Theorem dvdsrcl

Description: Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014)

	Ref	Expression
Hypotheses	dvdsr.1	⊢ 𝐵 = ( Base ‘ 𝑅 )
	dvdsr.2	⊢ ∥ = ( ∥r ‘ 𝑅 )
Assertion	dvdsrcl	⊢ ( 𝑋 ∥ 𝑌 → 𝑋 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		dvdsr.1	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		dvdsr.2	⊢ ∥ = ( ∥r ‘ 𝑅 )
3		eqid	⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
4	1 2 3	dvdsr	⊢ ( 𝑋 ∥ 𝑌 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( .r ‘ 𝑅 ) 𝑋 ) = 𝑌 ) )
5	4	simplbi	⊢ ( 𝑋 ∥ 𝑌 → 𝑋 ∈ 𝐵 )