Step |
Hyp |
Ref |
Expression |
1 |
|
leibpi.1 |
⊢ 𝐹 = ( 𝑛 ∈ ℕ0 ↦ ( ( - 1 ↑ 𝑛 ) / ( ( 2 · 𝑛 ) + 1 ) ) ) |
2 |
|
leibpilem2.2 |
⊢ 𝐺 = ( 𝑘 ∈ ℕ0 ↦ if ( ( 𝑘 = 0 ∨ 2 ∥ 𝑘 ) , 0 , ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) ) ) |
3 |
|
leibpilem2.3 |
⊢ 𝐴 ∈ V |
4 |
|
2cn |
⊢ 2 ∈ ℂ |
5 |
|
nn0cn |
⊢ ( 𝑛 ∈ ℕ0 → 𝑛 ∈ ℂ ) |
6 |
|
mulcl |
⊢ ( ( 2 ∈ ℂ ∧ 𝑛 ∈ ℂ ) → ( 2 · 𝑛 ) ∈ ℂ ) |
7 |
4 5 6
|
sylancr |
⊢ ( 𝑛 ∈ ℕ0 → ( 2 · 𝑛 ) ∈ ℂ ) |
8 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
9 |
|
pncan |
⊢ ( ( ( 2 · 𝑛 ) ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( ( 2 · 𝑛 ) + 1 ) − 1 ) = ( 2 · 𝑛 ) ) |
10 |
7 8 9
|
sylancl |
⊢ ( 𝑛 ∈ ℕ0 → ( ( ( 2 · 𝑛 ) + 1 ) − 1 ) = ( 2 · 𝑛 ) ) |
11 |
10
|
oveq1d |
⊢ ( 𝑛 ∈ ℕ0 → ( ( ( ( 2 · 𝑛 ) + 1 ) − 1 ) / 2 ) = ( ( 2 · 𝑛 ) / 2 ) ) |
12 |
|
2ne0 |
⊢ 2 ≠ 0 |
13 |
|
divcan3 |
⊢ ( ( 𝑛 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0 ) → ( ( 2 · 𝑛 ) / 2 ) = 𝑛 ) |
14 |
4 12 13
|
mp3an23 |
⊢ ( 𝑛 ∈ ℂ → ( ( 2 · 𝑛 ) / 2 ) = 𝑛 ) |
15 |
5 14
|
syl |
⊢ ( 𝑛 ∈ ℕ0 → ( ( 2 · 𝑛 ) / 2 ) = 𝑛 ) |
16 |
11 15
|
eqtrd |
⊢ ( 𝑛 ∈ ℕ0 → ( ( ( ( 2 · 𝑛 ) + 1 ) − 1 ) / 2 ) = 𝑛 ) |
17 |
16
|
oveq2d |
⊢ ( 𝑛 ∈ ℕ0 → ( - 1 ↑ ( ( ( ( 2 · 𝑛 ) + 1 ) − 1 ) / 2 ) ) = ( - 1 ↑ 𝑛 ) ) |
18 |
17
|
oveq1d |
⊢ ( 𝑛 ∈ ℕ0 → ( ( - 1 ↑ ( ( ( ( 2 · 𝑛 ) + 1 ) − 1 ) / 2 ) ) / ( ( 2 · 𝑛 ) + 1 ) ) = ( ( - 1 ↑ 𝑛 ) / ( ( 2 · 𝑛 ) + 1 ) ) ) |
19 |
18
|
mpteq2ia |
⊢ ( 𝑛 ∈ ℕ0 ↦ ( ( - 1 ↑ ( ( ( ( 2 · 𝑛 ) + 1 ) − 1 ) / 2 ) ) / ( ( 2 · 𝑛 ) + 1 ) ) ) = ( 𝑛 ∈ ℕ0 ↦ ( ( - 1 ↑ 𝑛 ) / ( ( 2 · 𝑛 ) + 1 ) ) ) |
20 |
1 19
|
eqtr4i |
⊢ 𝐹 = ( 𝑛 ∈ ℕ0 ↦ ( ( - 1 ↑ ( ( ( ( 2 · 𝑛 ) + 1 ) − 1 ) / 2 ) ) / ( ( 2 · 𝑛 ) + 1 ) ) ) |
21 |
|
seqeq3 |
⊢ ( 𝐹 = ( 𝑛 ∈ ℕ0 ↦ ( ( - 1 ↑ ( ( ( ( 2 · 𝑛 ) + 1 ) − 1 ) / 2 ) ) / ( ( 2 · 𝑛 ) + 1 ) ) ) → seq 0 ( + , 𝐹 ) = seq 0 ( + , ( 𝑛 ∈ ℕ0 ↦ ( ( - 1 ↑ ( ( ( ( 2 · 𝑛 ) + 1 ) − 1 ) / 2 ) ) / ( ( 2 · 𝑛 ) + 1 ) ) ) ) ) |
22 |
20 21
|
ax-mp |
