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Metamath Proof Explorer

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 )