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Theorem invghm 16715
 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.)
Hypotheses
Ref Expression
invghm.b
invghm.m
Assertion
Ref Expression
invghm

Proof of Theorem invghm
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 invghm.b . . 3
2 eqid 2467 . . 3
3 ablgrp 16676 . . 3
4 invghm.m . . . . 5
51, 4grpinvf 15966 . . . 4
63, 5syl 16 . . 3
71, 2, 4ablinvadd 16693 . . . 4
873expb 1197 . . 3
91, 1, 2, 2, 3, 3, 6, 8isghmd 16148 . 2
10 ghmgrp1 16141 . . 3
1110adantr 465 . . . . . . . 8
12 simprr 756 . . . . . . . 8
13 simprl 755 . . . . . . . 8
141, 2, 4grpinvadd 15988 . . . . . . . 8
1511, 12, 13, 14syl3anc 1228 . . . . . . 7
1615fveq2d 5876 . . . . . 6
17 simpl 457 . . . . . . 7
181, 4grpinvcl 15967 . . . . . . . 8
1911, 13, 18syl2anc 661 . . . . . . 7
201, 4grpinvcl 15967 . . . . . . . 8
2111, 12, 20syl2anc 661 . . . . . . 7
221, 2, 2ghmlin 16144 . . . . . . 7
2317, 19, 21, 22syl3anc 1228 . . . . . 6
241, 4grpinvinv 15977 . . . . . . . 8
2511, 13, 24syl2anc 661 . . . . . . 7
261, 4grpinvinv 15977 . . . . . . . 8
2711, 12, 26syl2anc 661 . . . . . . 7
2825, 27oveq12d 6313 . . . . . 6
2916, 23, 283eqtrd 2512 . . . . 5
301, 2grpcl 15935 . . . . . . 7
3111, 12, 13, 30syl3anc 1228 . . . . . 6
321, 4grpinvinv 15977 . . . . . 6
3311, 31, 32syl2anc 661 . . . . 5
3429, 33eqtr3d 2510 . . . 4
3534ralrimivva 2888 . . 3
361, 2isabl2 16679 . . 3
3710, 35, 36sylanbrc 664 . 2
389, 37impbii 188 1
 Colors of variables: wff setvar class Syntax hints:   wb 184   wa 369   wceq 1379   wcel 1767  wral 2817  wf 5590  cfv 5594  (class class class)co 6295  cbs 14507   cplusg 14572  cgrp 15925  cminusg 15926   cghm 16136  cabl 16672 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-ghm 16137  df-cmn 16673  df-abl 16674 This theorem is referenced by:  gsuminv  16844  gsuminvOLD  16846  invlmhm  17559  tsmsinv  20518
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