Metamath Proof Explorer


Theorem srgcl

Description: Closure of the multiplication operation of a semiring. (Contributed by NM, 26-Aug-2011) (Revised by Mario Carneiro, 6-Jan-2015) (Revised by Thierry Arnoux, 1-Apr-2018)

Ref Expression
Hypotheses srgcl.b 𝐵 = ( Base ‘ 𝑅 )
srgcl.t · = ( .r𝑅 )
Assertion srgcl ( ( 𝑅 ∈ SRing ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 · 𝑌 ) ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 srgcl.b 𝐵 = ( Base ‘ 𝑅 )
2 srgcl.t · = ( .r𝑅 )
3 eqid ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 )
4 3 srgmgp ( 𝑅 ∈ SRing → ( mulGrp ‘ 𝑅 ) ∈ Mnd )
5 3 1 mgpbas 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑅 ) )
6 3 2 mgpplusg · = ( +g ‘ ( mulGrp ‘ 𝑅 ) )
7 5 6 mndcl ( ( ( mulGrp ‘ 𝑅 ) ∈ Mnd ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 · 𝑌 ) ∈ 𝐵 )
8 4 7 syl3an1 ( ( 𝑅 ∈ SRing ∧ 𝑋𝐵𝑌𝐵 ) → ( 𝑋 · 𝑌 ) ∈ 𝐵 )