This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )