Step |
Hyp |
Ref |
Expression |
1 |
|
prdsbasmpt.y |
|- Y = ( S Xs_ R ) |
2 |
|
prdsbasmpt.b |
|- B = ( Base ` Y ) |
3 |
|
prdsvscaval.t |
|- .x. = ( .s ` Y ) |
4 |
|
prdsvscaval.k |
|- K = ( Base ` S ) |
5 |
|
prdsvscaval.s |
|- ( ph -> S e. V ) |
6 |
|
prdsvscaval.i |
|- ( ph -> I e. W ) |
7 |
|
prdsvscaval.r |
|- ( ph -> R Fn I ) |
8 |
|
prdsvscaval.f |
|- ( ph -> F e. K ) |
9 |
|
prdsvscaval.g |
|- ( ph -> G e. B ) |
10 |
|
fnex |
|- ( ( R Fn I /\ I e. W ) -> R e. _V ) |
11 |
7 6 10
|
syl2anc |
|- ( ph -> R e. _V ) |
12 |
7
|
fndmd |
|- ( ph -> dom R = I ) |
13 |
1 5 11 2 12 4 3
|
prdsvsca |
|- ( ph -> .x. = ( y e. K , z e. B |-> ( x e. I |-> ( y ( .s ` ( R ` x ) ) ( z ` x ) ) ) ) ) |
14 |
|
id |
|- ( y = F -> y = F ) |
15 |
|
fveq1 |
|- ( z = G -> ( z ` x ) = ( G ` x ) ) |
16 |
14 15
|
oveqan12d |
|- ( ( y = F /\ z = G ) -> ( y ( .s ` ( R ` x ) ) ( z ` x ) ) = ( F ( .s ` ( R ` x ) ) ( G ` x ) ) ) |
17 |
16
|
adantl |
|- ( ( ph /\ ( y = F /\ z = G ) ) -> ( y ( .s ` ( R ` x ) ) ( z ` x ) ) = ( F ( .s ` ( R ` x ) ) ( G ` x ) ) ) |
18 |
17
|
mpteq2dv |
|- ( ( ph /\ ( y = F /\ z = G ) ) -> ( x e. I |-> ( y ( .s ` ( R ` x ) ) ( z ` x ) ) ) = ( x e. I |-> ( F ( .s ` ( R ` x ) ) ( G ` x ) ) ) ) |
19 |
6
|
mptexd |
|- ( ph -> ( x e. I |-> ( F ( .s ` ( R ` x ) ) ( G ` x ) ) ) e. _V ) |
20 |
13 18 8 9 19
|
ovmpod |
|- ( ph -> ( F .x. G ) = ( x e. I |-> ( F ( .s ` ( R ` x ) ) ( G ` x ) ) ) ) |