Step |
Hyp |
Ref |
Expression |
1 |
|
tsmssub.b |
|- B = ( Base ` G ) |
2 |
|
tsmssub.p |
|- .- = ( -g ` G ) |
3 |
|
tsmssub.1 |
|- ( ph -> G e. CMnd ) |
4 |
|
tsmssub.2 |
|- ( ph -> G e. TopGrp ) |
5 |
|
tsmssub.a |
|- ( ph -> A e. V ) |
6 |
|
tsmssub.f |
|- ( ph -> F : A --> B ) |
7 |
|
tsmssub.h |
|- ( ph -> H : A --> B ) |
8 |
|
tsmssub.x |
|- ( ph -> X e. ( G tsums F ) ) |
9 |
|
tsmssub.y |
|- ( ph -> Y e. ( G tsums H ) ) |
10 |
|
eqid |
|- ( +g ` G ) = ( +g ` G ) |
11 |
|
tgptmd |
|- ( G e. TopGrp -> G e. TopMnd ) |
12 |
4 11
|
syl |
|- ( ph -> G e. TopMnd ) |
13 |
|
tgpgrp |
|- ( G e. TopGrp -> G e. Grp ) |
14 |
|
eqid |
|- ( invg ` G ) = ( invg ` G ) |
15 |
1 14
|
grpinvf |
|- ( G e. Grp -> ( invg ` G ) : B --> B ) |
16 |
4 13 15
|
3syl |
|- ( ph -> ( invg ` G ) : B --> B ) |
17 |
|
fco |
|- ( ( ( invg ` G ) : B --> B /\ H : A --> B ) -> ( ( invg ` G ) o. H ) : A --> B ) |
18 |
16 7 17
|
syl2anc |
|- ( ph -> ( ( invg ` G ) o. H ) : A --> B ) |
19 |
1 14 3 4 5 7 9
|
tsmsinv |
|- ( ph -> ( ( invg ` G ) ` Y ) e. ( G tsums ( ( invg ` G ) o. H ) ) ) |
20 |
1 10 3 12 5 6 18 8 19
|
tsmsadd |
|- ( ph -> ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) e. ( G tsums ( F oF ( +g ` G ) ( ( invg ` G ) o. H ) ) ) ) |
21 |
|
tgptps |
|- ( G e. TopGrp -> G e. TopSp ) |
22 |
4 21
|
syl |
|- ( ph -> G e. TopSp ) |
23 |
1 3 22 5 6
|
tsmscl |
|- ( ph -> ( G tsums F ) C_ B ) |
24 |
23 8
|
sseldd |
|- ( ph -> X e. B ) |
25 |
1 3 22 5 7
|
tsmscl |
|- ( ph -> ( G tsums H ) C_ B ) |
26 |
25 9
|
sseldd |
|- ( ph -> Y e. B ) |
27 |
1 10 14 2
|
grpsubval |
|- ( ( X e. B /\ Y e. B ) -> ( X .- Y ) = ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) ) |
28 |
24 26 27
|
syl2anc |
|- ( ph -> ( X .- Y ) = ( X ( +g ` G ) ( ( invg ` G ) ` Y ) ) ) |
29 |
6
|
ffvelrnda |
|- ( ( ph /\ k e. A ) -> ( F ` k ) e. B ) |
30 |
7
|
ffvelrnda |
|- ( ( ph /\ k e. A ) -> ( H ` k ) e. B ) |
31 |
1 10 14 2
|
grpsubval |
|- ( ( ( F ` k ) e. B /\ ( H ` k ) e. B ) -> ( ( F ` k ) .- ( H ` k ) ) = ( ( F ` k ) ( +g ` G ) ( ( invg ` G ) ` ( H ` k ) ) ) ) |
32 |
29 30 31
|
syl2anc |
|- ( ( ph /\ k e. A ) -> ( ( F ` k ) .- ( H ` k ) ) = ( ( F ` k ) ( +g ` G ) ( ( invg ` G ) ` ( H ` k ) ) ) ) |
33 |
32
|
mpteq2dva |
|- ( ph -> ( k e. A |-> ( ( F ` k ) .- ( H ` k ) ) ) = ( k e. A |-> ( ( F ` k ) ( +g ` G ) ( ( invg ` G ) ` ( H ` k ) ) ) ) ) |
34 |
6
|
feqmptd |
|- ( ph -> F = ( k e. A |-> ( F ` k ) ) ) |
35 |
7
|
feqmptd |
|- ( ph -> H = ( k e. A |-> ( H ` k ) ) ) |
36 |
5 29 30 34 35
|
offval2 |
|- ( ph -> ( F oF .- H ) = ( k e. A |-> ( ( F ` k ) .- ( H ` k ) ) ) ) |
37 |
|
fvexd |
|- ( ( ph /\ k e. A ) -> ( ( invg ` G ) ` ( H ` k ) ) e. _V ) |
38 |
16
|
feqmptd |
|- ( ph -> ( invg ` G ) = ( x e. B |-> ( ( invg ` G ) ` x ) ) ) |
39 |
|
fveq2 |
|- ( x = ( H ` k ) -> ( ( invg ` G ) ` x ) = ( ( invg ` G ) ` ( H ` k ) ) ) |
40 |
30 35 38 39
|
fmptco |
|- ( ph -> ( ( invg ` G ) o. H ) = ( k e. A |-> ( ( invg ` G ) ` ( H ` k ) ) ) ) |
41 |
5 29 37 34 40
|
offval2 |
|- ( ph -> ( F oF ( +g ` G ) ( ( invg ` G ) o. H ) ) = ( k e. A |-> ( ( F ` k ) ( +g ` G ) ( ( invg ` G ) ` ( H ` k ) ) ) ) ) |
42 |
33 36 41
|
3eqtr4d |
|- ( ph -> ( F oF .- H ) = ( F oF ( +g ` G ) ( ( invg ` G ) o. H ) ) ) |
43 |
42
|
oveq2d |
|- ( ph -> ( G tsums ( F oF .- H ) ) = ( G tsums ( F oF ( +g ` G ) ( ( invg ` G ) o. H ) ) ) ) |
44 |
20 28 43
|
3eltr4d |
|- ( ph -> ( X .- Y ) e. ( G tsums ( F oF .- H ) ) ) |