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Theorem mamufval 20191
Description: Functional value of the matrix multiplication operator. (Contributed by Stefan O'Rear, 2-Sep-2015.)
Hypotheses
Ref Expression
mamufval.f |- F = ( R maMul <. M , N , P >. )
mamufval.b |- B = ( Base ` R )
mamufval.t |- .x. = ( .r ` R )
mamufval.r |- ( ph -> R e. V )
mamufval.m |- ( ph -> M e. Fin )
mamufval.n |- ( ph -> N e. Fin )
mamufval.p |- ( ph -> P e. Fin )
Assertion
Ref Expression
mamufval |- ( ph -> F = ( x e. ( B ^m ( M X. N ) ) , y e. ( B ^m ( N X. P ) ) |-> ( i e. M , k e. P |-> ( R gsumg ( j e. N |-> ( ( i x j ) .x. ( j y k ) ) ) ) ) ) )
Distinct variable groups:   i, j, k, x, y, M   i, N, j, k, x, y   P, i, j, k, x, y   R, i, j, k, x, y   ph, i, j, k, x, y   x, B, y   x, .x. , y, i, k
Allowed substitution hints:   B( i, j, k)   .x. ( j)   F( x, y, i, j, k)   V( x, y, i, j, k)
Proof of Theorem mamufval
Dummy variables m n o p r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mamufval.f . 2 |- F = ( R maMul <. M , N , P >. )
2 df-mamu 20190 . . . 4 |- maMul = ( r e. _V , o e. _V |-> [_ ( 1st ` ( 1st ` o ) ) / m ]_ [_ ( 2nd ` ( 1st ` o ) ) / n ]_ [_ ( 2nd ` o ) / p ]_ ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m ( n X. p ) ) |-> ( i e. m , k e. p |-> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) ) ) )
32a1i 11 . . 3 |- ( ph -> maMul = ( r e. _V , o e. _V |-> [_ ( 1st ` ( 1st ` o ) ) / m ]_ [_ ( 2nd ` ( 1st ` o ) ) / n ]_ [_ ( 2nd ` o ) / p ]_ ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m ( n X. p ) ) |-> ( i e. m , k e. p |-> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) ) ) ) )
4 fvex 6201 . . . . 5 |- ( 1st ` ( 1st ` o ) ) e. _V
5 fvex 6201 . . . . 5 |- ( 2nd ` ( 1st ` o ) ) e. _V
6 fvex 6201 . . . . . . 7 |- ( 2nd ` o ) e. _V
7 eqidd 2623 . . . . . . . 8 |- ( p = ( 2nd ` o ) -> ( ( Base ` r ) ^m ( m X. n ) ) = ( ( Base ` r ) ^m ( m X. n ) ) )
8 xpeq2 5129 . . . . . . . . 9 |- ( p = ( 2nd ` o ) -> ( n X. p ) = ( n X. ( 2nd ` o ) ) )
98oveq2d 6666 . . . . . . . 8 |- ( p = ( 2nd ` o ) -> ( ( Base ` r ) ^m ( n X. p ) ) = ( ( Base ` r ) ^m ( n X. ( 2nd ` o ) ) ) )
10 eqidd 2623 . . . . . . . . 9 |- ( p = ( 2nd ` o ) -> m = m )
11 id 22 . . . . . . . . 9 |- ( p = ( 2nd ` o ) -> p = ( 2nd ` o ) )
12 eqidd 2623 . . . . . . . . 9 |- ( p = ( 2nd ` o ) -> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) = ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) )
1310, 11, 12mpt2eq123dv 6717 . . . . . . . 8 |- ( p = ( 2nd ` o ) -> ( i e. m , k e. p |-> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) ) = ( i e. m , k e. ( 2nd ` o ) |-> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) ) )
147, 9, 13mpt2eq123dv 6717 . . . . . . 7 |- ( p = ( 2nd ` o ) -> ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m ( n X. p ) ) |-> ( i e. m , k e. p |-> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) ) ) = ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m ( n X. ( 2nd ` o ) ) ) |-> ( i e. m , k e. ( 2nd ` o ) |-> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) ) ) )
