Step |
Hyp |
Ref |
Expression |
1 |
|
grplcan.1 |
|- X = ran G |
2 |
|
oveq2 |
|- ( ( C G A ) = ( C G B ) -> ( ( ( inv ` G ) ` C ) G ( C G A ) ) = ( ( ( inv ` G ) ` C ) G ( C G B ) ) ) |
3 |
2
|
adantl |
|- ( ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) /\ ( C G A ) = ( C G B ) ) -> ( ( ( inv ` G ) ` C ) G ( C G A ) ) = ( ( ( inv ` G ) ` C ) G ( C G B ) ) ) |
4 |
|
eqid |
|- ( GId ` G ) = ( GId ` G ) |
5 |
|
eqid |
|- ( inv ` G ) = ( inv ` G ) |
6 |
1 4 5
|
grpolinv |
|- ( ( G e. GrpOp /\ C e. X ) -> ( ( ( inv ` G ) ` C ) G C ) = ( GId ` G ) ) |
7 |
6
|
adantlr |
|- ( ( ( G e. GrpOp /\ A e. X ) /\ C e. X ) -> ( ( ( inv ` G ) ` C ) G C ) = ( GId ` G ) ) |
8 |
7
|
oveq1d |
|- ( ( ( G e. GrpOp /\ A e. X ) /\ C e. X ) -> ( ( ( ( inv ` G ) ` C ) G C ) G A ) = ( ( GId ` G ) G A ) ) |
9 |
1 5
|
grpoinvcl |
|- ( ( G e. GrpOp /\ C e. X ) -> ( ( inv ` G ) ` C ) e. X ) |
10 |
9
|
adantrl |
|- ( ( G e. GrpOp /\ ( A e. X /\ C e. X ) ) -> ( ( inv ` G ) ` C ) e. X ) |
11 |
|
simprr |
|- ( ( G e. GrpOp /\ ( A e. X /\ C e. X ) ) -> C e. X ) |
12 |
|
simprl |
|- ( ( G e. GrpOp /\ ( A e. X /\ C e. X ) ) -> A e. X ) |
13 |
10 11 12
|
3jca |
|- ( ( G e. GrpOp /\ ( A e. X /\ C e. X ) ) -> ( ( ( inv ` G ) ` C ) e. X /\ C e. X /\ A e. X ) ) |
14 |
1
|
grpoass |
|- ( ( G e. GrpOp /\ ( ( ( inv ` G ) ` C ) e. X /\ C e. X /\ A e. X ) ) -> ( ( ( ( inv ` G ) ` C ) G C ) G A ) = ( ( ( inv ` G ) ` C ) G ( C G A ) ) ) |
15 |
13 14
|
syldan |
|- ( ( G e. GrpOp /\ ( A e. X /\ C e. X ) ) -> ( ( ( ( inv ` G ) ` C ) G C ) G A ) = ( ( ( inv ` G ) ` C ) G ( C G A ) ) ) |
16 |
15
|
anassrs |
|- ( ( ( G e. GrpOp /\ A e. X ) /\ C e. X ) -> ( ( ( ( inv ` G ) ` C ) G C ) G A ) = ( ( ( inv ` G ) ` C ) G ( C G A ) ) ) |
17 |
1 4
|
grpolid |
|- ( ( G e. GrpOp /\ A e. X ) -> ( ( GId ` G ) G A ) = A ) |
18 |
17
|
adantr |
|- ( ( ( G e. GrpOp /\ A e. X ) /\ C e. X ) -> ( ( GId ` G ) G A ) = A ) |
19 |
8 16 18
|
3eqtr3d |
|- ( ( ( G e. GrpOp /\ A e. X ) /\ C e. X ) -> ( ( ( inv ` G ) ` C ) G ( C G A ) ) = A ) |
20 |
19
|
adantrl |
|- ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) -> ( ( ( inv ` G ) ` C ) G ( C G A ) ) = A ) |
21 |
20
|
adantr |
|- ( ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) /\ ( C G A ) = ( C G B ) ) -> ( ( ( inv ` G ) ` C ) G ( C G A ) ) = A ) |
22 |
6
|
adantrl |
|- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> ( ( ( inv ` G ) ` C ) G C ) = ( GId ` G ) ) |
23 |
22
|
oveq1d |
|- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> ( ( ( ( inv ` G ) ` C ) G C ) G B ) = ( ( GId ` G ) G B ) ) |
24 |
9
|
adantrl |
|- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> ( ( inv ` G ) ` C ) e. X ) |
25 |
|
simprr |
|- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> C e. X ) |
26 |
|
simprl |
|- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> B e. X ) |
27 |
24 25 26
|
3jca |
|- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> ( ( ( inv ` G ) ` C ) e. X /\ C e. X /\ B e. X ) ) |
28 |
1
|
grpoass |
|- ( ( G e. GrpOp /\ ( ( ( inv ` G ) ` C ) e. X /\ C e. X /\ B e. X ) ) -> ( ( ( ( inv ` G ) ` C ) G C ) G B ) = ( ( ( inv ` G ) ` C ) G ( C G B ) ) ) |
29 |
27 28
|
syldan |
|- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> ( ( ( ( inv ` G ) ` C ) G C ) G B ) = ( ( ( inv ` G ) ` C ) G ( C G B ) ) ) |
30 |
1 4
|
grpolid |
|- ( ( G e. GrpOp /\ B e. X ) -> ( ( GId ` G ) G B ) = B ) |
31 |
30
|
adantrr |
|- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> ( ( GId ` G ) G B ) = B ) |
32 |
23 29 31
|
3eqtr3d |
|- ( ( G e. GrpOp /\ ( B e. X /\ C e. X ) ) -> ( ( ( inv ` G ) ` C ) G ( C G B ) ) = B ) |
33 |
32
|
adantlr |
|- ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) -> ( ( ( inv ` G ) ` C ) G ( C G B ) ) = B ) |
34 |
33
|
adantr |
|- ( ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) /\ ( C G A ) = ( C G B ) ) -> ( ( ( inv ` G ) ` C ) G ( C G B ) ) = B ) |
35 |
3 21 34
|
3eqtr3d |
|- ( ( ( ( G e. GrpOp /\ A e. X ) /\ ( B e. X /\ C e. X ) ) /\ ( C G A ) = ( C G B ) ) -> A = B ) |
36 |
35
|
exp53 |
|- ( G e. GrpOp -> ( A e. X -> ( B e. X -> ( C e. X -> ( ( C G A ) = ( C G B ) -> A = B ) ) ) ) ) |
37 |
36
|
3imp2 |
|- ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( C G A ) = ( C G B ) -> A = B ) ) |
38 |
|
oveq2 |
|- ( A = B -> ( C G A ) = ( C G B ) ) |
39 |
37 38
|
impbid1 |
|- ( ( G e. GrpOp /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( C G A ) = ( C G B ) <-> A = B ) ) |