Step |
Hyp |
Ref |
Expression |
1 |
|
fzfid |
⊢ ( 𝑁 ∈ ℕ0 → ( 1 ... 𝑁 ) ∈ Fin ) |
2 |
|
2cnd |
⊢ ( 𝑘 ∈ ( 1 ... 𝑁 ) → 2 ∈ ℂ ) |
3 |
|
elfznn |
⊢ ( 𝑘 ∈ ( 1 ... 𝑁 ) → 𝑘 ∈ ℕ ) |
4 |
3
|
nncnd |
⊢ ( 𝑘 ∈ ( 1 ... 𝑁 ) → 𝑘 ∈ ℂ ) |
5 |
2 4
|
mulcld |
⊢ ( 𝑘 ∈ ( 1 ... 𝑁 ) → ( 2 · 𝑘 ) ∈ ℂ ) |
6 |
5
|
adantl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ( 1 ... 𝑁 ) ) → ( 2 · 𝑘 ) ∈ ℂ ) |
7 |
|
1cnd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ( 1 ... 𝑁 ) ) → 1 ∈ ℂ ) |
8 |
1 6 7
|
fsumsub |
⊢ ( 𝑁 ∈ ℕ0 → Σ 𝑘 ∈ ( 1 ... 𝑁 ) ( ( 2 · 𝑘 ) − 1 ) = ( Σ 𝑘 ∈ ( 1 ... 𝑁 ) ( 2 · 𝑘 ) − Σ 𝑘 ∈ ( 1 ... 𝑁 ) 1 ) ) |
9 |
|
arisum |
⊢ ( 𝑁 ∈ ℕ0 → Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝑘 = ( ( ( 𝑁 ↑ 2 ) + 𝑁 ) / 2 ) ) |
10 |
9
|
oveq2d |
⊢ ( 𝑁 ∈ ℕ0 → ( 2 · Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝑘 ) = ( 2 · ( ( ( 𝑁 ↑ 2 ) + 𝑁 ) / 2 ) ) ) |
11 |
|
2cnd |
⊢ ( 𝑁 ∈ ℕ0 → 2 ∈ ℂ ) |
12 |
4
|
adantl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ( 1 ... 𝑁 ) ) → 𝑘 ∈ ℂ ) |
13 |
1 11 12
|
fsummulc2 |
⊢ ( 𝑁 ∈ ℕ0 → ( 2 · Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝑘 ) = Σ 𝑘 ∈ ( 1 ... 𝑁 ) ( 2 · 𝑘 ) ) |
14 |
|
nn0cn |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ ) |
15 |
14
|
sqcld |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 ↑ 2 ) ∈ ℂ ) |
16 |
15 14
|
addcld |
⊢ ( 𝑁 ∈ ℕ0 → ( ( 𝑁 ↑ 2 ) + 𝑁 ) ∈ ℂ ) |
17 |
|
2ne0 |
⊢ 2 ≠ 0 |
18 |
17
|
a1i |
⊢ ( 𝑁 ∈ ℕ0 → 2 ≠ 0 ) |
19 |
16 11 18
|
divcan2d |
⊢ ( 𝑁 ∈ ℕ0 → ( 2 · ( ( ( 𝑁 ↑ 2 ) + 𝑁 ) / 2 ) ) = ( ( 𝑁 ↑ 2 ) + 𝑁 ) ) |
20 |
10 13 19
|
3eqtr3d |
⊢ ( 𝑁 ∈ ℕ0 → Σ 𝑘 ∈ ( 1 ... 𝑁 ) ( 2 · 𝑘 ) = ( ( 𝑁 ↑ 2 ) + 𝑁 ) ) |
21 |
|
id |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0 ) |
22 |
|
1cnd |
⊢ ( 𝑁 ∈ ℕ0 → 1 ∈ ℂ ) |
23 |
21 22
|
fz1sumconst |
⊢ ( 𝑁 ∈ ℕ0 → Σ 𝑘 ∈ ( 1 ... 𝑁 ) 1 = ( 𝑁 · 1 ) ) |
24 |
14
|
mulridd |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 · 1 ) = 𝑁 ) |
25 |
23 24
|
eqtrd |
⊢ ( 𝑁 ∈ ℕ0 → Σ 𝑘 ∈ ( 1 ... 𝑁 ) 1 = 𝑁 ) |
26 |
20 25
|
oveq12d |
⊢ ( 𝑁 ∈ ℕ0 → ( Σ 𝑘 ∈ ( 1 ... 𝑁 ) ( 2 · 𝑘 ) − Σ 𝑘 ∈ ( 1 ... 𝑁 ) 1 ) = ( ( ( 𝑁 ↑ 2 ) + 𝑁 ) − 𝑁 ) ) |
27 |
15 14
|
pncand |
⊢ ( 𝑁 ∈ ℕ0 → ( ( ( 𝑁 ↑ 2 ) + 𝑁 ) − 𝑁 ) = ( 𝑁 ↑ 2 ) ) |
28 |
8 26 27
|
3eqtrd |
⊢ ( 𝑁 ∈ ℕ0 → Σ 𝑘 ∈ ( 1 ... 𝑁 ) ( ( 2 · 𝑘 ) − 1 ) = ( 𝑁 ↑ 2 ) ) |