156, 14csbie 3559 . . . . . 6 |- [_ ( 2nd ` o ) / p ]_ ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m ( n X. p ) ) |-> ( i e. m , k e. p |-> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) ) ) = ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m ( n X. ( 2nd ` o ) ) ) |-> ( i e. m , k e. ( 2nd ` o ) |-> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) ) )
16 xpeq12 5134 . . . . . . . 8 |- ( ( m = ( 1st ` ( 1st ` o ) ) /\ n = ( 2nd ` ( 1st ` o ) ) ) -> ( m X. n ) = ( ( 1st ` ( 1st ` o ) ) X. ( 2nd ` ( 1st ` o ) ) ) )
1716oveq2d 6666 . . . . . . 7 |- ( ( m = ( 1st ` ( 1st ` o ) ) /\ n = ( 2nd ` ( 1st ` o ) ) ) -> ( ( Base ` r ) ^m ( m X. n ) ) = ( ( Base ` r ) ^m ( ( 1st ` ( 1st ` o ) ) X. ( 2nd ` ( 1st ` o ) ) ) ) )
18 simpr 477 . . . . . . . . 9 |- ( ( m = ( 1st ` ( 1st ` o ) ) /\ n = ( 2nd ` ( 1st ` o ) ) ) -> n = ( 2nd ` ( 1st ` o ) ) )
1918xpeq1d 5138 . . . . . . . 8 |- ( ( m = ( 1st ` ( 1st ` o ) ) /\ n = ( 2nd ` ( 1st ` o ) ) ) -> ( n X. ( 2nd ` o ) ) = ( ( 2nd ` ( 1st ` o ) ) X. ( 2nd ` o ) ) )
2019oveq2d 6666 . . . . . . 7 |- ( ( m = ( 1st ` ( 1st ` o ) ) /\ n = ( 2nd ` ( 1st ` o ) ) ) -> ( ( Base ` r ) ^m ( n X. ( 2nd ` o ) ) ) = ( ( Base ` r ) ^m ( ( 2nd ` ( 1st ` o ) ) X. ( 2nd ` o ) ) ) )
21 id 22 . . . . . . . . 9 |- ( m = ( 1st ` ( 1st ` o ) ) -> m = ( 1st ` ( 1st ` o ) ) )
2221adantr 481 . . . . . . . 8 |- ( ( m = ( 1st ` ( 1st ` o ) ) /\ n = ( 2nd ` ( 1st ` o ) ) ) -> m = ( 1st ` ( 1st ` o ) ) )
23 eqidd 2623 . . . . . . . 8 |- ( ( m = ( 1st ` ( 1st ` o ) ) /\ n = ( 2nd ` ( 1st ` o ) ) ) -> ( 2nd ` o ) = ( 2nd ` o ) )
24 eqidd 2623 . . . . . . . . . 10 |- ( ( m = ( 1st ` ( 1st ` o ) ) /\ n = ( 2nd ` ( 1st ` o ) ) ) -> ( ( i x j ) ( .r ` r ) ( j y k ) ) = ( ( i x j ) ( .r ` r ) ( j y k ) ) )
2518, 24mpteq12dv 4733 . . . . . . . . 9 |- ( ( m = ( 1st ` ( 1st ` o ) ) /\ n = ( 2nd ` ( 1st ` o ) ) ) -> ( j e. n |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) = ( j e. ( 2nd ` ( 1st ` o ) ) |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) )
2625oveq2d 6666 . . . . . . . 8 |- ( ( m = ( 1st ` ( 1st ` o ) ) /\ n = ( 2nd ` ( 1st ` o ) ) ) -> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) = ( r gsumg ( j e. ( 2nd ` ( 1st ` o ) ) |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) )
2722, 23, 26mpt2eq123dv 6717 . . . . . . 7 |- ( ( m = ( 1st ` ( 1st ` o ) ) /\ n = ( 2nd ` ( 1st ` o ) ) ) -> ( i e. m , k e. ( 2nd ` o ) |-> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) ) = ( i e. ( 1st ` ( 1st ` o ) ) , k e. ( 2nd ` o ) |-> ( r gsumg ( j e. ( 2nd ` ( 1st ` o ) ) |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) ) )
