| Step |
Hyp |
Ref |
Expression |
| 1 |
|
nnnn0 |
⊢ ( ( 𝑎 / 2 ) ∈ ℕ → ( 𝑎 / 2 ) ∈ ℕ0 ) |
| 2 |
|
blennn0em1 |
⊢ ( ( 𝑎 ∈ ℕ ∧ ( 𝑎 / 2 ) ∈ ℕ0 ) → ( #b ‘ ( 𝑎 / 2 ) ) = ( ( #b ‘ 𝑎 ) − 1 ) ) |
| 3 |
1 2
|
sylan2 |
⊢ ( ( 𝑎 ∈ ℕ ∧ ( 𝑎 / 2 ) ∈ ℕ ) → ( #b ‘ ( 𝑎 / 2 ) ) = ( ( #b ‘ 𝑎 ) − 1 ) ) |
| 4 |
|
fveqeq2 |
⊢ ( 𝑥 = ( 𝑎 / 2 ) → ( ( #b ‘ 𝑥 ) = 𝑦 ↔ ( #b ‘ ( 𝑎 / 2 ) ) = 𝑦 ) ) |
| 5 |
|
id |
⊢ ( 𝑥 = ( 𝑎 / 2 ) → 𝑥 = ( 𝑎 / 2 ) ) |
| 6 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝑎 / 2 ) → ( 𝑘 ( digit ‘ 2 ) 𝑥 ) = ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) ) |
| 7 |
6
|
oveq1d |
⊢ ( 𝑥 = ( 𝑎 / 2 ) → ( ( 𝑘 ( digit ‘ 2 ) 𝑥 ) · ( 2 ↑ 𝑘 ) ) = ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) ) |
| 8 |
7
|
adantr |
⊢ ( ( 𝑥 = ( 𝑎 / 2 ) ∧ 𝑘 ∈ ( 0 ..^ 𝑦 ) ) → ( ( 𝑘 ( digit ‘ 2 ) 𝑥 ) · ( 2 ↑ 𝑘 ) ) = ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) ) |
| 9 |
8
|
sumeq2dv |
⊢ ( 𝑥 = ( 𝑎 / 2 ) → Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑥 ) · ( 2 ↑ 𝑘 ) ) = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) ) |
| 10 |
5 9
|
eqeq12d |
⊢ ( 𝑥 = ( 𝑎 / 2 ) → ( 𝑥 = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑥 ) · ( 2 ↑ 𝑘 ) ) ↔ ( 𝑎 / 2 ) = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) ) ) |
| 11 |
4 10
|
imbi12d |
⊢ ( 𝑥 = ( 𝑎 / 2 ) → ( ( ( #b ‘ 𝑥 ) = 𝑦 → 𝑥 = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑥 ) · ( 2 ↑ 𝑘 ) ) ) ↔ ( ( #b ‘ ( 𝑎 / 2 ) ) = 𝑦 → ( 𝑎 / 2 ) = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) ) ) ) |
| 12 |
11
|
rspcva |
⊢ ( ( ( 𝑎 / 2 ) ∈ ℕ0 ∧ ∀ 𝑥 ∈ ℕ0 ( ( #b ‘ 𝑥 ) = 𝑦 → 𝑥 = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑥 ) · ( 2 ↑ 𝑘 ) ) ) ) → ( ( #b ‘ ( 𝑎 / 2 ) ) = 𝑦 → ( 𝑎 / 2 ) = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) ) ) |
| 13 |
|
simpr |
⊢ ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) → ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) |
| 14 |
13
|
oveq1d |
⊢ ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) → ( ( #b ‘ 𝑎 ) − 1 ) = ( ( 𝑦 + 1 ) − 1 ) ) |
| 15 |
|
nncn |
⊢ ( 𝑦 ∈ ℕ → 𝑦 ∈ ℂ ) |
| 16 |
|
pncan1 |
⊢ ( 𝑦 ∈ ℂ → ( ( 𝑦 + 1 ) − 1 ) = 𝑦 ) |
| 17 |
15 16
|
syl |
⊢ ( 𝑦 ∈ ℕ → ( ( 𝑦 + 1 ) − 1 ) = 𝑦 ) |
| 18 |
14 17
|
sylan9eq |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → ( ( #b ‘ 𝑎 ) − 1 ) = 𝑦 ) |
| 19 |
18
|
eqeq2d |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → ( ( #b ‘ ( 𝑎 / 2 ) ) = ( ( #b ‘ 𝑎 ) − 1 ) ↔ ( #b ‘ ( 𝑎 / 2 ) ) = 𝑦 ) ) |
| 20 |
|
nnz |
⊢ ( 𝑦 ∈ ℕ → 𝑦 ∈ ℤ ) |
| 21 |
20
|
adantl |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → 𝑦 ∈ ℤ ) |
| 22 |
|
fzval3 |
⊢ ( 𝑦 ∈ ℤ → ( 0 ... 𝑦 ) = ( 0 ..^ ( 𝑦 + 1 ) ) ) |
| 23 |
21 22
|
syl |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → ( 0 ... 𝑦 ) = ( 0 ..^ ( 𝑦 + 1 ) ) ) |
| 24 |
23
|
eqcomd |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → ( 0 ..^ ( 𝑦 + 1 ) ) = ( 0 ... 𝑦 ) ) |
| 25 |
24
|
sumeq1d |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) = Σ 𝑘 ∈ ( 0 ... 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) |
| 26 |
|
nnnn0 |
⊢ ( 𝑦 ∈ ℕ → 𝑦 ∈ ℕ0 ) |
| 27 |
|
elnn0uz |
⊢ ( 𝑦 ∈ ℕ0 ↔ 𝑦 ∈ ( ℤ≥ ‘ 0 ) ) |
| 28 |
26 27
|
sylib |
