| Step |
Hyp |
Ref |
Expression |
| 1 |
|
mamufval.f |
|- F = ( R maMul <. M , N , P >. ) |
| 2 |
|
mamufval.b |
|- B = ( Base ` R ) |
| 3 |
|
mamufval.t |
|- .x. = ( .r ` R ) |
| 4 |
|
mamufval.r |
|- ( ph -> R e. V ) |
| 5 |
|
mamufval.m |
|- ( ph -> M e. Fin ) |
| 6 |
|
mamufval.n |
|- ( ph -> N e. Fin ) |
| 7 |
|
mamufval.p |
|- ( ph -> P e. Fin ) |
| 8 |
|
mamuval.x |
|- ( ph -> X e. ( B ^m ( M X. N ) ) ) |
| 9 |
|
mamuval.y |
|- ( ph -> Y e. ( B ^m ( N X. P ) ) ) |
| 10 |
|
mamufv.i |
|- ( ph -> I e. M ) |
| 11 |
|
mamufv.k |
|- ( ph -> K e. P ) |
| 12 |
1 2 3 4 5 6 7 8 9
|
mamuval |
|- ( ph -> ( X F Y ) = ( i e. M , k e. P |-> ( R gsum ( j e. N |-> ( ( i X j ) .x. ( j Y k ) ) ) ) ) ) |
| 13 |
|
oveq1 |
|- ( i = I -> ( i X j ) = ( I X j ) ) |
| 14 |
|
oveq2 |
|- ( k = K -> ( j Y k ) = ( j Y K ) ) |
| 15 |
13 14
|
oveqan12d |
|- ( ( i = I /\ k = K ) -> ( ( i X j ) .x. ( j Y k ) ) = ( ( I X j ) .x. ( j Y K ) ) ) |
| 16 |
15
|
adantl |
|- ( ( ph /\ ( i = I /\ k = K ) ) -> ( ( i X j ) .x. ( j Y k ) ) = ( ( I X j ) .x. ( j Y K ) ) ) |
| 17 |
16
|
mpteq2dv |
|- ( ( ph /\ ( i = I /\ k = K ) ) -> ( j e. N |-> ( ( i X j ) .x. ( j Y k ) ) ) = ( j e. N |-> ( ( I X j ) .x. ( j Y K ) ) ) ) |
| 18 |
17
|
oveq2d |
|- ( ( ph /\ ( i = I /\ k = K ) ) -> ( R gsum ( j e. N |-> ( ( i X j ) .x. ( j Y k ) ) ) ) = ( R gsum ( j e. N |-> ( ( I X j ) .x. ( j Y K ) ) ) ) ) |
| 19 |
|
ovexd |
|- ( ph -> ( R gsum ( j e. N |-> ( ( I X j ) .x. ( j Y K ) ) ) ) e. _V ) |
| 20 |
12 18 10 11 19
|
ovmpod |
|- ( ph -> ( I ( X F Y ) K ) = ( R gsum ( j e. N |-> ( ( I X j ) .x. ( j Y K ) ) ) ) ) |