This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem lmodmcl

Description: Closure of ring multiplication for a left module. (Contributed by NM, 14-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014)

Ref Expression
Hypotheses lmodmcl.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
lmodmcl.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
lmodmcl.t ⊢ · = ( .r ‘ 𝐹 )
Assertion lmodmcl ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ) → ( 𝑋 · 𝑌 ) ∈ 𝐾 )

Proof

Step Hyp Ref Expression
1 lmodmcl.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2 lmodmcl.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
3 lmodmcl.t ⊢ · = ( .r ‘ 𝐹 )
4 1 lmodring ⊢ ( 𝑊 ∈ LMod → 𝐹 ∈ Ring )
5 2 3 ringcl ⊢ ( ( 𝐹 ∈ Ring ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ) → ( 𝑋 · 𝑌 ) ∈ 𝐾 )
6 4 5 syl3an1 ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ) → ( 𝑋 · 𝑌 ) ∈ 𝐾 )