⊢ seq 0 ( + , 𝐹 ) = seq 0 ( + , ( 𝑛 ∈ ℕ0 ↦ ( ( - 1 ↑ ( ( ( ( 2 · 𝑛 ) + 1 ) − 1 ) / 2 ) ) / ( ( 2 · 𝑛 ) + 1 ) ) ) ) |
23 |
22
|
breq1i |
⊢ ( seq 0 ( + , 𝐹 ) ⇝ 𝐴 ↔ seq 0 ( + , ( 𝑛 ∈ ℕ0 ↦ ( ( - 1 ↑ ( ( ( ( 2 · 𝑛 ) + 1 ) − 1 ) / 2 ) ) / ( ( 2 · 𝑛 ) + 1 ) ) ) ) ⇝ 𝐴 ) |
24 |
|
neg1rr |
⊢ - 1 ∈ ℝ |
25 |
|
reexpcl |
⊢ ( ( - 1 ∈ ℝ ∧ 𝑛 ∈ ℕ0 ) → ( - 1 ↑ 𝑛 ) ∈ ℝ ) |
26 |
24 25
|
mpan |
⊢ ( 𝑛 ∈ ℕ0 → ( - 1 ↑ 𝑛 ) ∈ ℝ ) |
27 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
28 |
|
nn0mulcl |
⊢ ( ( 2 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ) → ( 2 · 𝑛 ) ∈ ℕ0 ) |
29 |
27 28
|
mpan |
⊢ ( 𝑛 ∈ ℕ0 → ( 2 · 𝑛 ) ∈ ℕ0 ) |
30 |
|
nn0p1nn |
⊢ ( ( 2 · 𝑛 ) ∈ ℕ0 → ( ( 2 · 𝑛 ) + 1 ) ∈ ℕ ) |
31 |
29 30
|
syl |
⊢ ( 𝑛 ∈ ℕ0 → ( ( 2 · 𝑛 ) + 1 ) ∈ ℕ ) |
32 |
26 31
|
nndivred |
⊢ ( 𝑛 ∈ ℕ0 → ( ( - 1 ↑ 𝑛 ) / ( ( 2 · 𝑛 ) + 1 ) ) ∈ ℝ ) |
33 |
32
|
recnd |
⊢ ( 𝑛 ∈ ℕ0 → ( ( - 1 ↑ 𝑛 ) / ( ( 2 · 𝑛 ) + 1 ) ) ∈ ℂ ) |
34 |
18 33
|
eqeltrd |
⊢ ( 𝑛 ∈ ℕ0 → ( ( - 1 ↑ ( ( ( ( 2 · 𝑛 ) + 1 ) − 1 ) / 2 ) ) / ( ( 2 · 𝑛 ) + 1 ) ) ∈ ℂ ) |
35 |
34
|
adantl |
⊢ ( ( ⊤ ∧ 𝑛 ∈ ℕ0 ) → ( ( - 1 ↑ ( ( ( ( 2 · 𝑛 ) + 1 ) − 1 ) / 2 ) ) / ( ( 2 · 𝑛 ) + 1 ) ) ∈ ℂ ) |
36 |
|
oveq1 |
⊢ ( 𝑘 = ( ( 2 · 𝑛 ) + 1 ) → ( 𝑘 − 1 ) = ( ( ( 2 · 𝑛 ) + 1 ) − 1 ) ) |
37 |
36
|
oveq1d |
⊢ ( 𝑘 = ( ( 2 · 𝑛 ) + 1 ) → ( ( 𝑘 − 1 ) / 2 ) = ( ( ( ( 2 · 𝑛 ) + 1 ) − 1 ) / 2 ) ) |
38 |
37
|
oveq2d |
⊢ ( 𝑘 = ( ( 2 · 𝑛 ) + 1 ) → ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) = ( - 1 ↑ ( ( ( ( 2 · 𝑛 ) + 1 ) − 1 ) / 2 ) ) ) |
39 |
|
id |
⊢ ( 𝑘 = ( ( 2 · 𝑛 ) + 1 ) → 𝑘 = ( ( 2 · 𝑛 ) + 1 ) ) |
40 |
38 39
|
oveq12d |
⊢ ( 𝑘 = ( ( 2 · 𝑛 ) + 1 ) → ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) = ( ( - 1 ↑ ( ( ( ( 2 · 𝑛 ) + 1 ) − 1 ) / 2 ) ) / ( ( 2 · 𝑛 ) + 1 ) ) ) |
41 |
35 40
|
iserodd |
⊢ ( ⊤ → ( seq 0 ( + , ( 𝑛 ∈ ℕ0 ↦ ( ( - 1 ↑ ( ( ( ( 2 · 𝑛 ) + 1 ) − 1 ) / 2 ) ) / ( ( 2 · 𝑛 ) + 1 ) ) ) ) ⇝ 𝐴 ↔ seq 1 ( + , ( 𝑘 ∈ ℕ ↦ if ( 2 ∥ 𝑘 , 0 , ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) ) ) ) ⇝ 𝐴 ) ) |
