Theorem ring0cl

Description: The zero element of a ring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014)

	Ref	Expression
Hypotheses	ring0cl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	ring0cl.z	⊢ 0 = ( 0g ‘ 𝑅 )
Assertion	ring0cl	⊢ ( 𝑅 ∈ Ring → 0 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ring0cl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		ring0cl.z	⊢ 0 = ( 0g ‘ 𝑅 )
3		ringgrp	⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Grp )
4	1 2	grpidcl	⊢ ( 𝑅 ∈ Grp → 0 ∈ 𝐵 )
5	3 4	syl	⊢ ( 𝑅 ∈ Ring → 0 ∈ 𝐵 )