Theorem rng0cl

Description: The zero element of a non-unital ring belongs to its base set. (Contributed by AV, 16-Feb-2025)

	Ref	Expression
Hypotheses	rng0cl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	rng0cl.z	⊢ 0 = ( 0g ‘ 𝑅 )
Assertion	rng0cl	⊢ ( 𝑅 ∈ Rng → 0 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		rng0cl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		rng0cl.z	⊢ 0 = ( 0g ‘ 𝑅 )
3		rnggrp	⊢ ( 𝑅 ∈ Rng → 𝑅 ∈ Grp )
4	1 2	grpidcl	⊢ ( 𝑅 ∈ Grp → 0 ∈ 𝐵 )
5	3 4	syl	⊢ ( 𝑅 ∈ Rng → 0 ∈ 𝐵 )