let K be Field; :: thesis: for a, b, c, d, e, f, g, h, i being Element of K

for M being Matrix of 3,K st M = <*<*a,b,c*>,<*d,e,f*>,<*g,h,i*>*> holds

Det M = ((((((a * e) * i) - ((c * e) * g)) - ((a * f) * h)) + ((b * f) * g)) - ((b * d) * i)) + ((c * d) * h)

reconsider a3 = <*1,3,2*>, a4 = <*2,3,1*>, a5 = <*2,1,3*>, a6 = <*3,1,2*> as Element of Permutations 3 by Th27;

reconsider id3 = idseq 3 as Permutation of (Seg 3) ;

reconsider Id3 = idseq 3 as Element of Permutations 3 by MATRIX_1:def 12;

A1: id3 is even by MATRIX_1:16;

reconsider rid3 = Rev (idseq 3) as Element of Permutations 3 by Th4;

let a, b, c, d, e, f, g, h, i be Element of K; :: thesis: for M being Matrix of 3,K st M = <*<*a,b,c*>,<*d,e,f*>,<*g,h,i*>*> holds

Det M = ((((((a * e) * i) - ((c * e) * g)) - ((a * f) * h)) + ((b * f) * g)) - ((b * d) * i)) + ((c * d) * h)

let M be Matrix of 3,K; :: thesis: ( M = <*<*a,b,c*>,<*d,e,f*>,<*g,h,i*>*> implies Det M = ((((((a * e) * i) - ((c * e) * g)) - ((a * f) * h)) + ((b * f) * g)) - ((b * d) * i)) + ((c * d) * h) )

assume A2: M = <*<*a,b,c*>,<*d,e,f*>,<*g,h,i*>*> ; :: thesis: Det M = ((((((a * e) * i) - ((c * e) * g)) - ((a * f) * h)) + ((b * f) * g)) - ((b * d) * i)) + ((c * d) * h)

Permutations 3 in Fin (Permutations 3) by FINSUB_1:def 5;

then In ((Permutations 3),(Fin (Permutations 3))) = Permutations 3 by SUBSET_1:def 8;

then reconsider X = {Id3,rid3,a3,a4,a5,a6} as Element of Fin (Permutations 3) by Th15, Th19, FINSEQ_2:53;

reconsider B1 = {.Id3,rid3,a3.}, B2 = {.a4,a5,a6.} as Element of Fin (Permutations 3) ;

set F = the addF of K;

A3: ( In ((Permutations 3),(Fin (Permutations 3))) = X & X = {Id3,rid3,a3} \/ {a4,a5,a6} ) by Th15, Th19, ENUMSET1:13, FINSEQ_2:53;

for x being object st x in B1 \ B2 holds

x in B1 by XBOOLE_0:def 5;

then B1 \ B2 c= B1 by TARSKI:def 3;

then B1 \ B2 = B1 by A5, XBOOLE_0:def 10;

then A6: B1 misses B2 by XBOOLE_1:83;

set r = Path_product M;

A7: 3 = len (Permutations 3) by MATRIX_1:9;

then Id3 is even by MATRIX_1:16;

then reconsider r1 = (Path_product M) . id3, r2 = (Path_product M) . rid3, r3 = (Path_product M) . a3, r4 = (Path_product M) . a4, r5 = (Path_product M) . a5, r6 = (Path_product M) . a6 as Element of K by FUNCT_2:5;

A8: (Path_product M) . id3 = - (( the multF of K $$ (Path_matrix (Id3,M))),Id3) by MATRIX_3:def 8

.= the multF of K $$ (Path_matrix (Id3,M)) by A1, A7, MATRIX_1:def 16

.= the multF of K $$ <*a,e,i*> by A2, Th20, FINSEQ_2:53

.= (a * e) * i by Th26 ;

A9: (Path_product M) . a6 = - (( the multF of K $$ (Path_matrix (a6,M))),a6) by MATRIX_3:def 8

.= the multF of K $$ (Path_matrix (a6,M)) by A7, Lm14, MATRIX_1:def 16

.= the multF of K $$ <*c,d,h*> by A2, Th25

.= (c * d) * h by Th26 ;

A10: (Path_product M) . a5 = - (( the multF of K $$ (Path_matrix (a5,M))),a5) by MATRIX_3:def 8

.= - ( the multF of K $$ (Path_matrix (a5,M))) by A7, Th43, MATRIX_1:def 16

.= - ( the multF of K $$ <*b,d,i*>) by A2, Th24

.= - ((b * d) * i) by Th26 ;

A11: (Path_product M) . a4 = - (( the multF of K $$ (Path_matrix (a4,M))),a4) by MATRIX_3:def 8

.= the multF of K $$ (Path_matrix (a4,M)) by A7, Lm13, MATRIX_1:def 16

.= the multF of K $$ <*b,f,g*> by A2, Th23

.= (b * f) * g by Th26 ;

A12: (Path_product M) . a3 = - (( the multF of K $$ (Path_matrix (a3,M))),a3) by MATRIX_3:def 8