⊢ ( 𝑦 ∈ ℕ → 𝑦 ∈ ( ℤ≥ ‘ 0 ) ) |
| 29 |
28
|
adantl |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → 𝑦 ∈ ( ℤ≥ ‘ 0 ) ) |
| 30 |
|
2nn |
⊢ 2 ∈ ℕ |
| 31 |
30
|
a1i |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑘 ∈ ( 0 ... 𝑦 ) ) → 2 ∈ ℕ ) |
| 32 |
|
elfzelz |
⊢ ( 𝑘 ∈ ( 0 ... 𝑦 ) → 𝑘 ∈ ℤ ) |
| 33 |
32
|
adantl |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑘 ∈ ( 0 ... 𝑦 ) ) → 𝑘 ∈ ℤ ) |
| 34 |
|
nnnn0 |
⊢ ( 𝑎 ∈ ℕ → 𝑎 ∈ ℕ0 ) |
| 35 |
|
nn0rp0 |
⊢ ( 𝑎 ∈ ℕ0 → 𝑎 ∈ ( 0 [,) +∞ ) ) |
| 36 |
34 35
|
syl |
⊢ ( 𝑎 ∈ ℕ → 𝑎 ∈ ( 0 [,) +∞ ) ) |
| 37 |
36
|
ad4antlr |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑘 ∈ ( 0 ... 𝑦 ) ) → 𝑎 ∈ ( 0 [,) +∞ ) ) |
| 38 |
|
digvalnn0 |
⊢ ( ( 2 ∈ ℕ ∧ 𝑘 ∈ ℤ ∧ 𝑎 ∈ ( 0 [,) +∞ ) ) → ( 𝑘 ( digit ‘ 2 ) 𝑎 ) ∈ ℕ0 ) |
| 39 |
31 33 37 38
|
syl3anc |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑘 ∈ ( 0 ... 𝑦 ) ) → ( 𝑘 ( digit ‘ 2 ) 𝑎 ) ∈ ℕ0 ) |
| 40 |
39
|
nn0cnd |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑘 ∈ ( 0 ... 𝑦 ) ) → ( 𝑘 ( digit ‘ 2 ) 𝑎 ) ∈ ℂ ) |
| 41 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
| 42 |
41
|
a1i |
⊢ ( 𝑘 ∈ ( 0 ... 𝑦 ) → 2 ∈ ℕ0 ) |
| 43 |
|
elfznn0 |
⊢ ( 𝑘 ∈ ( 0 ... 𝑦 ) → 𝑘 ∈ ℕ0 ) |
| 44 |
42 43
|
nn0expcld |
⊢ ( 𝑘 ∈ ( 0 ... 𝑦 ) → ( 2 ↑ 𝑘 ) ∈ ℕ0 ) |
| 45 |
44
|
nn0cnd |
⊢ ( 𝑘 ∈ ( 0 ... 𝑦 ) → ( 2 ↑ 𝑘 ) ∈ ℂ ) |
| 46 |
45
|
adantl |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑘 ∈ ( 0 ... 𝑦 ) ) → ( 2 ↑ 𝑘 ) ∈ ℂ ) |
| 47 |
40 46
|
mulcld |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑘 ∈ ( 0 ... 𝑦 ) ) → ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ∈ ℂ ) |
| 48 |
|
oveq1 |
⊢ ( 𝑘 = 0 → ( 𝑘 ( digit ‘ 2 ) 𝑎 ) = ( 0 ( digit ‘ 2 ) 𝑎 ) ) |
| 49 |
|
oveq2 |
⊢ ( 𝑘 = 0 → ( 2 ↑ 𝑘 ) = ( 2 ↑ 0 ) ) |
| 50 |
48 49
|
oveq12d |
⊢ ( 𝑘 = 0 → ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) = ( ( 0 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 0 ) ) ) |
| 51 |
|
2cn |
⊢ 2 ∈ ℂ |
| 52 |
|
exp0 |
⊢ ( 2 ∈ ℂ → ( 2 ↑ 0 ) = 1 ) |
| 53 |
51 52
|
ax-mp |
⊢ ( 2 ↑ 0 ) = 1 |
| 54 |
53
|
oveq2i |
⊢ ( ( 0 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 0 ) ) = ( ( 0 ( digit ‘ 2 ) 𝑎 ) · 1 ) |
| 55 |
50 54
|
eqtrdi |
⊢ ( 𝑘 = 0 → ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) = ( ( 0 ( digit ‘ 2 ) 𝑎 ) · 1 ) ) |
| 56 |
29 47 55
|
fsum1p |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → Σ 𝑘 ∈ ( 0 ... 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) = ( ( ( 0 ( digit ‘ 2 ) 𝑎 ) · 1 ) + Σ 𝑘 ∈ ( ( 0 + 1 ) ... 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ) |
| 57 |
|
0dig2nn0e |
⊢ ( ( 𝑎 ∈ ℕ0 ∧ ( 𝑎 / 2 ) ∈ ℕ0 ) → ( 0 ( digit ‘ 2 ) 𝑎 ) = 0 ) |
| 58 |
34 1 57
|
syl2anr |
⊢ ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) → ( 0 ( digit ‘ 2 ) 𝑎 ) = 0 ) |
| 59 |
58
|
oveq1d |
⊢ ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) → ( ( 0 ( digit ‘ 2 ) 𝑎 ) · 1 ) = ( 0 · 1 ) ) |
| 60 |
|
1re |
⊢ 1 ∈ ℝ |
| 61 |
|
mul02lem2 |
⊢ ( 1 ∈ ℝ → ( 0 · 1 ) = 0 ) |
| 62 |
60 61
|
ax-mp |
⊢ ( 0 · 1 ) = 0 |
| 63 |
59 62
|
eqtrdi |
⊢ ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) → ( ( 0 ( digit ‘ 2 ) 𝑎 ) · 1 ) = 0 ) |
