Step |
Hyp |
Ref |
Expression |
1 |
|
prdsbasmpt.y |
|- Y = ( S Xs_ R ) |
2 |
|
prdsbasmpt.b |
|- B = ( Base ` Y ) |
3 |
|
prdsbasmpt.s |
|- ( ph -> S e. V ) |
4 |
|
prdsbasmpt.i |
|- ( ph -> I e. W ) |
5 |
|
prdsbasmpt.r |
|- ( ph -> R Fn I ) |
6 |
|
prdsplusgval.f |
|- ( ph -> F e. B ) |
7 |
|
prdsplusgval.g |
|- ( ph -> G e. B ) |
8 |
|
prdsmulrval.t |
|- .x. = ( .r ` Y ) |
9 |
|
prdsmulrfval.j |
|- ( ph -> J e. I ) |
10 |
1 2 3 4 5 6 7 8
|
prdsmulrval |
|- ( ph -> ( F .x. G ) = ( x e. I |-> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) ) ) |
11 |
10
|
fveq1d |
|- ( ph -> ( ( F .x. G ) ` J ) = ( ( x e. I |-> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) ) ` J ) ) |
12 |
|
2fveq3 |
|- ( x = J -> ( .r ` ( R ` x ) ) = ( .r ` ( R ` J ) ) ) |
13 |
|
fveq2 |
|- ( x = J -> ( F ` x ) = ( F ` J ) ) |
14 |
|
fveq2 |
|- ( x = J -> ( G ` x ) = ( G ` J ) ) |
15 |
12 13 14
|
oveq123d |
|- ( x = J -> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) = ( ( F ` J ) ( .r ` ( R ` J ) ) ( G ` J ) ) ) |
16 |
|
eqid |
|- ( x e. I |-> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) ) = ( x e. I |-> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) ) |
17 |
|
ovex |
|- ( ( F ` J ) ( .r ` ( R ` J ) ) ( G ` J ) ) e. _V |
18 |
15 16 17
|
fvmpt |
|- ( J e. I -> ( ( x e. I |-> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) ) ` J ) = ( ( F ` J ) ( .r ` ( R ` J ) ) ( G ` J ) ) ) |
19 |
9 18
|
syl |
|- ( ph -> ( ( x e. I |-> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) ) ` J ) = ( ( F ` J ) ( .r ` ( R ` J ) ) ( G ` J ) ) ) |
20 |
11 19
|
eqtrd |
|- ( ph -> ( ( F .x. G ) ` J ) = ( ( F ` J ) ( .r ` ( R ` J ) ) ( G ` J ) ) ) |