Theorem rngorn1

Description: In a unital ring the range of the addition equals the domain of the first variable of the multiplication. (Contributed by FL, 24-Jan-2010) (New usage is discouraged.)

	Ref	Expression
Hypotheses	rnplrnml0.1	⊢ 𝐻 = ( 2nd ‘ 𝑅 )
	rnplrnml0.2	⊢ 𝐺 = ( 1st ‘ 𝑅 )
Assertion	rngorn1	⊢ ( 𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐻 )

Proof

Step	Hyp	Ref	Expression
1		rnplrnml0.1	⊢ 𝐻 = ( 2nd ‘ 𝑅 )
2		rnplrnml0.2	⊢ 𝐺 = ( 1st ‘ 𝑅 )
3	2	rngogrpo	⊢ ( 𝑅 ∈ RingOps → 𝐺 ∈ GrpOp )
4		grporndm	⊢ ( 𝐺 ∈ GrpOp → ran 𝐺 = dom dom 𝐺 )
5	3 4	syl	⊢ ( 𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐺 )
6	1 2	rngodm1dm2	⊢ ( 𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻 )
7	5 6	eqtrd	⊢ ( 𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐻 )