Theorem ringlz

Description: The zero of a unital ring is a left-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	ringlz	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 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		ringrng	⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Rng )
5	1 2 3	rnglz	⊢ ( ( 𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵 ) → ( 0 · 𝑋 ) = 0 )
6	4 5	sylan	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ) → ( 0 · 𝑋 ) = 0 )