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Theorem invghm 19619
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 2737 . . 3 (+g𝐺) = (+g𝐺)
3 ablgrp 19574 . . 3 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
4 invghm.m . . . . 5 𝐼 = (invg𝐺)
51, 4grpinvf 18804 . . . 4 (𝐺 ∈ Grp → 𝐼:𝐵𝐵)
63, 5syl 17 . . 3 (𝐺 ∈ Abel → 𝐼:𝐵𝐵)
71, 2, 4ablinvadd 19595 . . . 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 19024 . 2 (𝐺 ∈ Abel → 𝐼 ∈ (𝐺 GrpHom 𝐺))
10 ghmgrp1 19017 . . 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 18832 . . . . . . . 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 6851 . . . . . 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 18805 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (𝐼𝑥) ∈ 𝐵)
1911, 13, 18syl2anc 585 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼𝑥) ∈ 𝐵)
201, 4grpinvcl 18805 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → (𝐼𝑦) ∈ 𝐵)
2111, 12, 20syl2anc 585 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼𝑦) ∈ 𝐵)
221, 2, 2ghmlin 19020 . . . . . . 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 18821 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (𝐼‘(𝐼𝑥)) = 𝑥)
2511, 13, 24syl2anc 585 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼𝑥)) = 𝑥)
261, 4grpinvinv 18821 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → (𝐼‘(𝐼𝑦)) = 𝑦)
2711, 12, 26syl2anc 585 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼𝑦)) = 𝑦)
2825, 27oveq12d 7380 . . . . . 6 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → ((𝐼‘(𝐼𝑥))(+g𝐺)(𝐼‘(𝐼𝑦))) = (𝑥(+g𝐺)𝑦))
2916, 23, 283eqtrd 2781 . . . . 5 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼‘(𝑦(+g𝐺)𝑥))) = (𝑥(+g𝐺)𝑦))
301, 2grpcl 18763 . . . . . . 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 18821 . . . . . 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 2779 . . . 4 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
3534ralrimivva 3198 . . 3 (𝐼 ∈ (𝐺 GrpHom 𝐺) → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
361, 2isabl2 19579 . . 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 3065  wf 6497  cfv 6501  (class class class)co 7362  Basecbs 17090  +gcplusg 17140  Grpcgrp 18755  invgcminusg 18756   GrpHom cghm 19012  Abelcabl 19570
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-0g 17330  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-grp 18758  df-minusg 18759  df-ghm 19013  df-cmn 19571  df-abl 19572
This theorem is referenced by:  gsuminv  19730  invlmhm  20519  tsmsinv  23515
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