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Theorem grpoinvcl 27718
 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 2771 . . 3 (GId‘𝐺) = (GId‘𝐺)
3 grpinvcl.2 . . 3 𝑁 = (inv‘𝐺)
41, 2, 3grpoinvval 27717 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝑦𝑋 (𝑦𝐺𝐴) = (GId‘𝐺)))
51, 2grpoinveu 27713 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃!𝑦𝑋 (𝑦𝐺𝐴) = (GId‘𝐺))
6 riotacl 6771 . . 3 (∃!𝑦𝑋 (𝑦𝐺𝐴) = (GId‘𝐺) → (𝑦𝑋 (𝑦𝐺𝐴) = (GId‘𝐺)) ∈ 𝑋)
75, 6syl 17 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑦𝑋 (𝑦𝐺𝐴) = (GId‘𝐺)) ∈ 𝑋)
84, 7eqeltrd 2850 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∃!wreu 3063  ran crn 5251  ‘cfv 6030  ℩crio 6756  (class class class)co 6796  GrpOpcgr 27683  GIdcgi 27684  invcgn 27685 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7100 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-grpo 27687  df-gid 27688  df-ginv 27689 This theorem is referenced by:  grpoinvid1  27722  grpoinvid2  27723  grpolcan  27724  grpo2inv  27725  grpoinvf  27726  grpoinvop  27727  grpodivinv  27730  grpoinvdiv  27731  grpodivf  27732  grpomuldivass  27735  grponpcan  27737  ablodivdiv4  27748  vcm  27771  rngonegcl  34058  isdrngo2  34089
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