Step |
Hyp |
Ref |
Expression |
1 |
|
2nn |
⊢ 2 ∈ ℕ |
2 |
1
|
a1i |
⊢ ( 𝑁 ∈ ℕ → 2 ∈ ℕ ) |
3 |
|
blennnelnn |
⊢ ( 𝑁 ∈ ℕ → ( #b ‘ 𝑁 ) ∈ ℕ ) |
4 |
|
nnm1nn0 |
⊢ ( ( #b ‘ 𝑁 ) ∈ ℕ → ( ( #b ‘ 𝑁 ) − 1 ) ∈ ℕ0 ) |
5 |
3 4
|
syl |
⊢ ( 𝑁 ∈ ℕ → ( ( #b ‘ 𝑁 ) − 1 ) ∈ ℕ0 ) |
6 |
|
nnre |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℝ ) |
7 |
|
nnnn0 |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℕ0 ) |
8 |
7
|
nn0ge0d |
⊢ ( 𝑁 ∈ ℕ → 0 ≤ 𝑁 ) |
9 |
|
elrege0 |
⊢ ( 𝑁 ∈ ( 0 [,) +∞ ) ↔ ( 𝑁 ∈ ℝ ∧ 0 ≤ 𝑁 ) ) |
10 |
6 8 9
|
sylanbrc |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ( 0 [,) +∞ ) ) |
11 |
|
nn0digval |
⊢ ( ( 2 ∈ ℕ ∧ ( ( #b ‘ 𝑁 ) − 1 ) ∈ ℕ0 ∧ 𝑁 ∈ ( 0 [,) +∞ ) ) → ( ( ( #b ‘ 𝑁 ) − 1 ) ( digit ‘ 2 ) 𝑁 ) = ( ( ⌊ ‘ ( 𝑁 / ( 2 ↑ ( ( #b ‘ 𝑁 ) − 1 ) ) ) ) mod 2 ) ) |
12 |
2 5 10 11
|
syl3anc |
⊢ ( 𝑁 ∈ ℕ → ( ( ( #b ‘ 𝑁 ) − 1 ) ( digit ‘ 2 ) 𝑁 ) = ( ( ⌊ ‘ ( 𝑁 / ( 2 ↑ ( ( #b ‘ 𝑁 ) − 1 ) ) ) ) mod 2 ) ) |
13 |
|
n2dvds1 |
⊢ ¬ 2 ∥ 1 |
14 |
|
blennn |
⊢ ( 𝑁 ∈ ℕ → ( #b ‘ 𝑁 ) = ( ( ⌊ ‘ ( 2 logb 𝑁 ) ) + 1 ) ) |
15 |
14
|
oveq1d |
⊢ ( 𝑁 ∈ ℕ → ( ( #b ‘ 𝑁 ) − 1 ) = ( ( ( ⌊ ‘ ( 2 logb 𝑁 ) ) + 1 ) − 1 ) ) |
16 |
|
2z |
⊢ 2 ∈ ℤ |
17 |
|
uzid |
⊢ ( 2 ∈ ℤ → 2 ∈ ( ℤ≥ ‘ 2 ) ) |
18 |
16 17
|
ax-mp |
⊢ 2 ∈ ( ℤ≥ ‘ 2 ) |
19 |
|
nnrp |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℝ+ ) |
20 |
|
relogbzcl |
⊢ ( ( 2 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝑁 ∈ ℝ+ ) → ( 2 logb 𝑁 ) ∈ ℝ ) |
21 |
18 19 20
|
sylancr |
⊢ ( 𝑁 ∈ ℕ → ( 2 logb 𝑁 ) ∈ ℝ ) |
22 |
21
|
flcld |
⊢ ( 𝑁 ∈ ℕ → ( ⌊ ‘ ( 2 logb 𝑁 ) ) ∈ ℤ ) |
23 |
22
|
zcnd |
⊢ ( 𝑁 ∈ ℕ → ( ⌊ ‘ ( 2 logb 𝑁 ) ) ∈ ℂ ) |
24 |
|
pncan1 |
⊢ ( ( ⌊ ‘ ( 2 logb 𝑁 ) ) ∈ ℂ → ( ( ( ⌊ ‘ ( 2 logb 𝑁 ) ) + 1 ) − 1 ) = ( ⌊ ‘ ( 2 logb 𝑁 ) ) ) |
25 |
23 24
|
syl |
⊢ ( 𝑁 ∈ ℕ → ( ( ( ⌊ ‘ ( 2 logb 𝑁 ) ) + 1 ) − 1 ) = ( ⌊ ‘ ( 2 logb 𝑁 ) ) ) |
26 |
15 25
|
eqtrd |
