Theorem invghm 19875

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 2740	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		ablgrp 19827	. . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
4		invghm.m	. . . . 5 ⊢ 𝐼 = (invg‘𝐺)
5	1, 4	grpinvf 19026	. . . 4 ⊢ (𝐺 ∈ Grp → 𝐼:𝐵⟶𝐵)
6	3, 5	syl 17	. . 3 ⊢ (𝐺 ∈ Abel → 𝐼:𝐵⟶𝐵)
7	1, 2, 4	ablinvadd 19849	. . . 4 ⊢ ((𝐺 ∈ Abel ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝐼‘(𝑥(+g‘𝐺)𝑦)) = ((𝐼‘𝑥)(+g‘𝐺)(𝐼‘𝑦)))
8	7	3expb 1120	. . 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 19265	. 2 ⊢ (𝐺 ∈ Abel → 𝐼 ∈ (𝐺 GrpHom 𝐺))
10		ghmgrp1 19258	. . 3 ⊢ (𝐼 ∈ (𝐺 GrpHom 𝐺) → 𝐺 ∈ Grp)
11	10	adantr 480	. . . . . . . 8 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐺 ∈ Grp)
12		simprr 772	. . . . . . . 8 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
13		simprl 770	. . . . . . . 8 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
14	1, 2, 4	grpinvadd 19058	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐼‘(𝑦(+g‘𝐺)𝑥)) = ((𝐼‘𝑥)(+g‘𝐺)(𝐼‘𝑦)))
15	11, 12, 13, 14	syl3anc 1371	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝑦(+g‘𝐺)𝑥)) = ((𝐼‘𝑥)(+g‘𝐺)(𝐼‘𝑦)))
16	15	fveq2d 6924	. . . . . 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 19027	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝐼‘𝑥) ∈ 𝐵)
19	11, 13, 18	syl2anc 583	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘𝑥) ∈ 𝐵)
20	1, 4	grpinvcl 19027	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → (𝐼‘𝑦) ∈ 𝐵)
21	11, 12, 20	syl2anc 583	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘𝑦) ∈ 𝐵)
22	1, 2, 2	ghmlin 19261	. . . . . . 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 1371	. . . . . 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 19045	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝐼‘(𝐼‘𝑥)) = 𝑥)
25	11, 13, 24	syl2anc 583	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝐼‘𝑥)) = 𝑥)
26	1, 4	grpinvinv 19045	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → (𝐼‘(𝐼‘𝑦)) = 𝑦)
27	11, 12, 26	syl2anc 583	. . . . . . 7 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝐼‘𝑦)) = 𝑦)
28	25, 27	oveq12d 7466	. . . . . 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 2784	. . . . 5 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝐼‘(𝑦(+g‘𝐺)𝑥))) = (𝑥(+g‘𝐺)𝑦))
30	1, 2	grpcl 18981	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦(+g‘𝐺)𝑥) ∈ 𝐵)
31	11, 12, 13, 30	syl3anc 1371	. . . . . 6 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦(+g‘𝐺)𝑥) ∈ 𝐵)
32	1, 4	grpinvinv 19045	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑦(+g‘𝐺)𝑥) ∈ 𝐵) → (𝐼‘(𝐼‘(𝑦(+g‘𝐺)𝑥))) = (𝑦(+g‘𝐺)𝑥))
33	11, 31, 32	syl2anc 583	. . . . 5 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝐼‘(𝑦(+g‘𝐺)𝑥))) = (𝑦(+g‘𝐺)𝑥))
34	29, 33	eqtr3d 2782	. . . 4 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
35	34	ralrimivva 3208	. . 3 ⊢ (𝐼 ∈ (𝐺 GrpHom 𝐺) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
36	1, 2	isabl2 19832	. . 3 ⊢ (𝐺 ∈ Abel ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))
37	10, 35, 36	sylanbrc 582	. 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 1537   ∈ wcel 2108  ∀wral 3067  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  +_gcplusg 17311  Grpcgrp 18973  inv_gcminusg 18974   GrpHom cghm 19252  Abelcabl 19823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-map 8886  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-minusg 18977  df-ghm 19253  df-cmn 19824  df-abl 19825

This theorem is referenced by:  gsuminv  19988  invlmhm  21064  tsmsinv  24177