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 ) ) )