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Metamath Proof Explorer

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 ) )