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Metamath Proof Explorer

Theorem ringrz

Description: The zero of a unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009) (Proof shortened by AV, 30-Mar-2025)

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

Proof

Step Hyp Ref Expression
1 ringz.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 ringz.t ⊢ · = ( .r ‘ 𝑅 )
3 ringz.z ⊢ 0 = ( 0g ‘ 𝑅 )
4 ringrng ⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Rng )
5 1 2 3 rngrz ⊢ ( ( 𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 · 0 ) = 0 )
6 4 5 sylan ⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 · 0 ) = 0 )