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Mirrors > Home > MPE Home > Th. List > ghminv | Structured version Visualization version Unicode version |
Description: A homomorphism of groups preserves inverses. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
Ref | Expression |
---|---|
ghminv.b | |
ghminv.y | |
ghminv.z |
Ref | Expression |
---|---|
ghminv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ghmgrp1 17662 | . . . . . 6 | |
2 | ghminv.b | . . . . . . 7 | |
3 | eqid 2622 | . . . . . . 7 | |
4 | eqid 2622 | . . . . . . 7 | |
5 | ghminv.y | . . . . . . 7 | |
6 | 2, 3, 4, 5 | grprinv 17469 | . . . . . 6 |
7 | 1, 6 | sylan 488 | . . . . 5 |
8 | 7 | fveq2d 6195 | . . . 4 |
9 | 2, 5 | grpinvcl 17467 | . . . . . 6 |
10 | 1, 9 | sylan 488 | . . . . 5 |
11 | eqid 2622 | . . . . . 6 | |
12 | 2, 3, 11 | ghmlin 17665 | . . . . 5 |
13 | 10, 12 | mpd3an3 1425 | . . . 4 |
14 | eqid 2622 | . . . . . 6 | |
15 | 4, 14 | ghmid 17666 | . . . . 5 |
16 | 15 | adantr 481 | . . . 4 |
17 | 8, 13, 16 | 3eqtr3d 2664 | . . 3 |
18 | ghmgrp2 17663 | . . . . 5 | |
19 | 18 | adantr 481 | . . . 4 |
20 | eqid 2622 | . . . . . 6 | |
21 | 2, 20 | ghmf 17664 | . . . . 5 |
22 | 21 | ffvelrnda 6359 | . . . 4 |
23 | 21 | adantr 481 | . . . . 5 |
24 | 23, 10 | ffvelrnd 6360 | . . . 4 |
25 | ghminv.z | . . . . 5 | |
26 | 20, 11, 14, 25 | grpinvid1 17470 | . . . 4 |
27 | 19, 22, 24, 26 | syl3anc 1326 | . . 3 |
28 | 17, 27 | mpbird 247 | . 2 |
29 | 28 | eqcomd 2628 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wf 5884 cfv 5888 (class class class)co 6650 cbs 15857 cplusg 15941 c0g 16100 cgrp 17422 cminusg 17423 cghm 17657 |
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 |
This theorem is referenced by: ghmsub 17668 ghmmulg 17672 ghmrn 17673 ghmpreima 17682 ghmeql 17683 frgpup3lem 18190 asclinvg 19341 mplind 19502 psgninv 19928 zrhpsgnodpm 19938 cpmatinvcl 20522 sum2dchr 24999 |
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