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Metamath Proof Explorer

Theorem ringid

Description: The multiplication operation of a unital ring has (one or more) identity elements. (Contributed by Steve Rodriguez, 9-Sep-2007) (Revised by Mario Carneiro, 22-Dec-2013) (Revised by AV, 24-Aug-2021)

Ref Expression
Hypotheses ringid.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
ringid.t ⊢ · = ( .r ‘ 𝑅 )
Assertion ringid ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ∃ 𝑢 ∈ 𝐵 ( ( 𝑢 · 𝑋 ) = 𝑋 ∧ ( 𝑋 · 𝑢 ) = 𝑋 ) )

Proof

Step Hyp Ref Expression
1 ringid.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 ringid.t ⊢ · = ( .r ‘ 𝑅 )
3 eqid ⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
4 1 3 ringidcl ⊢ ( 𝑅 ∈ Ring → ( 1r ‘ 𝑅 ) ∈ 𝐵 )
5 4 adantr ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 1r ‘ 𝑅 ) ∈ 𝐵 )
6 oveq1 ⊢ ( 𝑢 = ( 1r ‘ 𝑅 ) → ( 𝑢 · 𝑋 ) = ( ( 1r ‘ 𝑅 ) · 𝑋 ) )
7 6 eqeq1d ⊢ ( 𝑢 = ( 1r ‘ 𝑅 ) → ( ( 𝑢 · 𝑋 ) = 𝑋 ↔ ( ( 1r ‘ 𝑅 ) · 𝑋 ) = 𝑋 ) )
8 oveq2 ⊢ ( 𝑢 = ( 1r ‘ 𝑅 ) → ( 𝑋 · 𝑢 ) = ( 𝑋 · ( 1r ‘ 𝑅 ) ) )
9 8 eqeq1d ⊢ ( 𝑢 = ( 1r ‘ 𝑅 ) → ( ( 𝑋 · 𝑢 ) = 𝑋 ↔ ( 𝑋 · ( 1r ‘ 𝑅 ) ) = 𝑋 ) )
10 7 9 anbi12d ⊢ ( 𝑢 = ( 1r ‘ 𝑅 ) → ( ( ( 𝑢 · 𝑋 ) = 𝑋 ∧ ( 𝑋 · 𝑢 ) = 𝑋 ) ↔ ( ( ( 1r ‘ 𝑅 ) · 𝑋 ) = 𝑋 ∧ ( 𝑋 · ( 1r ‘ 𝑅 ) ) = 𝑋 ) ) )
11 10 adantl ⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) ∧ 𝑢 = ( 1r ‘ 𝑅 ) ) → ( ( ( 𝑢 · 𝑋 ) = 𝑋 ∧ ( 𝑋 · 𝑢 ) = 𝑋 ) ↔ ( ( ( 1r ‘ 𝑅 ) · 𝑋 ) = 𝑋 ∧ ( 𝑋 · ( 1r ‘ 𝑅 ) ) = 𝑋 ) ) )
12 1 2 3 ringidmlem ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( ( ( 1r ‘ 𝑅 ) · 𝑋 ) = 𝑋 ∧ ( 𝑋 · ( 1r ‘ 𝑅 ) ) = 𝑋 ) )
13 5 11 12 rspcedvd ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ∃ 𝑢 ∈ 𝐵 ( ( 𝑢 · 𝑋 ) = 𝑋 ∧ ( 𝑋 · 𝑢 ) = 𝑋 ) )