Metamath Proof Explorer
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 ) |