Step |
Hyp |
Ref |
Expression |
1 |
|
eldprdi.0 |
|- .0. = ( 0g ` G ) |
2 |
|
eldprdi.w |
|- W = { h e. X_ i e. I ( S ` i ) | h finSupp .0. } |
3 |
|
eldprdi.1 |
|- ( ph -> G dom DProd S ) |
4 |
|
eldprdi.2 |
|- ( ph -> dom S = I ) |
5 |
|
eldprdi.3 |
|- ( ph -> F e. W ) |
6 |
|
eqid |
|- ( G gsum F ) = ( G gsum F ) |
7 |
|
oveq2 |
|- ( f = F -> ( G gsum f ) = ( G gsum F ) ) |
8 |
7
|
rspceeqv |
|- ( ( F e. W /\ ( G gsum F ) = ( G gsum F ) ) -> E. f e. W ( G gsum F ) = ( G gsum f ) ) |
9 |
5 6 8
|
sylancl |
|- ( ph -> E. f e. W ( G gsum F ) = ( G gsum f ) ) |
10 |
1 2
|
eldprd |
|- ( dom S = I -> ( ( G gsum F ) e. ( G DProd S ) <-> ( G dom DProd S /\ E. f e. W ( G gsum F ) = ( G gsum f ) ) ) ) |
11 |
4 10
|
syl |
|- ( ph -> ( ( G gsum F ) e. ( G DProd S ) <-> ( G dom DProd S /\ E. f e. W ( G gsum F ) = ( G gsum f ) ) ) ) |
12 |
3 9 11
|
mpbir2and |
|- ( ph -> ( G gsum F ) e. ( G DProd S ) ) |