42 |
41
|
mptru |
⊢ ( seq 0 ( + , ( 𝑛 ∈ ℕ0 ↦ ( ( - 1 ↑ ( ( ( ( 2 · 𝑛 ) + 1 ) − 1 ) / 2 ) ) / ( ( 2 · 𝑛 ) + 1 ) ) ) ) ⇝ 𝐴 ↔ seq 1 ( + , ( 𝑘 ∈ ℕ ↦ if ( 2 ∥ 𝑘 , 0 , ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) ) ) ) ⇝ 𝐴 ) |
43 |
|
addid2 |
⊢ ( 𝑛 ∈ ℂ → ( 0 + 𝑛 ) = 𝑛 ) |
44 |
43
|
adantl |
⊢ ( ( ⊤ ∧ 𝑛 ∈ ℂ ) → ( 0 + 𝑛 ) = 𝑛 ) |
45 |
|
0cnd |
⊢ ( ⊤ → 0 ∈ ℂ ) |
46 |
|
1eluzge0 |
⊢ 1 ∈ ( ℤ≥ ‘ 0 ) |
47 |
46
|
a1i |
⊢ ( ⊤ → 1 ∈ ( ℤ≥ ‘ 0 ) ) |
48 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
49 |
|
0cnd |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( 𝑘 = 0 ∨ 2 ∥ 𝑘 ) ) → 0 ∈ ℂ ) |
50 |
|
ioran |
⊢ ( ¬ ( 𝑘 = 0 ∨ 2 ∥ 𝑘 ) ↔ ( ¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘 ) ) |
51 |
|
leibpilem1 |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘 ) ) → ( 𝑘 ∈ ℕ ∧ ( ( 𝑘 − 1 ) / 2 ) ∈ ℕ0 ) ) |
52 |
51
|
simprd |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘 ) ) → ( ( 𝑘 − 1 ) / 2 ) ∈ ℕ0 ) |
53 |
|
reexpcl |
⊢ ( ( - 1 ∈ ℝ ∧ ( ( 𝑘 − 1 ) / 2 ) ∈ ℕ0 ) → ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) ∈ ℝ ) |
54 |
24 52 53
|
sylancr |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘 ) ) → ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) ∈ ℝ ) |
55 |
51
|
simpld |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘 ) ) → 𝑘 ∈ ℕ ) |
56 |
54 55
|
nndivred |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘 ) ) → ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) ∈ ℝ ) |
57 |
56
|
recnd |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ( ¬ 𝑘 = 0 ∧ ¬ 2 ∥ 𝑘 ) ) → ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) ∈ ℂ ) |
58 |
50 57
|
sylan2b |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ ¬ ( 𝑘 = 0 ∨ 2 ∥ 𝑘 ) ) → ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) ∈ ℂ ) |
59 |
