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Metamath Proof Explorer

Theorem srglidm

Description: The unity element of a semiring is a left multiplicative identity. (Contributed by NM, 15-Sep-2011) (Revised by Thierry Arnoux, 1-Apr-2018)

Ref Expression
Hypotheses srgidm.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
srgidm.t ⊢ · = ( .r ‘ 𝑅 )
srgidm.u ⊢ 1 = ( 1r ‘ 𝑅 )
Assertion srglidm ( ( 𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ) → ( 1 · 𝑋 ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 srgidm.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 srgidm.t ⊢ · = ( .r ‘ 𝑅 )
3 srgidm.u ⊢ 1 = ( 1r ‘ 𝑅 )
4 1 2 3 srgidmlem ⊢ ( ( 𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ) → ( ( 1 · 𝑋 ) = 𝑋 ∧ ( 𝑋 · 1 ) = 𝑋 ) )
5 4 simpld ⊢ ( ( 𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ) → ( 1 · 𝑋 ) = 𝑋 )