Theorem fsummulc1

Description: A finite sum multiplied by a constant. (Contributed by NM, 13-Nov-2005) (Revised by Mario Carneiro, 24-Apr-2014)

	Ref	Expression
Hypotheses	fsummulc2.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	fsummulc2.2	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
	fsummulc2.3	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
Assertion	fsummulc1	⊢ ( 𝜑 → ( Σ 𝑘 ∈ 𝐴 𝐵 · 𝐶 ) = Σ 𝑘 ∈ 𝐴 ( 𝐵 · 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		fsummulc2.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		fsummulc2.2	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
3		fsummulc2.3	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
4	1 2 3	fsummulc2	⊢ ( 𝜑 → ( 𝐶 · Σ 𝑘 ∈ 𝐴 𝐵 ) = Σ 𝑘 ∈ 𝐴 ( 𝐶 · 𝐵 ) )
5	1 3	fsumcl	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 ∈ ℂ )
6	5 2	mulcomd	⊢ ( 𝜑 → ( Σ 𝑘 ∈ 𝐴 𝐵 · 𝐶 ) = ( 𝐶 · Σ 𝑘 ∈ 𝐴 𝐵 ) )
7	2	adantr	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 ∈ ℂ )
8	3 7	mulcomd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝐵 · 𝐶 ) = ( 𝐶 · 𝐵 ) )
9	8	sumeq2dv	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 ( 𝐵 · 𝐶 ) = Σ 𝑘 ∈ 𝐴 ( 𝐶 · 𝐵 ) )
10	4 6 9	3eqtr4d	⊢ ( 𝜑 → ( Σ 𝑘 ∈ 𝐴 𝐵 · 𝐶 ) = Σ 𝑘 ∈ 𝐴 ( 𝐵 · 𝐶 ) )