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Theorem grpinvcnv 18167
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 2824 . . . 4 (𝑥𝐵 ↦ (𝑁𝑥)) = (𝑥𝐵 ↦ (𝑁𝑥))
2 grpinvinv.b . . . . 5 𝐵 = (Base‘𝐺)
3 grpinvinv.n . . . . 5 𝑁 = (invg𝐺)
42, 3grpinvcl 18151 . . . 4 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (𝑁𝑥) ∈ 𝐵)
52, 3grpinvcl 18151 . . . 4 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → (𝑁𝑦) ∈ 𝐵)
6 eqid 2824 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
7 eqid 2824 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
82, 6, 7, 3grpinvid1 18154 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑦𝐵𝑥𝐵) → ((𝑁𝑦) = 𝑥 ↔ (𝑦(+g𝐺)𝑥) = (0g𝐺)))
983com23 1123 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → ((𝑁𝑦) = 𝑥 ↔ (𝑦(+g𝐺)𝑥) = (0g𝐺)))
102, 6, 7, 3grpinvid2 18155 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → ((𝑁𝑥) = 𝑦 ↔ (𝑦(+g𝐺)𝑥) = (0g𝐺)))
119, 10bitr4d 285 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → ((𝑁𝑦) = 𝑥 ↔ (𝑁𝑥) = 𝑦))
12113expb 1117 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((𝑁𝑦) = 𝑥 ↔ (𝑁𝑥) = 𝑦))
13 eqcom 2831 . . . . 5 (𝑥 = (𝑁𝑦) ↔ (𝑁𝑦) = 𝑥)
14 eqcom 2831 . . . . 5 (𝑦 = (𝑁𝑥) ↔ (𝑁𝑥) = 𝑦)
1512, 13, 143bitr4g 317 . . . 4 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 = (𝑁𝑦) ↔ 𝑦 = (𝑁𝑥)))
161, 4, 5, 15f1ocnv2d 7392 . . 3 (𝐺 ∈ Grp → ((𝑥𝐵 ↦ (𝑁𝑥)):𝐵1-1-onto𝐵(𝑥𝐵 ↦ (𝑁𝑥)) = (𝑦𝐵 ↦ (𝑁𝑦))))
1716simprd 499 . 2 (𝐺 ∈ Grp → (𝑥𝐵 ↦ (𝑁𝑥)) = (𝑦𝐵 ↦ (𝑁𝑦)))
182, 3grpinvf 18150 . . . 4 (𝐺 ∈ Grp → 𝑁:𝐵𝐵)
1918feqmptd 6724 . . 3 (𝐺 ∈ Grp → 𝑁 = (𝑥𝐵 ↦ (𝑁𝑥)))
2019cnveqd 5733 . 2 (𝐺 ∈ Grp → 𝑁 = (𝑥𝐵 ↦ (𝑁𝑥)))
2118feqmptd 6724 . 2 (𝐺 ∈ Grp → 𝑁 = (𝑦𝐵 ↦ (𝑁𝑦)))
2217, 20, 213eqtr4d 2869 1 (𝐺 ∈ Grp → 𝑁 = 𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  cmpt 5132  ccnv 5541  1-1-ontowf1o 6342  cfv 6343  (class class class)co 7149  Basecbs 16483  +gcplusg 16565  0gc0g 16713  Grpcgrp 18103  invgcminusg 18104
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-0g 16715  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-grp 18106  df-minusg 18107
This theorem is referenced by:  grpinvf1o  18169  grpinvhmeo  22694
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