| 64 |
63
|
adantr |
⊢ ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) → ( ( 0 ( digit ‘ 2 ) 𝑎 ) · 1 ) = 0 ) |
| 65 |
64
|
adantr |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → ( ( 0 ( digit ‘ 2 ) 𝑎 ) · 1 ) = 0 ) |
| 66 |
|
1z |
⊢ 1 ∈ ℤ |
| 67 |
66
|
a1i |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → 1 ∈ ℤ ) |
| 68 |
|
0p1e1 |
⊢ ( 0 + 1 ) = 1 |
| 69 |
68 66
|
eqeltri |
⊢ ( 0 + 1 ) ∈ ℤ |
| 70 |
69
|
a1i |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → ( 0 + 1 ) ∈ ℤ ) |
| 71 |
30
|
a1i |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑘 ∈ ( ( 0 + 1 ) ... 𝑦 ) ) → 2 ∈ ℕ ) |
| 72 |
|
elfzelz |
⊢ ( 𝑘 ∈ ( ( 0 + 1 ) ... 𝑦 ) → 𝑘 ∈ ℤ ) |
| 73 |
72
|
adantl |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑘 ∈ ( ( 0 + 1 ) ... 𝑦 ) ) → 𝑘 ∈ ℤ ) |
| 74 |
36
|
ad4antlr |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑘 ∈ ( ( 0 + 1 ) ... 𝑦 ) ) → 𝑎 ∈ ( 0 [,) +∞ ) ) |
| 75 |
71 73 74 38
|
syl3anc |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑘 ∈ ( ( 0 + 1 ) ... 𝑦 ) ) → ( 𝑘 ( digit ‘ 2 ) 𝑎 ) ∈ ℕ0 ) |
| 76 |
75
|
nn0cnd |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑘 ∈ ( ( 0 + 1 ) ... 𝑦 ) ) → ( 𝑘 ( digit ‘ 2 ) 𝑎 ) ∈ ℂ ) |
| 77 |
|
2cnd |
⊢ ( 𝑘 ∈ ( ( 0 + 1 ) ... 𝑦 ) → 2 ∈ ℂ ) |
| 78 |
|
elfznn |
⊢ ( 𝑘 ∈ ( 1 ... 𝑦 ) → 𝑘 ∈ ℕ ) |
| 79 |
78
|
nnnn0d |
⊢ ( 𝑘 ∈ ( 1 ... 𝑦 ) → 𝑘 ∈ ℕ0 ) |
| 80 |
68
|
oveq1i |
⊢ ( ( 0 + 1 ) ... 𝑦 ) = ( 1 ... 𝑦 ) |
| 81 |
79 80
|
eleq2s |
⊢ ( 𝑘 ∈ ( ( 0 + 1 ) ... 𝑦 ) → 𝑘 ∈ ℕ0 ) |
| 82 |
77 81
|
expcld |
⊢ ( 𝑘 ∈ ( ( 0 + 1 ) ... 𝑦 ) → ( 2 ↑ 𝑘 ) ∈ ℂ ) |
| 83 |
82
|
adantl |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑘 ∈ ( ( 0 + 1 ) ... 𝑦 ) ) → ( 2 ↑ 𝑘 ) ∈ ℂ ) |
| 84 |
76 83
|
mulcld |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑘 ∈ ( ( 0 + 1 ) ... 𝑦 ) ) → ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ∈ ℂ ) |
| 85 |
|
oveq1 |
⊢ ( 𝑘 = ( 𝑖 + 1 ) → ( 𝑘 ( digit ‘ 2 ) 𝑎 ) = ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) ) |
| 86 |
|
oveq2 |
⊢ ( 𝑘 = ( 𝑖 + 1 ) → ( 2 ↑ 𝑘 ) = ( 2 ↑ ( 𝑖 + 1 ) ) ) |
| 87 |
85 86
|
oveq12d |
⊢ ( 𝑘 = ( 𝑖 + 1 ) → ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) = ( ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ ( 𝑖 + 1 ) ) ) ) |
| 88 |
67 70 21 84 87
|
fsumshftm |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → Σ 𝑘 ∈ ( ( 0 + 1 ) ... 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) = Σ 𝑖 ∈ ( ( ( 0 + 1 ) − 1 ) ... ( 𝑦 − 1 ) ) ( ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ ( 𝑖 + 1 ) ) ) ) |
| 89 |
65 88
|
oveq12d |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → ( ( ( 0 ( digit ‘ 2 ) 𝑎 ) · 1 ) + Σ 𝑘 ∈ ( ( 0 + 1 ) ... 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) = ( 0 + Σ 𝑖 ∈ ( ( ( 0 + 1 ) − 1 ) ... ( 𝑦 − 1 ) ) ( ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ ( 𝑖 + 1 ) ) ) ) ) |
| 90 |
1
|
ad4antr |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( 𝑎 / 2 ) ∈ ℕ0 ) |
| 91 |
34
|
ad4antlr |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → 𝑎 ∈ ℕ0 ) |
| 92 |
|
elfzonn0 |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑦 ) → 𝑖 ∈ ℕ0 ) |
| 93 |
