Theorem rngonegcl 37934

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 37917	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		ringnegcl.2	. . 3 ⊢ 𝑋 = ran 𝐺
4		ringnegcl.3	. . 3 ⊢ 𝑁 = (inv‘𝐺)
5	3, 4	grpoinvcl 30543	. 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 2108  ran crn 5686  ‘cfv 6561  1^st c1st 8012  GrpOpcgr 30508  invcgn 30510  RingOpscrngo 37901

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-1st 8014  df-2nd 8015  df-grpo 30512  df-gid 30513  df-ginv 30514  df-ablo 30564  df-rngo 37902

This theorem is referenced by:  rngonegmn1l  37948  rngonegmn1r  37949  rngoneglmul  37950  rngonegrmul  37951  rngosubdi  37952  rngosubdir  37953  idlnegcl  38029