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Metamath Proof Explorer

Theorem dvdsrid

Description: An element in a (unital) ring divides itself. (Contributed by Mario Carneiro, 1-Dec-2014) (Revised by Mario Carneiro, 30-Apr-2015)

Ref Expression
Hypotheses dvdsr.1 ⊢ 𝐵 = ( Base ‘ 𝑅 )
dvdsr.2 ⊢ ∥ = ( ∥r ‘ 𝑅 )
Assertion dvdsrid ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∥ 𝑋 )

Proof

Step Hyp Ref Expression
1 dvdsr.1 ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 dvdsr.2 ⊢ ∥ = ( ∥r ‘ 𝑅 )
3 id ⊢ ( 𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐵 )
4 eqid ⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
5 1 4 ringidcl ⊢ ( 𝑅 ∈ Ring → ( 1r ‘ 𝑅 ) ∈ 𝐵 )
6 eqid ⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
7 1 2 6 dvdsrmul ⊢ ( ( 𝑋 ∈ 𝐵 ∧ ( 1r ‘ 𝑅 ) ∈ 𝐵 ) → 𝑋 ∥ ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑅 ) 𝑋 ) )
8 3 5 7 syl2anr ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∥ ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑅 ) 𝑋 ) )
9 1 6 4 ringlidm ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( ( 1r ‘ 𝑅 ) ( .r ‘ 𝑅 ) 𝑋 ) = 𝑋 )
10 8 9 breqtrd ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∥ 𝑋 )