Step |
Hyp |
Ref |
Expression |
1 |
|
gsummptmhm.b |
|- B = ( Base ` G ) |
2 |
|
gsummptmhm.z |
|- .0. = ( 0g ` G ) |
3 |
|
gsummptmhm.g |
|- ( ph -> G e. CMnd ) |
4 |
|
gsummptmhm.h |
|- ( ph -> H e. Mnd ) |
5 |
|
gsummptmhm.a |
|- ( ph -> A e. V ) |
6 |
|
gsummptmhm.k |
|- ( ph -> K e. ( G MndHom H ) ) |
7 |
|
gsummptmhm.c |
|- ( ( ph /\ x e. A ) -> C e. B ) |
8 |
|
gsummptmhm.w |
|- ( ph -> ( x e. A |-> C ) finSupp .0. ) |
9 |
|
eqidd |
|- ( ph -> ( x e. A |-> C ) = ( x e. A |-> C ) ) |
10 |
|
eqid |
|- ( Base ` H ) = ( Base ` H ) |
11 |
1 10
|
mhmf |
|- ( K e. ( G MndHom H ) -> K : B --> ( Base ` H ) ) |
12 |
|
ffn |
|- ( K : B --> ( Base ` H ) -> K Fn B ) |
13 |
6 11 12
|
3syl |
|- ( ph -> K Fn B ) |
14 |
|
dffn5 |
|- ( K Fn B <-> K = ( y e. B |-> ( K ` y ) ) ) |
15 |
13 14
|
sylib |
|- ( ph -> K = ( y e. B |-> ( K ` y ) ) ) |
16 |
|
fveq2 |
|- ( y = C -> ( K ` y ) = ( K ` C ) ) |
17 |
7 9 15 16
|
fmptco |
|- ( ph -> ( K o. ( x e. A |-> C ) ) = ( x e. A |-> ( K ` C ) ) ) |
18 |
17
|
oveq2d |
|- ( ph -> ( H gsum ( K o. ( x e. A |-> C ) ) ) = ( H gsum ( x e. A |-> ( K ` C ) ) ) ) |
19 |
7
|
fmpttd |
|- ( ph -> ( x e. A |-> C ) : A --> B ) |
20 |
1 2 3 4 5 6 19 8
|
gsummhm |
|- ( ph -> ( H gsum ( K o. ( x e. A |-> C ) ) ) = ( K ` ( G gsum ( x e. A |-> C ) ) ) ) |
21 |
18 20
|
eqtr3d |
|- ( ph -> ( H gsum ( x e. A |-> ( K ` C ) ) ) = ( K ` ( G gsum ( x e. A |-> C ) ) ) ) |