Theorem dvdsrcl2

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	dvdsrcl2	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∥ 𝑌 ) → 𝑌 ∈ 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	1 3	ringcl	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑥 ( .r ‘ 𝑅 ) 𝑋 ) ∈ 𝐵 )
6	5	3expa	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝑥 ( .r ‘ 𝑅 ) 𝑋 ) ∈ 𝐵 )
7	6	an32s	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) → ( 𝑥 ( .r ‘ 𝑅 ) 𝑋 ) ∈ 𝐵 )
8		eleq1	⊢ ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑋 ) = 𝑌 → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑋 ) ∈ 𝐵 ↔ 𝑌 ∈ 𝐵 ) )
9	7 8	syl5ibcom	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) → ( ( 𝑥 ( .r ‘ 𝑅 ) 𝑋 ) = 𝑌 → 𝑌 ∈ 𝐵 ) )
10	9	rexlimdva	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( .r ‘ 𝑅 ) 𝑋 ) = 𝑌 → 𝑌 ∈ 𝐵 ) )
11	10	impr	⊢ ( ( 𝑅 ∈ Ring ∧ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑥 ∈ 𝐵 ( 𝑥 ( .r ‘ 𝑅 ) 𝑋 ) = 𝑌 ) ) → 𝑌 ∈ 𝐵 )
12	4 11	sylan2b	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∥ 𝑌 ) → 𝑌 ∈ 𝐵 )