Description: The motions of a geometry form a group with respect to function composition, called the Isometry group. (Contributed by Thierry Arnoux, 15-Dec-2019)
Ref | Expression | ||
---|---|---|---|
Hypotheses | ismot.p | |
|
ismot.m | |
||
motgrp.1 | |
||
motgrp.i | |
||
Assertion | motgrp | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismot.p | |
|
2 | ismot.m | |
|
3 | motgrp.1 | |
|
4 | motgrp.i | |
|
5 | ovex | |
|
6 | 4 | grpbase | |
7 | 5 6 | mp1i | |
8 | eqidd | |
|
9 | 3 | 3ad2ant1 | |
10 | simp2 | |
|
11 | simp3 | |
|
12 | 1 2 9 4 10 11 | motplusg | |
13 | 1 2 9 10 11 | motco | |
14 | 12 13 | eqeltrd | |
15 | coass | |
|
16 | 12 | 3adant3r3 | |
17 | 16 | oveq1d | |
18 | 3 | adantr | |
19 | 13 | 3adant3r3 | |
20 | simpr3 | |
|
21 | 1 2 18 4 19 20 | motplusg | |
22 | 17 21 | eqtrd | |
23 | simpr2 | |
|
24 | 1 2 18 4 23 20 | motplusg | |
25 | 24 | oveq2d | |
26 | simpr1 | |
|
27 | 1 2 18 23 20 | motco | |
28 | 1 2 18 4 26 27 | motplusg | |
29 | 25 28 | eqtrd | |
30 | 15 22 29 | 3eqtr4a | |
31 | 1 2 3 | idmot | |
32 | 3 | adantr | |
33 | 31 | adantr | |
34 | simpr | |
|
35 | 1 2 32 4 33 34 | motplusg | |
36 | 1 2 | ismot | |
37 | 36 | simprbda | |
38 | 3 37 | sylan | |
39 | f1of | |
|
40 | fcoi2 | |
|
41 | 38 39 40 | 3syl | |
42 | 35 41 | eqtrd | |
43 | 1 2 32 34 | cnvmot | |
44 | 1 2 32 4 43 34 | motplusg | |
45 | f1ococnv1 | |
|
46 | 38 45 | syl | |
47 | 44 46 | eqtrd | |
48 | 7 8 14 30 31 42 43 47 | isgrpd | |