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 )