| Step |
Hyp |
Ref |
Expression |
| 1 |
|
gsummulg.b |
|- B = ( Base ` G ) |
| 2 |
|
gsummulg.z |
|- .0. = ( 0g ` G ) |
| 3 |
|
gsummulg.t |
|- .x. = ( .g ` G ) |
| 4 |
|
gsummulg.a |
|- ( ph -> A e. V ) |
| 5 |
|
gsummulg.f |
|- ( ( ph /\ k e. A ) -> X e. B ) |
| 6 |
|
gsummulg.w |
|- ( ph -> ( k e. A |-> X ) finSupp .0. ) |
| 7 |
|
gsummulglem.g |
|- ( ph -> G e. CMnd ) |
| 8 |
|
gsummulglem.n |
|- ( ph -> N e. ZZ ) |
| 9 |
|
gsummulglem.o |
|- ( ph -> ( G e. Abel \/ N e. NN0 ) ) |
| 10 |
|
cmnmnd |
|- ( G e. CMnd -> G e. Mnd ) |
| 11 |
7 10
|
syl |
|- ( ph -> G e. Mnd ) |
| 12 |
1 3
|
mulgghm |
|- ( ( G e. Abel /\ N e. ZZ ) -> ( x e. B |-> ( N .x. x ) ) e. ( G GrpHom G ) ) |
| 13 |
|
ghmmhm |
|- ( ( x e. B |-> ( N .x. x ) ) e. ( G GrpHom G ) -> ( x e. B |-> ( N .x. x ) ) e. ( G MndHom G ) ) |
| 14 |
12 13
|
syl |
|- ( ( G e. Abel /\ N e. ZZ ) -> ( x e. B |-> ( N .x. x ) ) e. ( G MndHom G ) ) |
| 15 |
14
|
expcom |
|- ( N e. ZZ -> ( G e. Abel -> ( x e. B |-> ( N .x. x ) ) e. ( G MndHom G ) ) ) |
| 16 |
8 15
|
syl |
|- ( ph -> ( G e. Abel -> ( x e. B |-> ( N .x. x ) ) e. ( G MndHom G ) ) ) |
| 17 |
1 3
|
mulgmhm |
|- ( ( G e. CMnd /\ N e. NN0 ) -> ( x e. B |-> ( N .x. x ) ) e. ( G MndHom G ) ) |
| 18 |
17
|
ex |
|- ( G e. CMnd -> ( N e. NN0 -> ( x e. B |-> ( N .x. x ) ) e. ( G MndHom G ) ) ) |
| 19 |
7 18
|
syl |
|- ( ph -> ( N e. NN0 -> ( x e. B |-> ( N .x. x ) ) e. ( G MndHom G ) ) ) |
| 20 |
16 19 9
|
mpjaod |
|- ( ph -> ( x e. B |-> ( N .x. x ) ) e. ( G MndHom G ) ) |
| 21 |
|
oveq2 |
|- ( x = X -> ( N .x. x ) = ( N .x. X ) ) |
| 22 |
|
oveq2 |
|- ( x = ( G gsum ( k e. A |-> X ) ) -> ( N .x. x ) = ( N .x. ( G gsum ( k e. A |-> X ) ) ) ) |
| 23 |
1 2 7 11 4 20 5 6 21 22
|
gsummhm2 |
|- ( ph -> ( G gsum ( k e. A |-> ( N .x. X ) ) ) = ( N .x. ( G gsum ( k e. A |-> X ) ) ) ) |