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Metamath Proof Explorer

Theorem ringlzd

Description: The zero of a unital ring is a left-absorbing element. (Contributed by SN, 7-Mar-2025)

Ref Expression
Hypotheses ringz.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
ringz.t ⊢ · = ( .r ‘ 𝑅 )
ringz.z ⊢ 0 = ( 0g ‘ 𝑅 )
ringlzd.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
ringlzd.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion ringlzd ( 𝜑 → ( 0 · 𝑋 ) = 0 )

Proof

Step Hyp Ref Expression
1 ringz.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 ringz.t ⊢ · = ( .r ‘ 𝑅 )
3 ringz.z ⊢ 0 = ( 0g ‘ 𝑅 )
4 ringlzd.r ⊢ ( 𝜑 → 𝑅 ∈ Ring )
5 ringlzd.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 1 2 3 ringlz ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 0 · 𝑋 ) = 0 )
7 4 5 6 syl2anc ⊢ ( 𝜑 → ( 0 · 𝑋 ) = 0 )