((Line (A,i)),(Col (B,j))) is Element of m -tuples_on the carrier of K by A1, FINSEQ_2:120; then len (mlt ((Line (A,i)),(Col (B,j)))) = m by CARD_1:def_7; then A60: dom (mlt ((Line (A,i)),(Col (B,j)))) = Seg m by FINSEQ_1:def_3; j in NAT by ORDINAL1:def_12; then A61: j in Seg w by A54, A58, A59; then ex LC being FinSequence of the carrier of K st ( LC = (mlt ((Line (A,i)),(Col (B,j)))) | (Seg m) & the addF of K "**" LC = Fm . j ) by A9, A56; then A62: Fm . j = (Line (A,i)) "*" (Col (B,j)) by A60, RELAT_1:69; A63: len (A * B) = len A by A1, MATRIX_3:def_4; A64: width (A * B) = w by A1, MATRIX_3:def_4; len A <> 0 by A3; then A * B is Matrix of len A,w,K by A63, A64, MATRIX_1:20; then Indices (A * B) = [:(Seg (len A)),(Seg w):] by A64, MATRIX_1:25; then [i,j] in Indices (A * B) by A3, A61, ZFMISC_1:87; then (A * B) * (i,j) = (Line (A,i)) "*" (Col (B,j)) by A1, MATRIX_3:def_4; hence (Line ((A * B),i)) . j = Fm . j by A61, A62, A64, MATRIX_1:def_7; ::_thesis: verum end; take P ; ::_thesis: ( len P = len B & Line ((A * B),i) = (addFinS K) "**" P & ( for j being Nat st j in Seg (len B) holds P . j = (A * (i,j)) * (Line (B,j)) ) ) thus len P = len B by A4; ::_thesis: ( Line ((A * B),i) = (addFinS K) "**" P & ( for j being Nat st j in Seg (len B) holds P . j = (A * (i,j)) * (Line (B,j)) ) ) len Fm = w by A9, A56; hence Line ((A * B),i) = (addFinS K) "**" P by A4, A12, A54, A57, FINSEQ_1:14; ::_thesis: for j being Nat st j in Seg (len B) holds P . j = (A * (i,j)) * (Line (B,j)) let j be Nat; ::_thesis: ( j in Seg (len B) implies P . j = (A * (i,j)) * (Line (B,j)) ) assume j in Seg (len B) ; ::_thesis: P . j = (A * (i,j)) * (Line (B,j)) hence P . j = (A * (i,j)) * (Line (B,j)) by A5, A6; ::_thesis: verum end; theorem Th56: :: MATRIX11:56 for n being Nat for K being Field for A, B, C being Matrix of n,K for i being Nat st i in Seg n holds ex P being FinSequence of K st ( len P = n & Det (RLine (C,i,(Line ((A * B),i)))) = the addF of K "**" P & ( for j being Nat st j in Seg n holds P . j = (A * (i,j)) * (Det (RLine (C,i,(Line (B,j))))) ) ) proof let n be Nat; ::_thesis: for K being Field for A, B, C being Matrix of n,K for i being Nat st i in Seg n holds ex P being FinSequence of K st ( len P = n & Det (RLine (C,i,(Line ((A * B),i)))) = the addF of K "**" P & ( for j being Nat st j in Seg n holds P . j = (A * (i,j)) * (Det (RLine (C,i,(Line (B,j))))) ) ) let K be Field; ::_thesis: for A, B, C being Matrix of n,K for i being Nat st i in Seg n holds ex P being FinSequence of K st ( len P = n & Det (RLine (C,i,(Line ((A * B),i)))) = the addF of K "**" P & ( for j being Nat st j in Seg n holds P . j = (A * (i,j)) * (Det (RLine (C,i,(Line (B,j))))) ) ) let A, B, C be Matrix of n,K; ::_thesis: for i being Nat st i in Seg n holds ex P being FinSequence of K st ( len P = n & Det (RLine (C,i,(Line ((A * B),i)))) = the addF of K "**" P & ( for j being Nat st j in Seg n holds P . j = (A * (i,j)) * (Det (RLine (C,i,(Line (B,j))))) ) ) let i be Nat; ::_thesis: ( i in Seg n implies ex P being FinSequence of K st ( len P = n & Det (RLine (C,i,(Line ((A * B),i)))) = the addF of K "**" P & ( for j being Nat st j in Seg n holds P . j = (A * (i,j)) * (Det (RLine (C,i,(Line (B,j))))) ) ) ) assume A1: i in Seg n ; ::_thesis: ex P being FinSequence of K st ( len P = n & Det (RLine (C,i,(Line ((A * B),i)))) = the addF of K "**" P & ( for j being Nat st j in Seg n holds P . j = (A * (i,j)) * (Det (RLine (C,i,(Line (B,j))))) ) ) Seg n <> {} by A1; then A2: n <> 0 ; set a = addFinS K; A3: len B = n by MATRIX_1:24; deffunc H1( Nat) -> Element of the carrier of K = (A * (i,$1)) * (Det (RLine (C,i,(Line (B,$1))))); set aa = the addF of K; set KK = the carrier of K; A4: len A = n by MATRIX_1:24; consider D being FinSequence of the carrier of K such that A5: len D = len A and A6: for j being Nat st j in dom D holds D . j = H1(j) from FINSEQ_2:sch_1(); A7: n <> 0 by A1; then len D >= 1 by A4, A5, NAT_1:14; then consider Fd being Function of NAT, the carrier of K such that A8: Fd . 1 = D . 1 and A9: for k being Element of NAT st 0 <> k & k < n holds Fd . (k + 1) = the addF of K . ((Fd . k),(D . (k + 1))) and A10: the addF of K "**" D = Fd . n by A4, A5, FINSOP_1:def_1; A11: dom D = Seg (len A) by A5, FINSEQ_1:def_3; width A = n by MATRIX_1:24; then consider P being FinSequence of the carrier of K * such that A12: len P = n and A13: Line ((A * B),i) = (addFinS K) "**" P and A14: for j being Nat st j in Seg (len B) holds P . j = (A * (i,j)) * (Line (B,j)) by A1, A7, A3, A4, Th55; len P >= 1 by A12, A2, NAT_1:14; then consider Fp being Function of NAT,( the carrier of K *) such that A15: Fp . 1 = P . 1 and A16: for k being Element of NAT st 0 <> k & k < n holds Fp . (k + 1) = (addFinS K) . ((Fp . k),(P . (k + 1))) and A17: Line ((A * B),i) = Fp . n by A12, A13, FINSOP_1:def_1; defpred S1[ Nat] means ( 1 <= $1 & $1 <= n implies for pK being FinSequence of K st pK = Fp . $1 holds ( len pK = n & Fd . $1 = Det (RLine (C,i,pK)) ) ); A18: width B = n by MATRIX_1:24; A19: for k being Element of NAT st S1[k] holds S1[k + 1] proof let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] ) assume A20: S1[k] ; ::_thesis: S1[k + 1] set k1 = k + 1; set A9 = A * (i,(k + 1)); set L = Line (B,(k + 1)); assume that A21: 1 <= k + 1 and A22: k + 1 <= n ; ::_thesis: for pK being FinSequence of K st pK = Fp . (k + 1) holds ( len pK = n & Fd . (k + 1) = Det (RLine (C,i,pK)) ) A23: k + 1 in Seg n by A21, A22; let Fpk1 be FinSequence of the carrier of K; ::_thesis: ( Fpk1 = Fp . (k + 1) implies ( len Fpk1 = n & Fd . (k + 1) = Det (RLine (C,i,Fpk1)) ) ) assume A24: Fpk1 = Fp . (k + 1) ; ::_thesis: ( len Fpk1 = n & Fd . (k + 1) = Det (RLine (C,i,Fpk1)) ) percases ( k = 0 or k > 0 ) ; supposeA25: k = 0 ; ::_thesis: ( len Fpk1 = n & Fd . (k + 1) = Det (RLine (C,i,Fpk1)) ) A26: P . (k + 1) = (A * (i,(k + 1))) * (Line (B,(k + 1))) by A3, A14, A23; A27: len (Line (B,(k + 1))) = n by A18, MATRIX_1:def_7; D . (k + 1) = H1(1) by A4, A6, A11, A23, A25; hence ( len Fpk1 = n & Fd . (k + 1) = Det (RLine (C,i,Fpk1)) ) by A1, A15, A8, A24, A25, A26, A27, Lm5, Th34; ::_thesis: verum end; supposeA28: k > 0 ; ::_thesis: ( len Fpk1 = n & Fd . (k + 1) = Det (RLine (C,i,Fpk1)) ) k + 1 in dom P by A12, A23, FINSEQ_1:def_3; then A29: P . (k + 1) in rng P by FUNCT_1:def_3; rng P c= the carrier of K * by FINSEQ_1:def_4; then reconsider Pk1 = P . (k + 1), Fpk = Fp . k as Element of the carrier of K * by A29; A30: k + 0 < k + 1 by XREAL_1:8; then A31: Fd . k = Det (RLine (C,i,Fpk)) by A20, A22, A28, NAT_1:14, XXREAL_0:2; A32: len Fpk = n by A20, A22, A28, A30, NAT_1:14, XXREAL_0:2; A33: k < n by A22, A30, XXREAL_0:2; then A34: Fd . (k + 1) = the addF of K . ((Fd . k),(D . (k + 1))) by A9, A28; A35: P . (k + 1) = (A * (i,(k + 1))) * (Line (B,(k + 1))) by A3, A14, A23; Fpk1 = (addFinS K) . (Fpk,Pk1) by A16, A24, A28, A33; then A36: Fpk1 = Fpk + ((A * (i,(k + 1))) * (Line (B,(k + 1)))) by A35, Def5; A37: len (Line (B,(k + 1))) = n by A18, MATRIX_1:def_7; then A38: len ((A * (i,(k + 1))) * (Line (B,(k + 1)))) = n by Lm5; Det (RLine (C,i,((A * (i,(k + 1))) * (Line (B,(k + 1)))))) = (A * (i,(k + 1))) * (Det (RLine (C,i,(Line (B,(k + 1)))))) by A1, A37, Th34; then Det (RLine (C,i,Fpk1)) = (Det (RLine (C,i,Fpk))) + ((A * (i,(k + 1))) * (Det (RLine (C,i,(Line (B,(k + 1))))))) by A1, A32, A36, A38, Th36; hence ( len Fpk1 = n & Fd . (k + 1) = Det (RLine (C,i,Fpk1)) ) by A4, A6, A11, A23, A31, A34, A32, A36, A38, Lm6; ::_thesis: verum end; end; end; take D ; ::_thesis: ( len D = n & Det (RLine (C,i,(Line ((A * B),i)))) = the addF of K "**" D & ( for j being Nat st j in Seg n holds D . j = (A * (i,j)) * (Det (RLine (C,i,(Line (B,j))))) ) ) thus len D = n by A5, MATRIX_1:24; ::_thesis: ( Det (RLine (C,i,(Line ((A * B),i)))) = the addF of K "**" D & ( for j being Nat st j in Seg n holds D . j = (A * (i,j)) * (Det (RLine (C,i,(Line (B,j))))) ) ) A39: S1[ 0 ] ; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A39, A19); then S1[ len P] ; hence ( Det (RLine (C,i,(Line ((A * B),i)))) = the addF of K "**" D & ( for j being Nat st j in Seg n holds D . j = (A * (i,j)) * (Det (RLine (C,i,(Line (B,j))))) ) ) by A4, A12, A6, A11, A17, A10, A2, NAT_1:14; ::_thesis: verum end; theorem Th57: :: MATRIX11:57 for X being set for Y being non empty set for x being set st not x in X holds ex BIJECT being Function of [:(Funcs (X,Y)),Y:],(Funcs ((X \/ {x}),Y)) st ( BIJECT is bijective & ( for f being Function of X,Y for F being Function of (X \/ {x}),Y st F | X = f holds BIJECT . (f,(F . x)) = F ) ) proof let X be set ; ::_thesis: for Y being non empty set for x being set st not x in X holds ex BIJECT being Function of [:(Funcs (X,Y)),Y:],(Funcs ((X \/ {x}),Y)) st ( BIJECT is bijective & ( for f being Function of X,Y for F being Function of (X \/ {x}),Y st F | X = f holds BIJECT . (f,(F . x)) = F ) ) let Y be non empty set ; ::_thesis: for x being set st not x in X holds ex BIJECT being Function of [:(Funcs (X,Y)),Y:],(Funcs ((X \/ {x}),Y)) st ( BIJECT is bijective & ( for f being Function of X,Y for F being Function of (X \/ {x}),Y st F | X = f holds BIJECT . (f,(F . x)) = F ) ) let x be set ; ::_thesis: ( not x in X implies ex BIJECT being Function of [:(Funcs (X,Y)),Y:],(Funcs ((X \/ {x}),Y)) st ( BIJECT is bijective & ( for f being Function of X,Y for F being Function