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Metamath Proof Explorer

Theorem prdsmulrval

Description: Value of a componentwise ring product in a structure product. (Contributed by Mario Carneiro, 11-Jan-2015)

Ref Expression
Hypotheses prdsbasmpt.y 
|- Y = ( S Xs_ R )
prdsbasmpt.b 
|- B = ( Base ` Y )
prdsbasmpt.s 
|- ( ph -> S e. V )
prdsbasmpt.i 
|- ( ph -> I e. W )
prdsbasmpt.r 
|- ( ph -> R Fn I )
prdsplusgval.f 
|- ( ph -> F e. B )
prdsplusgval.g 
|- ( ph -> G e. B )
prdsmulrval.t 
|- .x. = ( .r ` Y )
Assertion prdsmulrval 
|- ( ph -> ( F .x. G ) = ( x e. I |-> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) ) )

Proof

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 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 prdsmulr 
 |-  ( ph -> .x. = ( y e. B , z e. B |-> ( x e. I |-> ( ( y ` x ) ( .r ` ( 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 ) ( .r ` ( R ` x ) ) ( z ` x ) ) = ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) )
16 15 adantl 
 |-  ( ( ph /\ ( y = F /\ z = G ) ) -> ( ( y ` x ) ( .r ` ( R ` x ) ) ( z ` x ) ) = ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) )
17 16 mpteq2dv 
 |-  ( ( ph /\ ( y = F /\ z = G ) ) -> ( x e. I |-> ( ( y ` x ) ( .r ` ( R ` x ) ) ( z ` x ) ) ) = ( x e. I |-> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) ) )
18 4 mptexd 
 |-  ( ph -> ( x e. I |-> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) ) e. _V )
19 12 17 6 7 18 ovmpod 
 |-  ( ph -> ( F .x. G ) = ( x e. I |-> ( ( F ` x ) ( .r ` ( R ` x ) ) ( G ` x ) ) ) )