Metamath Proof Explorer


Theorem dvdsrmul

Description: A left-multiple of X is divisible by X . (Contributed by Mario Carneiro, 1-Dec-2014)

Ref Expression
Hypotheses dvdsr.1 𝐵 = ( Base ‘ 𝑅 )
dvdsr.2 = ( ∥r𝑅 )
dvdsr.3 · = ( .r𝑅 )
Assertion dvdsrmul ( ( 𝑋𝐵𝑌𝐵 ) → 𝑋 ( 𝑌 · 𝑋 ) )

Proof

Step Hyp Ref Expression
1 dvdsr.1 𝐵 = ( Base ‘ 𝑅 )
2 dvdsr.2 = ( ∥r𝑅 )
3 dvdsr.3 · = ( .r𝑅 )
4 simpl ( ( 𝑋𝐵𝑌𝐵 ) → 𝑋𝐵 )
5 simpr ( ( 𝑋𝐵𝑌𝐵 ) → 𝑌𝐵 )
6 eqid ( 𝑌 · 𝑋 ) = ( 𝑌 · 𝑋 )
7 oveq1 ( 𝑧 = 𝑌 → ( 𝑧 · 𝑋 ) = ( 𝑌 · 𝑋 ) )
8 7 eqeq1d ( 𝑧 = 𝑌 → ( ( 𝑧 · 𝑋 ) = ( 𝑌 · 𝑋 ) ↔ ( 𝑌 · 𝑋 ) = ( 𝑌 · 𝑋 ) ) )
9 8 rspcev ( ( 𝑌𝐵 ∧ ( 𝑌 · 𝑋 ) = ( 𝑌 · 𝑋 ) ) → ∃ 𝑧𝐵 ( 𝑧 · 𝑋 ) = ( 𝑌 · 𝑋 ) )
10 5 6 9 sylancl ( ( 𝑋𝐵𝑌𝐵 ) → ∃ 𝑧𝐵 ( 𝑧 · 𝑋 ) = ( 𝑌 · 𝑋 ) )
11 1 2 3 dvdsr ( 𝑋 ( 𝑌 · 𝑋 ) ↔ ( 𝑋𝐵 ∧ ∃ 𝑧𝐵 ( 𝑧 · 𝑋 ) = ( 𝑌 · 𝑋 ) ) )
12 4 10 11 sylanbrc ( ( 𝑋𝐵𝑌𝐵 ) → 𝑋 ( 𝑌 · 𝑋 ) )