92
|
adantl |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → 𝑖 ∈ ℕ0 ) |
| 94 |
|
dignn0ehalf |
⊢ ( ( ( 𝑎 / 2 ) ∈ ℕ0 ∧ 𝑎 ∈ ℕ0 ∧ 𝑖 ∈ ℕ0 ) → ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) = ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) ) |
| 95 |
90 91 93 94
|
syl3anc |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) = ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) ) |
| 96 |
|
2cnd |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑦 ) → 2 ∈ ℂ ) |
| 97 |
96 92
|
expp1d |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑦 ) → ( 2 ↑ ( 𝑖 + 1 ) ) = ( ( 2 ↑ 𝑖 ) · 2 ) ) |
| 98 |
97
|
adantl |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( 2 ↑ ( 𝑖 + 1 ) ) = ( ( 2 ↑ 𝑖 ) · 2 ) ) |
| 99 |
95 98
|
oveq12d |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ ( 𝑖 + 1 ) ) ) = ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( ( 2 ↑ 𝑖 ) · 2 ) ) ) |
| 100 |
30
|
a1i |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → 2 ∈ ℕ ) |
| 101 |
|
elfzoelz |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑦 ) → 𝑖 ∈ ℤ ) |
| 102 |
101
|
adantl |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → 𝑖 ∈ ℤ ) |
| 103 |
|
nn0rp0 |
⊢ ( ( 𝑎 / 2 ) ∈ ℕ0 → ( 𝑎 / 2 ) ∈ ( 0 [,) +∞ ) ) |
| 104 |
1 103
|
syl |
⊢ ( ( 𝑎 / 2 ) ∈ ℕ → ( 𝑎 / 2 ) ∈ ( 0 [,) +∞ ) ) |
| 105 |
104
|
ad4antr |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( 𝑎 / 2 ) ∈ ( 0 [,) +∞ ) ) |
| 106 |
|
digvalnn0 |
⊢ ( ( 2 ∈ ℕ ∧ 𝑖 ∈ ℤ ∧ ( 𝑎 / 2 ) ∈ ( 0 [,) +∞ ) ) → ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) ∈ ℕ0 ) |
| 107 |
100 102 105 106
|
syl3anc |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) ∈ ℕ0 ) |
| 108 |
107
|
nn0cnd |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) ∈ ℂ ) |
| 109 |
|
2re |
⊢ 2 ∈ ℝ |
| 110 |
109
|
a1i |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑦 ) → 2 ∈ ℝ ) |
| 111 |
110 92
|
reexpcld |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑦 ) → ( 2 ↑ 𝑖 ) ∈ ℝ ) |
| 112 |
111
|
recnd |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑦 ) → ( 2 ↑ 𝑖 ) ∈ ℂ ) |
| 113 |
112
|
adantl |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( 2 ↑ 𝑖 ) ∈ ℂ ) |
| 114 |
|
2cnd |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → 2 ∈ ℂ ) |
| 115 |
|
mulass |
⊢ ( ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) ∈ ℂ ∧ ( 2 ↑ 𝑖 ) ∈ ℂ ∧ 2 ∈ ℂ ) → ( ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑖 ) ) · 2 ) = ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( ( 2 ↑ 𝑖 ) · 2 ) ) ) |
| 116 |
115
|
eqcomd |
⊢ ( ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) ∈ ℂ ∧ ( 2 ↑ 𝑖 ) ∈ ℂ ∧ 2 ∈ ℂ ) → ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( ( 2 ↑ 𝑖 ) · 2 ) ) = ( ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑖 ) ) · 2 ) ) |
| 117 |
108 113 114 116
|
syl3anc |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( ( 2 ↑ 𝑖 ) · 2 ) ) = ( ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑖 ) ) · 2 ) ) |
| 118 |
99 117
|
eqtrd |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ ( 𝑖 + 1 ) ) ) = ( ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑖 ) ) · 2 ) ) |