of (X \/ {x}),Y st F | X = f holds BIJECT . (f,(F . x)) = F ) ) ) assume A1: not x in X ; ::_thesis: ex BIJECT being Function of [:(Funcs (X,Y)),Y:],(Funcs ((X \/ {x}),Y)) st ( BIJECT is bijective & ( for f being Function of X,Y for F being Function of (X \/ {x}),Y st F | X = f holds BIJECT . (f,(F . x)) = F ) ) set Xx = X \/ {x}; set FXY = Funcs (X,Y); set FXxY = Funcs ((X \/ {x}),Y); defpred S1[ set , set ] means for f being Function of X,Y for F being Function of (X \/ {x}),Y for y being set st [f,y] = $1 & F . x = y & F | X = f holds F = $2; A2: for x being Element of [:(Funcs (X,Y)),Y:] ex y being Element of Funcs ((X \/ {x}),Y) st S1[x,y] proof let x9 be Element of [:(Funcs (X,Y)),Y:]; ::_thesis: ex y being Element of Funcs ((X \/ {x}),Y) st S1[x9,y] consider f, y being set such that A3: f in Funcs (X,Y) and A4: y in Y and A5: x9 = [f,y] by ZFMISC_1:def_2; reconsider f = f as Function of X,Y by A3, FUNCT_2:66; Y \/ {y} = Y by A4, ZFMISC_1:40; then consider F being Function of (X \/ {x}),Y such that A6: F | X = f and A7: F . x = y by A1, STIRL2_1:57; reconsider F9 = F as Element of Funcs ((X \/ {x}),Y) by FUNCT_2:8; take F9 ; ::_thesis: S1[x9,F9] let g be Function of X,Y; ::_thesis: for F being Function of (X \/ {x}),Y for y being set st [g,y] = x9 & F . x = y & F | X = g holds F = F9 let G be Function of (X \/ {x}),Y; ::_thesis: for y being set st [g,y] = x9 & G . x = y & G | X = g holds G = F9 let y9 be set ; ::_thesis: ( [g,y9] = x9 & G . x = y9 & G | X = g implies G = F9 ) assume that A8: [g,y9] = x9 and A9: G . x = y9 and A10: G | X = g ; ::_thesis: G = F9 now__::_thesis:_for_xx_being_set_st_xx_in_X_\/_{x}_holds_ G_._xx_=_F_._xx let xx be set ; ::_thesis: ( xx in X \/ {x} implies G . xx = F . xx ) assume xx in X \/ {x} ; ::_thesis: G . xx = F . xx then A11: ( xx in X or xx in {x} ) by XBOOLE_0:def_3; A12: dom f = X by FUNCT_2:def_1; dom g = X by FUNCT_2:def_1; then ( ( G . xx = g . xx & F . xx = f . xx ) or xx = x ) by A6, A10, A11, A12, FUNCT_1:47, TARSKI:def_1; hence G . xx = F . xx by A5, A7, A8, A9, XTUPLE_0:1; ::_thesis: verum end; hence G = F9 by FUNCT_2:12; ::_thesis: verum end; consider H being Function of [:(Funcs (X,Y)),Y:],(Funcs ((X \/ {x}),Y)) such that A13: for x being Element of [:(Funcs (X,Y)),Y:] holds S1[x,H . x] from FUNCT_2:sch_3(A2); A14: now__::_thesis:_for_x1,_x2_being_set_st_x1_in_[:(Funcs_(X,Y)),Y:]_&_x2_in_[:(Funcs_(X,Y)),Y:]_&_H_._x1_=_H_._x2_holds_ x1_=_x2 let x1, x2 be set ; ::_thesis: ( x1 in [:(Funcs (X,Y)),Y:] & x2 in [:(Funcs (X,Y)),Y:] & H . x1 = H . x2 implies x1 = x2 ) assume that A15: x1 in [:(Funcs (X,Y)),Y:] and A16: x2 in [:(Funcs (X,Y)),Y:] and A17: H . x1 = H . x2 ; ::_thesis: x1 = x2 consider f2, y2 being set such that A18: f2 in Funcs (X,Y) and A19: y2 in Y and A20: x2 = [f2,y2] by A16, ZFMISC_1:def_2; consider f1, y1 being set such that A21: f1 in Funcs (X,Y) and A22: y1 in Y and A23: x1 = [f1,y1] by A15, ZFMISC_1:def_2; reconsider f1 = f1, f2 = f2 as Function of X,Y by A21, A18, FUNCT_2:66; Y \/ {y2} = Y by A19, ZFMISC_1:40; then consider F2 being Function of (X \/ {x}),Y such that A24: F2 | X = f2 and A25: F2 . x = y2 by A1, STIRL2_1:57; A26: H . x2 = F2 by A13, A16, A20, A24, A25; Y \/ {y1} = Y by A22, ZFMISC_1:40; then consider F1 being Function of (X \/ {x}),Y such that A27: F1 | X = f1 and A28: F1 . x = y1 by A1, STIRL2_1:57; H . x1 = F1 by A13, A15, A23, A27, A28; hence x1 = x2 by A17, A23, A20, A27, A28, A24, A25, A26; ::_thesis: verum end; take H ; ::_thesis: ( H is bijective & ( for f being Function of X,Y for F being Function of (X \/ {x}),Y st F | X = f holds H . (f,(F . x)) = F ) ) x in {x} by TARSKI:def_1; then A29: x in X \/ {x} by XBOOLE_0:def_3; A30: Funcs ((X \/ {x}),Y) c= rng H proof let f9 be set ; :: according to TARSKI:def_3 ::_thesis: ( not f9 in Funcs ((X \/ {x}),Y) or f9 in rng H ) assume f9 in Funcs ((X \/ {x}),Y) ; ::_thesis: f9 in rng H then reconsider f = f9 as Function of (X \/ {x}),Y by FUNCT_2:66; dom f = X \/ {x} by FUNCT_2:def_1; then A31: dom (f | X) = X by RELAT_1:62, XBOOLE_1:7; rng (f | X) c= Y by RELAT_1:def_19; then reconsider fX = f | X as Function of X,Y by A31, FUNCT_2:2; A32: fX in Funcs (X,Y) by FUNCT_2:8; x in {x} by TARSKI:def_1; then A33: x in X \/ {x} by XBOOLE_0:def_3; X \/ {x} = dom f by FUNCT_2:def_1; then A34: f . x in rng f by A33, FUNCT_1:def_3; rng f c= Y by RELAT_1:def_19; then A35: [fX,(f . x)] in [:(Funcs (X,Y)),Y:] by A34, A32, ZFMISC_1:87; [:(Funcs (X,Y)),Y:] = dom H by FUNCT_2:def_1; then H . [fX,(f . x)] in rng H by A35, FUNCT_1:def_3; hence f9 in rng H by A13, A35; ::_thesis: verum end; rng H c= Funcs ((X \/ {x}),Y) by RELAT_1:def_19; then Funcs ((X \/ {x}),Y) = rng H by A30, XBOOLE_0:def_10; then ( H is one-to-one & H is onto ) by A14, FUNCT_2:19, FUNCT_2:def_3; hence H is bijective ; ::_thesis: for f being Function of X,Y for F being Function of (X \/ {x}),Y st F | X = f holds H . (f,(F . x)) = F let f be Function of X,Y; ::_thesis: for F being Function of (X \/ {x}),Y st F | X = f holds H . (f,(F . x)) = F let F be Function of (X \/ {x}),Y; ::_thesis: ( F | X = f implies H . (f,(F . x)) = F ) assume A36: F | X = f ; ::_thesis: H . (f,(F . x)) = F X \/ {x} = dom F by FUNCT_2:def_1; then A37: F . x in rng F by A29, FUNCT_1:def_3; A38: f in Funcs (X,Y) by FUNCT_2:8; rng F c= Y by RELAT_1:def_19; then [f,(F . x)] in [:(Funcs (X,Y)),Y:] by A37, A38, ZFMISC_1:87; hence H . (f,(F . x)) = F by A13, A36; ::_thesis: verum end; theorem Th58: :: MATRIX11:58 for D being non empty set for X being finite set for Y being non empty finite set for x being set st not x in X holds for F being BinOp of D st F is having_a_unity & F is commutative & F is associative holds for f being Function of (Funcs (X,Y)),D for g being Function of (Funcs ((X \/ {x}),Y)),D st ( for H being Function of X,Y for SF being Element of Fin (Funcs ((X \/ {x}),Y)) st SF = { h where h is Function of (X \/ {x}),Y : h | X = H } holds F $$ (SF,g) = f . H ) holds F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g) proof let D be non empty set ; ::_thesis: for X being finite set for Y being non empty finite set for x being set st not x in X holds for F being BinOp of D st F is having_a_unity & F is commutative & F is associative holds for f being Function of (Funcs (X,Y)),D for g being Function of (Funcs ((X \/ {x}),Y)),D st ( for H being Function of X,Y for SF being Element of Fin (Funcs ((X \/ {x}),Y)) st SF = { h where h is Function of (X \/ {x}),Y : h | X = H } holds F $$ (SF,g) = f . H ) holds F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g) let X be finite set ; ::_thesis: for Y being non empty finite set for x being set st not x in X holds for F being BinOp of D st F is having_a_unity & F is commutative & F is associative holds for f being Function of (Funcs (X,Y)),D for g being Function of (Funcs ((X \/ {x}),Y)),D st ( for H being Function of X,Y for SF being Element of Fin (Funcs ((X \/ {x}),Y)) st SF = { h where h is Function of (X \/ {x}),Y : h | X = H } holds F $$ (SF,g) = f . H ) holds F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g) let Y be non empty finite set ; ::_thesis: for x being set st not x in X holds for F being BinOp of D st F is having_a_unity & F is commutative & F is associative holds for f being Function of (Funcs (X,Y)),D for g being Function of (Funcs ((X \/ {x}),Y)),D st ( for H being Function of X,Y for SF being Element of Fin (Funcs ((X \/ {x}),Y)) st SF = { h where h is Function of (X \/ {x}),Y : h | X = H } holds F $$ (SF,g) = f . H ) holds F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g) let x be set ; ::_thesis: ( not x in X implies for F being BinOp of D st F is having_a_unity & F is commutative & F is associative holds for f being Function of (Funcs (X,Y)),D for g being Function of (Funcs ((X \/ {x}),Y)),D st ( for H being Function of X,Y for SF being Element of Fin (Funcs ((X \/ {x}),Y)) st SF = { h where h is Function of (X \/ {x}),Y : h | X = H } holds F $$ (SF,g) = f . H ) holds F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g) ) assume A1: not x in X ; ::_thesis: for F being BinOp of D st F is having_a_unity & F is commutative & F is associative holds for f being Function of (Funcs (X,Y)),D for g being Function of (Funcs ((X \/ {x}),Y)),D st ( for H being Function of X,Y for SF being Element of Fin (Funcs ((X \/ {x}),Y)) st SF = { h where h is Function of (X \/ {x}),Y : h | X = H } holds F $$ (SF,g) = f . H ) holds F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g) set Xx = X \/ {x}; set FXY = Funcs (X,Y); set FXxY = Funcs ((X \/ {x}),Y); consider B being Function of [:(Funcs (X,Y)),Y:],(Funcs ((X \/ {x}),Y)) such that A2: B is bijective and A3: for f being Function of X,Y for F being Function of (X \/ {x}),Y st F | X = f holds B . (f,(F . x)) = F by A1, Th57; A4: Funcs (X,Y) is finite by FRAENKEL:6; dom B = [:(Funcs (X,Y)),Y:] by FUNCT_2:def_1; then reconsider domB = dom B as Element of Fin [:(Funcs (X,Y)),Y:] by A4, FINSUB_1:def_5; Funcs (X,Y) is finite by FRAENKEL:6; then reconsider FXY9 = Funcs (X,Y) as Element of Fin (Funcs (X,Y)) by FINSUB_1:def_5; A5: FinOmega (Funcs ((X \/ {x}),Y)) = Funcs ((X \/ {x}),Y) by FRAENKEL:6, MATRIX_2:def_14; reconsider Y9 = Y as Element of Fin Y by FINSUB_1:def_5; let F be BinOp of D; ::_thesis: ( F is having_a_unity & F is commutative & F is associative