2817, 20, 27mpt2eq123dv 6717 . . . . . 6 |- ( ( m = ( 1st ` ( 1st ` o ) ) /\ n = ( 2nd ` ( 1st ` o ) ) ) -> ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m ( n X. ( 2nd ` o ) ) ) |-> ( i e. m , k e. ( 2nd ` o ) |-> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) ) ) = ( x e. ( ( Base ` r ) ^m ( ( 1st ` ( 1st ` o ) ) X. ( 2nd ` ( 1st ` o ) ) ) ) , y e. ( ( Base ` r ) ^m ( ( 2nd ` ( 1st ` o ) ) X. ( 2nd ` o ) ) ) |-> ( i e. ( 1st ` ( 1st ` o ) ) , k e. ( 2nd ` o ) |-> ( r gsumg ( j e. ( 2nd ` ( 1st ` o ) ) |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) ) ) )
2915, 28syl5eq 2668 . . . . 5 |- ( ( m = ( 1st ` ( 1st ` o ) ) /\ n = ( 2nd ` ( 1st ` o ) ) ) -> [_ ( 2nd ` o ) / p ]_ ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m ( n X. p ) ) |-> ( i e. m , k e. p |-> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) ) ) = ( x e. ( ( Base ` r ) ^m ( ( 1st ` ( 1st ` o ) ) X. ( 2nd ` ( 1st ` o ) ) ) ) , y e. ( ( Base ` r ) ^m ( ( 2nd ` ( 1st ` o ) ) X. ( 2nd ` o ) ) ) |-> ( i e. ( 1st ` ( 1st ` o ) ) , k e. ( 2nd ` o ) |-> ( r gsumg ( j e. ( 2nd ` ( 1st ` o ) ) |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) ) ) )
304, 5, 29csbie2 3563 . . . 4 |- [_ ( 1st ` ( 1st ` o ) ) / m ]_ [_ ( 2nd ` ( 1st ` o ) ) / n ]_ [_ ( 2nd ` o ) / p ]_ ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m ( n X. p ) ) |-> ( i e. m , k e. p |-> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) ) ) = ( x e. ( ( Base ` r ) ^m ( ( 1st ` ( 1st ` o ) ) X. ( 2nd ` ( 1st ` o ) ) ) ) , y e. ( ( Base ` r ) ^m ( ( 2nd ` ( 1st ` o ) ) X. ( 2nd ` o ) ) ) |-> ( i e. ( 1st ` ( 1st ` o ) ) , k e. ( 2nd ` o ) |-> ( r gsumg ( j e. ( 2nd ` ( 1st ` o ) ) |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) ) )
31 simprl 794 . . . . . . . 8 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> r = R )
3231fveq2d 6195 . . . . . . 7 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> ( Base ` r ) = ( Base ` R ) )
33 mamufval.b . . . . . . 7 |- B = ( Base ` R )
3432, 33syl6eqr 2674 . . . . . 6 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> ( Base ` r ) = B )
35 fveq2 6191 . . . . . . . . . 10 |- ( o = <. M , N , P >. -> ( 1st ` o ) = ( 1st ` <. M , N , P >. ) )
3635fveq2d 6195 . . . . . . . . 9 |- ( o = <. M , N , P >. -> ( 1st ` ( 1st ` o ) ) = ( 1st ` ( 1st ` <. M , N , P >. ) ) )
3736ad2antll 765 . . . . . . . 8 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> ( 1st ` ( 1st ` o ) ) = ( 1st ` ( 1st ` <. M , N , P >. ) ) )
38 mamufval.m . . . . . . . . . 10 |- ( ph -> M e. Fin )
39 mamufval.n . . . . . . . . . 10 |- ( ph -> N e. Fin )
40 mamufval.p . . . . . . . . . 10 |- ( ph -> P e. Fin )
41 ot1stg 7182 . . . . . . . . . 10 |- ( ( M e. Fin /\ N e. Fin /\ P e. Fin ) -> ( 1st ` ( 1st ` <. M , N , P >. ) ) = M )
4238, 39, 40, 41syl3anc 1326 . . . . . . . . 9 |- ( ph -> ( 1st ` ( 1st ` <. M , N , P >. ) ) = M )
4342adantr 481 . . . . . . . 8 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> ( 1st ` ( 1st ` <. M , N , P >. ) ) = M )
4437, 43eqtrd 2656 . . . . . . 7 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> ( 1st ` ( 1st ` o ) ) = M )
