Theorem rngosub

Description: Subtraction in a ring, in terms of addition and negation. (Contributed by Jeff Madsen, 19-Jun-2010)

	Ref	Expression
Hypotheses	ringnegcl.1	⊢ 𝐺 = ( 1st ‘ 𝑅 )
	ringnegcl.2	⊢ 𝑋 = ran 𝐺
	ringnegcl.3	⊢ 𝑁 = ( inv ‘ 𝐺 )
	ringsub.4	⊢ 𝐷 = ( /𝑔 ‘ 𝐺 )
Assertion	rngosub	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 𝐴 𝐺 ( 𝑁 ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ringnegcl.1	⊢ 𝐺 = ( 1st ‘ 𝑅 )
2		ringnegcl.2	⊢ 𝑋 = ran 𝐺
3		ringnegcl.3	⊢ 𝑁 = ( inv ‘ 𝐺 )
4		ringsub.4	⊢ 𝐷 = ( /𝑔 ‘ 𝐺 )
5	1	rngogrpo	⊢ ( 𝑅 ∈ RingOps → 𝐺 ∈ GrpOp )
6	2 3 4	grpodivval	⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 𝐴 𝐺 ( 𝑁 ‘ 𝐵 ) ) )
7	5 6	syl3an1	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 𝐷 𝐵 ) = ( 𝐴 𝐺 ( 𝑁 ‘ 𝐵 ) ) )