Step |
Hyp |
Ref |
Expression |
1 |
|
gsumsubmcl.z |
|- .0. = ( 0g ` G ) |
2 |
|
gsumsubmcl.g |
|- ( ph -> G e. CMnd ) |
3 |
|
gsumsubmcl.a |
|- ( ph -> A e. V ) |
4 |
|
gsumsubmcl.s |
|- ( ph -> S e. ( SubMnd ` G ) ) |
5 |
|
gsumsubmcl.f |
|- ( ph -> F : A --> S ) |
6 |
|
gsumsubmcl.w |
|- ( ph -> F finSupp .0. ) |
7 |
|
eqid |
|- ( Cntz ` G ) = ( Cntz ` G ) |
8 |
|
cmnmnd |
|- ( G e. CMnd -> G e. Mnd ) |
9 |
2 8
|
syl |
|- ( ph -> G e. Mnd ) |
10 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
11 |
10
|
submss |
|- ( S e. ( SubMnd ` G ) -> S C_ ( Base ` G ) ) |
12 |
4 11
|
syl |
|- ( ph -> S C_ ( Base ` G ) ) |
13 |
5 12
|
fssd |
|- ( ph -> F : A --> ( Base ` G ) ) |
14 |
10 7 2 13
|
cntzcmnf |
|- ( ph -> ran F C_ ( ( Cntz ` G ) ` ran F ) ) |
15 |
1 7 9 3 4 5 14 6
|
gsumzsubmcl |
|- ( ph -> ( G gsum F ) e. S ) |