4535fveq2d 6195 . . . . . . . . 9 |- ( o = <. M , N , P >. -> ( 2nd ` ( 1st ` o ) ) = ( 2nd ` ( 1st ` <. M , N , P >. ) ) )
4645ad2antll 765 . . . . . . . 8 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> ( 2nd ` ( 1st ` o ) ) = ( 2nd ` ( 1st ` <. M , N , P >. ) ) )
47 ot2ndg 7183 . . . . . . . . . 10 |- ( ( M e. Fin /\ N e. Fin /\ P e. Fin ) -> ( 2nd ` ( 1st ` <. M , N , P >. ) ) = N )
4838, 39, 40, 47syl3anc 1326 . . . . . . . . 9 |- ( ph -> ( 2nd ` ( 1st ` <. M , N , P >. ) ) = N )
4948adantr 481 . . . . . . . 8 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> ( 2nd ` ( 1st ` <. M , N , P >. ) ) = N )
5046, 49eqtrd 2656 . . . . . . 7 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> ( 2nd ` ( 1st ` o ) ) = N )
5144, 50xpeq12d 5140 . . . . . 6 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> ( ( 1st ` ( 1st ` o ) ) X. ( 2nd ` ( 1st ` o ) ) ) = ( M X. N ) )
5234, 51oveq12d 6668 . . . . 5 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> ( ( Base ` r ) ^m ( ( 1st ` ( 1st ` o ) ) X. ( 2nd ` ( 1st ` o ) ) ) ) = ( B ^m ( M X. N ) ) )
53 fveq2 6191 . . . . . . . . 9 |- ( o = <. M , N , P >. -> ( 2nd ` o ) = ( 2nd ` <. M , N , P >. ) )
5453ad2antll 765 . . . . . . . 8 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> ( 2nd ` o ) = ( 2nd ` <. M , N , P >. ) )
55 ot3rdg 7184 . . . . . . . . . 10 |- ( P e. Fin -> ( 2nd ` <. M , N , P >. ) = P )
5640, 55syl 17 . . . . . . . . 9 |- ( ph -> ( 2nd ` <. M , N , P >. ) = P )
5756adantr 481 . . . . . . . 8 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> ( 2nd ` <. M , N , P >. ) = P )
5854, 57eqtrd 2656 . . . . . . 7 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> ( 2nd ` o ) = P )
5950, 58xpeq12d 5140 . . . . . 6 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> ( ( 2nd ` ( 1st ` o ) ) X. ( 2nd ` o ) ) = ( N X. P ) )
6034, 59oveq12d 6668 . . . . 5 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> ( ( Base ` r ) ^m ( ( 2nd ` ( 1st ` o ) ) X. ( 2nd ` o ) ) ) = ( B ^m ( N X. P ) ) )
6131fveq2d 6195 . . . . . . . . . 10 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> ( .r ` r ) = ( .r ` R ) )
62 mamufval.t . . . . . . . . . 10 |- .x. = ( .r ` R )
6361, 62syl6eqr 2674 . . . . . . . . 9 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> ( .r ` r ) = .x. )
6463oveqd 6667 . . . . . . . 8 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> ( ( i x j ) ( .r ` r ) ( j y k ) ) = ( ( i x j ) .x. ( j y k ) ) )
6550, 64mpteq12dv 4733 . . . . . . 7 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> ( j e. ( 2nd ` ( 1st ` o ) ) |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) = ( j e. N |-> ( ( i x j ) .x. ( j y k ) ) ) )
6631, 65oveq12d 6668 . . . . . 6 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> ( r gsumg ( j e. ( 2nd ` ( 1st ` o ) ) |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) = ( R gsumg ( j e. N |-> ( ( i x j ) .x. ( j y k ) ) ) ) )
