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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 ⊢ ∥ = ( ∥r ‘ 𝑅 )
dvdsr.3 ⊢ · = ( .r ‘ 𝑅 )
Assertion dvdsrmul ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∥ ( 𝑌 · 𝑋 ) )

Proof

Step Hyp Ref Expression
1 dvdsr.1 ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 dvdsr.2 ⊢ ∥ = ( ∥r ‘ 𝑅 )
3 dvdsr.3 ⊢ · = ( .r ‘ 𝑅 )
4 simpl ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 )
5 simpr ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 )
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 ⊢ ( ( 𝑌 ∈ 𝐵 ∧ ( 𝑌 · 𝑋 ) = ( 𝑌 · 𝑋 ) ) → ∃ 𝑧 ∈ 𝐵 ( 𝑧 · 𝑋 ) = ( 𝑌 · 𝑋 ) )
10 5 6 9 sylancl ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ∃ 𝑧 ∈ 𝐵 ( 𝑧 · 𝑋 ) = ( 𝑌 · 𝑋 ) )
11 1 2 3 dvdsr ⊢ ( 𝑋 ∥ ( 𝑌 · 𝑋 ) ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑧 ∈ 𝐵 ( 𝑧 · 𝑋 ) = ( 𝑌 · 𝑋 ) ) )
12 4 10 11 sylanbrc ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∥ ( 𝑌 · 𝑋 ) )