| 119 |
118
|
sumeq2dv |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → Σ 𝑖 ∈ ( 0 ..^ 𝑦 ) ( ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ ( 𝑖 + 1 ) ) ) = Σ 𝑖 ∈ ( 0 ..^ 𝑦 ) ( ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑖 ) ) · 2 ) ) |
| 120 |
|
0cn |
⊢ 0 ∈ ℂ |
| 121 |
|
pncan1 |
⊢ ( 0 ∈ ℂ → ( ( 0 + 1 ) − 1 ) = 0 ) |
| 122 |
120 121
|
ax-mp |
⊢ ( ( 0 + 1 ) − 1 ) = 0 |
| 123 |
122
|
a1i |
⊢ ( 𝑦 ∈ ℕ → ( ( 0 + 1 ) − 1 ) = 0 ) |
| 124 |
123
|
oveq1d |
⊢ ( 𝑦 ∈ ℕ → ( ( ( 0 + 1 ) − 1 ) ... ( 𝑦 − 1 ) ) = ( 0 ... ( 𝑦 − 1 ) ) ) |
| 125 |
|
fzoval |
⊢ ( 𝑦 ∈ ℤ → ( 0 ..^ 𝑦 ) = ( 0 ... ( 𝑦 − 1 ) ) ) |
| 126 |
125
|
eqcomd |
⊢ ( 𝑦 ∈ ℤ → ( 0 ... ( 𝑦 − 1 ) ) = ( 0 ..^ 𝑦 ) ) |
| 127 |
20 126
|
syl |
⊢ ( 𝑦 ∈ ℕ → ( 0 ... ( 𝑦 − 1 ) ) = ( 0 ..^ 𝑦 ) ) |
| 128 |
124 127
|
eqtrd |
⊢ ( 𝑦 ∈ ℕ → ( ( ( 0 + 1 ) − 1 ) ... ( 𝑦 − 1 ) ) = ( 0 ..^ 𝑦 ) ) |
| 129 |
128
|
adantl |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → ( ( ( 0 + 1 ) − 1 ) ... ( 𝑦 − 1 ) ) = ( 0 ..^ 𝑦 ) ) |
| 130 |
129
|
sumeq1d |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → Σ 𝑖 ∈ ( ( ( 0 + 1 ) − 1 ) ... ( 𝑦 − 1 ) ) ( ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ ( 𝑖 + 1 ) ) ) = Σ 𝑖 ∈ ( 0 ..^ 𝑦 ) ( ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ ( 𝑖 + 1 ) ) ) ) |
| 131 |
130
|
oveq2d |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → ( 0 + Σ 𝑖 ∈ ( ( ( 0 + 1 ) − 1 ) ... ( 𝑦 − 1 ) ) ( ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ ( 𝑖 + 1 ) ) ) ) = ( 0 + Σ 𝑖 ∈ ( 0 ..^ 𝑦 ) ( ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ ( 𝑖 + 1 ) ) ) ) ) |
| 132 |
|
fzofi |
⊢ ( 0 ..^ 𝑦 ) ∈ Fin |
| 133 |
132
|
a1i |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → ( 0 ..^ 𝑦 ) ∈ Fin ) |
| 134 |
101
|
peano2zd |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑦 ) → ( 𝑖 + 1 ) ∈ ℤ ) |
| 135 |
134
|
adantl |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( 𝑖 + 1 ) ∈ ℤ ) |
| 136 |
36
|
ad4antlr |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → 𝑎 ∈ ( 0 [,) +∞ ) ) |
| 137 |
|
digvalnn0 |
⊢ ( ( 2 ∈ ℕ ∧ ( 𝑖 + 1 ) ∈ ℤ ∧ 𝑎 ∈ ( 0 [,) +∞ ) ) → ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) ∈ ℕ0 ) |
| 138 |
100 135 136 137
|
syl3anc |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) ∈ ℕ0 ) |
| 139 |
138
|
nn0cnd |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) ∈ ℂ ) |
| 140 |
41
|
a1i |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑦 ) → 2 ∈ ℕ0 ) |
| 141 |
|
peano2nn0 |
⊢ ( 𝑖 ∈ ℕ0 → ( 𝑖 + 1 ) ∈ ℕ0 ) |
| 142 |
92 141
|
syl |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑦 ) → ( 𝑖 + 1 ) ∈ ℕ0 ) |
| 143 |
140 142
|
nn0expcld |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑦 ) → ( 2 ↑ ( 𝑖 + 1 ) ) ∈ ℕ0 ) |
| 144 |
143
|
nn0cnd |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑦 ) → ( 2 ↑ ( 𝑖 + 1 ) ) ∈ ℂ ) |
| 145 |
144
|
adantl |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( 2 ↑ ( 𝑖 + 1 ) ) ∈ ℂ ) |
| 146 |
139 145
|
mulcld |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ ( 𝑖 + 1 ) ) ) ∈ ℂ ) |
| 147 |
133 146
|
