Step |
Hyp |
Ref |
Expression |
1 |
|
gsumzinv.b |
|- B = ( Base ` G ) |
2 |
|
gsumzinv.0 |
|- .0. = ( 0g ` G ) |
3 |
|
gsumzinv.z |
|- Z = ( Cntz ` G ) |
4 |
|
gsumzinv.i |
|- I = ( invg ` G ) |
5 |
|
gsumzinv.g |
|- ( ph -> G e. Grp ) |
6 |
|
gsumzinv.a |
|- ( ph -> A e. V ) |
7 |
|
gsumzinv.f |
|- ( ph -> F : A --> B ) |
8 |
|
gsumzinv.c |
|- ( ph -> ran F C_ ( Z ` ran F ) ) |
9 |
|
gsumzinv.n |
|- ( ph -> F finSupp .0. ) |
10 |
|
eqid |
|- ( oppG ` G ) = ( oppG ` G ) |
11 |
|
grpmnd |
|- ( G e. Grp -> G e. Mnd ) |
12 |
5 11
|
syl |
|- ( ph -> G e. Mnd ) |
13 |
1 4
|
grpinvf |
|- ( G e. Grp -> I : B --> B ) |
14 |
5 13
|
syl |
|- ( ph -> I : B --> B ) |
15 |
|
fco |
|- ( ( I : B --> B /\ F : A --> B ) -> ( I o. F ) : A --> B ) |
16 |
14 7 15
|
syl2anc |
|- ( ph -> ( I o. F ) : A --> B ) |
17 |
10 4
|
invoppggim |
|- ( G e. Grp -> I e. ( G GrpIso ( oppG ` G ) ) ) |
18 |
|
gimghm |
|- ( I e. ( G GrpIso ( oppG ` G ) ) -> I e. ( G GrpHom ( oppG ` G ) ) ) |
19 |
|
ghmmhm |
|- ( I e. ( G GrpHom ( oppG ` G ) ) -> I e. ( G MndHom ( oppG ` G ) ) ) |
20 |
5 17 18 19
|
4syl |
|- ( ph -> I e. ( G MndHom ( oppG ` G ) ) ) |
21 |
|
eqid |
|- ( Cntz ` ( oppG ` G ) ) = ( Cntz ` ( oppG ` G ) ) |
22 |
3 21
|
cntzmhm2 |
|- ( ( I e. ( G MndHom ( oppG ` G ) ) /\ ran F C_ ( Z ` ran F ) ) -> ( I " ran F ) C_ ( ( Cntz ` ( oppG ` G ) ) ` ( I " ran F ) ) ) |
23 |
20 8 22
|
syl2anc |
|- ( ph -> ( I " ran F ) C_ ( ( Cntz ` ( oppG ` G ) ) ` ( I " ran F ) ) ) |
24 |
|
rnco2 |
|- ran ( I o. F ) = ( I " ran F ) |
25 |
24
|
fveq2i |
|- ( Z ` ran ( I o. F ) ) = ( Z ` ( I " ran F ) ) |
26 |
10 3
|
oppgcntz |
|- ( Z ` ( I " ran F ) ) = ( ( Cntz ` ( oppG ` G ) ) ` ( I " ran F ) ) |
27 |
25 26
|
eqtri |
|- ( Z ` ran ( I o. F ) ) = ( ( Cntz ` ( oppG ` G ) ) ` ( I " ran F ) ) |
28 |
23 24 27
|
3sstr4g |
|- ( ph -> ran ( I o. F ) C_ ( Z ` ran ( I o. F ) ) ) |
29 |
2
|
fvexi |
|- .0. e. _V |
30 |
29
|
a1i |
|- ( ph -> .0. e. _V ) |
31 |
1
|
fvexi |
|- B e. _V |
32 |
31
|
a1i |
|- ( ph -> B e. _V ) |
33 |
2 4
|
grpinvid |
|- ( G e. Grp -> ( I ` .0. ) = .0. ) |
34 |
5 33
|
syl |
|- ( ph -> ( I ` .0. ) = .0. ) |
35 |
30 7 14 6 32 9 34
|
fsuppco2 |
|- ( ph -> ( I o. F ) finSupp .0. ) |
36 |
1 2 3 10 12 6 16 28 35
|
gsumzoppg |
|- ( ph -> ( ( oppG ` G ) gsum ( I o. F ) ) = ( G gsum ( I o. F ) ) ) |
37 |
10
|
oppgmnd |
|- ( G e. Mnd -> ( oppG ` G ) e. Mnd ) |
38 |
12 37
|
syl |
|- ( ph -> ( oppG ` G ) e. Mnd ) |
39 |
1 3 12 38 6 20 7 8 2 9
|
gsumzmhm |
|- ( ph -> ( ( oppG ` G ) gsum ( I o. F ) ) = ( I ` ( G gsum F ) ) ) |
40 |
36 39
|
eqtr3d |
|- ( ph -> ( G gsum ( I o. F ) ) = ( I ` ( G gsum F ) ) ) |