Step |
Hyp |
Ref |
Expression |
1 |
|
gsumsubm.a |
|- ( ph -> A e. V ) |
2 |
|
gsumsubm.s |
|- ( ph -> S e. ( SubMnd ` G ) ) |
3 |
|
gsumsubm.f |
|- ( ph -> F : A --> S ) |
4 |
|
gsumsubm.h |
|- H = ( G |`s S ) |
5 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
6 |
|
eqid |
|- ( +g ` G ) = ( +g ` G ) |
7 |
|
submrcl |
|- ( S e. ( SubMnd ` G ) -> G e. Mnd ) |
8 |
2 7
|
syl |
|- ( ph -> G e. Mnd ) |
9 |
5
|
submss |
|- ( S e. ( SubMnd ` G ) -> S C_ ( Base ` G ) ) |
10 |
2 9
|
syl |
|- ( ph -> S C_ ( Base ` G ) ) |
11 |
|
eqid |
|- ( 0g ` G ) = ( 0g ` G ) |
12 |
11
|
subm0cl |
|- ( S e. ( SubMnd ` G ) -> ( 0g ` G ) e. S ) |
13 |
2 12
|
syl |
|- ( ph -> ( 0g ` G ) e. S ) |
14 |
5 6 11
|
mndlrid |
|- ( ( G e. Mnd /\ x e. ( Base ` G ) ) -> ( ( ( 0g ` G ) ( +g ` G ) x ) = x /\ ( x ( +g ` G ) ( 0g ` G ) ) = x ) ) |
15 |
8 14
|
sylan |
|- ( ( ph /\ x e. ( Base ` G ) ) -> ( ( ( 0g ` G ) ( +g ` G ) x ) = x /\ ( x ( +g ` G ) ( 0g ` G ) ) = x ) ) |
16 |
5 6 4 8 1 10 3 13 15
|
gsumress |
|- ( ph -> ( G gsum F ) = ( H gsum F ) ) |