implies for f being Function of (Funcs (X,Y)),D for g being Function of (Funcs ((X \/ {x}),Y)),D st ( for H being Function of X,Y for SF being Element of Fin (Funcs ((X \/ {x}),Y)) st SF = { h where h is Function of (X \/ {x}),Y : h | X = H } holds F $$ (SF,g) = f . H ) holds F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g) ) assume that A6: F is having_a_unity and A7: F is commutative and A8: F is associative ; ::_thesis: for f being Function of (Funcs (X,Y)),D for g being Function of (Funcs ((X \/ {x}),Y)),D st ( for H being Function of X,Y for SF being Element of Fin (Funcs ((X \/ {x}),Y)) st SF = { h where h is Function of (X \/ {x}),Y : h | X = H } holds F $$ (SF,g) = f . H ) holds F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g) let f be Function of (Funcs (X,Y)),D; ::_thesis: for g being Function of (Funcs ((X \/ {x}),Y)),D st ( for H being Function of X,Y for SF being Element of Fin (Funcs ((X \/ {x}),Y)) st SF = { h where h is Function of (X \/ {x}),Y : h | X = H } holds F $$ (SF,g) = f . H ) holds F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g) let g be Function of (Funcs ((X \/ {x}),Y)),D; ::_thesis: ( ( for H being Function of X,Y for SF being Element of Fin (Funcs ((X \/ {x}),Y)) st SF = { h where h is Function of (X \/ {x}),Y : h | X = H } holds F $$ (SF,g) = f . H ) implies F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g) ) assume A9: for H being Function of X,Y for SF being Element of Fin (Funcs ((X \/ {x}),Y)) st SF = { h where h is Function of (X \/ {x}),Y : h | X = H } holds F $$ (SF,g) = f . H ; ::_thesis: F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g) reconsider gB = g * B as Function of [:(Funcs (X,Y)),Y:],D ; for z being Element of Funcs (X,Y) holds f . z = F $$ (Y9,((curry gB) . z)) proof let z be Element of Funcs (X,Y); ::_thesis: f . z = F $$ (Y9,((curry gB) . z)) reconsider Z = z as Function of X,Y ; set SF = { h where h is Function of (X \/ {x}),Y : h | X = Z } ; deffunc H1( set ) -> set = [z,$1]; consider q being Function such that A10: ( dom q = Y & ( for x being set st x in Y holds q . x = H1(x) ) ) from FUNCT_1:sch_3(); A11: {z} c= Funcs (X,Y) by ZFMISC_1:31; then [:{Z},Y:] c= [:(Funcs (X,Y)),Y:] by ZFMISC_1:95; then reconsider ZY = [:{Z},Y:] as Element of Fin [:(Funcs (X,Y)),Y:] by FINSUB_1:def_5; for x9 being set holds ( x9 in ZY iff x9 in q .: Y ) proof let x9 be set ; ::_thesis: ( x9 in ZY iff x9 in q .: Y ) thus ( x9 in ZY implies x9 in q .: Y ) ::_thesis: ( x9 in q .: Y implies x9 in ZY ) proof assume x9 in ZY ; ::_thesis: x9 in q .: Y then consider z9, y9 being set such that A12: z9 in {Z} and A13: y9 in Y and A14: x9 = [z9,y9] by ZFMISC_1:def_2; A15: z = z9 by A12, TARSKI:def_1; q . y9 = [z,y9] by A10, A13; hence x9 in q .: Y by A10, A13, A14, A15, FUNCT_1:def_6; ::_thesis: verum end; assume x9 in q .: Y ; ::_thesis: x9 in ZY then consider y9 being set such that A16: y9 in dom q and A17: y9 in Y and A18: x9 = q . y9 by FUNCT_1:def_6; x9 = [z,y9] by A10, A16, A18; hence x9 in ZY by A17, ZFMISC_1:105; ::_thesis: verum end; then A19: q .: Y = ZY by TARSKI:1; then A20: rng q = ZY by A10, RELAT_1:113; now__::_thesis:_for_x1,_x2_being_set_st_x1_in_dom_q_&_x2_in_dom_q_&_q_._x1_=_q_._x2_holds_ x1_=_x2 let x1, x2 be set ; ::_thesis: ( x1 in dom q & x2 in dom q & q . x1 = q . x2 implies x1 = x2 ) assume that A21: x1 in dom q and A22: x2 in dom q and A23: q . x1 = q . x2 ; ::_thesis: x1 = x2 A24: q . x2 = [z,x2] by A10, A22; [z,x1] = q . x1 by A10, A21; hence x1 = x2 by A23, A24, XTUPLE_0:1; ::_thesis: verum end; then A25: q is one-to-one by FUNCT_1:def_4; ZY c= [:(Funcs (X,Y)),Y:] by A11, ZFMISC_1:95; then reconsider q = q as Function of Y,[:(Funcs (X,Y)),Y:] by A10, A20, FUNCT_2:2; reconsider gBq = gB * q as Function of Y,D ; dom gB = [:(Funcs (X,Y)),Y:] by FUNCT_2:def_1; then consider C being Function such that A26: (curry gB) . z = C and dom C = Y and rng C c= rng gB and A27: for y9 being set st y9 in Y holds C . y9 = gB . (z,y9) by FUNCT_5:29; reconsider C = C as Function of Y,D by A26; now__::_thesis:_for_x9_being_set_st_x9_in_Y_holds_ C_._x9_=_gBq_._x9 let x9 be set ; ::_thesis: ( x9 in Y implies C . x9 = gBq . x9 ) assume A28: x9 in Y ; ::_thesis: C . x9 = gBq . x9 A29: q . x9 = [z,x9] by A10, A28; C . x9 = gB . (z,x9) by A27, A28; hence C . x9 = gBq . x9 by A28, A29, FUNCT_2:15; ::_thesis: verum end; then A30: C = gBq by FUNCT_2:12; A31: B .: ZY c= { h where h is Function of (X \/ {x}),Y : h | X = Z } proof let b be set ; :: according to TARSKI:def_3 ::_thesis: ( not b in B .: ZY or b in { h where h is Function of (X \/ {x}),Y : h | X = Z } ) assume b in B .: ZY ; ::_thesis: b in { h where h is Function of (X \/ {x}),Y : h | X = Z } then consider zy being set such that zy in dom B and A32: zy in ZY and A33: b = B . zy by FUNCT_1:def_6; consider z9, y9 being set such that A34: z9 in {Z} and A35: y9 in Y and A36: zy = [z9,y9] by A32, ZFMISC_1:def_2; Y \/ {y9} = Y by A35, ZFMISC_1:40; then consider F1 being Function of (X \/ {x}),Y such that A37: F1 | X = Z and A38: F1 . x = y9 by A1, STIRL2_1:57; z9 = Z by A34, TARSKI:def_1; then B . (z9,y9) = F1 by A3, A37, A38; hence b in { h where h is Function of (X \/ {x}),Y : h | X = Z } by A33, A36, A37; ::_thesis: verum end; A39: { h where h is Function of (X \/ {x}),Y : h | X = Z } c= B .: ZY proof x in {x} by TARSKI:def_1; then A40: x in X \/ {x} by XBOOLE_0:def_3; let sf be set ; :: according to TARSKI:def_3 ::_thesis: ( not sf in { h where h is Function of (X \/ {x}),Y : h | X = Z } or sf in B .: ZY ) assume sf in { h where h is Function of (X \/ {x}),Y : h | X = Z } ; ::_thesis: sf in B .: ZY then consider h being Function of (X \/ {x}),Y such that A41: h = sf and A42: h | X = Z ; A43: [:(Funcs (X,Y)),Y:] = dom B by FUNCT_2:def_1; dom h = X \/ {x} by FUNCT_2:def_1; then A44: h . x in rng h by A40, FUNCT_1:def_3; A45: rng h c= Y by RELAT_1:def_19; then A46: [z,(h . x)] in [:(Funcs (X,Y)),Y:] by A44, ZFMISC_1:87; z in {Z} by TARSKI:def_1; then [z,(h . x)] in ZY by A44, A45, ZFMISC_1:87; then B . (z,(h . x)) in B .: ZY by A46, A43, FUNCT_1:def_6; hence sf in B .: ZY by A3, A41, A42; ::_thesis: verum end; then reconsider SF = { h where h is Function of (X \/ {x}),Y : h | X = Z } as Element of Fin (Funcs ((X \/ {x}),Y)) by A31, XBOOLE_0:def_10; B .: ZY = SF by A31, A39, XBOOLE_0:def_10; then A47: F $$ ((B .: ZY),g) = f . Z by A9; F $$ ((B .: ZY),g) = F $$ (ZY,gB) by A7, A8, A2, SETWOP_2:6; hence f . z = F $$ (Y9,((curry gB) . z)) by A7, A8, A47, A25, A19, A26, A30, SETWOP_2:6; ::_thesis: verum end; then F $$ ([:FXY9,Y9:],gB) = F $$ (FXY9,f) by A6, A7, A8, MATRIX_3:30; then A48: F $$ (domB,gB) = F $$ (FXY9,f) by FUNCT_2:def_1; A49: rng B = Funcs ((X \/ {x}),Y) by A2, FUNCT_2:def_3; F $$ ((B .: domB),g) = F $$ (domB,(g * B)) by A7, A8, A2, SETWOP_2:6; then F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g) = F $$ (domB,(g * B)) by A49, A5, RELAT_1:113; hence F $$ ((FinOmega (Funcs (X,Y))),f) = F $$ ((FinOmega (Funcs ((X \/ {x}),Y))),g) by A48, MATRIX_2:def_14; ::_thesis: verum end; theorem Th59: :: MATRIX11:59 for n, m being Nat for D being non empty set for A, B being Matrix of n,m,D for i being Nat st i <= n & 0 < n holds for F being Function of (Seg i),(Seg n) ex M being Matrix of n,m,D st ( M = A +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = A . j ) ) ) ) proof let n, m be Nat; ::_thesis: for D being non empty set for A, B being Matrix of n,m,D for i being Nat st i <= n & 0 < n holds for F being Function of (Seg i),(Seg n) ex M being Matrix of n,m,D st ( M = A +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = A . j ) ) ) ) let D be non empty set ; ::_thesis: for A, B being Matrix of n,m,D for i being Nat st i <= n & 0 < n holds for F being Function of (Seg i),(Seg n) ex M being Matrix of n,m,D st ( M = A +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = A . j ) ) ) ) let A, B be Matrix of n,m,D; ::_thesis: for i being Nat st i <= n & 0 < n holds for F being Function of (Seg i),(Seg n) ex M being Matrix of n,m,D st ( M = A +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = A . j ) ) ) ) let i be Nat; ::_thesis: ( i <= n & 0 < n implies for F being Function of (Seg i),(Seg n) ex M being Matrix of n,m,D st ( M = A +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = A . j ) ) ) ) ) assume that A1: i <= n and A2: 0 < n ; ::_thesis: for F being Function of (Seg i),(Seg n) ex M being Matrix of n,m,D st ( M = A +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = A . j ) ) ) ) set I = idseq n; let F be Function of (Seg i),(Seg n); ::_thesis: ex M being Matrix of n,m,D st ( M = A +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = A . j ) ) ) ) set IF = (idseq n) +* F; A3: dom (idseq n) = Seg n by RELAT_1:45; A4: rng (idseq n) = Seg n by RELAT_1:45; rng F c= Seg n by RELAT_1:def_19; then (rng F) \/ (Seg n) = Seg n by XBOOLE_1:12; then A5: rng ((idseq n) +* F) c= Seg n by A4, FUNCT_4:17; A6: Seg i c= Seg n by A1, FINSEQ_1:5; then A7: (Seg i) \/ (Seg n) = Seg n by XBOOLE_1:12; A8: dom F = Seg i by A2, FUNCT_2:def_1; then dom F c= Seg n by A1, FINSEQ_1:5; then (dom F) \/ (Seg n) = Seg n by XBOOLE_1:12; then dom ((idseq n) +* F) = Seg n by A3, FUNCT_4:def_1; then reconsider IF = (idseq n) +* F as Function of (Seg n),(Seg n) by A5, FUNCT_2:2; reconsider BIF = B * IF as Matrix of n,m,D ; set