Theorem rngonegcl

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

	Ref	Expression
Hypotheses	ringnegcl.1	⊢ 𝐺 = ( 1st ‘ 𝑅 )
	ringnegcl.2	⊢ 𝑋 = ran 𝐺
	ringnegcl.3	⊢ 𝑁 = ( inv ‘ 𝐺 )
Assertion	rngonegcl	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐴 ) ∈ 𝑋 )

Proof

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