Description: Left cancellation law for groups. (Contributed by NM, 27-Oct-2006) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypothesis | grplcan.1 | |
|
Assertion | grpolcan | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grplcan.1 | |
|
2 | oveq2 | |
|
3 | 2 | adantl | |
4 | eqid | |
|
5 | eqid | |
|
6 | 1 4 5 | grpolinv | |
7 | 6 | adantlr | |
8 | 7 | oveq1d | |
9 | 1 5 | grpoinvcl | |
10 | 9 | adantrl | |
11 | simprr | |
|
12 | simprl | |
|
13 | 10 11 12 | 3jca | |
14 | 1 | grpoass | |
15 | 13 14 | syldan | |
16 | 15 | anassrs | |
17 | 1 4 | grpolid | |
18 | 17 | adantr | |
19 | 8 16 18 | 3eqtr3d | |
20 | 19 | adantrl | |
21 | 20 | adantr | |
22 | 6 | adantrl | |
23 | 22 | oveq1d | |
24 | 9 | adantrl | |
25 | simprr | |
|
26 | simprl | |
|
27 | 24 25 26 | 3jca | |
28 | 1 | grpoass | |
29 | 27 28 | syldan | |
30 | 1 4 | grpolid | |
31 | 30 | adantrr | |
32 | 23 29 31 | 3eqtr3d | |
33 | 32 | adantlr | |
34 | 33 | adantr | |
35 | 3 21 34 | 3eqtr3d | |
36 | 35 | exp53 | |
37 | 36 | 3imp2 | |
38 | oveq2 | |
|
39 | 37 38 | impbid1 | |