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 )