49 58
|
ifclda |
⊢ ( 𝑘 ∈ ℕ0 → if ( ( 𝑘 = 0 ∨ 2 ∥ 𝑘 ) , 0 , ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) ) ∈ ℂ ) |
60 |
2 59
|
fmpti |
⊢ 𝐺 : ℕ0 ⟶ ℂ |
61 |
60
|
ffvelrni |
⊢ ( 1 ∈ ℕ0 → ( 𝐺 ‘ 1 ) ∈ ℂ ) |
62 |
48 61
|
mp1i |
⊢ ( ⊤ → ( 𝐺 ‘ 1 ) ∈ ℂ ) |
63 |
|
simpr |
⊢ ( ( ⊤ ∧ 𝑛 ∈ ( 0 ... ( 1 − 1 ) ) ) → 𝑛 ∈ ( 0 ... ( 1 − 1 ) ) ) |
64 |
|
1m1e0 |
⊢ ( 1 − 1 ) = 0 |
65 |
64
|
oveq2i |
⊢ ( 0 ... ( 1 − 1 ) ) = ( 0 ... 0 ) |
66 |
63 65
|
eleqtrdi |
⊢ ( ( ⊤ ∧ 𝑛 ∈ ( 0 ... ( 1 − 1 ) ) ) → 𝑛 ∈ ( 0 ... 0 ) ) |
67 |
|
elfz1eq |
⊢ ( 𝑛 ∈ ( 0 ... 0 ) → 𝑛 = 0 ) |
68 |
66 67
|
syl |
⊢ ( ( ⊤ ∧ 𝑛 ∈ ( 0 ... ( 1 − 1 ) ) ) → 𝑛 = 0 ) |
69 |
68
|
fveq2d |
⊢ ( ( ⊤ ∧ 𝑛 ∈ ( 0 ... ( 1 − 1 ) ) ) → ( 𝐺 ‘ 𝑛 ) = ( 𝐺 ‘ 0 ) ) |
70 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
71 |
|
iftrue |
⊢ ( ( 𝑘 = 0 ∨ 2 ∥ 𝑘 ) → if ( ( 𝑘 = 0 ∨ 2 ∥ 𝑘 ) , 0 , ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) ) = 0 ) |
72 |
71
|
orcs |
⊢ ( 𝑘 = 0 → if ( ( 𝑘 = 0 ∨ 2 ∥ 𝑘 ) , 0 , ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) ) = 0 ) |
73 |
|
c0ex |
⊢ 0 ∈ V |
74 |
72 2 73
|
fvmpt |
⊢ ( 0 ∈ ℕ0 → ( 𝐺 ‘ 0 ) = 0 ) |
75 |
70 74
|
ax-mp |
⊢ ( 𝐺 ‘ 0 ) = 0 |
76 |
69 75
|
eqtrdi |
⊢ ( ( ⊤ ∧ 𝑛 ∈ ( 0 ... ( 1 − 1 ) ) ) → ( 𝐺 ‘ 𝑛 ) = 0 ) |
77 |
44 45 47 62 76
|
seqid |
⊢ ( ⊤ → ( seq 0 ( + , 𝐺 ) ↾ ( ℤ≥ ‘ 1 ) ) = seq 1 ( + , 𝐺 ) ) |
78 |
|
1zzd |
⊢ ( ⊤ → 1 ∈ ℤ ) |
79 |
|
simpr |
⊢ ( ( ⊤ ∧ 𝑛 ∈ ( ℤ≥ ‘ 1 ) ) → 𝑛 ∈ ( ℤ≥ ‘ 1 ) ) |
80 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
81 |
79 80
|
eleqtrrdi |
⊢ ( ( ⊤ ∧ 𝑛 ∈ ( ℤ≥ ‘ 1 ) ) → 𝑛 ∈ ℕ ) |
82 |
|
nnne0 |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ≠ 0 ) |
83 |
82
|
neneqd |
⊢ ( 𝑛 ∈ ℕ → ¬ 𝑛 = 0 ) |
84 |
|
biorf |
⊢ ( ¬ 𝑛 = 0 → ( 2 ∥ 𝑛 ↔ ( 𝑛 = 0 ∨ 2 ∥ 𝑛 ) ) ) |
85 |
83 84
|
syl |
⊢ ( 𝑛 ∈ ℕ → ( 2 ∥ 𝑛 ↔ ( 𝑛 = 0 ∨ 2 ∥ 𝑛 ) ) ) |
86 |
85
|
ifbid |
