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 |
|
prdsplusgval.p |
|- .+ = ( +g ` Y ) |
9 |
|
fnex |
|- ( ( R Fn I /\ I e. W ) -> R e. _V ) |
10 |
5 4 9
|
syl2anc |
|- ( ph -> R e. _V ) |
11 |
5
|
fndmd |
|- ( ph -> dom R = I ) |
12 |
1 3 10 2 11 8
|
prdsplusg |
|- ( ph -> .+ = ( y e. B , z e. B |-> ( x e. I |-> ( ( y ` x ) ( +g ` ( R ` x ) ) ( z ` x ) ) ) ) ) |
13 |
|
fveq1 |
|- ( y = F -> ( y ` x ) = ( F ` x ) ) |
14 |
|
fveq1 |
|- ( z = G -> ( z ` x ) = ( G ` x ) ) |
15 |
13 14
|
oveqan12d |
|- ( ( y = F /\ z = G ) -> ( ( y ` x ) ( +g ` ( R ` x ) ) ( z ` x ) ) = ( ( F ` x ) ( +g ` ( R ` x ) ) ( G ` x ) ) ) |
16 |
15
|
adantl |
|- ( ( ph /\ ( y = F /\ z = G ) ) -> ( ( y ` x ) ( +g ` ( R ` x ) ) ( z ` x ) ) = ( ( F ` x ) ( +g ` ( R ` x ) ) ( G ` x ) ) ) |
17 |
16
|
mpteq2dv |
|- ( ( ph /\ ( y = F /\ z = G ) ) -> ( x e. I |-> ( ( y ` x ) ( +g ` ( R ` x ) ) ( z ` x ) ) ) = ( x e. I |-> ( ( F ` x ) ( +g ` ( R ` x ) ) ( G ` x ) ) ) ) |
18 |
4
|
mptexd |
|- ( ph -> ( x e. I |-> ( ( F ` x ) ( +g ` ( R ` x ) ) ( G ` x ) ) ) e. _V ) |
19 |
12 17 6 7 18
|
ovmpod |
|- ( ph -> ( F .+ G ) = ( x e. I |-> ( ( F ` x ) ( +g ` ( R ` x ) ) ( G ` x ) ) ) ) |