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Mirrors > Home > MPE Home > Th. List > invghm | Structured version Visualization version Unicode version |
Description: The inversion map is a group automorphism if and only if the group is abelian. (In general it is only a group homomorphism into the opposite group, but in an abelian group the opposite group coincides with the group itself.) (Contributed by Mario Carneiro, 4-May-2015.) |
Ref | Expression |
---|---|
invghm.b | |
invghm.m |
Ref | Expression |
---|---|
invghm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invghm.b | . . 3 | |
2 | eqid 2622 | . . 3 | |
3 | ablgrp 18198 | . . 3 | |
4 | invghm.m | . . . . 5 | |
5 | 1, 4 | grpinvf 17466 | . . . 4 |
6 | 3, 5 | syl 17 | . . 3 |
7 | 1, 2, 4 | ablinvadd 18215 | . . . 4 |
8 | 7 | 3expb 1266 | . . 3 |
9 | 1, 1, 2, 2, 3, 3, 6, 8 | isghmd 17669 | . 2 |
10 | ghmgrp1 17662 | . . 3 | |
11 | 10 | adantr 481 | . . . . . . . 8 |
12 | simprr 796 | . . . . . . . 8 | |
13 | simprl 794 | . . . . . . . 8 | |
14 | 1, 2, 4 | grpinvadd 17493 | . . . . . . . 8 |
15 | 11, 12, 13, 14 | syl3anc 1326 | . . . . . . 7 |
16 | 15 | fveq2d 6195 | . . . . . 6 |
17 | simpl 473 | . . . . . . 7 | |
18 | 1, 4 | grpinvcl 17467 | . . . . . . . 8 |
19 | 11, 13, 18 | syl2anc 693 | . . . . . . 7 |
20 | 1, 4 | grpinvcl 17467 | . . . . . . . 8 |
21 | 11, 12, 20 | syl2anc 693 | . . . . . . 7 |
22 | 1, 2, 2 | ghmlin 17665 | . . . . . . 7 |
23 | 17, 19, 21, 22 | syl3anc 1326 | . . . . . 6 |
24 | 1, 4 | grpinvinv 17482 | . . . . . . . 8 |
25 | 11, 13, 24 | syl2anc 693 | . . . . . . 7 |
26 | 1, 4 | grpinvinv 17482 | . . . . . . . 8 |
27 | 11, 12, 26 | syl2anc 693 | . . . . . . 7 |
28 | 25, 27 | oveq12d 6668 | . . . . . 6 |
29 | 16, 23, 28 | 3eqtrd 2660 | . . . . 5 |
30 | 1, 2 | grpcl 17430 | . . . . . . 7 |
31 | 11, 12, 13, 30 | syl3anc 1326 | . . . . . 6 |
32 | 1, 4 | grpinvinv 17482 | . . . . . 6 |
33 | 11, 31, 32 | syl2anc 693 | . . . . 5 |
34 | 29, 33 | eqtr3d 2658 | . . . 4 |
35 | 34 | ralrimivva 2971 | . . 3 |
36 | 1, 2 | isabl2 18201 | . . 3 |
37 | 10, 35, 36 | sylanbrc 698 | . 2 |
38 | 9, 37 | impbii 199 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 wf 5884 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 cgrp 17422 cminusg 17423 cghm 17657 cabl 18194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-ghm 17658 df-cmn 18195 df-abl 18196 |
This theorem is referenced by: gsuminv 18346 invlmhm 19042 tsmsinv 21951 |
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