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Metamath Proof Explorer

Theorem clmvsdi

Description: Distributive law for scalar product (left-distributivity). ( lmodvsdi analog.) (Contributed by NM, 3-Nov-2006) (Revised by AV, 28-Sep-2021)

Ref Expression
Hypotheses clmvscl.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
clmvscl.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
clmvscl.s ⊢ · = ( ·𝑠 ‘ 𝑊 )
clmvscl.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
clmvsdir.a ⊢ + = ( +g ‘ 𝑊 )
Assertion clmvsdi ( ( 𝑊 ∈ ℂMod ∧ ( 𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝐴 · ( 𝑋 + 𝑌 ) ) = ( ( 𝐴 · 𝑋 ) + ( 𝐴 · 𝑌 ) ) )

Proof

Step Hyp Ref Expression
1 clmvscl.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
2 clmvscl.f ⊢ 𝐹 = ( Scalar ‘ 𝑊 )
3 clmvscl.s ⊢ · = ( ·𝑠 ‘ 𝑊 )
4 clmvscl.k ⊢ 𝐾 = ( Base ‘ 𝐹 )
5 clmvsdir.a ⊢ + = ( +g ‘ 𝑊 )
6 clmlmod ⊢ ( 𝑊 ∈ ℂMod → 𝑊 ∈ LMod )
7 1 5 2 3 4 lmodvsdi ⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝐴 · ( 𝑋 + 𝑌 ) ) = ( ( 𝐴 · 𝑋 ) + ( 𝐴 · 𝑌 ) ) )
8 6 7 sylan ⊢ ( ( 𝑊 ∈ ℂMod ∧ ( 𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝐴 · ( 𝑋 + 𝑌 ) ) = ( ( 𝐴 · 𝑋 ) + ( 𝐴 · 𝑌 ) ) )