Step |
Hyp |
Ref |
Expression |
1 |
|
fveqeq2 |
⊢ ( 𝑎 = 𝑥 → ( ( #b ‘ 𝑎 ) = 𝑦 ↔ ( #b ‘ 𝑥 ) = 𝑦 ) ) |
2 |
|
id |
⊢ ( 𝑎 = 𝑥 → 𝑎 = 𝑥 ) |
3 |
|
oveq2 |
⊢ ( 𝑎 = 𝑥 → ( 𝑘 ( digit ‘ 2 ) 𝑎 ) = ( 𝑘 ( digit ‘ 2 ) 𝑥 ) ) |
4 |
3
|
oveq1d |
⊢ ( 𝑎 = 𝑥 → ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) = ( ( 𝑘 ( digit ‘ 2 ) 𝑥 ) · ( 2 ↑ 𝑘 ) ) ) |
5 |
4
|
sumeq2sdv |
⊢ ( 𝑎 = 𝑥 → Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑥 ) · ( 2 ↑ 𝑘 ) ) ) |
6 |
2 5
|
eqeq12d |
⊢ ( 𝑎 = 𝑥 → ( 𝑎 = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ↔ 𝑥 = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑥 ) · ( 2 ↑ 𝑘 ) ) ) ) |
7 |
1 6
|
imbi12d |
⊢ ( 𝑎 = 𝑥 → ( ( ( #b ‘ 𝑎 ) = 𝑦 → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ↔ ( ( #b ‘ 𝑥 ) = 𝑦 → 𝑥 = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑥 ) · ( 2 ↑ 𝑘 ) ) ) ) ) |
8 |
7
|
cbvralvw |
⊢ ( ∀ 𝑎 ∈ ℕ0 ( ( #b ‘ 𝑎 ) = 𝑦 → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ↔ ∀ 𝑥 ∈ ℕ0 ( ( #b ‘ 𝑥 ) = 𝑦 → 𝑥 = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑥 ) · ( 2 ↑ 𝑘 ) ) ) ) |
9 |
|
elnn0 |
⊢ ( 𝑎 ∈ ℕ0 ↔ ( 𝑎 ∈ ℕ ∨ 𝑎 = 0 ) ) |
10 |
|
nn0sumshdiglemA |
⊢ ( ( ( 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 ) ) ) ) ) |
11 |
10
|
expimpd |
⊢ ( ( 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 ) ) ) ) ) |
12 |
|
nn0sumshdiglemB |
⊢ ( ( ( 𝑎 ∈ ℕ ∧ ( ( 𝑎 − 1 ) / 2 ) ∈ ℕ0 ) ∧ 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 ) ) ) ) ) |
13 |
12
|
expimpd |
⊢ ( ( 𝑎 ∈ ℕ ∧ ( ( 𝑎 − 1 ) / 2 ) ∈ ℕ0 ) → ( ( 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 ) ) ) ) ) |
14 |
|
nneom |
⊢ ( 𝑎 ∈ ℕ → ( ( 𝑎 / 2 ) ∈ ℕ ∨ ( ( 𝑎 − 1 ) / 2 ) ∈ ℕ0 ) ) |
15 |
11 13 14
|
mpjaodan |
⊢ ( 𝑎 ∈ 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 ) ) ) ) ) |
16 |
|
eqcom |
⊢ ( 1 = ( 𝑦 + 1 ) ↔ ( 𝑦 + 1 ) = 1 ) |
17 |
16
|
a1i |
⊢ ( 𝑦 ∈ ℕ → ( 1 = ( 𝑦 + 1 ) ↔ ( 𝑦 + 1 ) = 1 ) ) |
18 |
|
nncn |
⊢ ( 𝑦 ∈ ℕ → 𝑦 ∈ ℂ ) |
19 |
|
1cnd |
⊢ ( 𝑦 ∈ ℕ → 1 ∈ ℂ ) |
20 |
18 19 19
|
addlsub |
⊢ ( 𝑦 ∈ ℕ → ( ( 𝑦 + 1 ) = 1 ↔ 𝑦 = ( 1 − 1 ) ) ) |
21 |
|
1m1e0 |
⊢ ( 1 − 1 ) = 0 |
22 |
21
|
a1i |
⊢ ( 𝑦 ∈ ℕ → ( 1 − 1 ) = 0 ) |
23 |
22
|
eqeq2d |
⊢ ( 𝑦 ∈ ℕ → ( 𝑦 = ( 1 − 1 ) ↔ 𝑦 = 0 ) ) |
24 |
17 20 23
|
3bitrd |
⊢ ( 𝑦 ∈ ℕ → ( 1 = ( 𝑦 + 1 ) ↔ 𝑦 = 0 ) ) |
25 |
|
oveq1 |
⊢ ( 𝑦 = 0 → ( 𝑦 + 1 ) = ( 0 + 1 ) ) |
