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Theorem grpinvcnv 18936
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 2726 . . . 4 (𝑥𝐵 ↦ (𝑁𝑥)) = (𝑥𝐵 ↦ (𝑁𝑥))
2 grpinvinv.b . . . . 5 𝐵 = (Base‘𝐺)
3 grpinvinv.n . . . . 5 𝑁 = (invg𝐺)
42, 3grpinvcl 18917 . . . 4 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (𝑁𝑥) ∈ 𝐵)
52, 3grpinvcl 18917 . . . 4 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → (𝑁𝑦) ∈ 𝐵)
6 eqid 2726 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
7 eqid 2726 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
82, 6, 7, 3grpinvid1 18921 . . . . . . . 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 18922 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → ((𝑁𝑥) = 𝑦 ↔ (𝑦(+g𝐺)𝑥) = (0g𝐺)))
119, 10bitr4d 282 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → ((𝑁𝑦) = 𝑥 ↔ (𝑁𝑥) = 𝑦))
12113expb 1117 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((𝑁𝑦) = 𝑥 ↔ (𝑁𝑥) = 𝑦))
13 eqcom 2733 . . . . 5 (𝑥 = (𝑁𝑦) ↔ (𝑁𝑦) = 𝑥)
14 eqcom 2733 . . . . 5 (𝑦 = (𝑁𝑥) ↔ (𝑁𝑥) = 𝑦)
1512, 13, 143bitr4g 314 . . . 4 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 = (𝑁𝑦) ↔ 𝑦 = (𝑁𝑥)))
161, 4, 5, 15f1ocnv2d 7656 . . 3 (𝐺 ∈ Grp → ((𝑥𝐵 ↦ (𝑁𝑥)):𝐵1-1-onto𝐵(𝑥𝐵 ↦ (𝑁𝑥)) = (𝑦𝐵 ↦ (𝑁𝑦))))
1716simprd 495 . 2 (𝐺 ∈ Grp → (𝑥𝐵 ↦ (𝑁𝑥)) = (𝑦𝐵 ↦ (𝑁𝑦)))
182, 3grpinvf 18916 . . . 4 (𝐺 ∈ Grp → 𝑁:𝐵𝐵)
1918feqmptd 6954 . . 3 (𝐺 ∈ Grp → 𝑁 = (𝑥𝐵 ↦ (𝑁𝑥)))
2019cnveqd 5869 . 2 (𝐺 ∈ Grp → 𝑁 = (𝑥𝐵 ↦ (𝑁𝑥)))
2118feqmptd 6954 . 2 (𝐺 ∈ Grp → 𝑁 = (𝑦𝐵 ↦ (𝑁𝑦)))
2217, 20, 213eqtr4d 2776 1 (𝐺 ∈ Grp → 𝑁 = 𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  cmpt 5224  ccnv 5668  1-1-ontowf1o 6536  cfv 6537  (class class class)co 7405  Basecbs 17153  +gcplusg 17206  0gc0g 17394  Grpcgrp 18863  invgcminusg 18864
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-grp 18866  df-minusg 18867
This theorem is referenced by:  grpinvf1o  18938  grpinvhmeo  23945
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