Theorem dvdsr2

Description: Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014)

	Ref	Expression
Hypotheses	dvdsr.1	⊢ 𝐵 = ( Base ‘ 𝑅 )
	dvdsr.2	⊢ ∥ = ( ∥r ‘ 𝑅 )
	dvdsr.3	⊢ · = ( .r ‘ 𝑅 )
Assertion	dvdsr2	⊢ ( 𝑋 ∈ 𝐵 → ( 𝑋 ∥ 𝑌 ↔ ∃ 𝑧 ∈ 𝐵 ( 𝑧 · 𝑋 ) = 𝑌 ) )

Proof

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