| Step |
Hyp |
Ref |
Expression |
| 1 |
|
gsummulgc1.b |
|- B = ( Base ` M ) |
| 2 |
|
gsummulgc1.t |
|- .x. = ( .g ` M ) |
| 3 |
|
gsummulgc1.r |
|- ( ph -> M e. Grp ) |
| 4 |
|
gsummulgc1.a |
|- ( ph -> A e. Fin ) |
| 5 |
|
gsummulgc1.y |
|- ( ph -> Y e. B ) |
| 6 |
|
gsummulgc1.x |
|- ( ( ph /\ k e. A ) -> X e. ZZ ) |
| 7 |
|
zringbas |
|- ZZ = ( Base ` ZZring ) |
| 8 |
|
zring0 |
|- 0 = ( 0g ` ZZring ) |
| 9 |
|
zringring |
|- ZZring e. Ring |
| 10 |
|
ringcmn |
|- ( ZZring e. Ring -> ZZring e. CMnd ) |
| 11 |
9 10
|
mp1i |
|- ( ph -> ZZring e. CMnd ) |
| 12 |
3
|
grpmndd |
|- ( ph -> M e. Mnd ) |
| 13 |
|
eqid |
|- ( x e. ZZ |-> ( x .x. Y ) ) = ( x e. ZZ |-> ( x .x. Y ) ) |
| 14 |
2 13 1
|
mulgghm2 |
|- ( ( M e. Grp /\ Y e. B ) -> ( x e. ZZ |-> ( x .x. Y ) ) e. ( ZZring GrpHom M ) ) |
| 15 |
3 5 14
|
syl2anc |
|- ( ph -> ( x e. ZZ |-> ( x .x. Y ) ) e. ( ZZring GrpHom M ) ) |
| 16 |
|
ghmmhm |
|- ( ( x e. ZZ |-> ( x .x. Y ) ) e. ( ZZring GrpHom M ) -> ( x e. ZZ |-> ( x .x. Y ) ) e. ( ZZring MndHom M ) ) |
| 17 |
15 16
|
syl |
|- ( ph -> ( x e. ZZ |-> ( x .x. Y ) ) e. ( ZZring MndHom M ) ) |
| 18 |
|
eqid |
|- ( k e. A |-> X ) = ( k e. A |-> X ) |
| 19 |
|
0zd |
|- ( ph -> 0 e. ZZ ) |
| 20 |
18 4 6 19
|
fsuppmptdm |
|- ( ph -> ( k e. A |-> X ) finSupp 0 ) |
| 21 |
|
oveq1 |
|- ( x = X -> ( x .x. Y ) = ( X .x. Y ) ) |
| 22 |
|
oveq1 |
|- ( x = ( ZZring gsum ( k e. A |-> X ) ) -> ( x .x. Y ) = ( ( ZZring gsum ( k e. A |-> X ) ) .x. Y ) ) |
| 23 |
7 8 11 12 4 17 6 20 21 22
|
gsummhm2 |
|- ( ph -> ( M gsum ( k e. A |-> ( X .x. Y ) ) ) = ( ( ZZring gsum ( k e. A |-> X ) ) .x. Y ) ) |
| 24 |
4 6
|
gsumzrsum |
|- ( ph -> ( ZZring gsum ( k e. A |-> X ) ) = sum_ k e. A X ) |
| 25 |
24
|
oveq1d |
|- ( ph -> ( ( ZZring gsum ( k e. A |-> X ) ) .x. Y ) = ( sum_ k e. A X .x. Y ) ) |
| 26 |
23 25
|
eqtrd |
|- ( ph -> ( M gsum ( k e. A |-> ( X .x. Y ) ) ) = ( sum_ k e. A X .x. Y ) ) |