BIFi = BIF | (Seg i); set M = A +* (BIF | (Seg i)); A9: len B = n by A2, MATRIX_1:23; len BIF = len B by Def4; then dom BIF = Seg n by A9, FINSEQ_1:def_3; then A10: dom (BIF | (Seg i)) = Seg i by A6, RELAT_1:62; len A = n by A2, MATRIX_1:23; then dom A = Seg n by FINSEQ_1:def_3; then A11: dom (A +* (BIF | (Seg i))) = (Seg i) \/ (Seg n) by A10, FUNCT_4:def_1; then reconsider M = A +* (BIF | (Seg i)) as FinSequence by A7, FINSEQ_1:def_2; A12: for j being Nat holds ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = A . j ) ) proof let j be Nat; ::_thesis: ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = A . j ) ) thus ( j in Seg i implies M . j = B . (F . j) ) ::_thesis: ( not j in Seg i implies M . j = A . j ) proof A13: Seg i c= Seg n by A1, FINSEQ_1:5; assume A14: j in Seg i ; ::_thesis: M . j = B . (F . j) then A15: (BIF | (Seg i)) . j = BIF . j by A10, FUNCT_1:47; IF . j = F . j by A8, A14, FUNCT_4:13; then A16: Line (BIF,j) = B . (F . j) by A14, A13, Th38; Line (BIF,j) = BIF . j by A14, A13, MATRIX_2:8; hence M . j = B . (F . j) by A10, A14, A16, A15, FUNCT_4:13; ::_thesis: verum end; assume not j in Seg i ; ::_thesis: M . j = A . j hence M . j = A . j by A10, FUNCT_4:11; ::_thesis: verum end; A17: for x being set st x in rng M holds ex p being FinSequence of D st ( x = p & len p = m ) proof let x be set ; ::_thesis: ( x in rng M implies ex p being FinSequence of D st ( x = p & len p = m ) ) assume x in rng M ; ::_thesis: ex p being FinSequence of D st ( x = p & len p = m ) then consider k being set such that A18: k in dom M and A19: M . k = x by FUNCT_1:def_3; reconsider k = k as Nat by A18; percases ( k in Seg i or not k in Seg i ) ; supposeA20: k in Seg i ; ::_thesis: ex p being FinSequence of D st ( x = p & len p = m ) A21: rng F c= Seg n by RELAT_1:def_19; A22: F . k in rng F by A8, A20, FUNCT_1:def_3; then reconsider Fk = F . k as Element of NAT by A21, TARSKI:def_3; take L = Line (B,Fk); ::_thesis: ( x = L & len L = m ) A23: len L = width B by MATRIX_1:def_7; B . (F . k) = L by A22, A21, MATRIX_2:8; hence ( x = L & len L = m ) by A2, A12, A19, A20, A23, MATRIX_1:23; ::_thesis: verum end; supposeA24: not k in Seg i ; ::_thesis: ex p being FinSequence of D st ( x = p & len p = m ) take L = Line (A,k); ::_thesis: ( x = L & len L = m ) A25: len L = width A by MATRIX_1:def_7; M . k = A . k by A12, A24; hence ( x = L & len L = m ) by A2, A11, A7, A18, A19, A25, MATRIX_1:23, MATRIX_2:8; ::_thesis: verum end; end; end; then reconsider M = M as Matrix of D by MATRIX_1:9; n is Element of NAT by ORDINAL1:def_12; then A26: len M = n by A11, A7, FINSEQ_1:def_3; now__::_thesis:_for_p_being_FinSequence_of_D_st_p_in_rng_M_holds_ len_p_=_m let p be FinSequence of D; ::_thesis: ( p in rng M implies len p = m ) assume p in rng M ; ::_thesis: len p = m then ex q being FinSequence of D st ( p = q & len q = m ) by A17; hence len p = m ; ::_thesis: verum end; then reconsider M = M as Matrix of n,m,D by A26, MATRIX_1:def_2; take M ; ::_thesis: ( M = A +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = A . j ) ) ) ) thus ( M = A +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = A . j ) ) ) ) by A12; ::_thesis: verum end; Lm10: for n being Nat for K being Field for A, B being Matrix of n,n,K for i being Nat st i <= n & 0 < n holds ex P being Function of (Funcs ((Seg i),(Seg n))), the carrier of K st for F being Function of (Seg i),(Seg n) for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) holds ex Path being FinSequence of K st ( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det M) ) proof let n be Nat; ::_thesis: for K being Field for A, B being Matrix of n,n,K for i being Nat st i <= n & 0 < n holds ex P being Function of (Funcs ((Seg i),(Seg n))), the carrier of K st for F being Function of (Seg i),(Seg n) for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) holds ex Path being FinSequence of K st ( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det M) ) let K be Field; ::_thesis: for A, B being Matrix of n,n,K for i being Nat st i <= n & 0 < n holds ex P being Function of (Funcs ((Seg i),(Seg n))), the carrier of K st for F being Function of (Seg i),(Seg n) for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) holds ex Path being FinSequence of K st ( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det M) ) let A, B be Matrix of n,n,K; ::_thesis: for i being Nat st i <= n & 0 < n holds ex P being Function of (Funcs ((Seg i),(Seg n))), the carrier of K st for F being Function of (Seg i),(Seg n) for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) holds ex Path being FinSequence of K st ( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det M) ) let i be Nat; ::_thesis: ( i <= n & 0 < n implies ex P being Function of (Funcs ((Seg i),(Seg n))), the carrier of K st for F being Function of (Seg i),(Seg n) for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) holds ex Path being FinSequence of K st ( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det M) ) ) assume that A1: i <= n and A2: 0 < n ; ::_thesis: ex P being Function of (Funcs ((Seg i),(Seg n))), the carrier of K st for F being Function of (Seg i),(Seg n) for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) holds ex Path being FinSequence of K st ( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det M) ) set KK = the carrier of K; set I = idseq n; set Sn = Seg n; set Si = Seg i; set mm = the multF of K; set FF = Funcs ((Seg i),(Seg n)); reconsider Sn = Seg n as non empty set by A2; set AB = A * B; reconsider AB = A * B as Matrix of n,K ; defpred S1[ set , set ] means for F being Function of (Seg i),(Seg n) st F = $1 holds for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) holds ex Path being FinSequence of K st ( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds Path . j = A * (j,Fj) ) & $2 = ( the multF of K $$ Path) * (Det M) ); ex f being Function st ( dom f = Seg i & rng f c= Sn ) by FUNCT_1:8; then reconsider FF = Funcs ((Seg i),(Seg n)) as non empty set ; A3: for x being Element of FF ex y being Element of the carrier of K st S1[x,y] proof let x be Element of FF; ::_thesis: ex y being Element of the carrier of K st S1[x,y] reconsider F = x as Function of (Seg i),Sn by FUNCT_2:66; defpred S2[ set , set ] means for Fj, j being Nat st j = $1 & Fj = F . j holds $2 = A * (j,Fj); A4: i is Element of NAT by ORDINAL1:def_12; A5: for x9 being set st x9 in Seg i holds ex y being set st ( y in the carrier of K & S2[x9,y] ) proof let x9 be set ; ::_thesis: ( x9 in Seg i implies ex y being set st ( y in the carrier of K & S2[x9,y] ) ) assume A6: x9 in Seg i ; ::_thesis: ex y being set st ( y in the carrier of K & S2[x9,y] ) reconsider i = x9 as Nat by A6; A7: rng F c= Seg n by RELAT_1:def_19; Seg i = dom F by FUNCT_2:def_1; then F . i in rng F by A6, FUNCT_1:def_3; then F . i in Sn by A7; then reconsider Fi = F . i as Nat ; take A * (i,Fi) ; ::_thesis: ( A * (i,Fi) in the carrier of K & S2[x9,A * (i,Fi)] ) thus ( A * (i,Fi) in the carrier of K & S2[x9,A * (i,Fi)] ) ; ::_thesis: verum end; consider path being Function of (Seg i), the carrier of K such that A8: for x being set st x in Seg i holds S2[x,path . x] from FUNCT_2:sch_1(A5); dom path = Seg i by FUNCT_2:def_1; then reconsider p = path as FinSequence by FINSEQ_1:def_2; rng path c= the carrier of K by RELAT_1:def_19; then reconsider p = p as FinSequence of K by FINSEQ_1:def_4; consider M being Matrix of n,K such that M = AB +* ((B * ((idseq n) +* F)) | (Seg i)) and A9: for j being Nat holds ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = AB . j ) ) by A1, A2, Th59; take ( the multF of K $$ p) * (Det M) ; ::_thesis: S1[x,( the multF of K $$ p) * (Det M)] let F9 be Function of (Seg i),(Seg n); ::_thesis: ( F9 = x implies for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F9)) | (Seg i)) & ( for j being Nat holds ( ( j in Seg i implies M . j = B . (F9 . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) holds ex Path being FinSequence of K st ( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F9 . j holds Path . j = A * (j,Fj) ) & ( the multF of K $$ p) * (Det M) = ( the multF of K $$ Path) * (Det M) ) ) assume A10: F9 = x ; ::_thesis: for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F9)) | (Seg i)) & ( for j being Nat holds ( ( j in Seg i implies M . j = B . (F9 . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) holds ex Path being FinSequence of K st ( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F9 . j holds Path . j = A * (j,Fj) ) & ( the multF of K $$ p) * (Det M) = ( the multF of K $$ Path) * (Det M) ) let M9 be Matrix of n,n,K; ::_thesis: ( M9 = (A * B) +* ((B * ((idseq n) +* F9)) | (Seg i)) & ( for j being Nat holds ( ( j in Seg i implies M9 . j = B . (F9 . j) ) & ( not j in Seg i implies M9 . j = (A * B) . j ) ) ) implies ex Path being FinSequence of K st ( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F9 . j holds Path . j = A * (j,Fj) ) & ( the multF of K $$ p) * (Det M) = ( the multF of K $$ Path) * (Det M9) ) ) assume that M9 = (A * B) +* ((B * ((idseq n) +* F9)) | (Seg i)) and A11: for j being Nat holds ( ( j in Seg i implies M9 . j = B . (F9 . j) ) & ( not j in Seg i implies M9 . j = (A * B) . j ) ) ; ::_thesis: ex Path being FinSequence of K st ( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F9 . j holds Path . j = A * (j,Fj) ) & ( the multF of K $$ p) * (Det M) = ( the multF of K $$ Path) * (Det M9) ) take p ; ::_thesis: ( len p = i & ( for Fj, j being Nat st j in Seg i & Fj = F9 . j holds p . j = A * (j,Fj) ) & ( the multF of K $$ p) * (Det M) = ( the multF of K $$ p) * (Det M9) ) dom path = Seg i by FUNCT_2:def_1; hence ( len p = i & ( for Fj, j being Nat st j in Seg i & Fj = F9 . j holds p . j = A * (j,Fj) ) ) by A8, A10, A4, FINSEQ_1:def_3; ::_thesis: ( the multF of K $$ p) * (Det M) = ( the multF of K $$ p) * (Det M9) A12: len M9 = n by MATRIX_1:24; A13: now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_k_<=_len_M_holds_ M_._k_=_M9_._k let k be Nat; ::_thesis: ( 1 <= k & k <= len M implies M . b1 = M9 . b1 ) assume that 1 <= k and k <= len M ; ::_thesis: M . b1 = M9 . b1 percases ( k in Seg i or not k in Seg i ) ; supposeA14: k in Seg i ; ::_thesis: M . b1 = M9 . b1 then M . k = B . (F . k) by A9; hence M . k = M9 . k by A10, A11, A14; ::_thesis: verum end; supposeA15: not k in Seg i ; ::_thesis: M . b1 = M9 . b1 then M . k = AB . k by A9; hence M . k = M9 . k by A11, A15; ::_thesis: verum end; end; end; len M = n by MATRIX_1:24; hence ( the multF of K $$ p) * (Det M) = ( the multF of K $$ p) * (Det M9) by A12, A13, FINSEQ_1:14; ::_thesis: verum end; consider P being Function of FF, the carrier of K such that A16: for x being Element of FF holds S1[x,P . x] from FUNCT_2:sch_3(A3); reconsider P = P as Function of (Funcs ((Seg i),(Seg n))), the carrier of K ; take P ; ::_thesis: for F being Function of (Seg i),(Seg n) for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) holds ex Path being FinSequence of K st ( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det M) ) let F be Function of (Seg i),(Seg n); ::_thesis: for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) holds ex Path being FinSequence of K st ( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det M) ) reconsider F9 = F as Function of (Seg i),Sn ; let M be Matrix of n,K; ::_thesis: ( M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg i)) & ( for j being Nat holds ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ) implies ex Path being FinSequence of K st ( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det M) ) ) assume that A17: M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg i)) and A18: for j being Nat holds ( ( j in Seg i implies M . j = B . (F . j) ) & ( not j in Seg i implies M . j = (A * B) . j ) ) ; ::_thesis: ex Path being FinSequence of K st ( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det M) ) F9 in FF by FUNCT_2:8; hence ex Path being FinSequence of K st ( len Path = i & ( for Fj, j being Nat st j in Seg i & Fj = F . j holds Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det M) ) by A16, A17, A18; ::_thesis: verum end; theorem Th60: :: MATRIX11:60 for n being Nat for K being Field for A, B being Matrix of n,K st 0 < n holds ex P being Function of (Funcs ((Seg n),(Seg n))), the carrier of K st ( ( for F being Function of (Seg n),(Seg n) ex Path being FinSequence of K st ( len Path = n & ( for Fj, j being Nat st j in Seg n & Fj = F . j holds Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det (B * F)) ) ) & Det (A * B) = the addF of K $$ ((FinOmega (Funcs ((Seg n),(Seg n)))),P) ) proof let n be Nat; ::_thesis: for K being Field for A, B being Matrix of n,K st 0 < n holds ex P being Function of (Funcs ((Seg n),(Seg n))), the carrier of K st ( ( for F being Function of (Seg n),(Seg n) ex Path being FinSequence of K st ( len Path = n & ( for Fj, j being Nat st j in Seg n & Fj = F . j holds Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det (B * F)) ) ) & Det (A * B) = the addF of K $$ ((FinOmega (Funcs ((Seg n),(Seg n)))),P) ) let K be Field; ::_thesis: for A, B being Matrix of n,K st 0 < n holds ex P being Function of (Funcs ((Seg n),(Seg n))), the carrier of K st ( ( for F being Function of (Seg n),(Seg n) ex Path being FinSequence of K st ( len Path = n & ( for Fj, j being Nat st j in Seg n & Fj = F . j holds Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det (B * F)) ) ) & Det (A * B) = the addF of K $$ ((FinOmega (Funcs ((Seg n),(Seg n)))),P) ) let A, B be Matrix of n,K; ::_thesis: ( 0 < n implies ex P being Function of (Funcs ((Seg n),(Seg n))), the carrier of K st ( ( for F being Function of (Seg n),(Seg n) ex Path being FinSequence of K st ( len Path = n & ( for Fj, j being Nat st j in Seg n & Fj = F . j holds Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det (B * F)) ) ) & Det (A * B) = the addF of K $$ ((FinOmega (Funcs ((Seg n),(Seg n)))),P) ) ) assume A1: 0 < n ; ::_thesis: ex P being Function of (Funcs ((Seg n),(Seg n))), the carrier of K st ( ( for F being Function of (Seg n),(Seg n) ex Path being FinSequence of K st ( len Path = n & ( for Fj, j being Nat st j in Seg n & Fj = F . j holds Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det (B * F)) ) ) & Det (A * B) = the addF of K $$ ((FinOmega (Funcs ((Seg n),(Seg n)))),P) ) set AB = A * B; set aa = the addF of K; set I = idseq n; set mm = the multF of K; set KK = the carrier of K; defpred S1[ Function, Nat] means for F being Function of (Seg $2),(Seg n) for M being Matrix of n,n,K st M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg $2)) & ( for j being Nat holds ( ( j in Seg $2 implies M . j = B . (F . j) ) & ( not j in Seg $2 implies M . j = (A * B) . j ) ) ) holds ex Path being FinSequence of K st ( len Path = $2 & ( for Fj, j being Nat st j in Seg $2 & Fj = F . j holds Path . j = A * (j,Fj) ) & $1 . F = ( the multF of K $$ Path) * (Det M) ); defpred S2[ Nat] means ( $1 <= n implies for FUNC being non empty set st FUNC = Funcs ((Seg $1),(Seg n)) holds ex P being Function of FUNC, the carrier of K st ( S1[P,$1] & Det (A * B) = the addF of K $$ ((FinOmega FUNC),P) ) ); A2: for k being Element of NAT st S2[k] holds S2[k + 1] proof set Y = Seg n; let k be Element of NAT ; ::_thesis: ( S2[k] implies S2[k + 1] ) assume A3: S2[k] ; ::_thesis: S2[k + 1] set X = Seg k; reconsider FUNC = Funcs ((Seg k),(Seg n)) as non empty set by A1; set k1 = k + 1; assume A4: k + 1 <= n ; ::_thesis: for FUNC being non empty set st FUNC = Funcs ((Seg (k + 1)),(Seg n)) holds ex P being Function of FUNC, the carrier of K st ( S1[P,k + 1] & Det (A * B) = the addF of K $$ ((FinOmega FUNC),P) ) set Xx = (Seg k) \/ {(k + 1)}; let FUNC1 be non empty set ; ::_thesis: ( FUNC1 = Funcs ((Seg (k + 1)),(Seg n)) implies ex P being Function of FUNC1, the carrier of K st ( S1[P,k + 1] & Det (A * B) = the addF of K $$ ((FinOmega FUNC1),P) ) ) assume A5: FUNC1 = Funcs ((Seg (k + 1)),(Seg n)) ; ::_thesis: ex P being Function of FUNC1, the carrier of K st ( S1[P,k + 1] & Det (A * B) = the addF of K $$ ((FinOmega FUNC1),P) ) consider P1 being Function of FUNC1, the carrier of K such that A6: S1[P1,k + 1] by A4, A5, Lm10; reconsider FUNC19 = Funcs (((Seg k) \/ {(k + 1)}),(Seg n)) as non empty set by A5, FINSEQ_1:9; A7: FUNC1 = Funcs (((Seg k) \/ {(k + 1)}),(Seg n)) by A5, FINSEQ_1:9; then reconsider P19 = P1 as Function of FUNC19, the carrier of K ; A8: k + 0 <= k + 1 by XREAL_1:8; then consider P being Function of FUNC, the carrier of K such that A9: S1[P,k] and A10: Det (A * B) = the addF of K $$ ((FinOmega FUNC),P) by A3, A4, XXREAL_0:2; A11: not k + 1 in Seg k by FINSEQ_3:8; A12: for H being Function of (Seg k),(Seg n) for SF being Element of Fin FUNC19 st SF = { h where h is Function of ((Seg k) \/ {(k + 1)}),(Seg n) : h | (Seg k) = H } holds the addF of K $$ (SF,P19) = P . H proof reconsider YY = Seg n as non empty set by A1; let H be Function of (Seg k),(Seg n); ::_thesis: for SF being Element of Fin FUNC19 st SF = { h where h is Function of ((Seg k) \/ {(k + 1)}),(Seg n) : h | (Seg k) = H } holds the addF of K $$ (SF,P19) = P . H let SF be Element of Fin FUNC19; ::_thesis: ( SF = { h where h is Function of ((Seg k) \/ {(k + 1)}),(Seg n) : h | (Seg k) = H } implies the addF of K $$ (SF,P19) = P . H ) assume A13: SF = { h where h is Function of ((Seg k) \/ {(k + 1)}),(Seg n) : h | (Seg k) = H } ; ::_thesis: the addF of K $$ (SF,P19) = P . H defpred S3[ set , set ] means for h being Function of ((Seg k) \/ {(k + 1)}),(Seg n) st h | (Seg k) = H & h . (k + 1) = $1 holds h = $2; A14: for y being set st y in YY holds ex f9 being set st ( f9 in SF & S3[y,f9] ) proof let y be set ; ::_thesis: ( y in YY implies ex f9 being set st ( f9 in SF & S3[y,f9] ) ) assume y in YY ; ::_thesis: ex f9 being set st ( f9 in SF & S3[y,f9] ) then (Seg n) \/ {y} = Seg n by ZFMISC_1:40; then consider q being Function of ((Seg k) \/ {(k + 1)}),(Seg n) such that A15: q | (Seg k) = H and A16: q . (k + 1) = y by A11, STIRL2_1:57; take q ; ::_thesis: ( q in SF & S3[y,q] ) thus q in SF by A13, A15; ::_thesis: S3[y,q] let h be Function of ((Seg k) \/ {(k + 1)}),(Seg n); ::_thesis: ( h | (Seg k) = H & h . (k + 1) = y implies h = q ) assume that A17: h | (Seg k) = H and A18: h . (k + 1) = y ; ::_thesis: h = q now__::_thesis:_for_x_being_set_st_x_in_(Seg_k)_\/_{(k_+_1)}_holds_ h_._x_=_q_._x let x be set ; ::_thesis: ( x in (Seg k) \/ {(k + 1)} implies h . x = q . x ) assume x in (Seg k) \/ {(k + 1)} ; ::_thesis: h . x = q . x then A19: ( x in Seg k or x in {(k + 1)} ) by XBOOLE_0:def_3; dom H = Seg k by A1, FUNCT_2:def_1; then ( ( q . x = H . x & h . x = H . x ) or x = k + 1 ) by