6744, 58, 66mpt2eq123dv 6717 . . . . 5 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> ( i e. ( 1st ` ( 1st ` o ) ) , k e. ( 2nd ` o ) |-> ( r gsumg ( j e. ( 2nd ` ( 1st ` o ) ) |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) ) = ( i e. M , k e. P |-> ( R gsumg ( j e. N |-> ( ( i x j ) .x. ( j y k ) ) ) ) ) )
6852, 60, 67mpt2eq123dv 6717 . . . 4 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> ( x e. ( ( Base ` r ) ^m ( ( 1st ` ( 1st ` o ) ) X. ( 2nd ` ( 1st ` o ) ) ) ) , y e. ( ( Base ` r ) ^m ( ( 2nd ` ( 1st ` o ) ) X. ( 2nd ` o ) ) ) |-> ( i e. ( 1st ` ( 1st ` o ) ) , k e. ( 2nd ` o ) |-> ( r gsumg ( j e. ( 2nd ` ( 1st ` o ) ) |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) ) ) = ( x e. ( B ^m ( M X. N ) ) , y e. ( B ^m ( N X. P ) ) |-> ( i e. M , k e. P |-> ( R gsumg ( j e. N |-> ( ( i x j ) .x. ( j y k ) ) ) ) ) ) )
6930, 68syl5eq 2668 . . 3 |- ( ( ph /\ ( r = R /\ o = <. M , N , P >. ) ) -> [_ ( 1st ` ( 1st ` o ) ) / m ]_ [_ ( 2nd ` ( 1st ` o ) ) / n ]_ [_ ( 2nd ` o ) / p ]_ ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m ( n X. p ) ) |-> ( i e. m , k e. p |-> ( r gsumg ( j e. n |-> ( ( i x j ) ( .r ` r ) ( j y k ) ) ) ) ) ) = ( x e. ( B ^m ( M X. N ) ) , y e. ( B ^m ( N X. P ) ) |-> ( i e. M , k e. P |-> ( R gsumg ( j e. N |-> ( ( i x j ) .x. ( j y k ) ) ) ) ) ) )
70 mamufval.r . . . 4 |- ( ph -> R e. V )
71 elex 3212 . . . 4 |- ( R e. V -> R e. _V )
7270, 71syl 17 . . 3 |- ( ph -> R e. _V )
73 otex 4933 . . . 4 |- <. M , N , P >. e. _V
7473a1i 11 . . 3 |- ( ph -> <. M , N , P >. e. _V )
75 ovex 6678 . . . . 5 |- ( B ^m ( M X. N ) ) e. _V
76 ovex 6678 . . . . 5 |- ( B ^m ( N X. P ) ) e. _V
7775, 76mpt2ex 7247 . . . 4 |- ( x e. ( B ^m ( M X. N ) ) , y e. ( B ^m ( N X. P ) ) |-> ( i e. M , k e. P |-> ( R gsumg ( j e. N |-> ( ( i x j ) .x. ( j y k ) ) ) ) ) ) e. _V
7877a1i 11 . . 3 |- ( ph -> ( x e. ( B ^m ( M X. N ) ) , y e. ( B ^m ( N X. P ) ) |-> ( i e. M , k e. P |-> ( R gsumg ( j e. N |-> ( ( i x j ) .x. ( j y k ) ) ) ) ) ) e. _V )
793, 69, 72, 74, 78ovmpt2d 6788 . 2 |- ( ph -> ( R maMul <. M , N , P >. ) = ( x e. ( B ^m ( M X. N ) ) , y e. ( B ^m ( N X. P ) ) |-> ( i e. M , k e. P |-> ( R gsumg ( j e. N |-> ( ( i x j ) .x. ( j y k ) ) ) ) ) ) )
801, 79syl5eq 2668 1 |- ( ph -> F = ( x e. ( B ^m ( M X. N ) ) , y e. ( B ^m ( N X. P ) ) |-> ( i e. M , k e. P |-> ( R gsumg ( j e. N |-> ( ( i x j ) .x. ( j y k ) ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   <.cotp 4185   |-> cmpt 4729   X. cxp 5112   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   ^m cmap 7857   Fincfn 7955   Basecbs 15857   .rcmulr 15942   gsumg cgsu 16101   maMul cmmul 20189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-ot 4186  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-mamu 20190
This theorem is referenced by:  mamuval  20192  mamudm  20194
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