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Theorem grpinvcnv 17404
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 2621 . . . 4 (𝑥𝐵 ↦ (𝑁𝑥)) = (𝑥𝐵 ↦ (𝑁𝑥))
2 grpinvinv.b . . . . 5 𝐵 = (Base‘𝐺)
3 grpinvinv.n . . . . 5 𝑁 = (invg𝐺)
42, 3grpinvcl 17388 . . . 4 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (𝑁𝑥) ∈ 𝐵)
52, 3grpinvcl 17388 . . . 4 ((𝐺 ∈ Grp ∧ 𝑦𝐵) → (𝑁𝑦) ∈ 𝐵)
6 eqid 2621 . . . . . . . . 9 (+g𝐺) = (+g𝐺)
7 eqid 2621 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
82, 6, 7, 3grpinvid1 17391 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑦𝐵𝑥𝐵) → ((𝑁𝑦) = 𝑥 ↔ (𝑦(+g𝐺)𝑥) = (0g𝐺)))
983com23 1268 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → ((𝑁𝑦) = 𝑥 ↔ (𝑦(+g𝐺)𝑥) = (0g𝐺)))
102, 6, 7, 3grpinvid2 17392 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → ((𝑁𝑥) = 𝑦 ↔ (𝑦(+g𝐺)𝑥) = (0g𝐺)))
119, 10bitr4d 271 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥𝐵𝑦𝐵) → ((𝑁𝑦) = 𝑥 ↔ (𝑁𝑥) = 𝑦))
12113expb 1263 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((𝑁𝑦) = 𝑥 ↔ (𝑁𝑥) = 𝑦))
13 eqcom 2628 . . . . 5 (𝑥 = (𝑁𝑦) ↔ (𝑁𝑦) = 𝑥)
14 eqcom 2628 . . . . 5 (𝑦 = (𝑁𝑥) ↔ (𝑁𝑥) = 𝑦)
1512, 13, 143bitr4g 303 . . . 4 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 = (𝑁𝑦) ↔ 𝑦 = (𝑁𝑥)))
161, 4, 5, 15f1ocnv2d 6839 . . 3 (𝐺 ∈ Grp → ((𝑥𝐵 ↦ (𝑁𝑥)):𝐵1-1-onto𝐵(𝑥𝐵 ↦ (𝑁𝑥)) = (𝑦𝐵 ↦ (𝑁𝑦))))
1716simprd 479 . 2 (𝐺 ∈ Grp → (𝑥𝐵 ↦ (𝑁𝑥)) = (𝑦𝐵 ↦ (𝑁𝑦)))
182, 3grpinvf 17387 . . . 4 (𝐺 ∈ Grp → 𝑁:𝐵𝐵)
1918feqmptd 6206 . . 3 (𝐺 ∈ Grp → 𝑁 = (𝑥𝐵 ↦ (𝑁𝑥)))
2019cnveqd 5258 . 2 (𝐺 ∈ Grp → 𝑁 = (𝑥𝐵 ↦ (𝑁𝑥)))
2118feqmptd 6206 . 2 (𝐺 ∈ Grp → 𝑁 = (𝑦𝐵 ↦ (𝑁𝑦)))
2217, 20, 213eqtr4d 2665 1 (𝐺 ∈ Grp → 𝑁 = 𝑁)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  cmpt 4673  ccnv 5073  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  0gc0g 16021  Grpcgrp 17343  invgcminusg 17344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-minusg 17347
This theorem is referenced by:  grpinvf1o  17406  grpinvhmeo  21800
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