Step |
Hyp |
Ref |
Expression |
1 |
|
mvmulfval.x |
|- .X. = ( R maVecMul <. M , N >. ) |
2 |
|
mvmulfval.b |
|- B = ( Base ` R ) |
3 |
|
mvmulfval.t |
|- .x. = ( .r ` R ) |
4 |
|
mvmulfval.r |
|- ( ph -> R e. V ) |
5 |
|
mvmulfval.m |
|- ( ph -> M e. Fin ) |
6 |
|
mvmulfval.n |
|- ( ph -> N e. Fin ) |
7 |
|
df-mvmul |
|- maVecMul = ( r e. _V , o e. _V |-> [_ ( 1st ` o ) / m ]_ [_ ( 2nd ` o ) / n ]_ ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m n ) |-> ( i e. m |-> ( r gsum ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) ) ) |
8 |
7
|
a1i |
|- ( ph -> maVecMul = ( r e. _V , o e. _V |-> [_ ( 1st ` o ) / m ]_ [_ ( 2nd ` o ) / n ]_ ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m n ) |-> ( i e. m |-> ( r gsum ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) ) ) ) |
9 |
|
fvex |
|- ( 1st ` o ) e. _V |
10 |
|
fvex |
|- ( 2nd ` o ) e. _V |
11 |
|
xpeq12 |
|- ( ( m = ( 1st ` o ) /\ n = ( 2nd ` o ) ) -> ( m X. n ) = ( ( 1st ` o ) X. ( 2nd ` o ) ) ) |
12 |
11
|
oveq2d |
|- ( ( m = ( 1st ` o ) /\ n = ( 2nd ` o ) ) -> ( ( Base ` r ) ^m ( m X. n ) ) = ( ( Base ` r ) ^m ( ( 1st ` o ) X. ( 2nd ` o ) ) ) ) |
13 |
|
oveq2 |
|- ( n = ( 2nd ` o ) -> ( ( Base ` r ) ^m n ) = ( ( Base ` r ) ^m ( 2nd ` o ) ) ) |
14 |
13
|
adantl |
|- ( ( m = ( 1st ` o ) /\ n = ( 2nd ` o ) ) -> ( ( Base ` r ) ^m n ) = ( ( Base ` r ) ^m ( 2nd ` o ) ) ) |
15 |
|
simpl |
|- ( ( m = ( 1st ` o ) /\ n = ( 2nd ` o ) ) -> m = ( 1st ` o ) ) |
16 |
|
simpr |
|- ( ( m = ( 1st ` o ) /\ n = ( 2nd ` o ) ) -> n = ( 2nd ` o ) ) |
17 |
16
|
mpteq1d |
|- ( ( m = ( 1st ` o ) /\ n = ( 2nd ` o ) ) -> ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) = ( j e. ( 2nd ` o ) |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) |
18 |
17
|
oveq2d |
|- ( ( m = ( 1st ` o ) /\ n = ( 2nd ` o ) ) -> ( r gsum ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) = ( r gsum ( j e. ( 2nd ` o ) |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) |
19 |
15 18
|
mpteq12dv |
|- ( ( m = ( 1st ` o ) /\ n = ( 2nd ` o ) ) -> ( i e. m |-> ( r gsum ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) = ( i e. ( 1st ` o ) |-> ( r gsum ( j e. ( 2nd ` o ) |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) ) |
20 |
12 14 19
|
mpoeq123dv |
|- ( ( m = ( 1st ` o ) /\ n = ( 2nd ` o ) ) -> ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m n ) |-> ( i e. m |-> ( r gsum ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) ) = ( x e. ( ( Base ` r ) ^m ( ( 1st ` o ) X. ( 2nd ` o ) ) ) , y e. ( ( Base ` r ) ^m ( 2nd ` o ) ) |-> ( i e. ( 1st ` o ) |-> ( r gsum ( j e. ( 2nd ` o ) |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) ) ) |
21 |
9 10 20
|
csbie2 |
|- [_ ( 1st ` o ) / m ]_ [_ ( 2nd ` o ) / n ]_ ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m n ) |-> ( i e. m |-> ( r gsum ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) ) = ( x e. ( ( Base ` r ) ^m ( ( 1st ` o ) X. ( 2nd ` o ) ) ) , y e. ( ( Base ` r ) ^m ( 2nd ` o ) ) |-> ( i e. ( 1st ` o ) |-> ( r gsum ( j e. ( 2nd ` o ) |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) ) |
22 |
|
simprl |
|- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> r = R ) |
23 |
22
|
fveq2d |
|- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( Base ` r ) = ( Base ` R ) ) |
24 |
23 2
|
eqtr4di |
|- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( Base ` r ) = B ) |
25 |
|
fveq2 |
|- ( o = <. M , N >. -> ( 1st ` o ) = ( 1st ` <. M , N >. ) ) |
26 |
25
|
ad2antll |
|- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( 1st ` o ) = ( 1st ` <. M , N >. ) ) |
27 |
|
op1stg |
|- ( ( M e. Fin /\ N e. Fin ) -> ( 1st ` <. M , N >. ) = M ) |
28 |
5 6 27
|
syl2anc |
|- ( ph -> ( 1st ` <. M , N >. ) = M ) |
29 |
28
|
adantr |
|- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( 1st ` <. M , N >. ) = M ) |
30 |
26 29
|
eqtrd |
|- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( 1st ` o ) = M ) |
31 |
|
fveq2 |
|- ( o = <. M , N >. -> ( 2nd ` o ) = ( 2nd ` <. M , N >. ) ) |
32 |
31
|
ad2antll |
|- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( 2nd ` o ) = ( 2nd ` <. M , N >. ) ) |
33 |
|
op2ndg |
|- ( ( M e. Fin /\ N e. Fin ) -> ( 2nd ` <. M , N >. ) = N ) |
34 |
5 6 33
|
syl2anc |
|- ( ph -> ( 2nd ` <. M , N >. ) = N ) |
35 |
34
|
adantr |
|- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( 2nd ` <. M , N >. ) = N ) |
36 |
32 35
|
eqtrd |
|- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( 2nd ` o ) = N ) |
37 |
30 36
|
xpeq12d |
|- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( ( 1st ` o ) X. ( 2nd ` o ) ) = ( M X. N ) ) |
38 |
24 37
|
oveq12d |
|- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( ( Base ` r ) ^m ( ( 1st ` o ) X. ( 2nd ` o ) ) ) = ( B ^m ( M X. N ) ) ) |
39 |
24 36
|
oveq12d |
|- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( ( Base ` r ) ^m ( 2nd ` o ) ) = ( B ^m N ) ) |
40 |
|
fveq2 |
|- ( r = R -> ( .r ` r ) = ( .r ` R ) ) |
41 |
40
|
adantr |
|- ( ( r = R /\ o = <. M , N >. ) -> ( .r ` r ) = ( .r ` R ) ) |
42 |
41
|
adantl |
|- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( .r ` r ) = ( .r ` R ) ) |
43 |
42 3
|
eqtr4di |
|- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( .r ` r ) = .x. ) |
44 |
43
|
oveqd |
|- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( ( i x j ) ( .r ` r ) ( y ` j ) ) = ( ( i x j ) .x. ( y ` j ) ) ) |
45 |
36 44
|
mpteq12dv |
|- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( j e. ( 2nd ` o ) |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) = ( j e. N |-> ( ( i x j ) .x. ( y ` j ) ) ) ) |
46 |
22 45
|
oveq12d |
|- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( r gsum ( j e. ( 2nd ` o ) |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) = ( R gsum ( j e. N |-> ( ( i x j ) .x. ( y ` j ) ) ) ) ) |
47 |
30 46
|
mpteq12dv |
|- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( i e. ( 1st ` o ) |-> ( r gsum ( j e. ( 2nd ` o ) |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) = ( i e. M |-> ( R gsum ( j e. N |-> ( ( i x j ) .x. ( y ` j ) ) ) ) ) ) |
48 |
38 39 47
|
mpoeq123dv |
|- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> ( x e. ( ( Base ` r ) ^m ( ( 1st ` o ) X. ( 2nd ` o ) ) ) , y e. ( ( Base ` r ) ^m ( 2nd ` o ) ) |-> ( i e. ( 1st ` o ) |-> ( r gsum ( j e. ( 2nd ` o ) |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) ) = ( x e. ( B ^m ( M X. N ) ) , y e. ( B ^m N ) |-> ( i e. M |-> ( R gsum ( j e. N |-> ( ( i x j ) .x. ( y ` j ) ) ) ) ) ) ) |
49 |
21 48
|
eqtrid |
|- ( ( ph /\ ( r = R /\ o = <. M , N >. ) ) -> [_ ( 1st ` o ) / m ]_ [_ ( 2nd ` o ) / n ]_ ( x e. ( ( Base ` r ) ^m ( m X. n ) ) , y e. ( ( Base ` r ) ^m n ) |-> ( i e. m |-> ( r gsum ( j e. n |-> ( ( i x j ) ( .r ` r ) ( y ` j ) ) ) ) ) ) = ( x e. ( B ^m ( M X. N ) ) , y e. ( B ^m N ) |-> ( i e. M |-> ( R gsum ( j e. N |-> ( ( i x j ) .x. ( y ` j ) ) ) ) ) ) ) |
50 |
4
|
elexd |
|- ( ph -> R e. _V ) |
51 |
|
opex |
|- <. M , N >. e. _V |
52 |
51
|
a1i |
|- ( ph -> <. M , N >. e. _V ) |
53 |
|
ovex |
|- ( B ^m ( M X. N ) ) e. _V |
54 |
|
ovex |
|- ( B ^m N ) e. _V |
55 |
53 54
|
mpoex |
|- ( x e. ( B ^m ( M X. N ) ) , y e. ( B ^m N ) |-> ( i e. M |-> ( R gsum ( j e. N |-> ( ( i x j ) .x. ( y ` j ) ) ) ) ) ) e. _V |
56 |
55
|
a1i |
|- ( ph -> ( x e. ( B ^m ( M X. N ) ) , y e. ( B ^m N ) |-> ( i e. M |-> ( R gsum ( j e. N |-> ( ( i x j ) .x. ( y ` j ) ) ) ) ) ) e. _V ) |
57 |
8 49 50 52 56
|
ovmpod |
|- ( ph -> ( R maVecMul <. M , N >. ) = ( x e. ( B ^m ( M X. N ) ) , y e. ( B ^m N ) |-> ( i e. M |-> ( R gsum ( j e. N |-> ( ( i x j ) .x. ( y ` j ) ) ) ) ) ) ) |
58 |
1 57
|
eqtrid |
|- ( ph -> .X. = ( x e. ( B ^m ( M X. N ) ) , y e. ( B ^m N ) |-> ( i e. M |-> ( R gsum ( j e. N |-> ( ( i x j ) .x. ( y ` j ) ) ) ) ) ) ) |