fsumcl |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → Σ 𝑖 ∈ ( 0 ..^ 𝑦 ) ( ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ ( 𝑖 + 1 ) ) ) ∈ ℂ ) |
| 148 |
147
|
addid2d |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → ( 0 + Σ 𝑖 ∈ ( 0 ..^ 𝑦 ) ( ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ ( 𝑖 + 1 ) ) ) ) = Σ 𝑖 ∈ ( 0 ..^ 𝑦 ) ( ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ ( 𝑖 + 1 ) ) ) ) |
| 149 |
131 148
|
eqtrd |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → ( 0 + Σ 𝑖 ∈ ( ( ( 0 + 1 ) − 1 ) ... ( 𝑦 − 1 ) ) ( ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ ( 𝑖 + 1 ) ) ) ) = Σ 𝑖 ∈ ( 0 ..^ 𝑦 ) ( ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ ( 𝑖 + 1 ) ) ) ) |
| 150 |
|
2cnd |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → 2 ∈ ℂ ) |
| 151 |
140 92
|
nn0expcld |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑦 ) → ( 2 ↑ 𝑖 ) ∈ ℕ0 ) |
| 152 |
151
|
nn0cnd |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑦 ) → ( 2 ↑ 𝑖 ) ∈ ℂ ) |
| 153 |
152
|
adantl |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( 2 ↑ 𝑖 ) ∈ ℂ ) |
| 154 |
108 153
|
mulcld |
⊢ ( ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ∧ 𝑖 ∈ ( 0 ..^ 𝑦 ) ) → ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑖 ) ) ∈ ℂ ) |
| 155 |
133 150 154
|
fsummulc1 |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → ( Σ 𝑖 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑖 ) ) · 2 ) = Σ 𝑖 ∈ ( 0 ..^ 𝑦 ) ( ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑖 ) ) · 2 ) ) |
| 156 |
119 149 155
|
3eqtr4d |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → ( 0 + Σ 𝑖 ∈ ( ( ( 0 + 1 ) − 1 ) ... ( 𝑦 − 1 ) ) ( ( ( 𝑖 + 1 ) ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ ( 𝑖 + 1 ) ) ) ) = ( Σ 𝑖 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑖 ) ) · 2 ) ) |
| 157 |
89 156
|
eqtrd |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → ( ( ( 0 ( digit ‘ 2 ) 𝑎 ) · 1 ) + Σ 𝑘 ∈ ( ( 0 + 1 ) ... 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) = ( Σ 𝑖 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑖 ) ) · 2 ) ) |
| 158 |
25 56 157
|
3eqtrd |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) = ( Σ 𝑖 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑖 ) ) · 2 ) ) |
| 159 |
158
|
adantl |
⊢ ( ( ( 𝑎 / 2 ) = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) ∧ ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ) → Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) = ( Σ 𝑖 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑖 ) ) · 2 ) ) |
| 160 |
|
oveq1 |
⊢ ( 𝑘 = 𝑖 → ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) = ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) ) |
| 161 |
|
oveq2 |
⊢ ( 𝑘 = 𝑖 → ( 2 ↑ 𝑘 ) = ( 2 ↑ 𝑖 ) ) |
| 162 |
160 161
|
oveq12d |
⊢ ( 𝑘 = 𝑖 → ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) = ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑖 ) ) ) |
| 163 |
162
|
cbvsumv |
⊢ Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) = Σ 𝑖 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑖 ) ) |
| 164 |
163
|
