Theorem invghm 19802

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 = (invg‘𝐺)

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.

Step	Hyp	Ref	Expression
1		invghm.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		eqid 2741	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		ablgrp 19754	. . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
4		invghm.m	. . . . 5 ⊢ 𝐼 = (invg‘𝐺)
5	1, 4	grpinvf 18957	. . . 4 ⊢ (𝐺 ∈ Grp → 𝐼:𝐵⟶𝐵)
6	3, 5	syl 17	. . 3 ⊢ (𝐺 ∈ Abel → 𝐼:𝐵⟶𝐵)
7	1, 2, 4	ablinvadd 19776	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝐼‘(𝑥(+g‘𝐺)𝑦)) = ((𝐼‘𝑥)(+g‘𝐺)(𝐼‘𝑦)))
8	7	3expb 1127	. . 3 ⊢ ((𝐺 ∈ Abel ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝑥(+g‘𝐺)𝑦)) = ((𝐼‘𝑥)(+g‘𝐺)(𝐼‘𝑦)))
9	1, 1, 2, 2, 3, 3, 6, 8	isghmd 19194	. 2 ⊢ (𝐺 ∈ Abel → 𝐼 ∈ (𝐺 GrpHom 𝐺))
10		ghmgrp1 19187	. . 3 ⊢ (𝐼 ∈ (𝐺 GrpHom 𝐺) → 𝐺 ∈ Grp)
11	10	adantr 482	. . . . . . . 8 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐺 ∈ Grp)
12		simprr 779	. . . . . . . 8 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
13		simprl 777	. . . . . . . 8 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
14	1, 2, 4	grpinvadd 18989	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐼‘(𝑦(+g‘𝐺)𝑥)) = ((𝐼‘𝑥)(+g‘𝐺)(𝐼‘𝑦)))
15	11, 12, 13, 14	syl3anc 1380	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝑦(+g‘𝐺)𝑥)) = ((𝐼‘𝑥)(+g‘𝐺)(𝐼‘𝑦)))
16	15	fveq2d 6834	. . . . . 6 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝐼‘(𝑦(+g‘𝐺)𝑥))) = (𝐼‘((𝐼‘𝑥)(+g‘𝐺)(𝐼‘𝑦))))
17		simpl 484	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐼 ∈ (𝐺 GrpHom 𝐺))
18	1, 4	grpinvcl 18958	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑥) ∈ 𝐵)
19	11, 13, 18	syl2anc 591	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘𝑥) ∈ 𝐵)
20	1, 4	grpinvcl 18958	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → (𝐼‘𝑦) ∈ 𝐵)
21	11, 12, 20	syl2anc 591	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘𝑦) ∈ 𝐵)
22	1, 2, 2	ghmlin 19190	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝐼‘𝑥) ∈ 𝐵 ∧ (𝐼‘𝑦) ∈ 𝐵) → (𝐼‘((𝐼‘𝑥)(+g‘𝐺)(𝐼‘𝑦))) = ((𝐼‘(𝐼‘𝑥))(+g‘𝐺)(𝐼‘(𝐼‘𝑦))))
23	17, 19, 21, 22	syl3anc 1380	. . . . . 6 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘((𝐼‘𝑥)(+g‘𝐺)(𝐼‘𝑦))) = ((𝐼‘(𝐼‘𝑥))(+g‘𝐺)(𝐼‘(𝐼‘𝑦))))
24	1, 4	grpinvinv 18976	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝐼‘(𝐼‘𝑥)) = 𝑥)
25	11, 13, 24	syl2anc 591	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝐼‘𝑥)) = 𝑥)
26	1, 4	grpinvinv 18976	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → (𝐼‘(𝐼‘𝑦)) = 𝑦)
27	11, 12, 26	syl2anc 591	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝐼‘𝑦)) = 𝑦)
28	25, 27	oveq12d 7377	. . . . . 6 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐼‘(𝐼‘𝑥))(+g‘𝐺)(𝐼‘(𝐼‘𝑦))) = (𝑥(+g‘𝐺)𝑦))
29	16, 23, 28	3eqtrd 2780	. . . . 5 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝐼‘(𝑦(+g‘𝐺)𝑥))) = (𝑥(+g‘𝐺)𝑦))
30	1, 2	grpcl 18912	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦(+g‘𝐺)𝑥) ∈ 𝐵)
31	11, 12, 13, 30	syl3anc 1380	. . . . . 6 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦(+g‘𝐺)𝑥) ∈ 𝐵)
32	1, 4	grpinvinv 18976	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑦(+g‘𝐺)𝑥) ∈ 𝐵) → (𝐼‘(𝐼‘(𝑦(+g‘𝐺)𝑥))) = (𝑦(+g‘𝐺)𝑥))
33	11, 31, 32	syl2anc 591	. . . . 5 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝐼‘(𝑦(+g‘𝐺)𝑥))) = (𝑦(+g‘𝐺)𝑥))
34	29, 33	eqtr3d 2778	. . . 4 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
35	34	ralrimivva 3184	. . 3 ⊢ (𝐼 ∈ (𝐺 GrpHom 𝐺) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
36	1, 2	isabl2 19759	. . 3 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
37	10, 35, 36	sylanbrc 590	. 2 ⊢ (𝐼 ∈ (𝐺 GrpHom 𝐺) → 𝐺 ∈ Abel)
38	9, 37	impbii 211	1 ⊢ (𝐺 ∈ Abel ↔ 𝐼 ∈ (𝐺 GrpHom 𝐺))

Syntax hints:   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055  ⟶wf 6484  ‘cfv 6488  (class class class)co 7359  Basecbs 17174  +_gcplusg 17215  Grpcgrp 18904  inv_gcminusg 18905   GrpHom cghm 19182  Abelcabl 19750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-fv 6496  df-riota 7316  df-ov 7362  df-oprab 7363  df-mpo 7364  df-1st 7933  df-2nd 7934  df-map 8769  df-0g 17399  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-grp 18907  df-minusg 18908  df-ghm 19183  df-cmn 19751  df-abl 19752

This theorem is referenced by:  gsuminv  19915  invlmhm  21035  tsmsinv  24134