Theorem invghm 19770

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 2730	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		ablgrp 19722	. . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ Grp)
4		invghm.m	. . . . 5 ⊢ 𝐼 = (invg‘𝐺)
5	1, 4	grpinvf 18925	. . . 4 ⊢ (𝐺 ∈ Grp → 𝐼:𝐵⟶𝐵)
6	3, 5	syl 17	. . 3 ⊢ (𝐺 ∈ Abel → 𝐼:𝐵⟶𝐵)
7	1, 2, 4	ablinvadd 19744	. . . 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 19164	. 2 ⊢ (𝐺 ∈ Abel → 𝐼 ∈ (𝐺 GrpHom 𝐺))
10		ghmgrp1 19157	. . 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 18957	. . . . . . . 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 6865	. . . . . 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 18926	. . . . . . . 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 18926	. . . . . . . 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 19160	. . . . . . 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 18944	. . . . . . . 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 18944	. . . . . . . 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 7408	. . . . . 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 2769	. . . . 5 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝐼‘(𝐼‘(𝑦(+g‘𝐺)𝑥))) = (𝑥(+g‘𝐺)𝑦))
30	1, 2	grpcl 18880	. . . . . . 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 18944	. . . . . 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 2767	. . . 4 ⊢ ((𝐼 ∈ (𝐺 GrpHom 𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
35	34	ralrimivva 3181	. . 3 ⊢ (𝐼 ∈ (𝐺 GrpHom 𝐺) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
36	1, 2	isabl2 19727	. . 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 2109  ∀wral 3045  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  +_gcplusg 17227  Grpcgrp 18872  inv_gcminusg 18873   GrpHom cghm 19151  Abelcabl 19718

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-map 8804  df-0g 17411  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-grp 18875  df-minusg 18876  df-ghm 19152  df-cmn 19719  df-abl 19720

This theorem is referenced by:  gsuminv  19883  invlmhm  20956  tsmsinv  24042