Theorem rngoaddneg1

Description: Adding the negative in a ring gives zero. (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 )
	ringaddneg.4	⊢ 𝑍 = ( GId ‘ 𝐺 )
Assertion	rngoaddneg1	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐺 ( 𝑁 ‘ 𝐴 ) ) = 𝑍 )

Proof

Step	Hyp	Ref	Expression
1		ringnegcl.1	⊢ 𝐺 = ( 1st ‘ 𝑅 )
2		ringnegcl.2	⊢ 𝑋 = ran 𝐺
3		ringnegcl.3	⊢ 𝑁 = ( inv ‘ 𝐺 )
4		ringaddneg.4	⊢ 𝑍 = ( GId ‘ 𝐺 )
5	1	rngogrpo	⊢ ( 𝑅 ∈ RingOps → 𝐺 ∈ GrpOp )
6	2 4 3	grporinv	⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐺 ( 𝑁 ‘ 𝐴 ) ) = 𝑍 )
7	5 6	sylan	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐺 ( 𝑁 ‘ 𝐴 ) ) = 𝑍 )