Theorem invghm 19851

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 2737	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		ablgrp 19803	. . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
4		invghm.m	. . . . 5 ⊢ 𝐼 = (invg‘𝐺)
5	1, 4	grpinvf 19004	. . . 4 ⊢ (𝐺 ∈ Grp → 𝐼:𝐵⟶𝐵)
6	3, 5	syl 17	. . 3 ⊢ (𝐺 ∈ Abel → 𝐼:𝐵⟶𝐵)
7	1, 2, 4	ablinvadd 19825	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝐼‘(𝑥(+g‘𝐺)𝑦)) = ((𝐼‘𝑥)(+g‘𝐺)(𝐼‘𝑦)))
8	7	3expb 1121	. . 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 19243	. 2 ⊢ (𝐺 ∈ Abel → 𝐼 ∈ (𝐺 GrpHom 𝐺))
10		ghmgrp1 19236	. . 3 ⊢ (𝐼 ∈ (𝐺 GrpHom 𝐺) → 𝐺 ∈ Grp)
11	10	adantr 480	. . . . . . . 8 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐺 ∈ Grp)
12		simprr 773	. . . . . . . 8 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
13		simprl 771	. . . . . . . 8 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
14	1, 2, 4	grpinvadd 19036	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐼‘(𝑦(+g‘𝐺)𝑥)) = ((𝐼‘𝑥)(+g‘𝐺)(𝐼‘𝑦)))
15	11, 12, 13, 14	syl3anc 1373	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝑦(+g‘𝐺)𝑥)) = ((𝐼‘𝑥)(+g‘𝐺)(𝐼‘𝑦)))
16	15	fveq2d 6910	. . . . . 6 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝐼‘(𝑦(+g‘𝐺)𝑥))) = (𝐼‘((𝐼‘𝑥)(+g‘𝐺)(𝐼‘𝑦))))
17		simpl 482	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐼 ∈ (𝐺 GrpHom 𝐺))
18	1, 4	grpinvcl 19005	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑥) ∈ 𝐵)
19	11, 13, 18	syl2anc 584	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘𝑥) ∈ 𝐵)
20	1, 4	grpinvcl 19005	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → (𝐼‘𝑦) ∈ 𝐵)
21	11, 12, 20	syl2anc 584	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘𝑦) ∈ 𝐵)
22	1, 2, 2	ghmlin 19239	. . . . . . 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 1373	. . . . . 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 19023	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝐼‘(𝐼‘𝑥)) = 𝑥)
25	11, 13, 24	syl2anc 584	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝐼‘𝑥)) = 𝑥)
26	1, 4	grpinvinv 19023	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → (𝐼‘(𝐼‘𝑦)) = 𝑦)
27	11, 12, 26	syl2anc 584	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝐼‘𝑦)) = 𝑦)
28	25, 27	oveq12d 7449	. . . . . 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 2781	. . . . 5 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝐼‘(𝑦(+g‘𝐺)𝑥))) = (𝑥(+g‘𝐺)𝑦))
30	1, 2	grpcl 18959	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦(+g‘𝐺)𝑥) ∈ 𝐵)
31	11, 12, 13, 30	syl3anc 1373	. . . . . 6 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦(+g‘𝐺)𝑥) ∈ 𝐵)
32	1, 4	grpinvinv 19023	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑦(+g‘𝐺)𝑥) ∈ 𝐵) → (𝐼‘(𝐼‘(𝑦(+g‘𝐺)𝑥))) = (𝑦(+g‘𝐺)𝑥))
33	11, 31, 32	syl2anc 584	. . . . 5 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝐼‘(𝑦(+g‘𝐺)𝑥))) = (𝑦(+g‘𝐺)𝑥))
34	29, 33	eqtr3d 2779	. . . 4 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
35	34	ralrimivva 3202	. . 3 ⊢ (𝐼 ∈ (𝐺 GrpHom 𝐺) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
36	1, 2	isabl2 19808	. . 3 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
37	10, 35, 36	sylanbrc 583	. 2 ⊢ (𝐼 ∈ (𝐺 GrpHom 𝐺) → 𝐺 ∈ Abel)
38	9, 37	impbii 209	1 ⊢ (𝐺 ∈ Abel ↔ 𝐼 ∈ (𝐺 GrpHom 𝐺))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  +_gcplusg 17297  Grpcgrp 18951  inv_gcminusg 18952   GrpHom cghm 19230  Abelcabl 19799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-ghm 19231  df-cmn 19800  df-abl 19801

This theorem is referenced by:  gsuminv  19964  invlmhm  21041  tsmsinv  24156