Step |
Hyp |
Ref |
Expression |
1 |
|
grpsubinv.b |
|- B = ( Base ` G ) |
2 |
|
grpsubinv.p |
|- .+ = ( +g ` G ) |
3 |
|
grpsubinv.m |
|- .- = ( -g ` G ) |
4 |
|
grpsubinv.n |
|- N = ( invg ` G ) |
5 |
|
grpsubinv.g |
|- ( ph -> G e. Grp ) |
6 |
|
grpsubinv.x |
|- ( ph -> X e. B ) |
7 |
|
grpsubinv.y |
|- ( ph -> Y e. B ) |
8 |
1 4
|
grpinvcl |
|- ( ( G e. Grp /\ Y e. B ) -> ( N ` Y ) e. B ) |
9 |
5 7 8
|
syl2anc |
|- ( ph -> ( N ` Y ) e. B ) |
10 |
1 2 4 3
|
grpsubval |
|- ( ( X e. B /\ ( N ` Y ) e. B ) -> ( X .- ( N ` Y ) ) = ( X .+ ( N ` ( N ` Y ) ) ) ) |
11 |
6 9 10
|
syl2anc |
|- ( ph -> ( X .- ( N ` Y ) ) = ( X .+ ( N ` ( N ` Y ) ) ) ) |
12 |
1 4
|
grpinvinv |
|- ( ( G e. Grp /\ Y e. B ) -> ( N ` ( N ` Y ) ) = Y ) |
13 |
5 7 12
|
syl2anc |
|- ( ph -> ( N ` ( N ` Y ) ) = Y ) |
14 |
13
|
oveq2d |
|- ( ph -> ( X .+ ( N ` ( N ` Y ) ) ) = ( X .+ Y ) ) |
15 |
11 14
|
eqtrd |
|- ( ph -> ( X .- ( N ` Y ) ) = ( X .+ Y ) ) |