⊢ ( 𝑛 ∈ ℕ → if ( 2 ∥ 𝑛 , 0 , ( ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) / 𝑛 ) ) = if ( ( 𝑛 = 0 ∨ 2 ∥ 𝑛 ) , 0 , ( ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) / 𝑛 ) ) ) |
87 |
|
breq2 |
⊢ ( 𝑘 = 𝑛 → ( 2 ∥ 𝑘 ↔ 2 ∥ 𝑛 ) ) |
88 |
|
oveq1 |
⊢ ( 𝑘 = 𝑛 → ( 𝑘 − 1 ) = ( 𝑛 − 1 ) ) |
89 |
88
|
oveq1d |
⊢ ( 𝑘 = 𝑛 → ( ( 𝑘 − 1 ) / 2 ) = ( ( 𝑛 − 1 ) / 2 ) ) |
90 |
89
|
oveq2d |
⊢ ( 𝑘 = 𝑛 → ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) = ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) ) |
91 |
|
id |
⊢ ( 𝑘 = 𝑛 → 𝑘 = 𝑛 ) |
92 |
90 91
|
oveq12d |
⊢ ( 𝑘 = 𝑛 → ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) = ( ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) / 𝑛 ) ) |
93 |
87 92
|
ifbieq2d |
⊢ ( 𝑘 = 𝑛 → if ( 2 ∥ 𝑘 , 0 , ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) ) = if ( 2 ∥ 𝑛 , 0 , ( ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) / 𝑛 ) ) ) |
94 |
|
eqid |
⊢ ( 𝑘 ∈ ℕ ↦ if ( 2 ∥ 𝑘 , 0 , ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) ) ) = ( 𝑘 ∈ ℕ ↦ if ( 2 ∥ 𝑘 , 0 , ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) ) ) |
95 |
|
ovex |
⊢ ( ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) / 𝑛 ) ∈ V |
96 |
73 95
|
ifex |
⊢ if ( 2 ∥ 𝑛 , 0 , ( ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) / 𝑛 ) ) ∈ V |
97 |
93 94 96
|
fvmpt |
⊢ ( 𝑛 ∈ ℕ → ( ( 𝑘 ∈ ℕ ↦ if ( 2 ∥ 𝑘 , 0 , ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) ) ) ‘ 𝑛 ) = if ( 2 ∥ 𝑛 , 0 , ( ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) / 𝑛 ) ) ) |
98 |
|
nnnn0 |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℕ0 ) |
99 |
|
eqeq1 |
⊢ ( 𝑘 = 𝑛 → ( 𝑘 = 0 ↔ 𝑛 = 0 ) ) |
100 |
99 87
|
orbi12d |
⊢ ( 𝑘 = 𝑛 → ( ( 𝑘 = 0 ∨ 2 ∥ 𝑘 ) ↔ ( 𝑛 = 0 ∨ 2 ∥ 𝑛 ) ) ) |
101 |
100 92
|
ifbieq2d |
⊢ ( 𝑘 = 𝑛 → if ( ( 𝑘 = 0 ∨ 2 ∥ 𝑘 ) , 0 , ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) ) = if ( ( 𝑛 = 0 ∨ 2 ∥ 𝑛 ) , 0 , ( ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) / 𝑛 ) ) ) |