⊢ ( 𝑁 ∈ ℕ → ( ( #b ‘ 𝑁 ) − 1 ) = ( ⌊ ‘ ( 2 logb 𝑁 ) ) ) |
27 |
26
|
oveq2d |
⊢ ( 𝑁 ∈ ℕ → ( 2 ↑ ( ( #b ‘ 𝑁 ) − 1 ) ) = ( 2 ↑ ( ⌊ ‘ ( 2 logb 𝑁 ) ) ) ) |
28 |
27
|
oveq2d |
⊢ ( 𝑁 ∈ ℕ → ( 𝑁 / ( 2 ↑ ( ( #b ‘ 𝑁 ) − 1 ) ) ) = ( 𝑁 / ( 2 ↑ ( ⌊ ‘ ( 2 logb 𝑁 ) ) ) ) ) |
29 |
28
|
fveq2d |
⊢ ( 𝑁 ∈ ℕ → ( ⌊ ‘ ( 𝑁 / ( 2 ↑ ( ( #b ‘ 𝑁 ) − 1 ) ) ) ) = ( ⌊ ‘ ( 𝑁 / ( 2 ↑ ( ⌊ ‘ ( 2 logb 𝑁 ) ) ) ) ) ) |
30 |
|
fldivexpfllog2 |
⊢ ( 𝑁 ∈ ℝ+ → ( ⌊ ‘ ( 𝑁 / ( 2 ↑ ( ⌊ ‘ ( 2 logb 𝑁 ) ) ) ) ) = 1 ) |
31 |
19 30
|
syl |
⊢ ( 𝑁 ∈ ℕ → ( ⌊ ‘ ( 𝑁 / ( 2 ↑ ( ⌊ ‘ ( 2 logb 𝑁 ) ) ) ) ) = 1 ) |
32 |
29 31
|
eqtrd |
⊢ ( 𝑁 ∈ ℕ → ( ⌊ ‘ ( 𝑁 / ( 2 ↑ ( ( #b ‘ 𝑁 ) − 1 ) ) ) ) = 1 ) |
33 |
32
|
breq2d |
⊢ ( 𝑁 ∈ ℕ → ( 2 ∥ ( ⌊ ‘ ( 𝑁 / ( 2 ↑ ( ( #b ‘ 𝑁 ) − 1 ) ) ) ) ↔ 2 ∥ 1 ) ) |
34 |
13 33
|
mtbiri |
⊢ ( 𝑁 ∈ ℕ → ¬ 2 ∥ ( ⌊ ‘ ( 𝑁 / ( 2 ↑ ( ( #b ‘ 𝑁 ) − 1 ) ) ) ) ) |
35 |
|
2re |
⊢ 2 ∈ ℝ |
36 |
35
|
a1i |
⊢ ( 𝑁 ∈ ℕ → 2 ∈ ℝ ) |
37 |
36 5
|
reexpcld |
⊢ ( 𝑁 ∈ ℕ → ( 2 ↑ ( ( #b ‘ 𝑁 ) − 1 ) ) ∈ ℝ ) |
38 |
|
2cnd |
⊢ ( 𝑁 ∈ ℕ → 2 ∈ ℂ ) |
39 |
|
2ne0 |
⊢ 2 ≠ 0 |
40 |
39
|
a1i |
⊢ ( 𝑁 ∈ ℕ → 2 ≠ 0 ) |
41 |
5
|
nn0zd |
⊢ ( 𝑁 ∈ ℕ → ( ( #b ‘ 𝑁 ) − 1 ) ∈ ℤ ) |
42 |
38 40 41
|
expne0d |
⊢ ( 𝑁 ∈ ℕ → ( 2 ↑ ( ( #b ‘ 𝑁 ) − 1 ) ) ≠ 0 ) |
43 |
6 37 42
|
redivcld |
⊢ ( 𝑁 ∈ ℕ → ( 𝑁 / ( 2 ↑ ( ( #b ‘ 𝑁 ) − 1 ) ) ) ∈ ℝ ) |
44 |
43
|
flcld |
⊢ ( 𝑁 ∈ ℕ → ( ⌊ ‘ ( 𝑁 / ( 2 ↑ ( ( #b ‘ 𝑁 ) − 1 ) ) ) ) ∈ ℤ ) |
45 |
|
mod2eq1n2dvds |
⊢ ( ( ⌊ ‘ ( 𝑁 / ( 2 ↑ ( ( #b ‘ 𝑁 ) − 1 ) ) ) ) ∈ ℤ → ( ( ( ⌊ ‘ ( 𝑁 / ( 2 ↑ ( ( #b ‘ 𝑁 ) − 1 ) ) ) ) mod 2 ) = 1 ↔ ¬ 2 ∥ ( ⌊ ‘ ( 𝑁 / ( 2 ↑ ( ( #b ‘ 𝑁 ) − 1 ) ) ) ) ) ) |
46 |
44 45
|
syl |
⊢ ( 𝑁 ∈ ℕ → ( ( ( ⌊ ‘ ( 𝑁 / ( 2 ↑ ( ( #b ‘ 𝑁 ) − 1 ) ) ) ) mod 2 ) = 1 ↔ ¬ 2 ∥ ( ⌊ ‘ ( 𝑁 / ( 2 ↑ ( ( #b ‘ 𝑁 ) − 1 ) ) ) ) ) ) |
47 |
34 46
|
mpbird |
⊢ ( 𝑁 ∈ ℕ → ( ( ⌊ ‘ ( 𝑁 / ( 2 ↑ ( ( #b ‘ 𝑁 ) − 1 ) ) ) ) mod 2 ) = 1 ) |
48 |
12 47
|
eqtrd |
⊢ ( 𝑁 ∈ ℕ → ( ( ( #b ‘ 𝑁 ) − 1 ) ( digit ‘ 2 ) 𝑁 ) = 1 ) |