a1i |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) = Σ 𝑖 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑖 ) ) ) |
| 165 |
164
|
eqeq2d |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → ( ( 𝑎 / 2 ) = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) ↔ ( 𝑎 / 2 ) = Σ 𝑖 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑖 ) ) ) ) |
| 166 |
165
|
biimpac |
⊢ ( ( ( 𝑎 / 2 ) = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) ∧ ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ) → ( 𝑎 / 2 ) = Σ 𝑖 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑖 ) ) ) |
| 167 |
166
|
eqcomd |
⊢ ( ( ( 𝑎 / 2 ) = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) ∧ ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ) → Σ 𝑖 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑖 ) ) = ( 𝑎 / 2 ) ) |
| 168 |
167
|
oveq1d |
⊢ ( ( ( 𝑎 / 2 ) = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) ∧ ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ) → ( Σ 𝑖 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑖 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑖 ) ) · 2 ) = ( ( 𝑎 / 2 ) · 2 ) ) |
| 169 |
|
nncn |
⊢ ( 𝑎 ∈ ℕ → 𝑎 ∈ ℂ ) |
| 170 |
|
2cnd |
⊢ ( 𝑎 ∈ ℕ → 2 ∈ ℂ ) |
| 171 |
|
2ne0 |
⊢ 2 ≠ 0 |
| 172 |
171
|
a1i |
⊢ ( 𝑎 ∈ ℕ → 2 ≠ 0 ) |
| 173 |
169 170 172
|
divcan1d |
⊢ ( 𝑎 ∈ ℕ → ( ( 𝑎 / 2 ) · 2 ) = 𝑎 ) |
| 174 |
173
|
ad3antlr |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → ( ( 𝑎 / 2 ) · 2 ) = 𝑎 ) |
| 175 |
174
|
adantl |
⊢ ( ( ( 𝑎 / 2 ) = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) ∧ ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ) → ( ( 𝑎 / 2 ) · 2 ) = 𝑎 ) |
| 176 |
159 168 175
|
3eqtrrd |
⊢ ( ( ( 𝑎 / 2 ) = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) ∧ ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) |
| 177 |
176
|
ex |
⊢ ( ( 𝑎 / 2 ) = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) → ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ) |
| 178 |
177
|
imim2i |
⊢ ( ( ( #b ‘ ( 𝑎 / 2 ) ) = 𝑦 → ( 𝑎 / 2 ) = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) ) → ( ( #b ‘ ( 𝑎 / 2 ) ) = 𝑦 → ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ) ) |
| 179 |
178
|
com13 |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → ( ( #b ‘ ( 𝑎 / 2 ) ) = 𝑦 → ( ( ( #b ‘ ( 𝑎 / 2 ) ) = 𝑦 → ( 𝑎 / 2 ) = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ) ) |
| 180 |
19 179
|
sylbid |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → ( ( #b ‘ ( 𝑎 / 2 ) ) = ( ( #b ‘ 𝑎 ) − 1 ) → ( ( ( #b ‘ ( 𝑎 / 2 ) ) = 𝑦 → ( 𝑎 / 2 ) = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ) ) |
| 181 |
180
|
com23 |
⊢ ( ( ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) ∧ ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ) ∧ 𝑦 ∈ ℕ ) → ( ( ( #b ‘ ( 𝑎 / 2 ) ) = 𝑦 → ( 𝑎 / 2 ) = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) ) → ( ( #b ‘ ( 𝑎 / 2 ) ) = ( ( #b ‘ 𝑎 ) − 1 ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ) ) |
| 182 |
181
|
exp31 |
⊢ ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) → ( ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) → ( 𝑦 ∈ ℕ → ( ( ( #b ‘ ( 𝑎 / 2 ) ) = 𝑦 → ( 𝑎 / 2 ) = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) ) → ( ( #b ‘ ( 𝑎 / 2 ) ) = ( ( #b ‘ 𝑎 ) − 1 ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ) ) ) ) |
