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Theorem invghm 19701
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 2733 . . 3 (+g𝐺) = (+g𝐺)
3 ablgrp 19653 . . 3 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
4 invghm.m . . . . 5 𝐼 = (invg𝐺)
51, 4grpinvf 18871 . . . 4 (𝐺 ∈ Grp → 𝐼:𝐵𝐵)
63, 5syl 17 . . 3 (𝐺 ∈ Abel → 𝐼:𝐵𝐵)
71, 2, 4ablinvadd 19675 . . . 4 ((𝐺 ∈ Abel ∧ 𝑥𝐵𝑦𝐵) → (𝐼‘(𝑥(+g𝐺)𝑦)) = ((𝐼𝑥)(+g𝐺)(𝐼𝑦)))
873expb 1121 . . 3 ((𝐺 ∈ Abel ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝑥(+g𝐺)𝑦)) = ((𝐼𝑥)(+g𝐺)(𝐼𝑦)))
91, 1, 2, 2, 3, 3, 6, 8isghmd 19101 . 2 (𝐺 ∈ Abel → 𝐼 ∈ (𝐺 GrpHom 𝐺))
10 ghmgrp1 19094 . . 3 (𝐼 ∈ (𝐺 GrpHom 𝐺) → 𝐺 ∈ Grp)
1110adantr 482 . . . . . . . 8 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → 𝐺 ∈ Grp)
12 simprr 772 . . . . . . . 8 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
13 simprl 770 . . . . . . . 8 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
141, 2, 4grpinvadd 18901 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑦𝐵𝑥𝐵) → (𝐼‘(𝑦(+g𝐺)𝑥)) = ((𝐼𝑥)(+g𝐺)(𝐼𝑦)))
1511, 12, 13, 14syl3anc 1372 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝑦(+g𝐺)𝑥)) = ((𝐼𝑥)(+g𝐺)(𝐼𝑦)))
1615fveq2d 6896 . . . . . 6 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼‘(𝑦(+g𝐺)𝑥))) = (𝐼‘((𝐼𝑥)(+g𝐺)(𝐼𝑦))))
17 simpl 484 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → 𝐼 ∈ (𝐺 GrpHom 𝐺))
181, 4grpinvcl 18872 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (𝐼𝑥) ∈ 𝐵)
1911, 13, 18syl2anc 585 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼𝑥) ∈ 𝐵)
201, 4grpinvcl 18872 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → (𝐼𝑦) ∈ 𝐵)
2111, 12, 20syl2anc 585 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼𝑦) ∈ 𝐵)
221, 2, 2ghmlin 19097 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝐼𝑥) ∈ 𝐵 ∧ (𝐼𝑦) ∈ 𝐵) → (𝐼‘((𝐼𝑥)(+g𝐺)(𝐼𝑦))) = ((𝐼‘(𝐼𝑥))(+g𝐺)(𝐼‘(𝐼𝑦))))
2317, 19, 21, 22syl3anc 1372 . . . . . 6 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘((𝐼𝑥)(+g𝐺)(𝐼𝑦))) = ((𝐼‘(𝐼𝑥))(+g𝐺)(𝐼‘(𝐼𝑦))))
241, 4grpinvinv 18890 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (𝐼‘(𝐼𝑥)) = 𝑥)
2511, 13, 24syl2anc 585 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼𝑥)) = 𝑥)
261, 4grpinvinv 18890 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → (𝐼‘(𝐼𝑦)) = 𝑦)
2711, 12, 26syl2anc 585 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼𝑦)) = 𝑦)
2825, 27oveq12d 7427 . . . . . 6 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → ((𝐼‘(𝐼𝑥))(+g𝐺)(𝐼‘(𝐼𝑦))) = (𝑥(+g𝐺)𝑦))
2916, 23, 283eqtrd 2777 . . . . 5 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼‘(𝑦(+g𝐺)𝑥))) = (𝑥(+g𝐺)𝑦))
301, 2grpcl 18827 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑦𝐵𝑥𝐵) → (𝑦(+g𝐺)𝑥) ∈ 𝐵)
3111, 12, 13, 30syl3anc 1372 . . . . . 6 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝑦(+g𝐺)𝑥) ∈ 𝐵)
321, 4grpinvinv 18890 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑦(+g𝐺)𝑥) ∈ 𝐵) → (𝐼‘(𝐼‘(𝑦(+g𝐺)𝑥))) = (𝑦(+g𝐺)𝑥))
3311, 31, 32syl2anc 585 . . . . 5 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼‘(𝑦(+g𝐺)𝑥))) = (𝑦(+g𝐺)𝑥))
3429, 33eqtr3d 2775 . . . 4 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
3534ralrimivva 3201 . . 3 (𝐼 ∈ (𝐺 GrpHom 𝐺) → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
361, 2isabl2 19658 . . 3 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
3710, 35, 36sylanbrc 584 . 2 (𝐼 ∈ (𝐺 GrpHom 𝐺) → 𝐺 ∈ Abel)
389, 37impbii 208 1 (𝐺 ∈ Abel ↔ 𝐼 ∈ (𝐺 GrpHom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  wf 6540  cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  Grpcgrp 18819  invgcminusg 18820   GrpHom cghm 19089  Abelcabl 19649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-ghm 19090  df-cmn 19650  df-abl 19651
This theorem is referenced by:  gsuminv  19814  invlmhm  20653  tsmsinv  23652
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