Metamath Proof Explorer


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 = ( ∥r𝑅 )
Assertion dvdsrcl2 ( ( 𝑅 ∈ Ring ∧ 𝑋 𝑌 ) → 𝑌𝐵 )

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 1 3 ringcl ( ( 𝑅 ∈ Ring ∧ 𝑥𝐵𝑋𝐵 ) → ( 𝑥 ( .r𝑅 ) 𝑋 ) ∈ 𝐵 )
6 5 3expa ( ( ( 𝑅 ∈ Ring ∧ 𝑥𝐵 ) ∧ 𝑋𝐵 ) → ( 𝑥 ( .r𝑅 ) 𝑋 ) ∈ 𝐵 )
7 6 an32s ( ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) ∧ 𝑥𝐵 ) → ( 𝑥 ( .r𝑅 ) 𝑋 ) ∈ 𝐵 )
8 eleq1 ( ( 𝑥 ( .r𝑅 ) 𝑋 ) = 𝑌 → ( ( 𝑥 ( .r𝑅 ) 𝑋 ) ∈ 𝐵𝑌𝐵 ) )
9 7 8 syl5ibcom ( ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) ∧ 𝑥𝐵 ) → ( ( 𝑥 ( .r𝑅 ) 𝑋 ) = 𝑌𝑌𝐵 ) )
10 9 rexlimdva ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) → ( ∃ 𝑥𝐵 ( 𝑥 ( .r𝑅 ) 𝑋 ) = 𝑌𝑌𝐵 ) )
11 10 impr ( ( 𝑅 ∈ Ring ∧ ( 𝑋𝐵 ∧ ∃ 𝑥𝐵 ( 𝑥 ( .r𝑅 ) 𝑋 ) = 𝑌 ) ) → 𝑌𝐵 )
12 4 11 sylan2b ( ( 𝑅 ∈ Ring ∧ 𝑋 𝑌 ) → 𝑌𝐵 )