Theorem ringacl

Description: Closure of the addition operation of a ring. (Contributed by Mario Carneiro, 14-Jan-2014)

	Ref	Expression
Hypotheses	ringacl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	ringacl.p	⊢ + = ( +g ‘ 𝑅 )
Assertion	ringacl	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ringacl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		ringacl.p	⊢ + = ( +g ‘ 𝑅 )
3		ringgrp	⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Grp )
4	1 2	grpcl	⊢ ( ( 𝑅 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )
5	3 4	syl3an1	⊢ ( ( 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )