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Metamath Proof Explorer

Theorem fz1sump1

Description: Add one more term to a sum. Special case of fsump1 generalized to N e. NN0 . (Contributed by SN, 22-Mar-2025)

Ref Expression
Hypotheses fz1sump1.n ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
fz1sump1.a ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → 𝐴 ∈ ℂ )
fz1sump1.s ⊢ ( 𝑘 = ( 𝑁 + 1 ) → 𝐴 = 𝐵 )
Assertion fz1sump1 ( 𝜑 → Σ 𝑘 ∈ ( 1 ... ( 𝑁 + 1 ) ) 𝐴 = ( Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝐴 + 𝐵 ) )

Proof

Step Hyp Ref Expression
1 fz1sump1.n ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
2 fz1sump1.a ⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 1 ... ( 𝑁 + 1 ) ) ) → 𝐴 ∈ ℂ )
3 fz1sump1.s ⊢ ( 𝑘 = ( 𝑁 + 1 ) → 𝐴 = 𝐵 )
4 nn0p1nn ⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 + 1 ) ∈ ℕ )
5 1 4 syl ⊢ ( 𝜑 → ( 𝑁 + 1 ) ∈ ℕ )
6 nnuz ⊢ ℕ = ( ℤ≥ ‘ 1 )
7 5 6 eleqtrdi ⊢ ( 𝜑 → ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 1 ) )
8 7 2 3 fsumm1 ⊢ ( 𝜑 → Σ 𝑘 ∈ ( 1 ... ( 𝑁 + 1 ) ) 𝐴 = ( Σ 𝑘 ∈ ( 1 ... ( ( 𝑁 + 1 ) − 1 ) ) 𝐴 + 𝐵 ) )
9 1 nn0cnd ⊢ ( 𝜑 → 𝑁 ∈ ℂ )
10 1cnd ⊢ ( 𝜑 → 1 ∈ ℂ )
11 9 10 pncand ⊢ ( 𝜑 → ( ( 𝑁 + 1 ) − 1 ) = 𝑁 )
12 11 oveq2d ⊢ ( 𝜑 → ( 1 ... ( ( 𝑁 + 1 ) − 1 ) ) = ( 1 ... 𝑁 ) )
13 12 sumeq1d ⊢ ( 𝜑 → Σ 𝑘 ∈ ( 1 ... ( ( 𝑁 + 1 ) − 1 ) ) 𝐴 = Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝐴 )
14 13 oveq1d ⊢ ( 𝜑 → ( Σ 𝑘 ∈ ( 1 ... ( ( 𝑁 + 1 ) − 1 ) ) 𝐴 + 𝐵 ) = ( Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝐴 + 𝐵 ) )
15 8 14 eqtrd ⊢ ( 𝜑 → Σ 𝑘 ∈ ( 1 ... ( 𝑁 + 1 ) ) 𝐴 = ( Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝐴 + 𝐵 ) )