.= - ( the multF of K $$ (Path_matrix (a3,M))) by A7, Th44, MATRIX_1:def 16

.= - ( the multF of K $$ <*a,f,h*>) by A2, Th22

.= - ((a * f) * h) by Th26 ;

A13: (Path_product M) . rid3 = - (( the multF of K $$ (Path_matrix (rid3,M))),rid3) by MATRIX_3:def 8

.= - ( the multF of K $$ (Path_matrix (rid3,M))) by A7, Th15, Th42, MATRIX_1:def 16

.= - ( the multF of K $$ <*c,e,g*>) by A2, Th15, Th21

.= - ((c * e) * g) by Th26 ;

A14: ( the addF of K $$ (B1,(Path_product M)) = (r1 + r2) + r3 & the addF of K $$ (B2,(Path_product M)) = (r4 + r5) + r6 ) by Lm3, Lm4, Lm10, Lm11, Th15, FINSEQ_2:53, SETWOP_2:3;

Det M = the addF of K $$ ((In ((Permutations 3),(Fin (Permutations 3)))),(Path_product M)) by MATRIX_3:def 9

.= the addF of K . (( the addF of K $$ (B1,(Path_product M))),( the addF of K $$ (B2,(Path_product M)))) by A3, A6, SETWOP_2:4

.= ((r1 + r2) + r3) + (r4 + (r5 + r6)) by A14, RLVECT_1:def 3

.= (((r1 + r2) + r3) + r4) + (r5 + r6) by RLVECT_1:def 3

.= ((((((a * e) * i) - ((c * e) * g)) - ((a * f) * h)) + ((b * f) * g)) - ((b * d) * i)) + ((c * d) * h) by A8, A13, A12, A11, A10, A9, RLVECT_1:def 3 ;

hence Det M = ((((((a * e) * i) - ((c * e) * g)) - ((a * f) * h)) + ((b * f) * g)) - ((b * d) * i)) + ((c * d) * h) ; :: thesis: verum

for M being Matrix of 3,K st M = <*<*a,b,c*>,<*d,e,f*>,<*g,h,i*>*> holds

Det M = ((((((a * e) * i) - ((c * e) * g)) - ((a * f) * h)) + ((b * f) * g)) - ((b * d) * i)) + ((c * d) * h)

reconsider a3 = <*1,3,2*>, a4 = <*2,3,1*>, a5 = <*2,1,3*>, a6 = <*3,1,2*> as Element of Permutations 3 by Th27;

reconsider id3 = idseq 3 as Permutation of (Seg 3) ;

reconsider Id3 = idseq 3 as Element of Permutations 3 by MATRIX_1:def 12;

A1: id3 is even by MATRIX_1:16;

reconsider rid3 = Rev (idseq 3) as Element of Permutations 3 by Th4;

let a, b, c, d, e, f, g, h, i be Element of K; :: thesis: for M being Matrix of 3,K st M = <*<*a,b,c*>,<*d,e,f*>,<*g,h,i*>*> holds

Det M = ((((((a * e) * i) - ((c * e) * g)) - ((a * f) * h)) + ((b * f) * g)) - ((b * d) * i)) + ((c * d) * h)

let M be Matrix of 3,K; :: thesis: ( M = <*<*a,b,c*>,<*d,e,f*>,<*g,h,i*>*> implies Det M = ((((((a * e) * i) - ((c * e) * g)) - ((a * f) * h)) + ((b * f) * g)) - ((b * d) * i)) + ((c * d) * h) )

assume A2: M = <*<*a,b,c*>,<*d,e,f*>,<*g,h,i*>*> ; :: thesis: Det M = ((((((a * e) * i) - ((c * e) * g)) - ((a * f) * h)) + ((b * f) * g)) - ((b * d) * i)) + ((c * d) * h)

Permutations 3 in Fin (Permutations 3) by FINSUB_1:def 5;

then In ((Permutations 3),(Fin (Permutations 3))) = Permutations 3 by SUBSET_1:def 8;

then reconsider X = {Id3,rid3,a3,a4,a5,a6} as Element of Fin (Permutations 3) by Th15, Th19, FINSEQ_2:53;

reconsider B1 = {.Id3,rid3,a3.}, B2 = {.a4,a5,a6.} as Element of Fin (Permutations 3) ;

set F = the addF of K;