A15, A17, A19, FUNCT_1:47, TARSKI:def_1; hence h . x = q . x by A16, A18; ::_thesis: verum end; hence h = q by FUNCT_2:12; ::_thesis: verum end; consider QQ being Function of YY,SF such that A20: for y being set st y in YY holds S3[y,QQ . y] from FUNCT_2:sch_1(A14); A21: SF c= rng QQ proof let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in SF or y in rng QQ ) k + 1 in {(k + 1)} by TARSKI:def_1; then A22: k + 1 in (Seg k) \/ {(k + 1)} by XBOOLE_0:def_3; assume A23: y in SF ; ::_thesis: y in rng QQ then consider h being Function of ((Seg k) \/ {(k + 1)}),(Seg n) such that A24: y = h and A25: h | (Seg k) = H by A13; dom h = (Seg k) \/ {(k + 1)} by A1, FUNCT_2:def_1; then A26: h . (k + 1) in rng h by A22, FUNCT_1:def_3; A27: rng h c= Seg n by RELAT_1:def_19; dom QQ = YY by A23, FUNCT_2:def_1; then QQ . (h . (k + 1)) in rng QQ by A26, A27, FUNCT_1:def_3; hence y in rng QQ by A20, A24, A25, A26, A27; ::_thesis: verum end; rng QQ c= SF by RELAT_1:def_19; then A28: rng QQ = SF by A21, XBOOLE_0:def_10; (Seg n) \/ {n} = Seg n by A1, FINSEQ_1:3, ZFMISC_1:40; then consider h being Function of ((Seg k) \/ {(k + 1)}),(Seg n) such that A29: h | (Seg k) = H and h . (k + 1) = n by A11, STIRL2_1:57; A30: SF c= FUNC19 by FINSUB_1:def_5; k <= n by A4, A8, XXREAL_0:2; then consider Mh being Matrix of n,K such that A31: Mh = (A * B) +* ((B * ((idseq n) +* H)) | (Seg k)) and A32: for j being Nat holds ( ( j in Seg k implies Mh . j = B . (H . j) ) & ( not j in Seg k implies Mh . j = (A * B) . j ) ) by A1, Th59; consider Path being FinSequence of K such that A33: len Path = k and A34: for Hj, j being Nat st j in Seg k & Hj = H . j holds Path . j = A * (j,Hj) and A35: P . H = ( the multF of K $$ Path) * (Det Mh) by A9, A31, A32; A36: Mh . (k + 1) = (A * B) . (k + 1) by A11, A32; h in SF by A13, A29; then reconsider QQ = QQ as Function of YY,FUNC19 by A28, A30, FUNCT_2:6; A37: dom (P19 * QQ) = Seg n by FUNCT_2:def_1; A38: QQ .: (dom QQ) = SF by A28, RELAT_1:113; 1 + 0 <= k + 1 by XREAL_1:7; then A39: k + 1 in Seg n by A4; then A40: (A * B) . (k + 1) = Line ((A * B),(k + 1)) by MATRIX_2:8; Mh . (k + 1) = Line (Mh,(k + 1)) by A39, MATRIX_2:8; then Mh = RLine (Mh,(k + 1),(Line ((A * B),(k + 1)))) by A36, A40, Th30; then consider SUM1 being FinSequence of the carrier of K such that A41: len SUM1 = n and A42: Det Mh = the addF of K "**" SUM1 and A43: for j being Nat st j in Seg n holds SUM1 . j = (A * ((k + 1),j)) * (Det (RLine (Mh,(k + 1),(Line (B,j))))) by A39, Th56; A44: dom (id (Seg n)) = Seg n ; set PA = the multF of K "**" Path; set PS = ( the multF of K "**" Path) * SUM1; len (( the multF of K "**" Path) * SUM1) = n by A41, Lm5; then A45: dom (( the multF of K "**" Path) * SUM1) = Seg n by FINSEQ_1:def_3; set PSaa = [#] ((( the multF of K "**" Path) * SUM1),(the_unity_wrt the addF of K)); A46: for j being Nat st j in Seg n holds (P19 * QQ) . j = (( the multF of K "**" Path) * (A * ((k + 1),j))) * (Det (RLine (Mh,(k + 1),(Line (B,j))))) proof A47: width B = n by MATRIX_1:24; A48: len Mh = n by MATRIX_1:24; A49: dom (P19 * QQ) = Seg n by FUNCT_2:def_1; A50: width Mh = n by MATRIX_1:24; let j be Nat; ::_thesis: ( j in Seg n implies (P19 * QQ) . j = (( the multF of K "**" Path) * (A * ((k + 1),j))) * (Det (RLine (Mh,(k + 1),(Line (B,j))))) ) assume A51: j in Seg n ; ::_thesis: (P19 * QQ) . j = (( the multF of K "**" Path) * (A * ((k + 1),j))) * (Det (RLine (Mh,(k + 1),(Line (B,j))))) (Seg n) \/ {j} = Seg n by A51, ZFMISC_1:40; then consider hj being Function of ((Seg k) \/ {(k + 1)}),(Seg n) such that A52: hj | (Seg k) = H and A53: hj . (k + 1) = j by A11, STIRL2_1:57; set L = Line (B,j); set R = RLine (Mh,(k + 1),(Line (B,j))); (Seg k) \/ {(k + 1)} = Seg (k + 1) by FINSEQ_1:9; then reconsider hj9 = hj as Function of (Seg (k + 1)),(Seg n) ; consider Mhj being Matrix of n,K such that A54: Mhj = (A * B) +* ((B * ((idseq n) +* hj9)) | (Seg (k + 1))) and A55: for i being Nat holds ( ( i in Seg (k + 1) implies Mhj . i = B . (hj9 . i) ) & ( not i in Seg (k + 1) implies Mhj . i = (A * B) . i ) ) by A4, Th59; A56: len Mhj = n by MATRIX_1:24; A57: len (Line (B,j)) = width B by MATRIX_1:def_7; A58: now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_len_Mhj_holds_ Mhj_._i_=_(RLine_(Mh,(k_+_1),(Line_(B,j))))_._i A59: k + 0 < k + 1 by XREAL_1:8; let i be Nat; ::_thesis: ( 1 <= i & i <= len Mhj implies Mhj . b1 = (RLine (Mh,(k + 1),(Line (B,j)))) . b1 ) assume that A60: 1 <= i and A61: i <= len Mhj ; ::_thesis: Mhj . b1 = (RLine (Mh,(k + 1),(Line (B,j)))) . b1 A62: i in NAT by ORDINAL1:def_12; then A63: i in Seg n by A56, A60, A61; percases ( i <= k or ( i > k & i <= k + 1 ) or ( i > k & i > k + 1 ) ) ; supposeA64: i <= k ; ::_thesis: Mhj . b1 = (RLine (Mh,(k + 1),(Line (B,j)))) . b1 then i <= k + 1 by A59, XXREAL_0:2; then A65: i in Seg (k + 1) by A60, A62; A66: Line ((RLine (Mh,(k + 1),(Line (B,j)))),i) = (RLine (Mh,(k + 1),(Line (B,j)))) . i by A63, MATRIX_2:8; A67: i in Seg k by A60, A62, A64; then A68: Mh . i = B . (H . i) by A32; dom H = Seg k by A56, A60, A61, FUNCT_2:def_1; then A69: H . i = hj . i by A52, A67, FUNCT_1:47; A70: Line (Mh,i) = Mh . i by A63, MATRIX_2:8; Line ((RLine (Mh,(k + 1),(Line (B,j)))),i) = Line (Mh,i) by A63, A59, A64, Th28; hence Mhj . i = (RLine (Mh,(k + 1),(Line (B,j)))) . i by A55, A66, A70, A68, A69, A65; ::_thesis: verum end; supposeA71: ( i > k & i <= k + 1 ) ; ::_thesis: Mhj . b1 = (RLine (Mh,(k + 1),(Line (B,j)))) . b1 A72: k + 1 in Seg (k + 1) by FINSEQ_1:4; A73: Line (B,j) = B . j by A51, MATRIX_2:8; A74: (RLine (Mh,(k + 1),(Line (B,j)))) . i = Line ((RLine (Mh,(k + 1),(Line (B,j)))),i) by A63, MATRIX_2:8; A75: i = k + 1 by A71, NAT_1:9; then Line (B,j) = Line ((RLine (Mh,(k + 1),(Line (B,j)))),i) by A57, A47, A50, A63, Th28; hence Mhj . i = (RLine (Mh,(k + 1),(Line (B,j)))) . i by A53, A55, A75, A72, A74, A73; ::_thesis: verum end; supposeA76: ( i > k & i > k + 1 ) ; ::_thesis: Mhj . b1 = (RLine (Mh,(k + 1),(Line (B,j)))) . b1 then not i in Seg (k + 1) by FINSEQ_1:1; then A77: Mhj . i = (A * B) . i by A55; A78: Line ((RLine (Mh,(k + 1),(Line (B,j)))),i) = (RLine (Mh,(k + 1),(Line (B,j)))) . i by A63, MATRIX_2:8; A79: not i in Seg k by A76, FINSEQ_1:1; A80: Line (Mh,i) = Mh . i by A63, MATRIX_2:8; Line ((RLine (Mh,(k + 1),(Line (B,j)))),i) = Line (Mh,i) by A63, A76, Th28; hence Mhj . i = (RLine (Mh,(k + 1),(Line (B,j)))) . i by A32, A78, A80, A79, A77; ::_thesis: verum end; end; end; len (RLine (Mh,(k + 1),(Line (B,j)))) = len Mh by Lm4; then RLine (Mh,(k + 1),(Line (B,j))) = Mhj by A48, A56, A58, FINSEQ_1:14; then consider Pathj being FinSequence of K such that A81: len Pathj = k + 1 and A82: for m, j being Nat st j in Seg (k + 1) & m = hj . j holds Pathj . j = A * (j,m) and A83: P1 . hj = ( the multF of K "**" Pathj) * (Det (RLine (Mh,(k + 1),(Line (B,j))))) by A6, A54, A55; A84: Pathj . (k + 1) = A * ((k + 1),j) by A53, A82, FINSEQ_1:4; A85: now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_len_Path_holds_ Path_._i_=_(Pathj_|_(Seg_k))_._i A86: rng H c= Seg n by RELAT_1:def_19; A87: Seg k = dom H by A51, FUNCT_2:def_1; let i be Nat; ::_thesis: ( 1 <= i & i <= len Path implies Path . i = (Pathj | (Seg k)) . i ) assume that A88: 1 <= i and A89: i <= len Path ; ::_thesis: Path . i = (Pathj | (Seg k)) . i A90: (Pathj | k) . i = Pathj . i by A33, A89, FINSEQ_3:112; A91: i in NAT by ORDINAL1:def_12; then A92: i in Seg k by A33, A88, A89; then H . i in rng H by A87, FUNCT_1:def_3; then H . i in Seg n by A86; then reconsider Hi = H . i as Nat ; i <= k + 1 by A8, A33, A89, XXREAL_0:2; then A93: i in Seg (k + 1) by A88, A91; i in Seg k by A33, A88, A89, A91; then A94: Path . i = A * (i,Hi) by A34; H . i = hj . i by A52, A92, A87, FUNCT_1:47; hence Path . i = (Pathj | (Seg k)) . i by A82, A93, A94, A90; ::_thesis: verum end; len (Pathj | k) = k by A8, A81, FINSEQ_1:59; then Pathj = Path ^ <*(Pathj . (k + 1))*> by A33, A81, A85, FINSEQ_1:14, FINSEQ_3:55; then A95: the multF of K "**" Pathj = ( the multF of K "**" Path) * (A * ((k + 1),j)) by A84, FINSOP_1:4; QQ . j = hj by A20, A51, A52, A53; hence (P19 * QQ) . j = (( the multF of K "**" Path) * (A * ((k + 1),j))) * (Det (RLine (Mh,(k + 1),(Line (B,j))))) by A51, A83, A49, A95, FUNCT_1:12; ::_thesis: verum end; now__::_thesis:_for_y_being_set_st_y_in_dom_((_the_multF_of_K_"**"_Path)_*_SUM1)_holds_ ((_the_multF_of_K_"**"_Path)_*_SUM1)_._y_=_(P19_*_QQ)_._y let y be set ; ::_thesis: ( y in dom (( the multF of K "**" Path) * SUM1) implies (( the multF of K "**" Path) * SUM1) . y = (P19 * QQ) . y ) assume A96: y in dom (( the multF of K "**" Path) * SUM1) ; ::_thesis: (( the multF of K "**" Path) * SUM1) . y = (P19 * QQ) . y reconsider j = y as Nat by A96; SUM1 . j = (A * ((k + 1),j)) * (Det (RLine (Mh,(k + 1),(Line (B,j))))) by A43, A45, A96; hence (( the multF of K "**" Path) * SUM1) . y = ( the multF of K "**" Path) * ((A * ((k + 1),j)) * (Det (RLine (Mh,(k + 1),(Line (B,j)))))) by A96, FVSUM_1:50 .= (( the multF of K "**" Path) * (A * ((k + 1),j))) * (Det (RLine (Mh,(k + 1),(Line (B,j))))) by GROUP_1:def_3 .