| Step |
Hyp |
Ref |
Expression |
| 1 |
|
tsms0.z |
|- .0. = ( 0g ` G ) |
| 2 |
|
tsms0.1 |
|- ( ph -> G e. CMnd ) |
| 3 |
|
tsms0.2 |
|- ( ph -> G e. TopSp ) |
| 4 |
|
tsms0.a |
|- ( ph -> A e. V ) |
| 5 |
|
cmnmnd |
|- ( G e. CMnd -> G e. Mnd ) |
| 6 |
2 5
|
syl |
|- ( ph -> G e. Mnd ) |
| 7 |
1
|
gsumz |
|- ( ( G e. Mnd /\ A e. V ) -> ( G gsum ( x e. A |-> .0. ) ) = .0. ) |
| 8 |
6 4 7
|
syl2anc |
|- ( ph -> ( G gsum ( x e. A |-> .0. ) ) = .0. ) |
| 9 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
| 10 |
9 1
|
mndidcl |
|- ( G e. Mnd -> .0. e. ( Base ` G ) ) |
| 11 |
6 10
|
syl |
|- ( ph -> .0. e. ( Base ` G ) ) |
| 12 |
11
|
adantr |
|- ( ( ph /\ x e. A ) -> .0. e. ( Base ` G ) ) |
| 13 |
12
|
fmpttd |
|- ( ph -> ( x e. A |-> .0. ) : A --> ( Base ` G ) ) |
| 14 |
|
fconstmpt |
|- ( A X. { .0. } ) = ( x e. A |-> .0. ) |
| 15 |
1
|
fvexi |
|- .0. e. _V |
| 16 |
15
|
a1i |
|- ( ph -> .0. e. _V ) |
| 17 |
4 16
|
fczfsuppd |
|- ( ph -> ( A X. { .0. } ) finSupp .0. ) |
| 18 |
14 17
|
eqbrtrrid |
|- ( ph -> ( x e. A |-> .0. ) finSupp .0. ) |
| 19 |
9 1 2 3 4 13 18
|
tsmsid |
|- ( ph -> ( G gsum ( x e. A |-> .0. ) ) e. ( G tsums ( x e. A |-> .0. ) ) ) |
| 20 |
8 19
|
eqeltrrd |
|- ( ph -> .0. e. ( G tsums ( x e. A |-> .0. ) ) ) |