Step |
Hyp |
Ref |
Expression |
1 |
|
oveq2 |
⊢ ( 𝑥 = 0 → ( 1 ... 𝑥 ) = ( 1 ... 0 ) ) |
2 |
1
|
sumeq1d |
⊢ ( 𝑥 = 0 → Σ 𝑘 ∈ ( 1 ... 𝑥 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑘 ∈ ( 1 ... 0 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) ) |
3 |
1
|
sumeq1d |
⊢ ( 𝑥 = 0 → Σ 𝑘 ∈ ( 1 ... 𝑥 ) 𝑘 = Σ 𝑘 ∈ ( 1 ... 0 ) 𝑘 ) |
4 |
3
|
oveq2d |
⊢ ( 𝑥 = 0 → ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑥 ) 𝑘 ) = ( 1 ... Σ 𝑘 ∈ ( 1 ... 0 ) 𝑘 ) ) |
5 |
4
|
sumeq1d |
⊢ ( 𝑥 = 0 → Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑥 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 0 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ) |
6 |
2 5
|
eqeq12d |
⊢ ( 𝑥 = 0 → ( Σ 𝑘 ∈ ( 1 ... 𝑥 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑥 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ↔ Σ 𝑘 ∈ ( 1 ... 0 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 0 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ) ) |
7 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 1 ... 𝑥 ) = ( 1 ... 𝑦 ) ) |
8 |
7
|
sumeq1d |
⊢ ( 𝑥 = 𝑦 → Σ 𝑘 ∈ ( 1 ... 𝑥 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑘 ∈ ( 1 ... 𝑦 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) ) |
9 |
7
|
sumeq1d |
⊢ ( 𝑥 = 𝑦 → Σ 𝑘 ∈ ( 1 ... 𝑥 ) 𝑘 = Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) |
10 |
9
|
oveq2d |
⊢ ( 𝑥 = 𝑦 → ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑥 ) 𝑘 ) = ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ) |
11 |
10
|
sumeq1d |
⊢ ( 𝑥 = 𝑦 → Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑥 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ) |
12 |
8 11
|
eqeq12d |
⊢ ( 𝑥 = 𝑦 → ( Σ 𝑘 ∈ ( 1 ... 𝑥 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑥 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ↔ Σ 𝑘 ∈ ( 1 ... 𝑦 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ) ) |
13 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 1 ... 𝑥 ) = ( 1 ... ( 𝑦 + 1 ) ) ) |
14 |
13
|
sumeq1d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → Σ 𝑘 ∈ ( 1 ... 𝑥 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) ) |
15 |
13
|
sumeq1d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → Σ 𝑘 ∈ ( 1 ... 𝑥 ) 𝑘 = Σ 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) 𝑘 ) |
16 |
15
|
oveq2d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑥 ) 𝑘 ) = ( 1 ... Σ 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) 𝑘 ) ) |
17 |
16
|
sumeq1d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑥 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ) |
18 |
14 17
|
eqeq12d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( Σ 𝑘 ∈ ( 1 ... 𝑥 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑥 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ↔ Σ 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ) ) |
19 |
|
oveq2 |
⊢ ( 𝑥 = 𝑁 → ( 1 ... 𝑥 ) = ( 1 ... 𝑁 ) ) |
20 |
19
|
sumeq1d |
⊢ ( 𝑥 = 𝑁 → Σ 𝑘 ∈ ( 1 ... 𝑥 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑘 ∈ ( 1 ... 𝑁 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) ) |
21 |
19
|
sumeq1d |
⊢ ( 𝑥 = 𝑁 → Σ 𝑘 ∈ ( 1 ... 𝑥 ) 𝑘 = Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝑘 ) |
