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Theorem grpoinvcl 27348
Description: A group element's inverse is a group element. (Contributed by NM, 27-Oct-2006.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinvcl.1 𝑋 = ran 𝐺
grpinvcl.2 𝑁 = (inv‘𝐺)
Assertion
Ref Expression
grpoinvcl ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)

Proof of Theorem grpoinvcl
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 grpinvcl.1 . . 3 𝑋 = ran 𝐺
2 eqid 2620 . . 3 (GId‘𝐺) = (GId‘𝐺)
3 grpinvcl.2 . . 3 𝑁 = (inv‘𝐺)
41, 2, 3grpoinvval 27347 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝑦𝑋 (𝑦𝐺𝐴) = (GId‘𝐺)))
51, 2grpoinveu 27343 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃!𝑦𝑋 (𝑦𝐺𝐴) = (GId‘𝐺))
6 riotacl 6610 . . 3 (∃!𝑦𝑋 (𝑦𝐺𝐴) = (GId‘𝐺) → (𝑦𝑋 (𝑦𝐺𝐴) = (GId‘𝐺)) ∈ 𝑋)
75, 6syl 17 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑦𝑋 (𝑦𝐺𝐴) = (GId‘𝐺)) ∈ 𝑋)
84, 7eqeltrd 2699 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  ∃!wreu 2911  ran crn 5105  cfv 5876  crio 6595  (class class class)co 6635  GrpOpcgr 27313  GIdcgi 27314  invcgn 27315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-grpo 27317  df-gid 27318  df-ginv 27319
This theorem is referenced by:  grpoinvid1  27352  grpoinvid2  27353  grpolcan  27354  grpo2inv  27355  grpoinvf  27356  grpoinvop  27357  grpodivinv  27360  grpoinvdiv  27361  grpodivf  27362  grpomuldivass  27365  grponpcan  27367  ablodivdiv4  27378  vcm  27401  rngonegcl  33697  isdrngo2  33728
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