= (P19 * QQ) . y by A46, A45, A96 ; ::_thesis: verum end; then ( the multF of K "**" Path) * SUM1 = P19 * QQ by A37, A45, FUNCT_1:2; then A97: ([#] ((( the multF of K "**" Path) * SUM1),(the_unity_wrt the addF of K))) | (dom (( the multF of K "**" Path) * SUM1)) = P19 * QQ by SETWOP_2:21; now__::_thesis:_for_x1,_x2_being_set_st_x1_in_Seg_n_&_x2_in_Seg_n_&_QQ_._x1_=_QQ_._x2_holds_ x1_=_x2 let x1, x2 be set ; ::_thesis: ( x1 in Seg n & x2 in Seg n & QQ . x1 = QQ . x2 implies x1 = x2 ) assume that A98: x1 in Seg n and A99: x2 in Seg n and A100: QQ . x1 = QQ . x2 ; ::_thesis: x1 = x2 (Seg n) \/ {x2} = Seg n by A99, ZFMISC_1:40; then A101: ex h2 being Function of ((Seg k) \/ {(k + 1)}),(Seg n) st ( h2 | (Seg k) = H & h2 . (k + 1) = x2 ) by A11, STIRL2_1:57; (Seg n) \/ {x1} = Seg n by A98, ZFMISC_1:40; then consider h1 being Function of ((Seg k) \/ {(k + 1)}),(Seg n) such that A102: h1 | (Seg k) = H and A103: h1 . (k + 1) = x1 by A11, STIRL2_1:57; QQ . x1 = h1 by A20, A98, A102, A103; hence x1 = x2 by A20, A99, A100, A103, A101; ::_thesis: verum end; then A104: QQ is one-to-one by FUNCT_2:19; reconsider Y9 = Seg n as Element of Fin YY by FINSUB_1:def_5; A105: dom QQ = Y9 by FUNCT_2:def_1; A106: rng (id (Seg n)) = Seg n ; (P19 * QQ) * (id (Seg n)) = P19 * QQ by A37, RELAT_1:52; then the addF of K $$ (Y9,(P19 * QQ)) = the addF of K $$ ((findom (( the multF of K "**" Path) * SUM1)),([#] ((( the multF of K "**" Path) * SUM1),(the_unity_wrt the addF of K)))) by A45, A44, A106, A97, SETWOP_2:5 .= Sum (( the multF of K "**" Path) * SUM1) by FVSUM_1:8, SETWOP_2:def_2 .= ( the multF of K "**" Path) * (Sum SUM1) by FVSUM_1:73 .= P . H by A35, A42 ; hence the addF of K $$ (SF,P19) = P . H by A104, A38, A105, SETWOP_2:6; ::_thesis: verum end; the addF of K is having_a_unity by FVSUM_1:8; then Det (A * B) = the addF of K $$ ((FinOmega FUNC19),P19) by A1, A10, A11, A12, Th58; hence ex P being Function of FUNC1, the carrier of K st ( S1[P,k + 1] & Det (A * B) = the addF of K $$ ((FinOmega FUNC1),P) ) by A6, A7; ::_thesis: verum end; set FUN = Funcs ((Seg n),(Seg n)); A107: n is Element of NAT by ORDINAL1:def_12; A108: S2[ 0 ] proof reconsider E = {} as Function of (Seg 0),(Seg n) by XBOOLE_1:2; assume 0 <= n ; ::_thesis: for FUNC being non empty set st FUNC = Funcs ((Seg 0),(Seg n)) holds ex P being Function of FUNC, the carrier of K st ( S1[P, 0 ] & Det (A * B) = the addF of K $$ ((FinOmega FUNC),P) ) A109: the_unity_wrt the multF of K = 1_ K by GROUP_1:22; let FUNC be non empty set ; ::_thesis: ( FUNC = Funcs ((Seg 0),(Seg n)) implies ex P being Function of FUNC, the carrier of K st ( S1[P, 0 ] & Det (A * B) = the addF of K $$ ((FinOmega FUNC),P) ) ) assume A110: FUNC = Funcs ((Seg 0),(Seg n)) ; ::_thesis: ex P being Function of FUNC, the carrier of K st ( S1[P, 0 ] & Det (A * B) = the addF of K $$ ((FinOmega FUNC),P) ) consider P being Function of FUNC, the carrier of K such that A111: S1[P, 0 ] by A1, A110, Lm10; A112: FUNC = {E} by A110, FUNCT_5:57; then A113: E in FUNC by TARSKI:def_1; FinOmega FUNC = {E} by A112, MATRIX_2:def_14; then A114: the addF of K $$ ((FinOmega FUNC),P) = P . E by A113, SETWISEO:17; consider M being Matrix of n,K such that A115: M = (A * B) +* ((B * ((idseq n) +* E)) | (Seg 0)) and A116: for j being Nat holds ( ( j in Seg 0 implies M . j = B . (E . j) ) & ( not j in Seg 0 implies M . j = (A * B) . j ) ) by A1, Th59; A117: M = (A * B) +* {} by A115; consider Path being FinSequence of K such that A118: len Path = 0 and for Fj, j being Nat st j in Seg 0 & Fj = E . j holds Path . j = A * (j,Fj) and A119: P . E = ( the multF of K $$ Path) * (Det M) by A111, A115, A116; Path = <*> the carrier of K by A118; then the multF of K "**" Path = 1_ K by A109, FINSOP_1:10; then P . E = Det (A * B) by A119, A117, VECTSP_1:def_4; hence ex P being Function of FUNC, the carrier of K st ( S1[P, 0 ] & Det (A * B) = the addF of K $$ ((FinOmega FUNC),P) ) by A111, A114; ::_thesis: verum end; for k being Element of NAT holds S2[k] from NAT_1:sch_1(A108, A2); then consider P being Function of (Funcs ((Seg n),(Seg n))), the carrier of K such that A120: S1[P,n] and A121: Det (A * B) = the addF of K $$ ((FinOmega (Funcs ((Seg n),(Seg n)))),P) by A107; take P ; ::_thesis: ( ( for F being Function of (Seg n),(Seg n) ex Path being FinSequence of K st ( len Path = n & ( for Fj, j being Nat st j in Seg n & Fj = F . j holds Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det (B * F)) ) ) & Det (A * B) = the addF of K $$ ((FinOmega (Funcs ((Seg n),(Seg n)))),P) ) thus for F being Function of (Seg n),(Seg n) ex Path being FinSequence of K st ( len Path = n & ( for Fj, j being Nat st j in Seg n & Fj = F . j holds Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det (B * F)) ) ::_thesis: Det (A * B) = the addF of K $$ ((FinOmega (Funcs ((Seg n),(Seg n)))),P) proof len (A * B) = n by MATRIX_1:24; then A122: dom (A * B) = Seg n by FINSEQ_1:def_3; let F be Function of (Seg n),(Seg n); ::_thesis: ex Path being FinSequence of K st ( len Path = n & ( for Fj, j being Nat st j in Seg n & Fj = F . j holds Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det (B * F)) ) A123: dom (idseq n) = Seg n by FUNCT_2:52; dom F = Seg n by FUNCT_2:52; then A124: (idseq n) +* F = F by A123, FUNCT_4:19; A125: len B = n by MATRIX_1:24; len (B * F) = len B by Def4; then A126: dom (B * F) = Seg n by A125, FINSEQ_1:def_3; consider M being Matrix of n,K such that A127: M = (A * B) +* ((B * ((idseq n) +* F)) | (Seg n)) and A128: for j being Nat holds ( ( j in Seg n implies M . j = B . (F . j) ) & ( not j in Seg n implies M . j = (A * B) . j ) ) by A1, Th59; (B * F) | n = B * F ; then M = B * F by A127, A124, A126, A122, FUNCT_4:19; hence ex Path being FinSequence of K st ( len Path = n & ( for Fj, j being Nat st j in Seg n & Fj = F . j holds Path . j = A * (j,Fj) ) & P . F = ( the multF of K $$ Path) * (Det (B * F)) ) by A120, A127, A128; ::_thesis: verum end; thus Det (A * B) = the addF of K $$ ((FinOmega (Funcs ((Seg n),(Seg n)))),P) by A121; ::_thesis: verum end; theorem Th61: :: MATRIX11:61 for n being Nat for K being Field for A, B being Matrix of n,K st 0 < n holds ex P being Function of (Permutations n), the carrier of K st ( Det (A * B) = the addF of K $$ ((FinOmega (Permutations n)),P) & ( for perm being Element of Permutations n holds P . perm = ( the multF of K $$ (Path_matrix (perm,A))) * (- ((Det B),perm)) ) ) proof let n be Nat; ::_thesis: for K being Field for A, B being Matrix of n,K st 0 < n holds ex P being Function of (Permutations n), the carrier of K st ( Det (A * B) = the addF of K $$ ((FinOmega (Permutations n)),P) & ( for perm being Element of Permutations n holds P . perm = ( the multF of K $$ (Path_matrix (perm,A))) * (- ((Det B),perm)) ) ) let K be Field; ::_thesis: for A, B being Matrix of n,K st 0 < n holds ex P being Function of (Permutations n), the carrier of K st ( Det (A * B) = the addF of K $$ ((FinOmega (Permutations n)),P) & ( for perm being Element of Permutations n holds P . perm = ( the multF of K $$ (Path_matrix (perm,A))) * (- ((Det B),perm)) ) ) let A, B be Matrix of n,K; ::_thesis: ( 0 < n implies ex P being Function of (Permutations n), the carrier of K st ( Det (A * B) = the addF of K $$ ((FinOmega (Permutations n)),P) & ( for perm being Element of Permutations n holds P . perm = ( the multF of K $$ (Path_matrix (perm,A))) * (- ((Det B),perm)) ) ) ) assume A1: 0 < n ; ::_thesis: ex P being Function of (Permutations n), the carrier of K st ( Det (A * B) = the addF of K $$ ((FinOmega (Permutations n)),P) & ( for perm being Element of Permutations n holds P . perm = ( the multF of K $$ (Path_matrix (perm,A))) * (- ((Det B),perm)) ) ) set P = Permutations n; A2: dom (id (Permutations n)) = Permutations n ; set KK = the carrier of K; set mm = the multF of K; set aa = the addF of K; set AB = A * B; set X = Seg n; set F = Funcs ((Seg n),(Seg n)); consider SUM1 being Function of (Funcs ((Seg n),(Seg n))), the carrier of K such that A3: for F being Function of (Seg n),(Seg n) ex Path being FinSequence of K st ( len Path = n & ( for Fj, j being Nat st j in Seg n & Fj = F . j holds Path . j = A * (j,Fj) ) & SUM1 . F = ( the multF of K $$ Path) * (Det (B * F)) ) and A4: Det (A * B) = the addF of K $$ ((FinOmega (Funcs ((Seg n),(Seg n)))),SUM1) by A1, Th60; A5: Funcs ((Seg n),(Seg n)) is finite by FRAENKEL:6; then reconsider FP = (Funcs ((Seg n),(Seg n))) \ (Permutations n) as Element of Fin (Funcs ((Seg n),(Seg n))) by FINSUB_1:def_5; A6: Permutations n c= Funcs ((Seg n),(Seg n)) proof let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Permutations n or x in Funcs ((Seg n),(Seg n)) ) assume x in Permutations n ; ::_thesis: x in Funcs ((Seg n),(Seg n)) then reconsider p = x as Permutation of (Seg n) by MATRIX_2:def_9; p is Element of Funcs ((Seg n),(Seg n)) by FUNCT_2:9; hence x in Funcs ((Seg n),(Seg n)) ; ::_thesis: verum end; then reconsider P9 = Permutations n as Element of Fin (Funcs ((Seg n),(Seg n))) by A5, FINSUB_1:def_5; A7: Permutations n = FinOmega (Permutations n) by MATRIX_2:26, MATRIX_2:def_14; A8: FinOmega (Funcs ((Seg n),(Seg n))) = Funcs ((Seg n),(Seg n)) by FRAENKEL:6, MATRIX_2:def_14; A9: now__::_thesis:_Det_(A_*_B)_=_the_addF_of_K_$$_(P9,SUM1) percases ( FP = {} or FP <> {} ) ; suppose FP = {} ; ::_thesis: Det (A * B) = the addF of K $$ (P9,SUM1) then Funcs ((Seg n),(Seg n)) c= Permutations n by XBOOLE_1:37; hence Det (A * B) = the addF of K $$ (P9,SUM1) by A4, A8, A6, XBOOLE_0:def_10; ::_thesis: verum end; supposeA10: FP <> {} ; ::_thesis: Det (A * B) = the addF of K $$ (P9,SUM1) A11: 0. K = the_unity_wrt the addF of K by FVSUM_1:7; A12: SUM1 .: FP c= {(0. K)} proof let s be set ; :: according to TARSKI:def_3 ::_thesis: ( not s in SUM1 .: FP or s in {(0. K)} ) assume s in SUM1 .: FP ; ::_thesis: s in {(0. K)} then consider x being set such that x in dom SUM1 and A13: x in FP and A14: s = SUM1 . x by FUNCT_1:def_6; reconsider f = x as Function of (Seg n),(Seg n) by A13, FUNCT_2:66; not f in Permutations n by A13, XBOOLE_0:def_5; then A15: Det (B * f) = 0. K by Th54; ex Path being FinSequence of K st ( len Path = n & ( for Fj, j being Nat st j in Seg n & Fj = f . j holds Path . j = A * (j,Fj) ) & SUM1 . f = ( the multF of K $$ Path) * (Det (B * f)) ) by A3; then SUM1 . f = 0. K by A15, VECTSP_1:6; hence s in {(0. K)} by A14, TARSKI:def_1; ::_thesis: verum end; dom SUM1 = Funcs ((Seg n),(Seg n)) by FUNCT_2:def_1; then SUM1 .: FP = {(0. K)} by A10, A12, ZFMISC_1:33; then A16: the addF of K $$ (FP,SUM1) = 0. K by A11, FVSUM_1:8, SETWOP_2:8; A17: FP misses Permutations n by XBOOLE_1:79; A18: FP \/ (Permutations n) = (Funcs ((Seg n),(Seg n))) \/ (Permutations n) by XBOOLE_1:39; (Funcs ((Seg n),(Seg n))) \/ (Permutations n) = Funcs ((Seg n),(Seg n)) by A6, XBOOLE_1:12; hence Det (A * B) = ( the addF of K $$ (P9,SUM1)) + (0. K) by A4, A8, A16, A17, A18, FVSUM_1:8, SETWOP_2:4 .