22 |
21
|
oveq2d |
⊢ ( 𝑥 = 𝑁 → ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑥 ) 𝑘 ) = ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝑘 ) ) |
23 |
22
|
sumeq1d |
⊢ ( 𝑥 = 𝑁 → Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑥 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ) |
24 |
20 23
|
eqeq12d |
⊢ ( 𝑥 = 𝑁 → ( Σ 𝑘 ∈ ( 1 ... 𝑥 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑥 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ↔ Σ 𝑘 ∈ ( 1 ... 𝑁 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ) ) |
25 |
|
sum0 |
⊢ Σ 𝑘 ∈ ∅ Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = 0 |
26 |
|
sum0 |
⊢ Σ 𝑚 ∈ ∅ ( ( 2 · 𝑚 ) − 1 ) = 0 |
27 |
25 26
|
eqtr4i |
⊢ Σ 𝑘 ∈ ∅ Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ∅ ( ( 2 · 𝑚 ) − 1 ) |
28 |
|
fz10 |
⊢ ( 1 ... 0 ) = ∅ |
29 |
28
|
sumeq1i |
⊢ Σ 𝑘 ∈ ( 1 ... 0 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑘 ∈ ∅ Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) |
30 |
28
|
sumeq1i |
⊢ Σ 𝑘 ∈ ( 1 ... 0 ) 𝑘 = Σ 𝑘 ∈ ∅ 𝑘 |
31 |
|
sum0 |
⊢ Σ 𝑘 ∈ ∅ 𝑘 = 0 |
32 |
30 31
|
eqtri |
⊢ Σ 𝑘 ∈ ( 1 ... 0 ) 𝑘 = 0 |
33 |
32
|
oveq2i |
⊢ ( 1 ... Σ 𝑘 ∈ ( 1 ... 0 ) 𝑘 ) = ( 1 ... 0 ) |
34 |
33 28
|
eqtri |
⊢ ( 1 ... Σ 𝑘 ∈ ( 1 ... 0 ) 𝑘 ) = ∅ |
35 |
34
|
sumeq1i |
⊢ Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 0 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) = Σ 𝑚 ∈ ∅ ( ( 2 · 𝑚 ) − 1 ) |
36 |
27 29 35
|
3eqtr4i |
⊢ Σ 𝑘 ∈ ( 1 ... 0 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 0 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) |
37 |
|
simpr |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ Σ 𝑘 ∈ ( 1 ... 𝑦 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ) → Σ 𝑘 ∈ ( 1 ... 𝑦 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ) |
38 |
|
fzfid |
⊢ ( 𝑦 ∈ ℕ0 → ( 1 ... 𝑦 ) ∈ Fin ) |
39 |
|
elfznn |
⊢ ( 𝑘 ∈ ( 1 ... 𝑦 ) → 𝑘 ∈ ℕ ) |
40 |
39
|
adantl |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑘 ∈ ( 1 ... 𝑦 ) ) → 𝑘 ∈ ℕ ) |
41 |
40
|
nnnn0d |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑘 ∈ ( 1 ... 𝑦 ) ) → 𝑘 ∈ ℕ0 ) |
42 |
38 41
|
fsumnn0cl |
⊢ ( 𝑦 ∈ ℕ0 → Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ∈ ℕ0 ) |
43 |
42
|
nn0zd |
⊢ ( 𝑦 ∈ ℕ0 → Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ∈ ℤ ) |
44 |
|
nn0p1nn |
⊢ ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ∈ ℕ0 → ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ∈ ℕ ) |
45 |
42 44
|
syl |
⊢ ( 𝑦 ∈ ℕ0 → ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ∈ ℕ ) |
46 |
45
|
nnzd |
⊢ ( 𝑦 ∈ ℕ0 → ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ∈ ℤ ) |
47 |
|
peano2nn0 |
⊢ ( 𝑦 ∈ ℕ0 → ( 𝑦 + 1 ) ∈ ℕ0 ) |
48 |
47
|
nn0zd |
⊢ ( 𝑦 ∈ ℕ0 → ( 𝑦 + 1 ) ∈ ℤ ) |
49 |
43 48
|
zaddcld |
⊢ ( 𝑦 ∈ ℕ0 → ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ∈ ℤ ) |
50 |
|
2cnd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ) → 2 ∈ ℂ ) |
51 |
|
elfzelz |
⊢ ( 𝑚 ∈ ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) → 𝑚 ∈ ℤ ) |
52 |
51
|
zcnd |
⊢ ( 𝑚 ∈ ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) → 𝑚 ∈ ℂ ) |
53 |
52
|
