| Step |
Hyp |
Ref |
Expression |
| 1 |
|
gsumdifsnd.b |
|- B = ( Base ` G ) |
| 2 |
|
gsumdifsnd.p |
|- .+ = ( +g ` G ) |
| 3 |
|
gsumdifsnd.g |
|- ( ph -> G e. CMnd ) |
| 4 |
|
gsumdifsnd.a |
|- ( ph -> A e. W ) |
| 5 |
|
gsumdifsnd.f |
|- ( ph -> ( k e. A |-> X ) finSupp ( 0g ` G ) ) |
| 6 |
|
gsumdifsnd.e |
|- ( ( ph /\ k e. A ) -> X e. B ) |
| 7 |
|
gsumdifsnd.m |
|- ( ph -> M e. A ) |
| 8 |
|
gsumdifsnd.y |
|- ( ph -> Y e. B ) |
| 9 |
|
gsumdifsnd.s |
|- ( ( ph /\ k = M ) -> X = Y ) |
| 10 |
|
eqid |
|- ( 0g ` G ) = ( 0g ` G ) |
| 11 |
7
|
snssd |
|- ( ph -> { M } C_ A ) |
| 12 |
|
difin2 |
|- ( { M } C_ A -> ( { M } \ { M } ) = ( ( A \ { M } ) i^i { M } ) ) |
| 13 |
11 12
|
syl |
|- ( ph -> ( { M } \ { M } ) = ( ( A \ { M } ) i^i { M } ) ) |
| 14 |
|
difid |
|- ( { M } \ { M } ) = (/) |
| 15 |
13 14
|
eqtr3di |
|- ( ph -> ( ( A \ { M } ) i^i { M } ) = (/) ) |
| 16 |
|
difsnid |
|- ( M e. A -> ( ( A \ { M } ) u. { M } ) = A ) |
| 17 |
7 16
|
syl |
|- ( ph -> ( ( A \ { M } ) u. { M } ) = A ) |
| 18 |
17
|
eqcomd |
|- ( ph -> A = ( ( A \ { M } ) u. { M } ) ) |
| 19 |
1 10 2 3 4 6 5 15 18
|
gsumsplit2 |
|- ( ph -> ( G gsum ( k e. A |-> X ) ) = ( ( G gsum ( k e. ( A \ { M } ) |-> X ) ) .+ ( G gsum ( k e. { M } |-> X ) ) ) ) |
| 20 |
|
cmnmnd |
|- ( G e. CMnd -> G e. Mnd ) |
| 21 |
3 20
|
syl |
|- ( ph -> G e. Mnd ) |
| 22 |
1 21 7 8 9
|
gsumsnd |
|- ( ph -> ( G gsum ( k e. { M } |-> X ) ) = Y ) |
| 23 |
22
|
oveq2d |
|- ( ph -> ( ( G gsum ( k e. ( A \ { M } ) |-> X ) ) .+ ( G gsum ( k e. { M } |-> X ) ) ) = ( ( G gsum ( k e. ( A \ { M } ) |-> X ) ) .+ Y ) ) |
| 24 |
19 23
|
eqtrd |
|- ( ph -> ( G gsum ( k e. A |-> X ) ) = ( ( G gsum ( k e. ( A \ { M } ) |-> X ) ) .+ Y ) ) |