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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 = (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 2777 . . . 4 (𝑥𝐵 ↦ (𝑁𝑥)) = (𝑥𝐵 ↦ (𝑁𝑥))
2 grpinvinv.b . . . . 5 𝐵 = (Base‘𝐺)
3 grpinvinv.n . . . . 5 𝑁 = (invg𝐺)
42, 3grpinvcl 17854 . . . 4 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (𝑁𝑥) ∈ 𝐵)
52, 3grpinvcl 17854 . . . 4 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → (𝑁𝑦) ∈ 𝐵)
6 eqid 2777 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
7 eqid 2777 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
82, 6, 7, 3grpinvid1 17857 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑦𝐵𝑥𝐵) → ((𝑁𝑦) = 𝑥 ↔ (𝑦(+g𝐺)𝑥) = (0g𝐺)))
983com23 1117 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → ((𝑁𝑦) = 𝑥 ↔ (𝑦(+g𝐺)𝑥) = (0g𝐺)))
102, 6, 7, 3grpinvid2 17858 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → ((𝑁𝑥) = 𝑦 ↔ (𝑦(+g𝐺)𝑥) = (0g𝐺)))
119, 10bitr4d 274 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → ((𝑁𝑦) = 𝑥 ↔ (𝑁𝑥) = 𝑦))
12113expb 1110 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((𝑁𝑦) = 𝑥 ↔ (𝑁𝑥) = 𝑦))
13 eqcom 2784 . . . . 5 (𝑥 = (𝑁𝑦) ↔ (𝑁𝑦) = 𝑥)
14 eqcom 2784 . . . . 5 (𝑦 = (𝑁𝑥) ↔ (𝑁𝑥) = 𝑦)
1512, 13, 143bitr4g 306 . . . 4 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 = (𝑁𝑦) ↔ 𝑦 = (𝑁𝑥)))
161, 4, 5, 15f1ocnv2d 7163 . . 3 (𝐺 ∈ Grp → ((𝑥𝐵 ↦ (𝑁𝑥)):𝐵1-1-onto𝐵(𝑥𝐵 ↦ (𝑁𝑥)) = (𝑦𝐵 ↦ (𝑁𝑦))))
1716simprd 491 . 2 (𝐺 ∈ Grp → (𝑥𝐵 ↦ (𝑁𝑥)) = (𝑦𝐵 ↦ (𝑁𝑦)))
182, 3grpinvf 17853 . . . 4 (𝐺 ∈ Grp → 𝑁:𝐵𝐵)
1918feqmptd 6509 . . 3 (𝐺 ∈ Grp → 𝑁 = (𝑥𝐵 ↦ (𝑁𝑥)))
2019cnveqd 5543 . 2 (𝐺 ∈ Grp → 𝑁 = (𝑥𝐵 ↦ (𝑁𝑥)))
2118feqmptd 6509 . 2 (𝐺 ∈ Grp → 𝑁 = (𝑦𝐵 ↦ (𝑁𝑦)))
2217, 20, 213eqtr4d 2823 1 (𝐺 ∈ Grp → 𝑁 = 𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wcel 2106  cmpt 4965  ccnv 5354  1-1-ontowf1o 6134  cfv 6135  (class class class)co 6922  Basecbs 16255  +gcplusg 16338  0gc0g 16486  Grpcgrp 17809  invgcminusg 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
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