A3: ( In ((Permutations 3),(Fin (Permutations 3))) = X & X = {Id3,rid3,a3} \/ {a4,a5,a6} ) by Th15, Th19, ENUMSET1:13, FINSEQ_2:53;

now :: thesis: for x being object st x in B1 holds

x in B1 \ B2

then A5:
B1 c= B1 \ B2
by TARSKI:def 3;x in B1 \ B2

let x be object ; :: thesis: ( x in B1 implies x in B1 \ B2 )

assume A4: x in B1 ; :: thesis: x in B1 \ B2

then ( x = Id3 or x = rid3 or x = a3 ) by ENUMSET1:def 1;

then not x in B2 by Lm5, Lm6, Lm7, Lm8, Lm9, Th15, ENUMSET1:def 1, FINSEQ_2:53;

hence x in B1 \ B2 by A4, XBOOLE_0:def 5; :: thesis: verum

end;assume A4: x in B1 ; :: thesis: x in B1 \ B2

then ( x = Id3 or x = rid3 or x = a3 ) by ENUMSET1:def 1;

then not x in B2 by Lm5, Lm6, Lm7, Lm8, Lm9, Th15, ENUMSET1:def 1, FINSEQ_2:53;

hence x in B1 \ B2 by A4, XBOOLE_0:def 5; :: thesis: verum

for x being object st x in B1 \ B2 holds

x in B1 by XBOOLE_0:def 5;

then B1 \ B2 c= B1 by TARSKI:def 3;

then B1 \ B2 = B1 by A5, XBOOLE_0:def 10;

then A6: B1 misses B2 by XBOOLE_1:83;

set r = Path_product M;

A7: 3 = len (Permutations 3) by MATRIX_1:9;

then Id3 is even by MATRIX_1:16;

then reconsider r1 = (Path_product M) . id3, r2 = (Path_product M) . rid3, r3 = (Path_product M) . a3, r4 = (Path_product M) . a4, r5 = (Path_product M) . a5, r6 = (Path_product M) . a6 as Element of K by FUNCT_2:5;

A8: (Path_product M) . id3 = - (( the multF of K $$ (Path_matrix (Id3,M))),Id3) by MATRIX_3:def 8

.= the multF of K $$ (Path_matrix (Id3,M)) by A1, A7, MATRIX_1:def 16

.= the multF of K $$ <*a,e,i*> by A2, Th20, FINSEQ_2:53

.= (a * e) * i by Th26 ;

A9: (Path_product M) . a6 = - (( the multF of K $$ (Path_matrix (a6,M))),a6) by MATRIX_3:def 8

.= the multF of K $$ (Path_matrix (a6,M)) by A7, Lm14, MATRIX_1:def 16

.= the multF of K $$ <*c,d,h*> by A2, Th25

.= (c * d) * h by Th26 ;

A10: (Path_product M) . a5 = - (( the multF of K $$ (Path_matrix (a5,M))),a5) by MATRIX_3:def 8

.= - ( the multF of K $$ (Path_matrix (a5,M))) by A7, Th43, MATRIX_1:def 16

.= - ( the multF of K $$ <*b,d,i*>) by A2, Th24

.= - ((b * d) * i) by Th26 ;

A11: (Path_product M) . a4 = - (( the multF of K $$ (Path_matrix (a4,M))),a4) by MATRIX_3:def 8

.= the multF of K $$ (Path_matrix (a4,M)) by A7, Lm13, MATRIX_1:def 16

.= the multF of K $$ <*b,f,g*> by A2, Th23

.= (b * f) * g by Th26 ;

A12: (Path_product M) . a3 = - (( the multF of K $$ (Path_matrix (a3,M))),a3) by MATRIX_3:def 8

.= - ( the multF of K $$ (Path_matrix (a3,M))) by A7, Th44, MATRIX_1:def 16

.= - ( the multF of K $$ <*a,f,h*>) by A2, Th22

.= - ((a * f) * h) by Th26 ;

A13: (Path_product M) . rid3 = - (( the multF of K $$ (Path_matrix (rid3,M))),rid3) by MATRIX_3:def 8

.= - ( the multF of K $$ (Path_matrix (rid3,M))) by A7, Th15, Th42, MATRIX_1:def 16

.= - ( the multF of K $$ <*c,e,g*>) by A2, Th15, Th21

.= - ((c * e) * g) by Th26 ;

A14: ( the addF of K $$ (B1,(Path_product M)) = (r1 + r2) + r3 & the addF of K $$ (B2,(Path_product M)) = (r4 + r5) + r6 ) by Lm3, Lm4, Lm10, Lm11, Th15, FINSEQ_2:53, SETWOP_2:3;

Det M = the addF of K $$ ((In ((Permutations 3),(Fin (Permutations 3)))),(Path_product M)) by MATRIX_3:def 9

.= the addF of K . (( the addF of K $$ (B1,(Path_product M))),( the addF of K $$ (B2,(Path_product M)))) by A3, A6, SETWOP_2:4

.= ((r1 + r2) + r3) + (r4 + (r5 + r6)) by A14, RLVECT_1:def 3

.= (((r1 + r2) + r3) + r4) + (r5 + r6) by RLVECT_1:def 3

.= ((((((a * e) * i) - ((c * e) * g)) - ((a * f) * h)) + ((b * f) * g)) - ((b * d) * i)) + ((c * d) * h) by A8, A13, A12, A11, A10, A9, RLVECT_1:def 3 ;

hence Det M = ((((((a * e) * i) - ((c * e) * g)) - ((a * f) * h)) + ((b * f) * g)) - ((b * d) * i)) + ((c * d) * h) ; :: thesis: verum