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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 ⊢ ∥ = ( ∥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 )