Theorem srgacl

Description: Closure of the addition operation of a semiring. (Contributed by Mario Carneiro, 14-Jan-2014) (Revised by Thierry Arnoux, 1-Apr-2018)

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

Proof

Step	Hyp	Ref	Expression
1		srgacl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		srgacl.p	⊢ + = ( +g ‘ 𝑅 )
3		srgmnd	⊢ ( 𝑅 ∈ SRing → 𝑅 ∈ Mnd )
4	1 2	mndcl	⊢ ( ( 𝑅 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )
5	3 4	syl3an1	⊢ ( ( 𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 + 𝑌 ) ∈ 𝐵 )