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Metamath Proof Explorer

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 ) )