Theorem grpinvcnv 19046

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 2740	. . . 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 19027	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝑁‘𝑥) ∈ 𝐵)
5	2, 3	grpinvcl 19027	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵) → (𝑁‘𝑦) ∈ 𝐵)
6		eqid 2740	. . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝐺)
7		eqid 2740	. . . . . . . . 9 ⊢ (0g‘𝐺) = (0g‘𝐺)
8	2, 6, 7, 3	grpinvid1 19031	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑁‘𝑦) = 𝑥 ↔ (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
9	8	3com23 1126	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑁‘𝑦) = 𝑥 ↔ (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
10	2, 6, 7, 3	grpinvid2 19032	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑁‘𝑥) = 𝑦 ↔ (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
11	9, 10	bitr4d 282	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑁‘𝑦) = 𝑥 ↔ (𝑁‘𝑥) = 𝑦))
12	11	3expb 1120	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑁‘𝑦) = 𝑥 ↔ (𝑁‘𝑥) = 𝑦))
13		eqcom 2747	. . . . 5 ⊢ (𝑥 = (𝑁‘𝑦) ↔ (𝑁‘𝑦) = 𝑥)
14		eqcom 2747	. . . . 5 ⊢ (𝑦 = (𝑁‘𝑥) ↔ (𝑁‘𝑥) = 𝑦)
15	12, 13, 14	3bitr4g 314	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = (𝑁‘𝑦) ↔ 𝑦 = (𝑁‘𝑥)))
16	1, 4, 5, 15	f1ocnv2d 7703	. . 3 ⊢ (𝐺 ∈ Grp → ((𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥)):𝐵–1-1-onto→𝐵 ∧ ◡(𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥)) = (𝑦 ∈ 𝐵 ↦ (𝑁‘𝑦))))
17	16	simprd 495	. 2 ⊢ (𝐺 ∈ Grp → ◡(𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥)) = (𝑦 ∈ 𝐵 ↦ (𝑁‘𝑦)))
18	2, 3	grpinvf 19026	. . . 4 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵)
19	18	feqmptd 6990	. . 3 ⊢ (𝐺 ∈ Grp → 𝑁 = (𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥)))
20	19	cnveqd 5900	. 2 ⊢ (𝐺 ∈ Grp → ◡𝑁 = ◡(𝑥 ∈ 𝐵 ↦ (𝑁‘𝑥)))
21	18	feqmptd 6990	. 2 ⊢ (𝐺 ∈ Grp → 𝑁 = (𝑦 ∈ 𝐵 ↦ (𝑁‘𝑦)))
22	17, 20, 21	3eqtr4d 2790	1 ⊢ (𝐺 ∈ Grp → ◡𝑁 = 𝑁)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ↦ cmpt 5249  ^◡ccnv 5699  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  +_gcplusg 17311  0_gc0g 17499  Grpcgrp 18973  inv_gcminusg 18974

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-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-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-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-minusg 18977

This theorem is referenced by:  grpinvf1o  19049  grpinvhmeo  24115