Step |
Hyp |
Ref |
Expression |
1 |
|
2nn |
⊢ 2 ∈ ℕ |
2 |
1
|
a1i |
⊢ ( 𝑁 ∈ ℕ → 2 ∈ ℕ ) |
3 |
|
id |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℕ ) |
4 |
2 3
|
nnmulcld |
⊢ ( 𝑁 ∈ ℕ → ( 2 · 𝑁 ) ∈ ℕ ) |
5 |
|
blennn |
⊢ ( ( 2 · 𝑁 ) ∈ ℕ → ( #b ‘ ( 2 · 𝑁 ) ) = ( ( ⌊ ‘ ( 2 logb ( 2 · 𝑁 ) ) ) + 1 ) ) |
6 |
4 5
|
syl |
⊢ ( 𝑁 ∈ ℕ → ( #b ‘ ( 2 · 𝑁 ) ) = ( ( ⌊ ‘ ( 2 logb ( 2 · 𝑁 ) ) ) + 1 ) ) |
7 |
|
2cn |
⊢ 2 ∈ ℂ |
8 |
7
|
a1i |
⊢ ( 𝑁 ∈ ℕ → 2 ∈ ℂ ) |
9 |
|
nncn |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℂ ) |
10 |
8 9
|
mulcomd |
⊢ ( 𝑁 ∈ ℕ → ( 2 · 𝑁 ) = ( 𝑁 · 2 ) ) |
11 |
10
|
oveq2d |
⊢ ( 𝑁 ∈ ℕ → ( 2 logb ( 2 · 𝑁 ) ) = ( 2 logb ( 𝑁 · 2 ) ) ) |
12 |
|
2z |
⊢ 2 ∈ ℤ |
13 |
|
uzid |
⊢ ( 2 ∈ ℤ → 2 ∈ ( ℤ≥ ‘ 2 ) ) |
14 |
12 13
|
ax-mp |
⊢ 2 ∈ ( ℤ≥ ‘ 2 ) |
15 |
|
eluz2cnn0n1 |
⊢ ( 2 ∈ ( ℤ≥ ‘ 2 ) → 2 ∈ ( ℂ ∖ { 0 , 1 } ) ) |
16 |
14 15
|
mp1i |
⊢ ( 𝑁 ∈ ℕ → 2 ∈ ( ℂ ∖ { 0 , 1 } ) ) |
17 |
|
nnrp |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℝ+ ) |
18 |
|
2rp |
⊢ 2 ∈ ℝ+ |
19 |
18
|
a1i |
⊢ ( 𝑁 ∈ ℕ → 2 ∈ ℝ+ ) |
20 |
|
relogbmul |
⊢ ( ( 2 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ ( 𝑁 ∈ ℝ+ ∧ 2 ∈ ℝ+ ) ) → ( 2 logb ( 𝑁 · 2 ) ) = ( ( 2 logb 𝑁 ) + ( 2 logb 2 ) ) ) |
21 |
16 17 19 20
|
syl12anc |
⊢ ( 𝑁 ∈ ℕ → ( 2 logb ( 𝑁 · 2 ) ) = ( ( 2 logb 𝑁 ) + ( 2 logb 2 ) ) ) |
22 |
|
2ne0 |
⊢ 2 ≠ 0 |
23 |
|
1ne2 |
⊢ 1 ≠ 2 |
24 |
23
|
necomi |
⊢ 2 ≠ 1 |
25 |
7 22 24
|
3pm3.2i |
⊢ ( 2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1 ) |
26 |
|
logbid1 |
⊢ ( ( 2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1 ) → ( 2 logb 2 ) = 1 ) |
27 |
25 26
|
mp1i |
⊢ ( 𝑁 ∈ ℕ → ( 2 logb 2 ) = 1 ) |
28 |
27
|
oveq2d |
⊢ ( 𝑁 ∈ ℕ → ( ( 2 logb 𝑁 ) + ( 2 logb 2 ) ) = ( ( 2 logb 𝑁 ) + 1 ) ) |
29 |
11 21 28
|
3eqtrd |
⊢ ( 𝑁 ∈ ℕ → ( 2 logb ( 2 · 𝑁 ) ) = ( ( 2 logb 𝑁 ) + 1 ) ) |
30 |
29
|
fveq2d |
⊢ ( 𝑁 ∈ ℕ → ( ⌊ ‘ ( 2 logb ( 2 · 𝑁 ) ) ) = ( ⌊ ‘ ( ( 2 logb 𝑁 ) + 1 ) ) ) |
31 |
24
|
a1i |
⊢ ( 𝑁 ∈ ℕ → 2 ≠ 1 ) |
32 |
|
relogbcl |
⊢ ( ( 2 ∈ ℝ+ ∧ 𝑁 ∈ ℝ+ ∧ 2 ≠ 1 ) → ( 2 logb 𝑁 ) ∈ ℝ ) |
33 |
19 17 31 32
|
syl3anc |
⊢ ( 𝑁 ∈ ℕ → ( 2 logb 𝑁 ) ∈ ℝ ) |
34 |
|
1zzd |
⊢ ( 𝑁 ∈ ℕ → 1 ∈ ℤ ) |
35 |
|
fladdz |
⊢ ( ( ( 2 logb 𝑁 ) ∈ ℝ ∧ 1 ∈ ℤ ) → ( ⌊ ‘ ( ( 2 logb 𝑁 ) + 1 ) ) = ( ( ⌊ ‘ ( 2 logb 𝑁 ) ) + 1 ) ) |
36 |
33 34 35
|
syl2anc |
⊢ ( 𝑁 ∈ ℕ → ( ⌊ ‘ ( ( 2 logb 𝑁 ) + 1 ) ) = ( ( ⌊ ‘ ( 2 logb 𝑁 ) ) + 1 ) ) |
37 |
30 36
|
eqtrd |
⊢ ( 𝑁 ∈ ℕ → ( ⌊ ‘ ( 2 logb ( 2 · 𝑁 ) ) ) = ( ( ⌊ ‘ ( 2 logb 𝑁 ) ) + 1 ) ) |
38 |
37
|
oveq1d |
⊢ ( 𝑁 ∈ ℕ → ( ( ⌊ ‘ ( 2 logb ( 2 · 𝑁 ) ) ) + 1 ) = ( ( ( ⌊ ‘ ( 2 logb 𝑁 ) ) + 1 ) + 1 ) ) |
39 |
|
blennn |
⊢ ( 𝑁 ∈ ℕ → ( #b ‘ 𝑁 ) = ( ( ⌊ ‘ ( 2 logb 𝑁 ) ) + 1 ) ) |
40 |
39
|
eqcomd |
⊢ ( 𝑁 ∈ ℕ → ( ( ⌊ ‘ ( 2 logb 𝑁 ) ) + 1 ) = ( #b ‘ 𝑁 ) ) |
41 |
40
|
oveq1d |
⊢ ( 𝑁 ∈ ℕ → ( ( ( ⌊ ‘ ( 2 logb 𝑁 ) ) + 1 ) + 1 ) = ( ( #b ‘ 𝑁 ) + 1 ) ) |
42 |
6 38 41
|
3eqtrd |
⊢ ( 𝑁 ∈ ℕ → ( #b ‘ ( 2 · 𝑁 ) ) = ( ( #b ‘ 𝑁 ) + 1 ) ) |