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Metamath Proof Explorer

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 ) )