Theorem ringadd2

Description: A ring element plus itself is two times the element. (Contributed by Steve Rodriguez, 9-Sep-2007) (Revised by Mario Carneiro, 22-Dec-2013) (Revised by AV, 24-Aug-2021) (Proof shortened by AV, 1-Feb-2025)

	Ref	Expression
Hypotheses	ringadd2.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	ringadd2.p	⊢ + = ( +g ‘ 𝑅 )
	ringadd2.t	⊢ · = ( .r ‘ 𝑅 )
Assertion	ringadd2	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ∃ 𝑥 ∈ 𝐵 ( 𝑋 + 𝑋 ) = ( ( 𝑥 + 𝑥 ) · 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		ringadd2.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		ringadd2.p	⊢ + = ( +g ‘ 𝑅 )
3		ringadd2.t	⊢ · = ( .r ‘ 𝑅 )
4		eqid	⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
5	1 4	ringidcl	⊢ ( 𝑅 ∈ Ring → ( 1r ‘ 𝑅 ) ∈ 𝐵 )
6	5	adantr	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 1r ‘ 𝑅 ) ∈ 𝐵 )
7		simpr	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑥 = ( 1r ‘ 𝑅 ) ) → 𝑥 = ( 1r ‘ 𝑅 ) )
8	7 7	oveq12d	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑥 = ( 1r ‘ 𝑅 ) ) → ( 𝑥 + 𝑥 ) = ( ( 1r ‘ 𝑅 ) + ( 1r ‘ 𝑅 ) ) )
9	8	oveq1d	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑥 = ( 1r ‘ 𝑅 ) ) → ( ( 𝑥 + 𝑥 ) · 𝑋 ) = ( ( ( 1r ‘ 𝑅 ) + ( 1r ‘ 𝑅 ) ) · 𝑋 ) )
10	9	eqeq2d	⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑥 = ( 1r ‘ 𝑅 ) ) → ( ( 𝑋 + 𝑋 ) = ( ( 𝑥 + 𝑥 ) · 𝑋 ) ↔ ( 𝑋 + 𝑋 ) = ( ( ( 1r ‘ 𝑅 ) + ( 1r ‘ 𝑅 ) ) · 𝑋 ) ) )
11	1 2 3 4	ringo2times	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 + 𝑋 ) = ( ( ( 1r ‘ 𝑅 ) + ( 1r ‘ 𝑅 ) ) · 𝑋 ) )
12	6 10 11	rspcedvd	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ∃ 𝑥 ∈ 𝐵 ( 𝑋 + 𝑋 ) = ( ( 𝑥 + 𝑥 ) · 𝑋 ) )