adantl |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ) → 𝑚 ∈ ℂ ) |
54 |
50 53
|
mulcld |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ) → ( 2 · 𝑚 ) ∈ ℂ ) |
55 |
|
1cnd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ) → 1 ∈ ℂ ) |
56 |
54 55
|
subcld |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ) → ( ( 2 · 𝑚 ) − 1 ) ∈ ℂ ) |
57 |
|
oveq2 |
⊢ ( 𝑚 = ( 𝑙 + Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) → ( 2 · 𝑚 ) = ( 2 · ( 𝑙 + Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ) ) |
58 |
57
|
oveq1d |
⊢ ( 𝑚 = ( 𝑙 + Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) → ( ( 2 · 𝑚 ) − 1 ) = ( ( 2 · ( 𝑙 + Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ) − 1 ) ) |
59 |
43 46 49 56 58
|
fsumshftm |
⊢ ( 𝑦 ∈ ℕ0 → Σ 𝑚 ∈ ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ( ( 2 · 𝑚 ) − 1 ) = Σ 𝑙 ∈ ( ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) − Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ... ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) − Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ) ( ( 2 · ( 𝑙 + Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ) − 1 ) ) |
60 |
|
elfzelz |
⊢ ( 𝑘 ∈ ( 1 ... 𝑦 ) → 𝑘 ∈ ℤ ) |
61 |
60
|
adantl |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑘 ∈ ( 1 ... 𝑦 ) ) → 𝑘 ∈ ℤ ) |
62 |
61
|
zred |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑘 ∈ ( 1 ... 𝑦 ) ) → 𝑘 ∈ ℝ ) |
63 |
38 62
|
fsumrecl |
⊢ ( 𝑦 ∈ ℕ0 → Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ∈ ℝ ) |
64 |
63
|
recnd |
⊢ ( 𝑦 ∈ ℕ0 → Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ∈ ℂ ) |
65 |
|
1cnd |
⊢ ( 𝑦 ∈ ℕ0 → 1 ∈ ℂ ) |
66 |
64 65
|
pncan2d |
⊢ ( 𝑦 ∈ ℕ0 → ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) − Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) = 1 ) |
67 |
47
|
nn0cnd |
⊢ ( 𝑦 ∈ ℕ0 → ( 𝑦 + 1 ) ∈ ℂ ) |
68 |
64 67
|
pncan2d |
⊢ ( 𝑦 ∈ ℕ0 → ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) − Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) = ( 𝑦 + 1 ) ) |
69 |
66 68
|
oveq12d |
⊢ ( 𝑦 ∈ ℕ0 → ( ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) − Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ... ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) − Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ) = ( 1 ... ( 𝑦 + 1 ) ) ) |
70 |
|
elfzelz |
⊢ ( 𝑙 ∈ ( ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) − Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ... ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) − Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ) → 𝑙 ∈ ℤ ) |
71 |
70
|
zcnd |
⊢ ( 𝑙 ∈ ( ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) − Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ... ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) − Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ) → 𝑙 ∈ ℂ ) |
72 |
|
2cnd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ ) → 2 ∈ ℂ ) |
73 |
|
simpr |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ ) → 𝑙 ∈ ℂ ) |
74 |
64
|
adantr |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ ) → Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ∈ ℂ ) |
75 |
72 73 74
|
