Theorem fsumsub

Description: Split a finite sum over a subtraction. (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 )
	fsumsub.3	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 ∈ ℂ )
Assertion	fsumsub	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 ( 𝐵 − 𝐶 ) = ( Σ 𝑘 ∈ 𝐴 𝐵 − Σ 𝑘 ∈ 𝐴 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		fsumneg.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		fsumneg.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
3		fsumsub.3	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐶 ∈ ℂ )
4	3	negcld	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → - 𝐶 ∈ ℂ )
5	1 2 4	fsumadd	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 ( 𝐵 + - 𝐶 ) = ( Σ 𝑘 ∈ 𝐴 𝐵 + Σ 𝑘 ∈ 𝐴 - 𝐶 ) )
6	1 3	fsumneg	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 - 𝐶 = - Σ 𝑘 ∈ 𝐴 𝐶 )
7	6	oveq2d	⊢ ( 𝜑 → ( Σ 𝑘 ∈ 𝐴 𝐵 + Σ 𝑘 ∈ 𝐴 - 𝐶 ) = ( Σ 𝑘 ∈ 𝐴 𝐵 + - Σ 𝑘 ∈ 𝐴 𝐶 ) )
8	5 7	eqtrd	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 ( 𝐵 + - 𝐶 ) = ( Σ 𝑘 ∈ 𝐴 𝐵 + - Σ 𝑘 ∈ 𝐴 𝐶 ) )
9	2 3	negsubd	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( 𝐵 + - 𝐶 ) = ( 𝐵 − 𝐶 ) )
10	9	sumeq2dv	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 ( 𝐵 + - 𝐶 ) = Σ 𝑘 ∈ 𝐴 ( 𝐵 − 𝐶 ) )
11	1 2	fsumcl	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐵 ∈ ℂ )
12	1 3	fsumcl	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 𝐶 ∈ ℂ )
13	11 12	negsubd	⊢ ( 𝜑 → ( Σ 𝑘 ∈ 𝐴 𝐵 + - Σ 𝑘 ∈ 𝐴 𝐶 ) = ( Σ 𝑘 ∈ 𝐴 𝐵 − Σ 𝑘 ∈ 𝐴 𝐶 ) )
14	8 10 13	3eqtr3d	⊢ ( 𝜑 → Σ 𝑘 ∈ 𝐴 ( 𝐵 − 𝐶 ) = ( Σ 𝑘 ∈ 𝐴 𝐵 − Σ 𝑘 ∈ 𝐴 𝐶 ) )