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Metamath Proof Explorer

Theorem clmvsdir

Description: Distributive law for scalar product (right-distributivity). ( lmodvsdir analog.) (Contributed by Mario Carneiro, 16-Oct-2015)

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 clmvsdir ( ( 𝑊 ∈ ℂMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑄 + 𝑅 ) · 𝑋 ) = ( ( 𝑄 · 𝑋 ) + ( 𝑅 · 𝑋 ) ) )

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 2 clmadd ⊢ ( 𝑊 ∈ ℂMod → + = ( +g ‘ 𝐹 ) )
7 6 oveqd ⊢ ( 𝑊 ∈ ℂMod → ( 𝑄 + 𝑅 ) = ( 𝑄 ( +g ‘ 𝐹 ) 𝑅 ) )
8 7 oveq1d ⊢ ( 𝑊 ∈ ℂMod → ( ( 𝑄 + 𝑅 ) · 𝑋 ) = ( ( 𝑄 ( +g ‘ 𝐹 ) 𝑅 ) · 𝑋 ) )
9 8 adantr ⊢ ( ( 𝑊 ∈ ℂMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑄 + 𝑅 ) · 𝑋 ) = ( ( 𝑄 ( +g ‘ 𝐹 ) 𝑅 ) · 𝑋 ) )
10 clmlmod ⊢ ( 𝑊 ∈ ℂMod → 𝑊 ∈ LMod )
11 eqid ⊢ ( +g ‘ 𝐹 ) = ( +g ‘ 𝐹 )
12 1 5 2 3 4 11 lmodvsdir ⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑄 ( +g ‘ 𝐹 ) 𝑅 ) · 𝑋 ) = ( ( 𝑄 · 𝑋 ) + ( 𝑅 · 𝑋 ) ) )
13 10 12 sylan ⊢ ( ( 𝑊 ∈ ℂMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑄 ( +g ‘ 𝐹 ) 𝑅 ) · 𝑋 ) = ( ( 𝑄 · 𝑋 ) + ( 𝑅 · 𝑋 ) ) )
14 9 13 eqtrd ⊢ ( ( 𝑊 ∈ ℂMod ∧ ( 𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ) ) → ( ( 𝑄 + 𝑅 ) · 𝑋 ) = ( ( 𝑄 · 𝑋 ) + ( 𝑅 · 𝑋 ) ) )