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Theorem rngonegcl 32896
 Description: A ring is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
ringnegcl.1 𝐺 = (1st𝑅)
ringnegcl.2 𝑋 = ran 𝐺
ringnegcl.3 𝑁 = (inv‘𝐺)
Assertion
Ref Expression
rngonegcl ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)

Proof of Theorem rngonegcl
StepHypRef Expression
1 ringnegcl.1 . . 3 𝐺 = (1st𝑅)
21rngogrpo 32879 . 2 (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3 ringnegcl.2 . . 3 𝑋 = ran 𝐺
4 ringnegcl.3 . . 3 𝑁 = (inv‘𝐺)
53, 4grpoinvcl 26762 . 2 ((𝐺 ∈ GrpOp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
62, 5sylan 487 1 ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑁𝐴) ∈ 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ran crn 5039  ‘cfv 5804  1st c1st 7057  GrpOpcgr 26727  invcgn 26729  RingOpscrngo 32863 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-1st 7059  df-2nd 7060  df-grpo 26731  df-gid 26732  df-ginv 26733  df-ablo 26783  df-rngo 32864 This theorem is referenced by:  rngonegmn1l  32910  rngonegmn1r  32911  rngoneglmul  32912  rngonegrmul  32913  rngosubdi  32914  rngosubdir  32915  idlnegcl  32991
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