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 = ( ∥r𝑅 )
Assertion dvdsrcl ( 𝑋 𝑌𝑋𝐵 )

Proof

Step Hyp Ref Expression
1 dvdsr.1 𝐵 = ( Base ‘ 𝑅 )
2 dvdsr.2 = ( ∥r𝑅 )
3 eqid ( .r𝑅 ) = ( .r𝑅 )
4 1 2 3 dvdsr ( 𝑋 𝑌 ↔ ( 𝑋𝐵 ∧ ∃ 𝑥𝐵 ( 𝑥 ( .r𝑅 ) 𝑋 ) = 𝑌 ) )
5 4 simplbi ( 𝑋 𝑌𝑋𝐵 )