Theorem fzosump1

Description: Separate out the last term in a finite sum. (Contributed by Mario Carneiro, 13-Apr-2016)

	Ref	Expression
Hypotheses	fsumm1.1	⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
	fsumm1.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → 𝐴 ∈ ℂ )
	fsumm1.3	⊢ ( 𝑘 = 𝑁 → 𝐴 = 𝐵 )
Assertion	fzosump1	⊢ ( 𝜑 → Σ 𝑘 ∈ ( 𝑀 ..^ ( 𝑁 + 1 ) ) 𝐴 = ( Σ 𝑘 ∈ ( 𝑀 ..^ 𝑁 ) 𝐴 + 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		fsumm1.1	⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
2		fsumm1.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → 𝐴 ∈ ℂ )
3		fsumm1.3	⊢ ( 𝑘 = 𝑁 → 𝐴 = 𝐵 )
4		eluzelz	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑁 ∈ ℤ )
5	1 4	syl	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
6		fzoval	⊢ ( 𝑁 ∈ ℤ → ( 𝑀 ..^ 𝑁 ) = ( 𝑀 ... ( 𝑁 − 1 ) ) )
7	5 6	syl	⊢ ( 𝜑 → ( 𝑀 ..^ 𝑁 ) = ( 𝑀 ... ( 𝑁 − 1 ) ) )
8	7	sumeq1d	⊢ ( 𝜑 → Σ 𝑘 ∈ ( 𝑀 ..^ 𝑁 ) 𝐴 = Σ 𝑘 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) 𝐴 )
9	8	oveq1d	⊢ ( 𝜑 → ( Σ 𝑘 ∈ ( 𝑀 ..^ 𝑁 ) 𝐴 + 𝐵 ) = ( Σ 𝑘 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) 𝐴 + 𝐵 ) )
10	1 2 3	fsumm1	⊢ ( 𝜑 → Σ 𝑘 ∈ ( 𝑀 ... 𝑁 ) 𝐴 = ( Σ 𝑘 ∈ ( 𝑀 ... ( 𝑁 − 1 ) ) 𝐴 + 𝐵 ) )
11		fzval3	⊢ ( 𝑁 ∈ ℤ → ( 𝑀 ... 𝑁 ) = ( 𝑀 ..^ ( 𝑁 + 1 ) ) )
12	5 11	syl	⊢ ( 𝜑 → ( 𝑀 ... 𝑁 ) = ( 𝑀 ..^ ( 𝑁 + 1 ) ) )
13	12	sumeq1d	⊢ ( 𝜑 → Σ 𝑘 ∈ ( 𝑀 ... 𝑁 ) 𝐴 = Σ 𝑘 ∈ ( 𝑀 ..^ ( 𝑁 + 1 ) ) 𝐴 )
14	9 10 13	3eqtr2rd	⊢ ( 𝜑 → Σ 𝑘 ∈ ( 𝑀 ..^ ( 𝑁 + 1 ) ) 𝐴 = ( Σ 𝑘 ∈ ( 𝑀 ..^ 𝑁 ) 𝐴 + 𝐵 ) )