26 |
25
|
oveq2d |
⊢ ( 𝑦 = 0 → ( 0 ..^ ( 𝑦 + 1 ) ) = ( 0 ..^ ( 0 + 1 ) ) ) |
27 |
|
0p1e1 |
⊢ ( 0 + 1 ) = 1 |
28 |
27
|
oveq2i |
⊢ ( 0 ..^ ( 0 + 1 ) ) = ( 0 ..^ 1 ) |
29 |
|
fzo01 |
⊢ ( 0 ..^ 1 ) = { 0 } |
30 |
28 29
|
eqtri |
⊢ ( 0 ..^ ( 0 + 1 ) ) = { 0 } |
31 |
26 30
|
eqtrdi |
⊢ ( 𝑦 = 0 → ( 0 ..^ ( 𝑦 + 1 ) ) = { 0 } ) |
32 |
31
|
sumeq1d |
⊢ ( 𝑦 = 0 → Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 0 ) · ( 2 ↑ 𝑘 ) ) = Σ 𝑘 ∈ { 0 } ( ( 𝑘 ( digit ‘ 2 ) 0 ) · ( 2 ↑ 𝑘 ) ) ) |
33 |
|
0cn |
⊢ 0 ∈ ℂ |
34 |
|
oveq1 |
⊢ ( 𝑘 = 0 → ( 𝑘 ( digit ‘ 2 ) 0 ) = ( 0 ( digit ‘ 2 ) 0 ) ) |
35 |
|
2nn |
⊢ 2 ∈ ℕ |
36 |
|
0z |
⊢ 0 ∈ ℤ |
37 |
|
dig0 |
⊢ ( ( 2 ∈ ℕ ∧ 0 ∈ ℤ ) → ( 0 ( digit ‘ 2 ) 0 ) = 0 ) |
38 |
35 36 37
|
mp2an |
⊢ ( 0 ( digit ‘ 2 ) 0 ) = 0 |
39 |
34 38
|
eqtrdi |
⊢ ( 𝑘 = 0 → ( 𝑘 ( digit ‘ 2 ) 0 ) = 0 ) |
40 |
|
oveq2 |
⊢ ( 𝑘 = 0 → ( 2 ↑ 𝑘 ) = ( 2 ↑ 0 ) ) |
41 |
|
2cn |
⊢ 2 ∈ ℂ |
42 |
|
exp0 |
⊢ ( 2 ∈ ℂ → ( 2 ↑ 0 ) = 1 ) |
43 |
41 42
|
ax-mp |
⊢ ( 2 ↑ 0 ) = 1 |
44 |
40 43
|
eqtrdi |
⊢ ( 𝑘 = 0 → ( 2 ↑ 𝑘 ) = 1 ) |
45 |
39 44
|
oveq12d |
⊢ ( 𝑘 = 0 → ( ( 𝑘 ( digit ‘ 2 ) 0 ) · ( 2 ↑ 𝑘 ) ) = ( 0 · 1 ) ) |
46 |
|
1re |
⊢ 1 ∈ ℝ |
47 |
|
mul02lem2 |
⊢ ( 1 ∈ ℝ → ( 0 · 1 ) = 0 ) |
48 |
46 47
|
ax-mp |
⊢ ( 0 · 1 ) = 0 |
49 |
45 48
|
eqtrdi |
⊢ ( 𝑘 = 0 → ( ( 𝑘 ( digit ‘ 2 ) 0 ) · ( 2 ↑ 𝑘 ) ) = 0 ) |
50 |
49
|
sumsn |
⊢ ( ( 0 ∈ ℂ ∧ 0 ∈ ℂ ) → Σ 𝑘 ∈ { 0 } ( ( 𝑘 ( digit ‘ 2 ) 0 ) · ( 2 ↑ 𝑘 ) ) = 0 ) |
51 |
33 33 50
|
mp2an |
⊢ Σ 𝑘 ∈ { 0 } ( ( 𝑘 ( digit ‘ 2 ) 0 ) · ( 2 ↑ 𝑘 ) ) = 0 |
52 |
32 51
|
eqtr2di |
⊢ ( 𝑦 = 0 → 0 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 0 ) · ( 2 ↑ 𝑘 ) ) ) |
53 |
24 52
|
syl6bi |
⊢ ( 𝑦 ∈ ℕ → ( 1 = ( 𝑦 + 1 ) → 0 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 0 ) · ( 2 ↑ 𝑘 ) ) ) ) |
54 |
53
|
adantl |
⊢ ( ( 𝑎 = 0 ∧ 𝑦 ∈ ℕ ) → ( 1 = ( 𝑦 + 1 ) → 0 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 0 ) · ( 2 ↑ 𝑘 ) ) ) ) |
55 |
|
fveq2 |
⊢ ( 𝑎 = 0 → ( #b ‘ 𝑎 ) = ( #b ‘ 0 ) ) |
56 |
|
blen0 |
⊢ ( #b ‘ 0 ) = 1 |
57 |
55 56
|
eqtrdi |
⊢ ( 𝑎 = 0 → ( #b ‘ 𝑎 ) = 1 ) |
58 |
57
|
eqeq1d |
⊢ ( 𝑎 = 0 → ( ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) ↔ 1 = ( 𝑦 + 1 ) ) ) |
59 |
|
id |
⊢ ( 𝑎 = 0 → 𝑎 = 0 ) |
60 |
|
oveq2 |
⊢ ( 𝑎 = 0 → ( 𝑘 ( digit ‘ 2 ) 𝑎 ) = ( 𝑘 ( digit ‘ 2 ) 0 ) ) |
