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 ) ) )