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Theorem invghm 18933
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 2821 . . 3 (+g𝐺) = (+g𝐺)
3 ablgrp 18890 . . 3 (𝐺 ∈ Abel → 𝐺 ∈ Grp)
4 invghm.m . . . . 5 𝐼 = (invg𝐺)
51, 4grpinvf 18129 . . . 4 (𝐺 ∈ Grp → 𝐼:𝐵𝐵)
63, 5syl 17 . . 3 (𝐺 ∈ Abel → 𝐼:𝐵𝐵)
71, 2, 4ablinvadd 18909 . . . 4 ((𝐺 ∈ Abel ∧ 𝑥𝐵𝑦𝐵) → (𝐼‘(𝑥(+g𝐺)𝑦)) = ((𝐼𝑥)(+g𝐺)(𝐼𝑦)))
873expb 1117 . . 3 ((𝐺 ∈ Abel ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝑥(+g𝐺)𝑦)) = ((𝐼𝑥)(+g𝐺)(𝐼𝑦)))
91, 1, 2, 2, 3, 3, 6, 8isghmd 18346 . 2 (𝐺 ∈ Abel → 𝐼 ∈ (𝐺 GrpHom 𝐺))
10 ghmgrp1 18339 . . 3 (𝐼 ∈ (𝐺 GrpHom 𝐺) → 𝐺 ∈ Grp)
1110adantr 484 . . . . . . . 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 18156 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑦𝐵𝑥𝐵) → (𝐼‘(𝑦(+g𝐺)𝑥)) = ((𝐼𝑥)(+g𝐺)(𝐼𝑦)))
1511, 12, 13, 14syl3anc 1368 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝑦(+g𝐺)𝑥)) = ((𝐼𝑥)(+g𝐺)(𝐼𝑦)))
1615fveq2d 6647 . . . . . 6 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼‘(𝑦(+g𝐺)𝑥))) = (𝐼‘((𝐼𝑥)(+g𝐺)(𝐼𝑦))))
17 simpl 486 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → 𝐼 ∈ (𝐺 GrpHom 𝐺))
181, 4grpinvcl 18130 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (𝐼𝑥) ∈ 𝐵)
1911, 13, 18syl2anc 587 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼𝑥) ∈ 𝐵)
201, 4grpinvcl 18130 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → (𝐼𝑦) ∈ 𝐵)
2111, 12, 20syl2anc 587 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼𝑦) ∈ 𝐵)
221, 2, 2ghmlin 18342 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝐼𝑥) ∈ 𝐵 ∧ (𝐼𝑦) ∈ 𝐵) → (𝐼‘((𝐼𝑥)(+g𝐺)(𝐼𝑦))) = ((𝐼‘(𝐼𝑥))(+g𝐺)(𝐼‘(𝐼𝑦))))
2317, 19, 21, 22syl3anc 1368 . . . . . 6 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘((𝐼𝑥)(+g𝐺)(𝐼𝑦))) = ((𝐼‘(𝐼𝑥))(+g𝐺)(𝐼‘(𝐼𝑦))))
241, 4grpinvinv 18145 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (𝐼‘(𝐼𝑥)) = 𝑥)
2511, 13, 24syl2anc 587 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼𝑥)) = 𝑥)
261, 4grpinvinv 18145 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → (𝐼‘(𝐼𝑦)) = 𝑦)
2711, 12, 26syl2anc 587 . . . . . . 7 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼𝑦)) = 𝑦)
2825, 27oveq12d 7148 . . . . . 6 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → ((𝐼‘(𝐼𝑥))(+g𝐺)(𝐼‘(𝐼𝑦))) = (𝑥(+g𝐺)𝑦))
2916, 23, 283eqtrd 2860 . . . . 5 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼‘(𝑦(+g𝐺)𝑥))) = (𝑥(+g𝐺)𝑦))
301, 2grpcl 18090 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑦𝐵𝑥𝐵) → (𝑦(+g𝐺)𝑥) ∈ 𝐵)
3111, 12, 13, 30syl3anc 1368 . . . . . 6 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝑦(+g𝐺)𝑥) ∈ 𝐵)
321, 4grpinvinv 18145 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑦(+g𝐺)𝑥) ∈ 𝐵) → (𝐼‘(𝐼‘(𝑦(+g𝐺)𝑥))) = (𝑦(+g𝐺)𝑥))
3311, 31, 32syl2anc 587 . . . . 5 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝐼‘(𝐼‘(𝑦(+g𝐺)𝑥))) = (𝑦(+g𝐺)𝑥))
3429, 33eqtr3d 2858 . . . 4 ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
3534ralrimivva 3179 . . 3 (𝐼 ∈ (𝐺 GrpHom 𝐺) → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
361, 2isabl2 18894 . . 3 (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
3710, 35, 36sylanbrc 586 . 2 (𝐼 ∈ (𝐺 GrpHom 𝐺) → 𝐺 ∈ Abel)
389, 37impbii 212 1 (𝐺 ∈ Abel ↔ 𝐼 ∈ (𝐺 GrpHom 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3126  wf 6324  cfv 6328  (class class class)co 7130  Basecbs 16462  +gcplusg 16544  Grpcgrp 18082  invgcminusg 18083   GrpHom cghm 18334  Abelcabl 18886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-0g 16694  df-mgm 17831  df-sgrp 17880  df-mnd 17891  df-grp 18085  df-minusg 18086  df-ghm 18335  df-cmn 18887  df-abl 18888
This theorem is referenced by:  gsuminv  19045  invlmhm  19790  tsmsinv  22732
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