Step |
Hyp |
Ref |
Expression |
1 |
|
grpinvinv.b |
|- B = ( Base ` G ) |
2 |
|
grpinvinv.n |
|- N = ( invg ` G ) |
3 |
|
grpinv11.g |
|- ( ph -> G e. Grp ) |
4 |
|
grpinv11.x |
|- ( ph -> X e. B ) |
5 |
|
grpinv11.y |
|- ( ph -> Y e. B ) |
6 |
|
fveq2 |
|- ( ( N ` X ) = ( N ` Y ) -> ( N ` ( N ` X ) ) = ( N ` ( N ` Y ) ) ) |
7 |
6
|
adantl |
|- ( ( ph /\ ( N ` X ) = ( N ` Y ) ) -> ( N ` ( N ` X ) ) = ( N ` ( N ` Y ) ) ) |
8 |
1 2
|
grpinvinv |
|- ( ( G e. Grp /\ X e. B ) -> ( N ` ( N ` X ) ) = X ) |
9 |
3 4 8
|
syl2anc |
|- ( ph -> ( N ` ( N ` X ) ) = X ) |
10 |
9
|
adantr |
|- ( ( ph /\ ( N ` X ) = ( N ` Y ) ) -> ( N ` ( N ` X ) ) = X ) |
11 |
1 2
|
grpinvinv |
|- ( ( G e. Grp /\ Y e. B ) -> ( N ` ( N ` Y ) ) = Y ) |
12 |
3 5 11
|
syl2anc |
|- ( ph -> ( N ` ( N ` Y ) ) = Y ) |
13 |
12
|
adantr |
|- ( ( ph /\ ( N ` X ) = ( N ` Y ) ) -> ( N ` ( N ` Y ) ) = Y ) |
14 |
7 10 13
|
3eqtr3d |
|- ( ( ph /\ ( N ` X ) = ( N ` Y ) ) -> X = Y ) |
15 |
14
|
ex |
|- ( ph -> ( ( N ` X ) = ( N ` Y ) -> X = Y ) ) |
16 |
|
fveq2 |
|- ( X = Y -> ( N ` X ) = ( N ` Y ) ) |
17 |
15 16
|
impbid1 |
|- ( ph -> ( ( N ` X ) = ( N ` Y ) <-> X = Y ) ) |