Metamath Proof Explorer


Theorem ringlidm

Description: The unit element of a ring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011)

Ref Expression
Hypotheses rngidm.b 𝐵 = ( Base ‘ 𝑅 )
rngidm.t · = ( .r𝑅 )
rngidm.u 1 = ( 1r𝑅 )
Assertion ringlidm ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) → ( 1 · 𝑋 ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 rngidm.b 𝐵 = ( Base ‘ 𝑅 )
2 rngidm.t · = ( .r𝑅 )
3 rngidm.u 1 = ( 1r𝑅 )
4 1 2 3 ringidmlem ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) → ( ( 1 · 𝑋 ) = 𝑋 ∧ ( 𝑋 · 1 ) = 𝑋 ) )
5 4 simpld ( ( 𝑅 ∈ Ring ∧ 𝑋𝐵 ) → ( 1 · 𝑋 ) = 𝑋 )