Theorem fsumneg

Description: Negation of a finite sum. (Contributed by Scott Fenton, 12-Jun-2013) (Revised by Mario Carneiro, 24-Apr-2014)

	Ref	Expression
Hypotheses	fsumneg.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	fsumneg.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
Assertion	fsumneg	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 - 𝐵 = - Σ 𝑘 ∈ 𝐴 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		fsumneg.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		fsumneg.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
3		neg1cn	⊢ - 1 ∈ ℂ
4	3	a1i	⊢ ( 𝜑 → - 1 ∈ ℂ )
5	1 4 2	fsummulc2	⊢ ( 𝜑 → ( - 1 · Σ 𝑘 ∈ 𝐴 𝐵 ) = Σ 𝑘 ∈ 𝐴 ( - 1 · 𝐵 ) )
6	1 2	fsumcl	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 ∈ ℂ )
7	6	mulm1d	⊢ ( 𝜑 → ( - 1 · Σ 𝑘 ∈ 𝐴 𝐵 ) = - Σ 𝑘 ∈ 𝐴 𝐵 )
8	2	mulm1d	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( - 1 · 𝐵 ) = - 𝐵 )
9	8	sumeq2dv	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 ( - 1 · 𝐵 ) = Σ 𝑘 ∈ 𝐴 - 𝐵 )
10	5 7 9	3eqtr3rd	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 - 𝐵 = - Σ 𝑘 ∈ 𝐴 𝐵 )