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 ) )