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Theorem grpoinvcl 27955
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 2778 . . 3 (GId‘𝐺) = (GId‘𝐺)
3 grpinvcl.2 . . 3 𝑁 = (inv‘𝐺)
41, 2, 3grpoinvval 27954 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝑦𝑋 (𝑦𝐺𝐴) = (GId‘𝐺)))
51, 2grpoinveu 27950 . . 3 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → ∃!𝑦𝑋 (𝑦𝐺𝐴) = (GId‘𝐺))
6 riotacl 6899 . . 3 (∃!𝑦𝑋 (𝑦𝐺𝐴) = (GId‘𝐺) → (𝑦𝑋 (𝑦𝐺𝐴) = (GId‘𝐺)) ∈ 𝑋)
75, 6syl 17 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑦𝑋 (𝑦𝐺𝐴) = (GId‘𝐺)) ∈ 𝑋)
84, 7eqeltrd 2859 1 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  ∃!wreu 3092  ran crn 5358  cfv 6137  crio 6884  (class class class)co 6924  GrpOpcgr 27920  GIdcgi 27921  invcgn 27922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-grpo 27924  df-gid 27925  df-ginv 27926
This theorem is referenced by:  grpoinvid1  27959  grpoinvid2  27960  grpolcan  27961  grpo2inv  27962  grpoinvf  27963  grpoinvop  27964  grpodivinv  27967  grpoinvdiv  27968  grpodivf  27969  grpomuldivass  27972  grponpcan  27974  ablodivdiv4  27985  vcm  28007  rngonegcl  34355  isdrngo2  34386
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