Theorem invghm 13459

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 2196	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		ablgrp 13419	. . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
4		invghm.m	. . . . 5 ⊢ 𝐼 = (invg‘𝐺)
5	1, 4	grpinvf 13179	. . . 4 ⊢ (𝐺 ∈ Grp → 𝐼:𝐵⟶𝐵)
6	3, 5	syl 14	. . 3 ⊢ (𝐺 ∈ Abel → 𝐼:𝐵⟶𝐵)
7	1, 2, 4	ablinvadd 13440	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝐼‘(𝑥(+g‘𝐺)𝑦)) = ((𝐼‘𝑥)(+g‘𝐺)(𝐼‘𝑦)))
8	7	3expb 1206	. . 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 13382	. 2 ⊢ (𝐺 ∈ Abel → 𝐼 ∈ (𝐺 GrpHom 𝐺))
10		ghmgrp1 13375	. . 3 ⊢ (𝐼 ∈ (𝐺 GrpHom 𝐺) → 𝐺 ∈ Grp)
11	10	adantr 276	. . . . . . . 8 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐺 ∈ Grp)
12		simprr 531	. . . . . . . 8 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
13		simprl 529	. . . . . . . 8 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
14	1, 2, 4	grpinvadd 13210	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐼‘(𝑦(+g‘𝐺)𝑥)) = ((𝐼‘𝑥)(+g‘𝐺)(𝐼‘𝑦)))
15	11, 12, 13, 14	syl3anc 1249	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝑦(+g‘𝐺)𝑥)) = ((𝐼‘𝑥)(+g‘𝐺)(𝐼‘𝑦)))
16	15	fveq2d 5562	. . . . . 6 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝐼‘(𝑦(+g‘𝐺)𝑥))) = (𝐼‘((𝐼‘𝑥)(+g‘𝐺)(𝐼‘𝑦))))
17		simpl 109	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐼 ∈ (𝐺 GrpHom 𝐺))
18	1, 4	grpinvcl 13180	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑥) ∈ 𝐵)
19	11, 13, 18	syl2anc 411	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘𝑥) ∈ 𝐵)
20	1, 4	grpinvcl 13180	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → (𝐼‘𝑦) ∈ 𝐵)
21	11, 12, 20	syl2anc 411	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘𝑦) ∈ 𝐵)
22	1, 2, 2	ghmlin 13378	. . . . . . 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 1249	. . . . . 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 13199	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝐼‘(𝐼‘𝑥)) = 𝑥)
25	11, 13, 24	syl2anc 411	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝐼‘𝑥)) = 𝑥)
26	1, 4	grpinvinv 13199	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → (𝐼‘(𝐼‘𝑦)) = 𝑦)
27	11, 12, 26	syl2anc 411	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝐼‘𝑦)) = 𝑦)
28	25, 27	oveq12d 5940	. . . . . 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 2233	. . . . 5 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝐼‘(𝑦(+g‘𝐺)𝑥))) = (𝑥(+g‘𝐺)𝑦))
30	1, 2	grpcl 13140	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦(+g‘𝐺)𝑥) ∈ 𝐵)
31	11, 12, 13, 30	syl3anc 1249	. . . . . 6 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦(+g‘𝐺)𝑥) ∈ 𝐵)
32	1, 4	grpinvinv 13199	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑦(+g‘𝐺)𝑥) ∈ 𝐵) → (𝐼‘(𝐼‘(𝑦(+g‘𝐺)𝑥))) = (𝑦(+g‘𝐺)𝑥))
33	11, 31, 32	syl2anc 411	. . . . 5 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝐼‘(𝑦(+g‘𝐺)𝑥))) = (𝑦(+g‘𝐺)𝑥))
34	29, 33	eqtr3d 2231	. . . 4 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
35	34	ralrimivva 2579	. . 3 ⊢ (𝐼 ∈ (𝐺 GrpHom 𝐺) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
36	1, 2	isabl2 13424	. . 3 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
37	10, 35, 36	sylanbrc 417	. 2 ⊢ (𝐼 ∈ (𝐺 GrpHom 𝐺) → 𝐺 ∈ Abel)
38	9, 37	impbii 126	1 ⊢ (𝐺 ∈ Abel ↔ 𝐼 ∈ (𝐺 GrpHom 𝐺))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ⟶wf 5254  ‘cfv 5258  (class class class)co 5922  Basecbs 12678  +_gcplusg 12755  Grpcgrp 13132  inv_gcminusg 13133   GrpHom cghm 13370  Abelcabl 13415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-inn 8991  df-2 9049  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-0g 12929  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-ghm 13371  df-cmn 13416  df-abl 13417

This theorem is referenced by: (None)