102 |
73 95
|
ifex |
⊢ if ( ( 𝑛 = 0 ∨ 2 ∥ 𝑛 ) , 0 , ( ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) / 𝑛 ) ) ∈ V |
103 |
101 2 102
|
fvmpt |
⊢ ( 𝑛 ∈ ℕ0 → ( 𝐺 ‘ 𝑛 ) = if ( ( 𝑛 = 0 ∨ 2 ∥ 𝑛 ) , 0 , ( ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) / 𝑛 ) ) ) |
104 |
98 103
|
syl |
⊢ ( 𝑛 ∈ ℕ → ( 𝐺 ‘ 𝑛 ) = if ( ( 𝑛 = 0 ∨ 2 ∥ 𝑛 ) , 0 , ( ( - 1 ↑ ( ( 𝑛 − 1 ) / 2 ) ) / 𝑛 ) ) ) |
105 |
86 97 104
|
3eqtr4d |
⊢ ( 𝑛 ∈ ℕ → ( ( 𝑘 ∈ ℕ ↦ if ( 2 ∥ 𝑘 , 0 , ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) ) ) ‘ 𝑛 ) = ( 𝐺 ‘ 𝑛 ) ) |
106 |
81 105
|
syl |
⊢ ( ( ⊤ ∧ 𝑛 ∈ ( ℤ≥ ‘ 1 ) ) → ( ( 𝑘 ∈ ℕ ↦ if ( 2 ∥ 𝑘 , 0 , ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) ) ) ‘ 𝑛 ) = ( 𝐺 ‘ 𝑛 ) ) |
107 |
78 106
|
seqfeq |
⊢ ( ⊤ → seq 1 ( + , ( 𝑘 ∈ ℕ ↦ if ( 2 ∥ 𝑘 , 0 , ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) ) ) ) = seq 1 ( + , 𝐺 ) ) |
108 |
77 107
|
eqtr4d |
⊢ ( ⊤ → ( seq 0 ( + , 𝐺 ) ↾ ( ℤ≥ ‘ 1 ) ) = seq 1 ( + , ( 𝑘 ∈ ℕ ↦ if ( 2 ∥ 𝑘 , 0 , ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) ) ) ) ) |
109 |
108
|
mptru |
⊢ ( seq 0 ( + , 𝐺 ) ↾ ( ℤ≥ ‘ 1 ) ) = seq 1 ( + , ( 𝑘 ∈ ℕ ↦ if ( 2 ∥ 𝑘 , 0 , ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) ) ) ) |
110 |
109
|
breq1i |
⊢ ( ( seq 0 ( + , 𝐺 ) ↾ ( ℤ≥ ‘ 1 ) ) ⇝ 𝐴 ↔ seq 1 ( + , ( 𝑘 ∈ ℕ ↦ if ( 2 ∥ 𝑘 , 0 , ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) ) ) ) ⇝ 𝐴 ) |
111 |
|
1z |
⊢ 1 ∈ ℤ |
112 |
|
seqex |
⊢ seq 0 ( + , 𝐺 ) ∈ V |
113 |
|
climres |
⊢ ( ( 1 ∈ ℤ ∧ seq 0 ( + , 𝐺 ) ∈ V ) → ( ( seq 0 ( + , 𝐺 ) ↾ ( ℤ≥ ‘ 1 ) ) ⇝ 𝐴 ↔ seq 0 ( + , 𝐺 ) ⇝ 𝐴 ) ) |
114 |
111 112 113
|
mp2an |
⊢ ( ( seq 0 ( + , 𝐺 ) ↾ ( ℤ≥ ‘ 1 ) ) ⇝ 𝐴 ↔ seq 0 ( + , 𝐺 ) ⇝ 𝐴 ) |
115 |
110 114
|
bitr3i |
⊢ ( seq 1 ( + , ( 𝑘 ∈ ℕ ↦ if ( 2 ∥ 𝑘 , 0 , ( ( - 1 ↑ ( ( 𝑘 − 1 ) / 2 ) ) / 𝑘 ) ) ) ) ⇝ 𝐴 ↔ seq 0 ( + , 𝐺 ) ⇝ 𝐴 ) |
116 |
23 42 115
|
3bitri |
⊢ ( seq 0 ( + , 𝐹 ) ⇝ 𝐴 ↔ seq 0 ( + , 𝐺 ) ⇝ 𝐴 ) |