Theorem grpinvcnv 18965

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 2725	. . . 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 18946	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝑁‘𝑥) ∈ 𝐵)
5	2, 3	grpinvcl 18946	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → (𝑁‘𝑦) ∈ 𝐵)
6		eqid 2725	. . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝐺)
7		eqid 2725	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
8	2, 6, 7, 3	grpinvid1 18950	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑁‘𝑦) = 𝑥 ↔ (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
9	8	3com23 1123	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑁‘𝑦) = 𝑥 ↔ (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
10	2, 6, 7, 3	grpinvid2 18951	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑁‘𝑥) = 𝑦 ↔ (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
11	9, 10	bitr4d 281	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑁‘𝑦) = 𝑥 ↔ (𝑁‘𝑥) = 𝑦))
12	11	3expb 1117	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑁‘𝑦) = 𝑥 ↔ (𝑁‘𝑥) = 𝑦))
13		eqcom 2732	. . . . 5 ⊢ (𝑥 = (𝑁‘𝑦) ↔ (𝑁‘𝑦) = 𝑥)
14		eqcom 2732	. . . . 5 ⊢ (𝑦 = (𝑁‘𝑥) ↔ (𝑁‘𝑥) = 𝑦)
15	12, 13, 14	3bitr4g 313	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = (𝑁‘𝑦) ↔ 𝑦 = (𝑁‘𝑥)))
16	1, 4, 5, 15	f1ocnv2d 7669	. . 3 ⊢ (𝐺 ∈ Grp → ((𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥)):𝐵–1-1-onto→𝐵 ∧ ◡(𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥)) = (𝑦 ∈ 𝐵 ↦ (𝑁‘𝑦))))
17	16	simprd 494	. 2 ⊢ (𝐺 ∈ Grp → ◡(𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥)) = (𝑦 ∈ 𝐵 ↦ (𝑁‘𝑦)))
18	2, 3	grpinvf 18945	. . . 4 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵)
19	18	feqmptd 6960	. . 3 ⊢ (𝐺 ∈ Grp → 𝑁 = (𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥)))
20	19	cnveqd 5870	. 2 ⊢ (𝐺 ∈ Grp → ◡𝑁 = ◡(𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥)))
21	18	feqmptd 6960	. 2 ⊢ (𝐺 ∈ Grp → 𝑁 = (𝑦 ∈ 𝐵 ↦ (𝑁‘𝑦)))
22	17, 20, 21	3eqtr4d 2775	1 ⊢ (𝐺 ∈ Grp → ◡𝑁 = 𝑁)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ↦ cmpt 5224  ^◡ccnv 5669  –1-1-onto→wf1o 6540  ‘cfv 6541  (class class class)co 7414  Basecbs 17177  +_gcplusg 17230  0_gc0g 17418  Grpcgrp 18892  inv_gcminusg 18893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5292  ax-nul 5299  ax-pow 5357  ax-pr 5421  ax-un 7736

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4317  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7417  df-0g 17420  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-grp 18895  df-minusg 18896

This theorem is referenced by:  grpinvf1o  18967  grpinvhmeo  24006