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Theorem invghm 18440
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 𝐵 = (Base‘𝐺)
invghm.m 𝐼 = (invg𝐺)
Assertion
Ref Expression
invghm (𝐺 ∈ Abel ↔ 𝐼 ∈ (𝐺 GrpHom 𝐺))

Proof of Theorem invghm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 invghm.b . . 3 𝐵 = (Base‘𝐺)
2 eqid 2806 . . 3 (+g𝐺) = (+g𝐺)
3 ablgrp 18399 . . 3 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
4 invghm.m . . . . 5 𝐼 = (invg𝐺)
51, 4grpinvf 17671 . . . 4 (𝐺 ∈ Grp → 𝐼:𝐵𝐵)
63, 5syl 17 . . 3 (𝐺 ∈ Abel → 𝐼:𝐵𝐵)
71, 2, 4ablinvadd 18416 . . . 4 ((𝐺 ∈ Abel ∧ 𝑥𝐵𝑦𝐵) → (𝐼‘(𝑥(+g𝐺)𝑦)) = ((𝐼𝑥)(+g𝐺)(𝐼𝑦)))
873expb 1142 . . 3 ((𝐺 ∈ Abel ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝑥(+g𝐺)𝑦)) = ((𝐼𝑥)(+g𝐺)(𝐼𝑦)))
91, 1, 2, 2, 3, 3, 6, 8isghmd 17871 . 2 (𝐺 ∈ Abel → 𝐼 ∈ (𝐺 GrpHom 𝐺))
10 ghmgrp1 17864 . . 3 (𝐼 ∈ (𝐺 GrpHom 𝐺) → 𝐺 ∈ Grp)
1110adantr 468 . . . . . . . 8 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → 𝐺 ∈ Grp)
12 simprr 780 . . . . . . . 8 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
13 simprl 778 . . . . . . . 8 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
141, 2, 4grpinvadd 17698 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑦𝐵𝑥𝐵) → (𝐼‘(𝑦(+g𝐺)𝑥)) = ((𝐼𝑥)(+g𝐺)(𝐼𝑦)))
1511, 12, 13, 14syl3anc 1483 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝑦(+g𝐺)𝑥)) = ((𝐼𝑥)(+g𝐺)(𝐼𝑦)))
1615fveq2d 6412 . . . . . 6 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼‘(𝑦(+g𝐺)𝑥))) = (𝐼‘((𝐼𝑥)(+g𝐺)(𝐼𝑦))))
17 simpl 470 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → 𝐼 ∈ (𝐺 GrpHom 𝐺))
181, 4grpinvcl 17672 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (𝐼𝑥) ∈ 𝐵)
1911, 13, 18syl2anc 575 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼𝑥) ∈ 𝐵)
201, 4grpinvcl 17672 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → (𝐼𝑦) ∈ 𝐵)
2111, 12, 20syl2anc 575 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼𝑦) ∈ 𝐵)
221, 2, 2ghmlin 17867 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝐼𝑥) ∈ 𝐵 ∧ (𝐼𝑦) ∈ 𝐵) → (𝐼‘((𝐼𝑥)(+g𝐺)(𝐼𝑦))) = ((𝐼‘(𝐼𝑥))(+g𝐺)(𝐼‘(𝐼𝑦))))
2317, 19, 21, 22syl3anc 1483 . . . . . 6 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘((𝐼𝑥)(+g𝐺)(𝐼𝑦))) = ((𝐼‘(𝐼𝑥))(+g𝐺)(𝐼‘(𝐼𝑦))))
241, 4grpinvinv 17687 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (𝐼‘(𝐼𝑥)) = 𝑥)
2511, 13, 24syl2anc 575 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼𝑥)) = 𝑥)
261, 4grpinvinv 17687 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → (𝐼‘(𝐼𝑦)) = 𝑦)
2711, 12, 26syl2anc 575 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼𝑦)) = 𝑦)
2825, 27oveq12d 6892 . . . . . 6 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → ((𝐼‘(𝐼𝑥))(+g𝐺)(𝐼‘(𝐼𝑦))) = (𝑥(+g𝐺)𝑦))
2916, 23, 283eqtrd 2844 . . . . 5 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼‘(𝑦(+g𝐺)𝑥))) = (𝑥(+g𝐺)𝑦))
301, 2grpcl 17635 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑦𝐵𝑥𝐵) → (𝑦(+g𝐺)𝑥) ∈ 𝐵)
3111, 12, 13, 30syl3anc 1483 . . . . . 6 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝑦(+g𝐺)𝑥) ∈ 𝐵)
321, 4grpinvinv 17687 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑦(+g𝐺)𝑥) ∈ 𝐵) → (𝐼‘(𝐼‘(𝑦(+g𝐺)𝑥))) = (𝑦(+g𝐺)𝑥))
3311, 31, 32syl2anc 575 . . . . 5 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼‘(𝑦(+g𝐺)𝑥))) = (𝑦(+g𝐺)𝑥))
3429, 33eqtr3d 2842 . . . 4 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
3534ralrimivva 3159 . . 3 (𝐼 ∈ (𝐺 GrpHom 𝐺) → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
361, 2isabl2 18402 . . 3 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
3710, 35, 36sylanbrc 574 . 2 (𝐼 ∈ (𝐺 GrpHom 𝐺) → 𝐺 ∈ Abel)
389, 37impbii 200 1 (𝐺 ∈ Abel ↔ 𝐼 ∈ (𝐺 GrpHom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1637  wcel 2156  wral 3096  wf 6097  cfv 6101  (class class class)co 6874  Basecbs 16068  +gcplusg 16153  Grpcgrp 17627  invgcminusg 17628   GrpHom cghm 17859  Abelcabl 18395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-0g 16307  df-mgm 17447  df-sgrp 17489  df-mnd 17500  df-grp 17630  df-minusg 17631  df-ghm 17860  df-cmn 18396  df-abl 18397
This theorem is referenced by:  gsuminv  18547  invlmhm  19249  tsmsinv  22164
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