Step |
Hyp |
Ref |
Expression |
1 |
|
nn0gsumfz.b |
|- B = ( Base ` G ) |
2 |
|
nn0gsumfz.0 |
|- .0. = ( 0g ` G ) |
3 |
|
nn0gsumfz.g |
|- ( ph -> G e. CMnd ) |
4 |
|
nn0gsumfz.f |
|- ( ph -> F e. ( B ^m NN0 ) ) |
5 |
|
nn0gsumfz.y |
|- ( ph -> F finSupp .0. ) |
6 |
1 2 3 4 5
|
nn0gsumfz |
|- ( ph -> E. s e. NN0 E. f e. ( B ^m ( 0 ... s ) ) ( f = ( F |` ( 0 ... s ) ) /\ A. x e. NN0 ( s < x -> ( F ` x ) = .0. ) /\ ( G gsum F ) = ( G gsum f ) ) ) |
7 |
|
simp3 |
|- ( ( f = ( F |` ( 0 ... s ) ) /\ A. x e. NN0 ( s < x -> ( F ` x ) = .0. ) /\ ( G gsum F ) = ( G gsum f ) ) -> ( G gsum F ) = ( G gsum f ) ) |
8 |
7
|
reximi |
|- ( E. f e. ( B ^m ( 0 ... s ) ) ( f = ( F |` ( 0 ... s ) ) /\ A. x e. NN0 ( s < x -> ( F ` x ) = .0. ) /\ ( G gsum F ) = ( G gsum f ) ) -> E. f e. ( B ^m ( 0 ... s ) ) ( G gsum F ) = ( G gsum f ) ) |
9 |
8
|
reximi |
|- ( E. s e. NN0 E. f e. ( B ^m ( 0 ... s ) ) ( f = ( F |` ( 0 ... s ) ) /\ A. x e. NN0 ( s < x -> ( F ` x ) = .0. ) /\ ( G gsum F ) = ( G gsum f ) ) -> E. s e. NN0 E. f e. ( B ^m ( 0 ... s ) ) ( G gsum F ) = ( G gsum f ) ) |
10 |
6 9
|
syl |
|- ( ph -> E. s e. NN0 E. f e. ( B ^m ( 0 ... s ) ) ( G gsum F ) = ( G gsum f ) ) |