= the addF of K $$ (P9,SUM1) by RLVECT_1:4 ; ::_thesis: verum end; end; end; dom SUM1 = Funcs ((Seg n),(Seg n)) by FUNCT_2:def_1; then A19: dom (SUM1 | (Permutations n)) = Permutations n by A6, RELAT_1:62; rng (SUM1 | (Permutations n)) c= the carrier of K by RELAT_1:def_19; then reconsider SP = SUM1 | (Permutations n) as Function of (Permutations n), the carrier of K by A19, FUNCT_2:2; take SP ; ::_thesis: ( Det (A * B) = the addF of K $$ ((FinOmega (Permutations n)),SP) & ( for perm being Element of Permutations n holds SP . perm = ( the multF of K $$ (Path_matrix (perm,A))) * (- ((Det B),perm)) ) ) A20: rng (id (Permutations n)) = Permutations n ; SP * (id (Permutations n)) = SP by A19, RELAT_1:52; hence Det (A * B) = the addF of K $$ ((FinOmega (Permutations n)),SP) by A9, A2, A20, A7, SETWOP_2:5; ::_thesis: for perm being Element of Permutations n holds SP . perm = ( the multF of K $$ (Path_matrix (perm,A))) * (- ((Det B),perm)) let perm be Element of Permutations n; ::_thesis: SP . perm = ( the multF of K $$ (Path_matrix (perm,A))) * (- ((Det B),perm)) reconsider Perm = perm as Permutation of (Seg n) by MATRIX_2:def_9; SUM1 . Perm = SP . Perm by A19, FUNCT_1:47; then consider Path being FinSequence of K such that A21: len Path = n and A22: for Fj, j being Nat st j in Seg n & Fj = Perm . j holds Path . j = A * (j,Fj) and A23: SP . Perm = ( the multF of K $$ Path) * (Det (B * Perm)) by A3; set PM = Path_matrix (perm,A); A24: len (Path_matrix (perm,A)) = n by MATRIX_3:def_7; now__::_thesis:_for_i_being_Nat_st_1_<=_i_&_i_<=_len_Path_holds_ Path_._i_=_(Path_matrix_(perm,A))_._i A25: Seg n = dom Perm by FUNCT_2:52; let i be Nat; ::_thesis: ( 1 <= i & i <= len Path implies Path . i = (Path_matrix (perm,A)) . i ) assume that A26: 1 <= i and A27: i <= len Path ; ::_thesis: Path . i = (Path_matrix (perm,A)) . i A28: i in NAT by ORDINAL1:def_12; then A29: i in Seg n by A21, A26, A27; i in Seg n by A21, A26, A27, A28; then A30: Perm . i in rng Perm by A25, FUNCT_1:def_3; rng Perm c= Seg n by RELAT_1:def_19; then Perm . i in Seg n by A30; then reconsider Pi = Perm . i as Element of NAT ; dom (Path_matrix (perm,A)) = Seg n by A24, FINSEQ_1:def_3; then (Path_matrix (perm,A)) . i = A * (i,Pi) by A29, MATRIX_3:def_7; hence Path . i = (Path_matrix (perm,A)) . i by A22, A29; ::_thesis: verum end; then Path = Path_matrix (perm,A) by A21, A24, FINSEQ_1:14; hence SP . perm = ( the multF of K $$ (Path_matrix (perm,A))) * (- ((Det B),perm)) by A23, Th46; ::_thesis: verum end; theorem :: MATRIX11:62 for n being Nat for K being Field for A, B being Matrix of n,K st 0 < n holds Det (A * B) = (Det A) * (Det B) proof let n be Nat; ::_thesis: for K being Field for A, B being Matrix of n,K st 0 < n holds Det (A * B) = (Det A) * (Det B) let K be Field; ::_thesis: for A, B being Matrix of n,K st 0 < n holds Det (A * B) = (Det A) * (Det B) let A, B be Matrix of n,K; ::_thesis: ( 0 < n implies Det (A * B) = (Det A) * (Det B) ) assume A1: 0 < n ; ::_thesis: Det (A * B) = (Det A) * (Det B) set P = Permutations n; set KK = the carrier of K; set mm = the multF of K; set aa = the addF of K; set AB = A * B; consider SUM1 being Function of (Permutations n), the carrier of K such that A2: Det (A * B) = the addF of K $$ ((FinOmega (Permutations n)),SUM1) and A3: for perm being Element of Permutations n holds SUM1 . perm = ( the multF of K $$ (Path_matrix (perm,A))) * (- ((Det B),perm)) by A1, Th61; set Path = Path_product A; set F = FinOmega (Permutations n); A4: FinOmega (Permutations n) = Permutations n by MATRIX_2:26, MATRIX_2:def_14; then consider Ga being Function of (Fin (Permutations n)), the carrier of K such that A5: Det A = Ga . (FinOmega (Permutations n)) and A6: for e being Element of the carrier of K st e is_a_unity_wrt the addF of K holds Ga . {} = e and A7: for x being Element of Permutations n holds Ga . {x} = (Path_product A) . x and A8: for B9 being Element of Fin (Permutations n) st B9 c= FinOmega (Permutations n) & B9 <> {} holds for x being Element of Permutations n st x in (FinOmega (Permutations n)) \ B9 holds Ga . (B9 \/ {x}) = the addF of K . ((Ga . B9),((Path_product A) . x)) by SETWISEO:def_3; A9: Ga . {} = 0. K by A6, FVSUM_1:6; consider Gs being Function of (Fin (Permutations n)), the carrier of K such that A10: Det (A * B) = Gs . (FinOmega (Permutations n)) and A11: for e being Element of the carrier of K st e is_a_unity_wrt the addF of K holds Gs . {} = e and A12: for x being Element of Permutations n holds Gs . {x} = SUM1 . x and A13: for B9 being Element of Fin (Permutations n) st B9 c= FinOmega (Permutations n) & B9 <> {} holds for x being Element of Permutations n st x in (FinOmega (Permutations n)) \ B9 holds Gs . (B9 \/ {x}) = the addF of K . ((Gs . B9),(SUM1 . x)) by A2, A4, SETWISEO:def_3; defpred S1[ set ] means for B9 being Element of Fin (Permutations n) st B9 = $1 holds Gs . B9 = (Ga . B9) * (Det B); A14: for B9 being Element of Fin (Permutations n) for b being Element of Permutations n st S1[B9] & not b in B9 holds S1[B9 \/ {b}] proof let B9 be Element of Fin (Permutations n); ::_thesis: for b being Element of Permutations n st S1[B9] & not b in B9 holds S1[B9 \/ {b}] let b be Element of Permutations n; ::_thesis: ( S1[B9] & not b in B9 implies S1[B9 \/ {b}] ) assume that A15: S1[B9] and A16: not b in B9 ; ::_thesis: S1[B9 \/ {b}] set mA = the multF of K $$ (Path_matrix (b,A)); let Bb be Element of Fin (Permutations n); ::_thesis: ( Bb = B9 \/ {b} implies Gs . Bb = (Ga . Bb) * (Det B) ) assume A17: Bb = B9 \/ {b} ; ::_thesis: Gs . Bb = (Ga . Bb) * (Det B) A18: now__::_thesis:_SUM1_._b_=_((Path_product_A)_._b)_*_(Det_B) percases ( b is even or b is odd ) ; supposeA19: b is even ; ::_thesis: SUM1 . b = ((Path_product A) . b) * (Det B) then A20: - (( the multF of K $$ (Path_matrix (b,A))),b) = the multF of K $$ (Path_matrix (b,A)) by MATRIX_2:def_13; - ((Det B),b) = Det B by A19, MATRIX_2:def_13; hence SUM1 . b = (- (( the multF of K $$ (Path_matrix (b,A))),b)) * (Det B) by A3, A20 .= ((Path_product A) . b) * (Det B) by MATRIX_3:def_8 ; ::_thesis: verum end; supposeA21: b is odd ; ::_thesis: SUM1 . b = ((Path_product A) . b) * (Det B) then A22: - (( the multF of K $$ (Path_matrix (b,A))),b) = - ( the multF of K $$ (Path_matrix (b,A))) by MATRIX_2:def_13; - ((Det B),b) = - (Det B) by A21, MATRIX_2:def_13; then - ((- (( the multF of K $$ (Path_matrix (b,A))),b)) * (Det B)) = (- ( the multF of K $$ (Path_matrix (b,A)))) * (- ((Det B),b)) by A22, VECTSP_1:9 .= - (( the multF of K $$ (Path_matrix (b,A))) * (- ((Det B),b))) by VECTSP_1:9 ; then (( the multF of K $$ (Path_matrix (b,A))) * (- ((Det B),b))) - ((- (( the multF of K $$ (Path_matrix (b,A))),b)) * (Det B)) = 0. K by VECTSP_1:16; then A23: (- (( the multF of K $$ (Path_matrix (b,A))),b)) * (Det B) = ( the multF of K $$ (Path_matrix (b,A))) * (- ((Det B),b)) by VECTSP_1:19; - (( the multF of K $$ (Path_matrix (b,A))),b) = (Path_product A) . b by MATRIX_3:def_8; hence SUM1 . b = ((Path_product A) . b) * (Det B) by A3, A23; ::_thesis: verum end; end; end; percases ( B9 = {} or B9 <> {} ) ; supposeA24: B9 = {} ; ::_thesis: Gs . Bb = (Ga . Bb) * (Det B) then Ga . Bb = (Path_product A) . b by A7, A17; hence Gs . Bb = (Ga . Bb) * (Det B) by A12, A17, A18, A24; ::_thesis: verum end; supposeA25: B9 <> {} ; ::_thesis: Gs . Bb = (Ga . Bb) * (Det B) A26: B9 c= Permutations n by FINSUB_1:def_5; A27: b in (FinOmega (Permutations n)) \ B9 by A4, A16, XBOOLE_0:def_5; then Gs . Bb = the addF of K . ((Gs . B9),(SUM1 . b)) by A4, A13, A17, A25, A26; then A28: Gs . Bb = ((Ga . B9) * (Det B)) + (((Path_product A) . b) * (Det B)) by A15, A18; Ga . Bb = (Ga . B9) + ((Path_product A) . b) by A4, A8, A17, A25, A27, A26; hence Gs . Bb = (Ga . Bb) * (Det B) by A28, VECTSP_1:def_7; ::_thesis: verum end; end; end; Gs . {} = 0. K by A11, FVSUM_1:6; then A29: S1[ {}. (Permutations n)] by A9, VECTSP_1:7; for B being Element of Fin (Permutations n) holds S1[B] from SETWISEO:sch_2(A29, A14); hence Det (A * B) = (Det A) * (Det B) by A10, A5; ::_thesis: verum end;