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 e. SRing /\ X e. B /\ Y e. B ) -> ( X .+ Y ) = ( Y .+ X ) )

Proof

Step	Hyp	Ref	Expression
1		srgacl.b	|- B = ( Base ` R )
2		srgacl.p	|- .+ = ( +g ` R )
3		srgcmn	|- ( R e. SRing -> R e. CMnd )
4	1 2	cmncom	|- ( ( R e. CMnd /\ X e. B /\ Y e. B ) -> ( X .+ Y ) = ( Y .+ X ) )
5	3 4	syl3an1	|- ( ( R e. SRing /\ X e. B /\ Y e. B ) -> ( X .+ Y ) = ( Y .+ X ) )