adddid |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ ) → ( 2 · ( 𝑙 + Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ) = ( ( 2 · 𝑙 ) + ( 2 · Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ) ) |
76 |
75
|
oveq1d |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ ) → ( ( 2 · ( 𝑙 + Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ) − 1 ) = ( ( ( 2 · 𝑙 ) + ( 2 · Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ) − 1 ) ) |
77 |
72 73
|
mulcld |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ ) → ( 2 · 𝑙 ) ∈ ℂ ) |
78 |
72 74
|
mulcld |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ ) → ( 2 · Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ∈ ℂ ) |
79 |
|
1cnd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ ) → 1 ∈ ℂ ) |
80 |
77 78 79
|
addsubassd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ ) → ( ( ( 2 · 𝑙 ) + ( 2 · Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ) − 1 ) = ( ( 2 · 𝑙 ) + ( ( 2 · Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) − 1 ) ) ) |
81 |
77 78 79
|
addsub12d |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ ) → ( ( 2 · 𝑙 ) + ( ( 2 · Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) − 1 ) ) = ( ( 2 · Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) ) |
82 |
|
arisum |
⊢ ( 𝑦 ∈ ℕ0 → Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 = ( ( ( 𝑦 ↑ 2 ) + 𝑦 ) / 2 ) ) |
83 |
82
|
oveq2d |
⊢ ( 𝑦 ∈ ℕ0 → ( 2 · Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) = ( 2 · ( ( ( 𝑦 ↑ 2 ) + 𝑦 ) / 2 ) ) ) |
84 |
|
nn0cn |
⊢ ( 𝑦 ∈ ℕ0 → 𝑦 ∈ ℂ ) |
85 |
84
|
sqcld |
⊢ ( 𝑦 ∈ ℕ0 → ( 𝑦 ↑ 2 ) ∈ ℂ ) |
86 |
85 84
|
addcld |
⊢ ( 𝑦 ∈ ℕ0 → ( ( 𝑦 ↑ 2 ) + 𝑦 ) ∈ ℂ ) |
87 |
|
2cnd |
⊢ ( 𝑦 ∈ ℕ0 → 2 ∈ ℂ ) |
88 |
|
2ne0 |
⊢ 2 ≠ 0 |
89 |
88
|
a1i |
⊢ ( 𝑦 ∈ ℕ0 → 2 ≠ 0 ) |
90 |
86 87 89
|
divcan2d |
⊢ ( 𝑦 ∈ ℕ0 → ( 2 · ( ( ( 𝑦 ↑ 2 ) + 𝑦 ) / 2 ) ) = ( ( 𝑦 ↑ 2 ) + 𝑦 ) ) |
91 |
|
binom21 |
⊢ ( 𝑦 ∈ ℂ → ( ( 𝑦 + 1 ) ↑ 2 ) = ( ( ( 𝑦 ↑ 2 ) + ( 2 · 𝑦 ) ) + 1 ) ) |
92 |
84 91
|
syl |
⊢ ( 𝑦 ∈ ℕ0 → ( ( 𝑦 + 1 ) ↑ 2 ) = ( ( ( 𝑦 ↑ 2 ) + ( 2 · 𝑦 ) ) + 1 ) ) |
93 |
92
|
oveq1d |
⊢ ( 𝑦 ∈ ℕ0 → ( ( ( 𝑦 + 1 ) ↑ 2 ) − ( 𝑦 + 1 ) ) = ( ( ( ( 𝑦 ↑ 2 ) + ( 2 · 𝑦 ) ) + 1 ) − ( 𝑦 + 1 ) ) ) |
94 |
87 84
|
mulcld |
⊢ ( 𝑦 ∈ ℕ0 → ( 2 · 𝑦 ) ∈ ℂ ) |
95 |
85 94
|
addcld |
⊢ ( 𝑦 ∈ ℕ0 → ( ( 𝑦 ↑ 2 ) + ( 2 · 𝑦 ) ) ∈ ℂ ) |
96 |
95 84 65
|
pnpcan2d |
⊢ ( 𝑦 ∈ ℕ0 → ( ( ( ( 𝑦 ↑ 2 ) + ( 2 · 𝑦 ) ) + 1 ) − ( 𝑦 + 1 ) ) = ( ( ( 𝑦 ↑ 2 ) + ( 2 · 𝑦 ) ) − 𝑦 ) ) |
97 |
85 94 84
|
addsubassd |
⊢ ( 𝑦 ∈ ℕ0 → ( ( ( 𝑦 ↑ 2 ) + ( 2 · 𝑦 ) ) − 𝑦 ) = ( ( 𝑦 ↑ 2 ) + ( ( 2 · 𝑦 ) − 𝑦 ) ) ) |
98 |
84
|
2timesd |
⊢ ( 𝑦 ∈ ℕ0 → ( 2 · 𝑦 ) = ( 𝑦 + 𝑦 ) ) |
99 |
84 84 98
|
mvrladdd |
⊢ ( 𝑦 ∈ ℕ0 → ( ( 2 · 𝑦 ) − 𝑦 ) = 𝑦 ) |
100 |
99
|
oveq2d |
⊢ ( 𝑦 ∈ ℕ0 → ( ( 𝑦 ↑ 2 ) + ( ( 2 · 𝑦 ) − 𝑦 ) ) = ( ( 𝑦 ↑ 2 ) + 𝑦 ) ) |
101 |
97 100
|
eqtrd |
⊢ ( 𝑦 ∈ ℕ0 → ( ( ( 𝑦 ↑ 2 ) + ( 2 · 𝑦 ) ) − 𝑦 ) = ( ( 𝑦 ↑ 2 ) + 𝑦 ) ) |
102 |
93 96 101
|
3eqtrrd |
⊢ ( 𝑦 ∈ ℕ0 → ( ( 𝑦 ↑ 2 ) + 𝑦 ) = ( ( ( 𝑦 + 1 ) ↑ 2 ) − ( 𝑦 + 1 ) ) ) |
103 |
83 90 102
|
3eqtrd |
⊢ ( 𝑦 ∈ ℕ0 → ( 2 · Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) = ( ( ( 𝑦 + 1 ) ↑ 2 ) − ( 𝑦 + 1 ) ) ) |
104 |
103
|
