Step |
Hyp |
Ref |
Expression |
1 |
|
gsummptcl.b |
|- B = ( Base ` G ) |
2 |
|
gsummptcl.g |
|- ( ph -> G e. CMnd ) |
3 |
|
gsummptcl.n |
|- ( ph -> N e. Fin ) |
4 |
|
gsummptcl.e |
|- ( ph -> A. i e. N X e. B ) |
5 |
|
eqid |
|- ( 0g ` G ) = ( 0g ` G ) |
6 |
|
eqid |
|- ( i e. N |-> X ) = ( i e. N |-> X ) |
7 |
6
|
fmpt |
|- ( A. i e. N X e. B <-> ( i e. N |-> X ) : N --> B ) |
8 |
4 7
|
sylib |
|- ( ph -> ( i e. N |-> X ) : N --> B ) |
9 |
6
|
fnmpt |
|- ( A. i e. N X e. B -> ( i e. N |-> X ) Fn N ) |
10 |
4 9
|
syl |
|- ( ph -> ( i e. N |-> X ) Fn N ) |
11 |
|
fvexd |
|- ( ph -> ( 0g ` G ) e. _V ) |
12 |
10 3 11
|
fndmfifsupp |
|- ( ph -> ( i e. N |-> X ) finSupp ( 0g ` G ) ) |
13 |
1 5 2 3 8 12
|
gsumcl |
|- ( ph -> ( G gsum ( i e. N |-> X ) ) e. B ) |