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Metamath Proof Explorer

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 )