| 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 ) ) |