adantr |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ ) → ( 2 · Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) = ( ( ( 𝑦 + 1 ) ↑ 2 ) − ( 𝑦 + 1 ) ) ) |
105 |
104
|
oveq1d |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ ) → ( ( 2 · Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = ( ( ( ( 𝑦 + 1 ) ↑ 2 ) − ( 𝑦 + 1 ) ) + ( ( 2 · 𝑙 ) − 1 ) ) ) |
106 |
81 105
|
eqtrd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ ) → ( ( 2 · 𝑙 ) + ( ( 2 · Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) − 1 ) ) = ( ( ( ( 𝑦 + 1 ) ↑ 2 ) − ( 𝑦 + 1 ) ) + ( ( 2 · 𝑙 ) − 1 ) ) ) |
107 |
76 80 106
|
3eqtrd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ℂ ) → ( ( 2 · ( 𝑙 + Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ) − 1 ) = ( ( ( ( 𝑦 + 1 ) ↑ 2 ) − ( 𝑦 + 1 ) ) + ( ( 2 · 𝑙 ) − 1 ) ) ) |
108 |
71 107
|
sylan2 |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑙 ∈ ( ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) − Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ... ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) − Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ) ) → ( ( 2 · ( 𝑙 + Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ) − 1 ) = ( ( ( ( 𝑦 + 1 ) ↑ 2 ) − ( 𝑦 + 1 ) ) + ( ( 2 · 𝑙 ) − 1 ) ) ) |
109 |
69 108
|
sumeq12dv |
⊢ ( 𝑦 ∈ ℕ0 → Σ 𝑙 ∈ ( ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) − Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ... ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) − Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ) ( ( 2 · ( 𝑙 + Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ) − 1 ) = Σ 𝑙 ∈ ( 1 ... ( 𝑦 + 1 ) ) ( ( ( ( 𝑦 + 1 ) ↑ 2 ) − ( 𝑦 + 1 ) ) + ( ( 2 · 𝑙 ) − 1 ) ) ) |
110 |
59 109
|
eqtr2d |
⊢ ( 𝑦 ∈ ℕ0 → Σ 𝑙 ∈ ( 1 ... ( 𝑦 + 1 ) ) ( ( ( ( 𝑦 + 1 ) ↑ 2 ) − ( 𝑦 + 1 ) ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ( ( 2 · 𝑚 ) − 1 ) ) |
111 |
110
|
adantr |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ Σ 𝑘 ∈ ( 1 ... 𝑦 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ) → Σ 𝑙 ∈ ( 1 ... ( 𝑦 + 1 ) ) ( ( ( ( 𝑦 + 1 ) ↑ 2 ) − ( 𝑦 + 1 ) ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ( ( 2 · 𝑚 ) − 1 ) ) |
112 |
37 111
|
oveq12d |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ Σ 𝑘 ∈ ( 1 ... 𝑦 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ) → ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) + Σ 𝑙 ∈ ( 1 ... ( 𝑦 + 1 ) ) ( ( ( ( 𝑦 + 1 ) ↑ 2 ) − ( 𝑦 + 1 ) ) + ( ( 2 · 𝑙 ) − 1 ) ) ) = ( Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) + Σ 𝑚 ∈ ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ( ( 2 · 𝑚 ) − 1 ) ) ) |
113 |
|
id |
⊢ ( 𝑦 ∈ ℕ0 → 𝑦 ∈ ℕ0 ) |
114 |
|
fzfid |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) ) → ( 1 ... 𝑘 ) ∈ Fin ) |
115 |
|
elfzelz |
⊢ ( 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) → 𝑘 ∈ ℤ ) |
116 |
115
|
zcnd |
⊢ ( 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) → 𝑘 ∈ ℂ ) |
117 |
116
|
sqcld |
⊢ ( 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) → ( 𝑘 ↑ 2 ) ∈ ℂ ) |
118 |
117 116
|
subcld |
⊢ ( 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) → ( ( 𝑘 ↑ 2 ) − 𝑘 ) ∈ ℂ ) |
119 |
|
2cnd |
⊢ ( 𝑙 ∈ ( 1 ... 𝑘 ) → 2 ∈ ℂ ) |
