Description: A subgroup of a group must have the same identity as the group. (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 30-Apr-2015)
Ref | Expression | ||
---|---|---|---|
Hypotheses | subg0.h | |
|
subg0.i | |
||
Assertion | subg0 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subg0.h | |
|
2 | subg0.i | |
|
3 | eqid | |
|
4 | 1 3 | ressplusg | |
5 | 4 | oveqd | |
6 | 1 | subggrp | |
7 | eqid | |
|
8 | eqid | |
|
9 | 7 8 | grpidcl | |
10 | 6 9 | syl | |
11 | eqid | |
|
12 | 7 11 8 | grplid | |
13 | 6 10 12 | syl2anc | |
14 | 5 13 | eqtrd | |
15 | subgrcl | |
|
16 | eqid | |
|
17 | 16 | subgss | |
18 | 1 | subgbas | |
19 | 10 18 | eleqtrrd | |
20 | 17 19 | sseldd | |
21 | 16 3 2 | grpid | |
22 | 15 20 21 | syl2anc | |
23 | 14 22 | mpbid | |