| 183 |
182
|
com25 |
⊢ ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) → ( ( #b ‘ ( 𝑎 / 2 ) ) = ( ( #b ‘ 𝑎 ) − 1 ) → ( 𝑦 ∈ ℕ → ( ( ( #b ‘ ( 𝑎 / 2 ) ) = 𝑦 → ( 𝑎 / 2 ) = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) ) → ( ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ) ) ) ) |
| 184 |
183
|
com14 |
⊢ ( ( ( #b ‘ ( 𝑎 / 2 ) ) = 𝑦 → ( 𝑎 / 2 ) = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) ( 𝑎 / 2 ) ) · ( 2 ↑ 𝑘 ) ) ) → ( ( #b ‘ ( 𝑎 / 2 ) ) = ( ( #b ‘ 𝑎 ) − 1 ) → ( 𝑦 ∈ ℕ → ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) → ( ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ) ) ) ) |
| 185 |
12 184
|
syl |
⊢ ( ( ( 𝑎 / 2 ) ∈ ℕ0 ∧ ∀ 𝑥 ∈ ℕ0 ( ( #b ‘ 𝑥 ) = 𝑦 → 𝑥 = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑥 ) · ( 2 ↑ 𝑘 ) ) ) ) → ( ( #b ‘ ( 𝑎 / 2 ) ) = ( ( #b ‘ 𝑎 ) − 1 ) → ( 𝑦 ∈ ℕ → ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) → ( ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ) ) ) ) |
| 186 |
185
|
ex |
⊢ ( ( 𝑎 / 2 ) ∈ ℕ0 → ( ∀ 𝑥 ∈ ℕ0 ( ( #b ‘ 𝑥 ) = 𝑦 → 𝑥 = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑥 ) · ( 2 ↑ 𝑘 ) ) ) → ( ( #b ‘ ( 𝑎 / 2 ) ) = ( ( #b ‘ 𝑎 ) − 1 ) → ( 𝑦 ∈ ℕ → ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) → ( ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ) ) ) ) ) |
| 187 |
186
|
com25 |
⊢ ( ( 𝑎 / 2 ) ∈ ℕ0 → ( ( ( 𝑎 / 2 ) ∈ ℕ ∧ 𝑎 ∈ ℕ ) → ( ( #b ‘ ( 𝑎 / 2 ) ) = ( ( #b ‘ 𝑎 ) − 1 ) → ( 𝑦 ∈ ℕ → ( ∀ 𝑥 ∈ ℕ0 ( ( #b ‘ 𝑥 ) = 𝑦 → 𝑥 = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑥 ) · ( 2 ↑ 𝑘 ) ) ) → ( ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ) ) ) ) ) |
| 188 |
187
|
expdcom |
⊢ ( ( 𝑎 / 2 ) ∈ ℕ → ( 𝑎 ∈ ℕ → ( ( 𝑎 / 2 ) ∈ ℕ0 → ( ( #b ‘ ( 𝑎 / 2 ) ) = ( ( #b ‘ 𝑎 ) − 1 ) → ( 𝑦 ∈ ℕ → ( ∀ 𝑥 ∈ ℕ0 ( ( #b ‘ 𝑥 ) = 𝑦 → 𝑥 = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑥 ) · ( 2 ↑ 𝑘 ) ) ) → ( ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ) ) ) ) ) ) |
| 189 |
1 188
|
mpid |
⊢ ( ( 𝑎 / 2 ) ∈ ℕ → ( 𝑎 ∈ ℕ → ( ( #b ‘ ( 𝑎 / 2 ) ) = ( ( #b ‘ 𝑎 ) − 1 ) → ( 𝑦 ∈ ℕ → ( ∀ 𝑥 ∈ ℕ0 ( ( #b ‘ 𝑥 ) = 𝑦 → 𝑥 = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑥 ) · ( 2 ↑ 𝑘 ) ) ) → ( ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ) ) ) ) ) |
| 190 |
189
|
impcom |
⊢ ( ( 𝑎 ∈ ℕ ∧ ( 𝑎 / 2 ) ∈ ℕ ) → ( ( #b ‘ ( 𝑎 / 2 ) ) = ( ( #b ‘ 𝑎 ) − 1 ) → ( 𝑦 ∈ ℕ → ( ∀ 𝑥 ∈ ℕ0 ( ( #b ‘ 𝑥 ) = 𝑦 → 𝑥 = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑥 ) · ( 2 ↑ 𝑘 ) ) ) → ( ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ) ) ) ) |
| 191 |
3 190
|
mpd |
⊢ ( ( 𝑎 ∈ ℕ ∧ ( 𝑎 / 2 ) ∈ ℕ ) → ( 𝑦 ∈ ℕ → ( ∀ 𝑥 ∈ ℕ0 ( ( #b ‘ 𝑥 ) = 𝑦 → 𝑥 = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑥 ) · ( 2 ↑ 𝑘 ) ) ) → ( ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ) ) ) |
| 192 |
191
|
imp |
⊢ ( ( ( 𝑎 ∈ ℕ ∧ ( 𝑎 / 2 ) ∈ ℕ ) ∧ 𝑦 ∈ ℕ ) → ( ∀ 𝑥 ∈ ℕ0 ( ( #b ‘ 𝑥 ) = 𝑦 → 𝑥 = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑥 ) · ( 2 ↑ 𝑘 ) ) ) → ( ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ) ) |