| 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 |
|
mvmulval.x |
|- ( ph -> X e. ( B ^m ( M X. N ) ) ) |
| 8 |
|
mvmulval.y |
|- ( ph -> Y e. ( B ^m N ) ) |
| 9 |
1 2 3 4 5 6
|
mvmulfval |
|- ( 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 ) ) ) ) ) ) ) |
| 10 |
|
oveq |
|- ( x = X -> ( i x j ) = ( i X j ) ) |
| 11 |
|
fveq1 |
|- ( y = Y -> ( y ` j ) = ( Y ` j ) ) |
| 12 |
10 11
|
oveqan12d |
|- ( ( x = X /\ y = Y ) -> ( ( i x j ) .x. ( y ` j ) ) = ( ( i X j ) .x. ( Y ` j ) ) ) |
| 13 |
12
|
adantl |
|- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( ( i x j ) .x. ( y ` j ) ) = ( ( i X j ) .x. ( Y ` j ) ) ) |
| 14 |
13
|
mpteq2dv |
|- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( j e. N |-> ( ( i x j ) .x. ( y ` j ) ) ) = ( j e. N |-> ( ( i X j ) .x. ( Y ` j ) ) ) ) |
| 15 |
14
|
oveq2d |
|- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( R gsum ( j e. N |-> ( ( i x j ) .x. ( y ` j ) ) ) ) = ( R gsum ( j e. N |-> ( ( i X j ) .x. ( Y ` j ) ) ) ) ) |
| 16 |
15
|
mpteq2dv |
|- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( i e. M |-> ( R gsum ( j e. N |-> ( ( i x j ) .x. ( y ` j ) ) ) ) ) = ( i e. M |-> ( R gsum ( j e. N |-> ( ( i X j ) .x. ( Y ` j ) ) ) ) ) ) |
| 17 |
5
|
mptexd |
|- ( ph -> ( i e. M |-> ( R gsum ( j e. N |-> ( ( i X j ) .x. ( Y ` j ) ) ) ) ) e. _V ) |
| 18 |
9 16 7 8 17
|
ovmpod |
|- ( ph -> ( X .X. Y ) = ( i e. M |-> ( R gsum ( j e. N |-> ( ( i X j ) .x. ( Y ` j ) ) ) ) ) ) |