Theorem srgcom

Description: Commutativity of the additive group of a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018)

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

Proof

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