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	$$⊢B={\mathrm{Base}}_R$$
	srgacl.p	$$⊢\underset{\dot{}}{+}={+}_R$$
Assertion	srgacl	$$⊢\left(R\in \mathrm{SRing}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{+}Y\in B$$

Proof

Step	Hyp	Ref	Expression
1		srgacl.b	$$⊢B={\mathrm{Base}}_R$$
2		srgacl.p	$$⊢\underset{\dot{}}{+}={+}_R$$
3		srgmnd	$$⊢R\in \mathrm{SRing}\to R\in \mathrm{Mnd}$$
4	1 2	mndcl	$$⊢\left(R\in \mathrm{Mnd}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{+}Y\in B$$
5	3 4	syl3an1	$$⊢\left(R\in \mathrm{SRing}\wedge X\in B\wedge Y\in B\right)\to X\underset{\dot{}}{+}Y\in B$$