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 ) ) |