Theorem rngonegcl 35822

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 35805	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		ringnegcl.2	. . 3 ⊢ 𝑋 = ran 𝐺
4		ringnegcl.3	. . 3 ⊢ 𝑁 = (inv‘𝐺)
5	3, 4	grpoinvcl 28605	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
6	2, 5	sylan 583	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ran crn 5552  ‘cfv 6380  1^st c1st 7759  GrpOpcgr 28570  invcgn 28572  RingOpscrngo 35789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-1st 7761  df-2nd 7762  df-grpo 28574  df-gid 28575  df-ginv 28576  df-ablo 28626  df-rngo 35790

This theorem is referenced by:  rngonegmn1l  35836  rngonegmn1r  35837  rngoneglmul  35838  rngonegrmul  35839  rngosubdi  35840  rngosubdir  35841  idlnegcl  35917