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Theorem invghm 18171
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 2621 . . 3 (+g𝐺) = (+g𝐺)
3 ablgrp 18130 . . 3 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
4 invghm.m . . . . 5 𝐼 = (invg𝐺)
51, 4grpinvf 17398 . . . 4 (𝐺 ∈ Grp → 𝐼:𝐵𝐵)
63, 5syl 17 . . 3 (𝐺 ∈ Abel → 𝐼:𝐵𝐵)
71, 2, 4ablinvadd 18147 . . . 4 ((𝐺 ∈ Abel ∧ 𝑥𝐵𝑦𝐵) → (𝐼‘(𝑥(+g𝐺)𝑦)) = ((𝐼𝑥)(+g𝐺)(𝐼𝑦)))
873expb 1263 . . 3 ((𝐺 ∈ Abel ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝑥(+g𝐺)𝑦)) = ((𝐼𝑥)(+g𝐺)(𝐼𝑦)))
91, 1, 2, 2, 3, 3, 6, 8isghmd 17601 . 2 (𝐺 ∈ Abel → 𝐼 ∈ (𝐺 GrpHom 𝐺))
10 ghmgrp1 17594 . . 3 (𝐼 ∈ (𝐺 GrpHom 𝐺) → 𝐺 ∈ Grp)
1110adantr 481 . . . . . . . 8 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → 𝐺 ∈ Grp)
12 simprr 795 . . . . . . . 8 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
13 simprl 793 . . . . . . . 8 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
141, 2, 4grpinvadd 17425 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑦𝐵𝑥𝐵) → (𝐼‘(𝑦(+g𝐺)𝑥)) = ((𝐼𝑥)(+g𝐺)(𝐼𝑦)))
1511, 12, 13, 14syl3anc 1323 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝑦(+g𝐺)𝑥)) = ((𝐼𝑥)(+g𝐺)(𝐼𝑦)))
1615fveq2d 6157 . . . . . 6 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼‘(𝑦(+g𝐺)𝑥))) = (𝐼‘((𝐼𝑥)(+g𝐺)(𝐼𝑦))))
17 simpl 473 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → 𝐼 ∈ (𝐺 GrpHom 𝐺))
181, 4grpinvcl 17399 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (𝐼𝑥) ∈ 𝐵)
1911, 13, 18syl2anc 692 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼𝑥) ∈ 𝐵)
201, 4grpinvcl 17399 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → (𝐼𝑦) ∈ 𝐵)
2111, 12, 20syl2anc 692 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼𝑦) ∈ 𝐵)
221, 2, 2ghmlin 17597 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝐼𝑥) ∈ 𝐵 ∧ (𝐼𝑦) ∈ 𝐵) → (𝐼‘((𝐼𝑥)(+g𝐺)(𝐼𝑦))) = ((𝐼‘(𝐼𝑥))(+g𝐺)(𝐼‘(𝐼𝑦))))
2317, 19, 21, 22syl3anc 1323 . . . . . 6 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘((𝐼𝑥)(+g𝐺)(𝐼𝑦))) = ((𝐼‘(𝐼𝑥))(+g𝐺)(𝐼‘(𝐼𝑦))))
241, 4grpinvinv 17414 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (𝐼‘(𝐼𝑥)) = 𝑥)
2511, 13, 24syl2anc 692 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼𝑥)) = 𝑥)
261, 4grpinvinv 17414 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → (𝐼‘(𝐼𝑦)) = 𝑦)
2711, 12, 26syl2anc 692 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼𝑦)) = 𝑦)
2825, 27oveq12d 6628 . . . . . 6 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → ((𝐼‘(𝐼𝑥))(+g𝐺)(𝐼‘(𝐼𝑦))) = (𝑥(+g𝐺)𝑦))
2916, 23, 283eqtrd 2659 . . . . 5 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼‘(𝑦(+g𝐺)𝑥))) = (𝑥(+g𝐺)𝑦))
301, 2grpcl 17362 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑦𝐵𝑥𝐵) → (𝑦(+g𝐺)𝑥) ∈ 𝐵)
3111, 12, 13, 30syl3anc 1323 . . . . . 6 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝑦(+g𝐺)𝑥) ∈ 𝐵)
321, 4grpinvinv 17414 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑦(+g𝐺)𝑥) ∈ 𝐵) → (𝐼‘(𝐼‘(𝑦(+g𝐺)𝑥))) = (𝑦(+g𝐺)𝑥))
3311, 31, 32syl2anc 692 . . . . 5 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼‘(𝑦(+g𝐺)𝑥))) = (𝑦(+g𝐺)𝑥))
3429, 33eqtr3d 2657 . . . 4 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
3534ralrimivva 2966 . . 3 (𝐼 ∈ (𝐺 GrpHom 𝐺) → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
361, 2isabl2 18133 . . 3 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
3710, 35, 36sylanbrc 697 . 2 (𝐼 ∈ (𝐺 GrpHom 𝐺) → 𝐺 ∈ Abel)
389, 37impbii 199 1 (𝐺 ∈ Abel ↔ 𝐼 ∈ (𝐺 GrpHom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wf 5848  cfv 5852  (class class class)co 6610  Basecbs 15792  +gcplusg 15873  Grpcgrp 17354  invgcminusg 17355   GrpHom cghm 17589  Abelcabl 18126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-0g 16034  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-grp 17357  df-minusg 17358  df-ghm 17590  df-cmn 18127  df-abl 18128
This theorem is referenced by:  gsuminv  18278  invlmhm  18974  tsmsinv  21874
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