Theorem rngonegcl 37894

Description: A ring is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

Ref	Expression
ringnegcl.1	⊢ 𝐺 = (1st ‘𝑅)
ringnegcl.2	⊢ 𝑋 = ran 𝐺
ringnegcl.3	⊢ 𝑁 = (inv‘𝐺)

Assertion

Ref	Expression
rngonegcl	⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)

Proof of Theorem rngonegcl

Step	Hyp	Ref	Expression
1		ringnegcl.1	. . 3 ⊢ 𝐺 = (1st ‘𝑅)
2	1	rngogrpo 37877	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		ringnegcl.2	. . 3 ⊢ 𝑋 = ran 𝐺
4		ringnegcl.3	. . 3 ⊢ 𝑁 = (inv‘𝐺)
5	3, 4	grpoinvcl 30426	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
6	2, 5	sylan 580	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ran crn 5632  ‘cfv 6499  1^st c1st 7945  GrpOpcgr 30391  invcgn 30393  RingOpscrngo 37861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-1st 7947  df-2nd 7948  df-grpo 30395  df-gid 30396  df-ginv 30397  df-ablo 30447  df-rngo 37862

This theorem is referenced by:  rngonegmn1l  37908  rngonegmn1r  37909  rngoneglmul  37910  rngonegrmul  37911  rngosubdi  37912  rngosubdir  37913  idlnegcl  37989