120 |
|
elfzelz |
⊢ ( 𝑙 ∈ ( 1 ... 𝑘 ) → 𝑙 ∈ ℤ ) |
121 |
120
|
zcnd |
⊢ ( 𝑙 ∈ ( 1 ... 𝑘 ) → 𝑙 ∈ ℂ ) |
122 |
119 121
|
mulcld |
⊢ ( 𝑙 ∈ ( 1 ... 𝑘 ) → ( 2 · 𝑙 ) ∈ ℂ ) |
123 |
|
1cnd |
⊢ ( 𝑙 ∈ ( 1 ... 𝑘 ) → 1 ∈ ℂ ) |
124 |
122 123
|
subcld |
⊢ ( 𝑙 ∈ ( 1 ... 𝑘 ) → ( ( 2 · 𝑙 ) − 1 ) ∈ ℂ ) |
125 |
|
addcl |
⊢ ( ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) ∈ ℂ ∧ ( ( 2 · 𝑙 ) − 1 ) ∈ ℂ ) → ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) ∈ ℂ ) |
126 |
118 124 125
|
syl2an |
⊢ ( ( 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) ∧ 𝑙 ∈ ( 1 ... 𝑘 ) ) → ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) ∈ ℂ ) |
127 |
126
|
adantll |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) ) ∧ 𝑙 ∈ ( 1 ... 𝑘 ) ) → ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) ∈ ℂ ) |
128 |
114 127
|
fsumcl |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) ) → Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) ∈ ℂ ) |
129 |
|
oveq2 |
⊢ ( 𝑘 = ( 𝑦 + 1 ) → ( 1 ... 𝑘 ) = ( 1 ... ( 𝑦 + 1 ) ) ) |
130 |
|
oveq1 |
⊢ ( 𝑘 = ( 𝑦 + 1 ) → ( 𝑘 ↑ 2 ) = ( ( 𝑦 + 1 ) ↑ 2 ) ) |
131 |
|
id |
⊢ ( 𝑘 = ( 𝑦 + 1 ) → 𝑘 = ( 𝑦 + 1 ) ) |
132 |
130 131
|
oveq12d |
⊢ ( 𝑘 = ( 𝑦 + 1 ) → ( ( 𝑘 ↑ 2 ) − 𝑘 ) = ( ( ( 𝑦 + 1 ) ↑ 2 ) − ( 𝑦 + 1 ) ) ) |
133 |
132
|
oveq1d |
⊢ ( 𝑘 = ( 𝑦 + 1 ) → ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = ( ( ( ( 𝑦 + 1 ) ↑ 2 ) − ( 𝑦 + 1 ) ) + ( ( 2 · 𝑙 ) − 1 ) ) ) |
134 |
133
|
adantr |
⊢ ( ( 𝑘 = ( 𝑦 + 1 ) ∧ 𝑙 ∈ ( 1 ... 𝑘 ) ) → ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = ( ( ( ( 𝑦 + 1 ) ↑ 2 ) − ( 𝑦 + 1 ) ) + ( ( 2 · 𝑙 ) − 1 ) ) ) |
135 |
129 134
|
sumeq12dv |
⊢ ( 𝑘 = ( 𝑦 + 1 ) → Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑙 ∈ ( 1 ... ( 𝑦 + 1 ) ) ( ( ( ( 𝑦 + 1 ) ↑ 2 ) − ( 𝑦 + 1 ) ) + ( ( 2 · 𝑙 ) − 1 ) ) ) |
136 |
113 128 135
|
fz1sump1 |
⊢ ( 𝑦 ∈ ℕ0 → Σ 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) + Σ 𝑙 ∈ ( 1 ... ( 𝑦 + 1 ) ) ( ( ( ( 𝑦 + 1 ) ↑ 2 ) − ( 𝑦 + 1 ) ) + ( ( 2 · 𝑙 ) − 1 ) ) ) ) |
137 |
136
|
adantr |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ Σ 𝑘 ∈ ( 1 ... 𝑦 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ) → Σ 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) + Σ 𝑙 ∈ ( 1 ... ( 𝑦 + 1 ) ) ( ( ( ( 𝑦 + 1 ) ↑ 2 ) − ( 𝑦 + 1 ) ) + ( ( 2 · 𝑙 ) − 1 ) ) ) ) |
138 |
116
|
adantl |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) ) → 𝑘 ∈ ℂ ) |
139 |
113 138 131
|
fz1sump1 |
⊢ ( 𝑦 ∈ ℕ0 → Σ 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) 𝑘 = ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) |
140 |
139
|
adantr |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ Σ 𝑘 ∈ ( 1 ... 𝑦 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ) → Σ 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) 𝑘 = ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) |
141 |
140
|
oveq2d |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ Σ 𝑘 ∈ ( 1 ... 𝑦 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ) → ( 1 ... Σ 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) 𝑘 ) = ( 1 ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ) |
142 |
141
|
