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Theorem grpoinv 30407
Description: The properties of a group element's inverse. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinv.1 𝑋 = ran 𝐺
grpinv.2 𝑈 = (GId‘𝐺)
grpinv.3 𝑁 = (inv‘𝐺)
Assertion
Ref Expression
grpoinv ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑁𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁𝐴)) = 𝑈))

Proof of Theorem grpoinv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 grpinv.1 . . . . . 6 𝑋 = ran 𝐺
2 grpinv.2 . . . . . 6 𝑈 = (GId‘𝐺)
3 grpinv.3 . . . . . 6 𝑁 = (inv‘𝐺)
41, 2, 3grpoinvval 30405 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
51, 2grpoinveu 30401 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈)
6 riotacl2 7392 . . . . . 6 (∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈 → (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈) ∈ {𝑦𝑋 ∣ (𝑦𝐺𝐴) = 𝑈})
75, 6syl 17 . . . . 5 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑦𝑋 (𝑦𝐺𝐴) = 𝑈) ∈ {𝑦𝑋 ∣ (𝑦𝐺𝐴) = 𝑈})
84, 7eqeltrd 2825 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ {𝑦𝑋 ∣ (𝑦𝐺𝐴) = 𝑈})
9 simpl 481 . . . . . . . . 9 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈)
109rgenw 3054 . . . . . . . 8 𝑦𝑋 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈)
1110a1i 11 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∀𝑦𝑋 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈))
121, 2grpoidinv2 30397 . . . . . . . 8 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
1312simprd 494 . . . . . . 7 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈))
1411, 13, 53jca 1125 . . . . . 6 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (∀𝑦𝑋 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) ∧ ∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈))
15 reupick2 4320 . . . . . 6 (((∀𝑦𝑋 (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) → (𝑦𝐺𝐴) = 𝑈) ∧ ∃𝑦𝑋 ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) ∧ ∃!𝑦𝑋 (𝑦𝐺𝐴) = 𝑈) ∧ 𝑦𝑋) → ((𝑦𝐺𝐴) = 𝑈 ↔ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
1614, 15sylan 578 . . . . 5 (((𝐺 ∈ GrpOp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → ((𝑦𝐺𝐴) = 𝑈 ↔ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)))
1716rabbidva 3425 . . . 4 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → {𝑦𝑋 ∣ (𝑦𝐺𝐴) = 𝑈} = {𝑦𝑋 ∣ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)})
188, 17eleqtrd 2827 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ {𝑦𝑋 ∣ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)})
19 oveq1 7426 . . . . . 6 (𝑦 = (𝑁𝐴) → (𝑦𝐺𝐴) = ((𝑁𝐴)𝐺𝐴))
2019eqeq1d 2727 . . . . 5 (𝑦 = (𝑁𝐴) → ((𝑦𝐺𝐴) = 𝑈 ↔ ((𝑁𝐴)𝐺𝐴) = 𝑈))
21 oveq2 7427 . . . . . 6 (𝑦 = (𝑁𝐴) → (𝐴𝐺𝑦) = (𝐴𝐺(𝑁𝐴)))
2221eqeq1d 2727 . . . . 5 (𝑦 = (𝑁𝐴) → ((𝐴𝐺𝑦) = 𝑈 ↔ (𝐴𝐺(𝑁𝐴)) = 𝑈))
2320, 22anbi12d 630 . . . 4 (𝑦 = (𝑁𝐴) → (((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈) ↔ (((𝑁𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁𝐴)) = 𝑈)))
2423elrab 3679 . . 3 ((𝑁𝐴) ∈ {𝑦𝑋 ∣ ((𝑦𝐺𝐴) = 𝑈 ∧ (𝐴𝐺𝑦) = 𝑈)} ↔ ((𝑁𝐴) ∈ 𝑋 ∧ (((𝑁𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁𝐴)) = 𝑈)))
2518, 24sylib 217 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ((𝑁𝐴) ∈ 𝑋 ∧ (((𝑁𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁𝐴)) = 𝑈)))
2625simprd 494 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (((𝑁𝐴)𝐺𝐴) = 𝑈 ∧ (𝐴𝐺(𝑁𝐴)) = 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3050  wrex 3059  ∃!wreu 3361  {crab 3418  ran crn 5679  cfv 6549  crio 7374  (class class class)co 7419  GrpOpcgr 30371  GIdcgi 30372  invcgn 30373
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 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-grpo 30375  df-gid 30376  df-ginv 30377
This theorem is referenced by:  grpolinv  30408  grporinv  30409
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