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Metamath Proof Explorer

Theorem lmodacl

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

Ref Expression
Hypotheses lmodacl.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
lmodacl.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
lmodacl.p ⊢ + = ( +g ‘ 𝐹 )
Assertion lmodacl ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ) → ( 𝑋 + 𝑌 ) ∈ 𝐾 )

Proof

Step Hyp Ref Expression
1 lmodacl.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
2 lmodacl.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
3 lmodacl.p ⊢ + = ( +g ‘ 𝐹 )
4 1 lmodfgrp ⊢ ( 𝑊 ∈ LMod → 𝐹 ∈ Grp )
5 2 3 grpcl ⊢ ( ( 𝐹 ∈ Grp ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ) → ( 𝑋 + 𝑌 ) ∈ 𝐾 )
6 4 5 syl3an1 ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ) → ( 𝑋 + 𝑌 ) ∈ 𝐾 )