sumeq1d |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ Σ 𝑘 ∈ ( 1 ... 𝑦 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ) → Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) = Σ 𝑚 ∈ ( 1 ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ( ( 2 · 𝑚 ) − 1 ) ) |
143 |
63
|
ltp1d |
⊢ ( 𝑦 ∈ ℕ0 → Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 < ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ) |
144 |
|
fzdisj |
⊢ ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 < ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) → ( ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ∩ ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ) = ∅ ) |
145 |
143 144
|
syl |
⊢ ( 𝑦 ∈ ℕ0 → ( ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ∩ ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ) = ∅ ) |
146 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
147 |
45 146
|
eleqtrdi |
⊢ ( 𝑦 ∈ ℕ0 → ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ∈ ( ℤ≥ ‘ 1 ) ) |
148 |
43
|
uzidd |
⊢ ( 𝑦 ∈ ℕ0 → Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ∈ ( ℤ≥ ‘ Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ) |
149 |
|
uzaddcl |
⊢ ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ∈ ( ℤ≥ ‘ Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ∧ ( 𝑦 + 1 ) ∈ ℕ0 ) → ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ∈ ( ℤ≥ ‘ Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ) |
150 |
148 47 149
|
syl2anc |
⊢ ( 𝑦 ∈ ℕ0 → ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ∈ ( ℤ≥ ‘ Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ) |
151 |
|
fzsplit2 |
⊢ ( ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ∈ ( ℤ≥ ‘ 1 ) ∧ ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ∈ ( ℤ≥ ‘ Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ) → ( 1 ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) = ( ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ∪ ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ) ) |
152 |
147 150 151
|
syl2anc |
⊢ ( 𝑦 ∈ ℕ0 → ( 1 ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) = ( ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ∪ ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ) ) |
153 |
|
fzfid |
⊢ ( 𝑦 ∈ ℕ0 → ( 1 ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ∈ Fin ) |
154 |
|
2cnd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ( 1 ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ) → 2 ∈ ℂ ) |
155 |
|
elfzelz |
⊢ ( 𝑚 ∈ ( 1 ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) → 𝑚 ∈ ℤ ) |
156 |
155
|
zcnd |
⊢ ( 𝑚 ∈ ( 1 ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) → 𝑚 ∈ ℂ ) |
157 |
156
|
adantl |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ( 1 ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ) → 𝑚 ∈ ℂ ) |
158 |
154 157
|
mulcld |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ( 1 ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ) → ( 2 · 𝑚 ) ∈ ℂ ) |
159 |
|
1cnd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ( 1 ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ) → 1 ∈ ℂ ) |
160 |
158 159
|
subcld |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝑚 ∈ ( 1 ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ) → ( ( 2 · 𝑚 ) − 1 ) ∈ ℂ ) |
