Theorem fsum1p

Description: Separate out the first term in a finite sum. (Contributed by NM, 3-Jan-2006) (Revised by Mario Carneiro, 23-Apr-2014)

	Ref	Expression
Hypotheses	fsumm1.1	⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
	fsumm1.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → 𝐴 ∈ ℂ )
	fsum1p.3	⊢ ( 𝑘 = 𝑀 → 𝐴 = 𝐵 )
Assertion	fsum1p	⊢ ( 𝜑 → Σ 𝑘 ∈ ( 𝑀 ... 𝑁 ) 𝐴 = ( 𝐵 + Σ 𝑘 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		fsumm1.1	⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
2		fsumm1.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → 𝐴 ∈ ℂ )
3		fsum1p.3	⊢ ( 𝑘 = 𝑀 → 𝐴 = 𝐵 )
4		eluzel2	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ℤ )
5	1 4	syl	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
6		fzsn	⊢ ( 𝑀 ∈ ℤ → ( 𝑀 ... 𝑀 ) = { 𝑀 } )
7	5 6	syl	⊢ ( 𝜑 → ( 𝑀 ... 𝑀 ) = { 𝑀 } )
8	7	ineq1d	⊢ ( 𝜑 → ( ( 𝑀 ... 𝑀 ) ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) = ( { 𝑀 } ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) )
9	5	zred	⊢ ( 𝜑 → 𝑀 ∈ ℝ )
10	9	ltp1d	⊢ ( 𝜑 → 𝑀 < ( 𝑀 + 1 ) )
11		fzdisj	⊢ ( 𝑀 < ( 𝑀 + 1 ) → ( ( 𝑀 ... 𝑀 ) ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) = ∅ )
12	10 11	syl	⊢ ( 𝜑 → ( ( 𝑀 ... 𝑀 ) ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) = ∅ )
13	8 12	eqtr3d	⊢ ( 𝜑 → ( { 𝑀 } ∩ ( ( 𝑀 + 1 ) ... 𝑁 ) ) = ∅ )
14		eluzfz1	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ( 𝑀 ... 𝑁 ) )
15	1 14	syl	⊢ ( 𝜑 → 𝑀 ∈ ( 𝑀 ... 𝑁 ) )
16		fzsplit	⊢ ( 𝑀 ∈ ( 𝑀 ... 𝑁 ) → ( 𝑀 ... 𝑁 ) = ( ( 𝑀 ... 𝑀 ) ∪ ( ( 𝑀 + 1 ) ... 𝑁 ) ) )
17	15 16	syl	⊢ ( 𝜑 → ( 𝑀 ... 𝑁 ) = ( ( 𝑀 ... 𝑀 ) ∪ ( ( 𝑀 + 1 ) ... 𝑁 ) ) )
18	7	uneq1d	⊢ ( 𝜑 → ( ( 𝑀 ... 𝑀 ) ∪ ( ( 𝑀 + 1 ) ... 𝑁 ) ) = ( { 𝑀 } ∪ ( ( 𝑀 + 1 ) ... 𝑁 ) ) )
19	17 18	eqtrd	⊢ ( 𝜑 → ( 𝑀 ... 𝑁 ) = ( { 𝑀 } ∪ ( ( 𝑀 + 1 ) ... 𝑁 ) ) )
20		fzfid	⊢ ( 𝜑 → ( 𝑀 ... 𝑁 ) ∈ Fin )
21	13 19 20 2	fsumsplit	⊢ ( 𝜑 → Σ 𝑘 ∈ ( 𝑀 ... 𝑁 ) 𝐴 = ( Σ 𝑘 ∈ { 𝑀 } 𝐴 + Σ 𝑘 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) 𝐴 ) )
22	3	eleq1d	⊢ ( 𝑘 = 𝑀 → ( 𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ ) )
23	2	ralrimiva	⊢ ( 𝜑 → ∀ 𝑘 ∈ ( 𝑀 ... 𝑁 ) 𝐴 ∈ ℂ )
24	22 23 15	rspcdva	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
25	3	sumsn	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝐵 ∈ ℂ ) → Σ 𝑘 ∈ { 𝑀 } 𝐴 = 𝐵 )
26	5 24 25	syl2anc	⊢ ( 𝜑 → Σ 𝑘 ∈ { 𝑀 } 𝐴 = 𝐵 )
27	26	oveq1d	⊢ ( 𝜑 → ( Σ 𝑘 ∈ { 𝑀 } 𝐴 + Σ 𝑘 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) 𝐴 ) = ( 𝐵 + Σ 𝑘 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) 𝐴 ) )
28	21 27	eqtrd	⊢ ( 𝜑 → Σ 𝑘 ∈ ( 𝑀 ... 𝑁 ) 𝐴 = ( 𝐵 + Σ 𝑘 ∈ ( ( 𝑀 + 1 ) ... 𝑁 ) 𝐴 ) )