61 |
60
|
oveq1d |
⊢ ( 𝑎 = 0 → ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) = ( ( 𝑘 ( digit ‘ 2 ) 0 ) · ( 2 ↑ 𝑘 ) ) ) |
62 |
61
|
sumeq2sdv |
⊢ ( 𝑎 = 0 → Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 0 ) · ( 2 ↑ 𝑘 ) ) ) |
63 |
59 62
|
eqeq12d |
⊢ ( 𝑎 = 0 → ( 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ↔ 0 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 0 ) · ( 2 ↑ 𝑘 ) ) ) ) |
64 |
58 63
|
imbi12d |
⊢ ( 𝑎 = 0 → ( ( ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ↔ ( 1 = ( 𝑦 + 1 ) → 0 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 0 ) · ( 2 ↑ 𝑘 ) ) ) ) ) |
65 |
64
|
adantr |
⊢ ( ( 𝑎 = 0 ∧ 𝑦 ∈ ℕ ) → ( ( ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ↔ ( 1 = ( 𝑦 + 1 ) → 0 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 0 ) · ( 2 ↑ 𝑘 ) ) ) ) ) |
66 |
54 65
|
mpbird |
⊢ ( ( 𝑎 = 0 ∧ 𝑦 ∈ ℕ ) → ( ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ) |
67 |
66
|
a1d |
⊢ ( ( 𝑎 = 0 ∧ 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 ) ) ) ) ) |
68 |
67
|
expimpd |
⊢ ( 𝑎 = 0 → ( ( 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 ) ) ) ) ) |
69 |
15 68
|
jaoi |
⊢ ( ( 𝑎 ∈ ℕ ∨ 𝑎 = 0 ) → ( ( 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 ) ) ) ) ) |
70 |
9 69
|
sylbi |
⊢ ( 𝑎 ∈ ℕ0 → ( ( 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 ) ) ) ) ) |
71 |
70
|
com12 |
⊢ ( ( 𝑦 ∈ ℕ ∧ ∀ 𝑥 ∈ ℕ0 ( ( #b ‘ 𝑥 ) = 𝑦 → 𝑥 = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑥 ) · ( 2 ↑ 𝑘 ) ) ) ) → ( 𝑎 ∈ ℕ0 → ( ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ) ) |
72 |
71
|
ralrimiv |
⊢ ( ( 𝑦 ∈ ℕ ∧ ∀ 𝑥 ∈ ℕ0 ( ( #b ‘ 𝑥 ) = 𝑦 → 𝑥 = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑥 ) · ( 2 ↑ 𝑘 ) ) ) ) → ∀ 𝑎 ∈ ℕ0 ( ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ) |
73 |
72
|
ex |
⊢ ( 𝑦 ∈ ℕ → ( ∀ 𝑥 ∈ ℕ0 ( ( #b ‘ 𝑥 ) = 𝑦 → 𝑥 = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑥 ) · ( 2 ↑ 𝑘 ) ) ) → ∀ 𝑎 ∈ ℕ0 ( ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ) ) |
74 |
8 73
|
syl5bi |
⊢ ( 𝑦 ∈ ℕ → ( ∀ 𝑎 ∈ ℕ0 ( ( #b ‘ 𝑎 ) = 𝑦 → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ 𝑦 ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) → ∀ 𝑎 ∈ ℕ0 ( ( #b ‘ 𝑎 ) = ( 𝑦 + 1 ) → 𝑎 = Σ 𝑘 ∈ ( 0 ..^ ( 𝑦 + 1 ) ) ( ( 𝑘 ( digit ‘ 2 ) 𝑎 ) · ( 2 ↑ 𝑘 ) ) ) ) ) |