161 |
145 152 153 160
|
fsumsplit |
⊢ ( 𝑦 ∈ ℕ0 → Σ 𝑚 ∈ ( 1 ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ( ( 2 · 𝑚 ) − 1 ) = ( Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) + Σ 𝑚 ∈ ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ( ( 2 · 𝑚 ) − 1 ) ) ) |
162 |
161
|
adantr |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ Σ 𝑘 ∈ ( 1 ... 𝑦 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ) → Σ 𝑚 ∈ ( 1 ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ( ( 2 · 𝑚 ) − 1 ) = ( Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) + Σ 𝑚 ∈ ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ( ( 2 · 𝑚 ) − 1 ) ) ) |
163 |
142 162
|
eqtrd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ Σ 𝑘 ∈ ( 1 ... 𝑦 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ) → Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) = ( Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) + Σ 𝑚 ∈ ( ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + 1 ) ... ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 + ( 𝑦 + 1 ) ) ) ( ( 2 · 𝑚 ) − 1 ) ) ) |
164 |
112 137 163
|
3eqtr4d |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ Σ 𝑘 ∈ ( 1 ... 𝑦 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ) → Σ 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ) |
165 |
164
|
ex |
⊢ ( 𝑦 ∈ ℕ0 → ( Σ 𝑘 ∈ ( 1 ... 𝑦 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑦 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) → Σ 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... ( 𝑦 + 1 ) ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ) ) |
166 |
6 12 18 24 36 165
|
nn0ind |
⊢ ( 𝑁 ∈ ℕ0 → Σ 𝑘 ∈ ( 1 ... 𝑁 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) ) |
167 |
|
fz1ssnn |
⊢ ( 1 ... 𝑁 ) ⊆ ℕ |
168 |
|
nnssnn0 |
⊢ ℕ ⊆ ℕ0 |
169 |
167 168
|
sstri |
⊢ ( 1 ... 𝑁 ) ⊆ ℕ0 |
170 |
169
|
a1i |
⊢ ( 𝑁 ∈ ℕ0 → ( 1 ... 𝑁 ) ⊆ ℕ0 ) |
171 |
170
|
sselda |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ( 1 ... 𝑁 ) ) → 𝑘 ∈ ℕ0 ) |
172 |
|
nicomachus |
⊢ ( 𝑘 ∈ ℕ0 → Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = ( 𝑘 ↑ 3 ) ) |
173 |
171 172
|
syl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑘 ∈ ( 1 ... 𝑁 ) ) → Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = ( 𝑘 ↑ 3 ) ) |
174 |
173
|
sumeq2dv |
⊢ ( 𝑁 ∈ ℕ0 → Σ 𝑘 ∈ ( 1 ... 𝑁 ) Σ 𝑙 ∈ ( 1 ... 𝑘 ) ( ( ( 𝑘 ↑ 2 ) − 𝑘 ) + ( ( 2 · 𝑙 ) − 1 ) ) = Σ 𝑘 ∈ ( 1 ... 𝑁 ) ( 𝑘 ↑ 3 ) ) |
175 |
|
fzfid |
⊢ ( 𝑁 ∈ ℕ0 → ( 1 ... 𝑁 ) ∈ Fin ) |
176 |
175 171
|
fsumnn0cl |
⊢ ( 𝑁 ∈ ℕ0 → Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝑘 ∈ ℕ0 ) |
177 |
|
oddnumth |
⊢ ( Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝑘 ∈ ℕ0 → Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) = ( Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝑘 ↑ 2 ) ) |
178 |
176 177
|
syl |
⊢ ( 𝑁 ∈ ℕ0 → Σ 𝑚 ∈ ( 1 ... Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝑘 ) ( ( 2 · 𝑚 ) − 1 ) = ( Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝑘 ↑ 2 ) ) |
179 |
166 174 178
|
3eqtr3d |
⊢ ( 𝑁 ∈ ℕ0 → Σ 𝑘 ∈ ( 1 ... 𝑁 ) ( 𝑘 ↑ 3 ) = ( Σ 𝑘 ∈ ( 1 ... 𝑁 ) 𝑘 ↑ 2 ) ) |