Metamath Proof Explorer


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 + = ( +g𝑅 )
Assertion srgcom ( ( 𝑅 ∈ SRing ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 + 𝑌 ) = ( 𝑌 + 𝑋 ) )

Proof

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