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Theorem grpinvcnv 18815
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 = (invg𝐺)
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.
StepHypRef Expression
1 eqid 2736 . . . 4 (𝑥𝐵 ↦ (𝑁𝑥)) = (𝑥𝐵 ↦ (𝑁𝑥))
2 grpinvinv.b . . . . 5 𝐵 = (Base‘𝐺)
3 grpinvinv.n . . . . 5 𝑁 = (invg𝐺)
42, 3grpinvcl 18798 . . . 4 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (𝑁𝑥) ∈ 𝐵)
52, 3grpinvcl 18798 . . . 4 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → (𝑁𝑦) ∈ 𝐵)
6 eqid 2736 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
7 eqid 2736 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
82, 6, 7, 3grpinvid1 18802 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑦𝐵𝑥𝐵) → ((𝑁𝑦) = 𝑥 ↔ (𝑦(+g𝐺)𝑥) = (0g𝐺)))
983com23 1126 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → ((𝑁𝑦) = 𝑥 ↔ (𝑦(+g𝐺)𝑥) = (0g𝐺)))
102, 6, 7, 3grpinvid2 18803 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → ((𝑁𝑥) = 𝑦 ↔ (𝑦(+g𝐺)𝑥) = (0g𝐺)))
119, 10bitr4d 281 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → ((𝑁𝑦) = 𝑥 ↔ (𝑁𝑥) = 𝑦))
12113expb 1120 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((𝑁𝑦) = 𝑥 ↔ (𝑁𝑥) = 𝑦))
13 eqcom 2743 . . . . 5 (𝑥 = (𝑁𝑦) ↔ (𝑁𝑦) = 𝑥)
14 eqcom 2743 . . . . 5 (𝑦 = (𝑁𝑥) ↔ (𝑁𝑥) = 𝑦)
1512, 13, 143bitr4g 313 . . . 4 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 = (𝑁𝑦) ↔ 𝑦 = (𝑁𝑥)))
161, 4, 5, 15f1ocnv2d 7606 . . 3 (𝐺 ∈ Grp → ((𝑥𝐵 ↦ (𝑁𝑥)):𝐵1-1-onto𝐵(𝑥𝐵 ↦ (𝑁𝑥)) = (𝑦𝐵 ↦ (𝑁𝑦))))
1716simprd 496 . 2 (𝐺 ∈ Grp → (𝑥𝐵 ↦ (𝑁𝑥)) = (𝑦𝐵 ↦ (𝑁𝑦)))
182, 3grpinvf 18797 . . . 4 (𝐺 ∈ Grp → 𝑁:𝐵𝐵)
1918feqmptd 6910 . . 3 (𝐺 ∈ Grp → 𝑁 = (𝑥𝐵 ↦ (𝑁𝑥)))
2019cnveqd 5831 . 2 (𝐺 ∈ Grp → 𝑁 = (𝑥𝐵 ↦ (𝑁𝑥)))
2118feqmptd 6910 . 2 (𝐺 ∈ Grp → 𝑁 = (𝑦𝐵 ↦ (𝑁𝑦)))
2217, 20, 213eqtr4d 2786 1 (𝐺 ∈ Grp → 𝑁 = 𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  cmpt 5188  ccnv 5632  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  Basecbs 17083  +gcplusg 17133  0gc0g 17321  Grpcgrp 18748  invgcminusg 18749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752
This theorem is referenced by:  grpinvf1o  18817  grpinvhmeo  23437
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