Theorem grpinvcnv 17870

Description: The group inverse is its own inverse function. (Contributed by Mario Carneiro, 14-Aug-2015.)

Hypotheses

Ref	Expression
grpinvinv.b	⊢ 𝐵 = (Base‘𝐺)
grpinvinv.n	⊢ 𝑁 = (invg‘𝐺)

Assertion

Ref	Expression
grpinvcnv	⊢ (𝐺 ∈ Grp → ◡𝑁 = 𝑁)

Proof of Theorem grpinvcnv

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2777	. . . 4 ⊢ (𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥))
2		grpinvinv.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
3		grpinvinv.n	. . . . 5 ⊢ 𝑁 = (invg‘𝐺)
4	2, 3	grpinvcl 17854	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝑁‘𝑥) ∈ 𝐵)
5	2, 3	grpinvcl 17854	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → (𝑁‘𝑦) ∈ 𝐵)
6		eqid 2777	. . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝐺)
7		eqid 2777	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
8	2, 6, 7, 3	grpinvid1 17857	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑁‘𝑦) = 𝑥 ↔ (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
9	8	3com23 1117	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑁‘𝑦) = 𝑥 ↔ (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
10	2, 6, 7, 3	grpinvid2 17858	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑁‘𝑥) = 𝑦 ↔ (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
11	9, 10	bitr4d 274	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑁‘𝑦) = 𝑥 ↔ (𝑁‘𝑥) = 𝑦))
12	11	3expb 1110	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑁‘𝑦) = 𝑥 ↔ (𝑁‘𝑥) = 𝑦))
13		eqcom 2784	. . . . 5 ⊢ (𝑥 = (𝑁‘𝑦) ↔ (𝑁‘𝑦) = 𝑥)
14		eqcom 2784	. . . . 5 ⊢ (𝑦 = (𝑁‘𝑥) ↔ (𝑁‘𝑥) = 𝑦)
15	12, 13, 14	3bitr4g 306	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = (𝑁‘𝑦) ↔ 𝑦 = (𝑁‘𝑥)))
16	1, 4, 5, 15	f1ocnv2d 7163	. . 3 ⊢ (𝐺 ∈ Grp → ((𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥)):𝐵–1-1-onto→𝐵 ∧ ◡(𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥)) = (𝑦 ∈ 𝐵 ↦ (𝑁‘𝑦))))
17	16	simprd 491	. 2 ⊢ (𝐺 ∈ Grp → ◡(𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥)) = (𝑦 ∈ 𝐵 ↦ (𝑁‘𝑦)))
18	2, 3	grpinvf 17853	. . . 4 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵)
19	18	feqmptd 6509	. . 3 ⊢ (𝐺 ∈ Grp → 𝑁 = (𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥)))
20	19	cnveqd 5543	. 2 ⊢ (𝐺 ∈ Grp → ◡𝑁 = ◡(𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥)))
21	18	feqmptd 6509	. 2 ⊢ (𝐺 ∈ Grp → 𝑁 = (𝑦 ∈ 𝐵 ↦ (𝑁‘𝑦)))
22	17, 20, 21	3eqtr4d 2823	1 ⊢ (𝐺 ∈ Grp → ◡𝑁 = 𝑁)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2106   ↦ cmpt 4965  ^◡ccnv 5354  –1-1-onto→wf1o 6134  ‘cfv 6135  (class class class)co 6922  Basecbs 16255  +_gcplusg 16338  0_gc0g 16486  Grpcgrp 17809  inv_gcminusg 17810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-0g 16488  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-grp 17812  df-minusg 17813

This theorem is referenced by:  grpinvf1o  17872  grpinvhmeo  22298