Step |
Hyp |
Ref |
Expression |
1 |
|
aks4d1p1p3.1 |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
2 |
|
aks4d1p1p3.2 |
⊢ 𝐵 = ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) |
3 |
|
aks4d1p1p3.3 |
⊢ ( 𝜑 → 3 ≤ 𝑁 ) |
4 |
|
2re |
⊢ 2 ∈ ℝ |
5 |
4
|
a1i |
⊢ ( 𝜑 → 2 ∈ ℝ ) |
6 |
|
2pos |
⊢ 0 < 2 |
7 |
6
|
a1i |
⊢ ( 𝜑 → 0 < 2 ) |
8 |
1
|
nnred |
⊢ ( 𝜑 → 𝑁 ∈ ℝ ) |
9 |
1
|
nngt0d |
⊢ ( 𝜑 → 0 < 𝑁 ) |
10 |
|
1red |
⊢ ( 𝜑 → 1 ∈ ℝ ) |
11 |
|
1lt2 |
⊢ 1 < 2 |
12 |
11
|
a1i |
⊢ ( 𝜑 → 1 < 2 ) |
13 |
10 12
|
ltned |
⊢ ( 𝜑 → 1 ≠ 2 ) |
14 |
13
|
necomd |
⊢ ( 𝜑 → 2 ≠ 1 ) |
15 |
5 7 8 9 14
|
relogbcld |
⊢ ( 𝜑 → ( 2 logb 𝑁 ) ∈ ℝ ) |
16 |
|
5nn0 |
⊢ 5 ∈ ℕ0 |
17 |
16
|
a1i |
⊢ ( 𝜑 → 5 ∈ ℕ0 ) |
18 |
15 17
|
reexpcld |
⊢ ( 𝜑 → ( ( 2 logb 𝑁 ) ↑ 5 ) ∈ ℝ ) |
19 |
|
ceilcl |
⊢ ( ( ( 2 logb 𝑁 ) ↑ 5 ) ∈ ℝ → ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) ∈ ℤ ) |
20 |
18 19
|
syl |
⊢ ( 𝜑 → ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) ∈ ℤ ) |
21 |
20
|
zred |
⊢ ( 𝜑 → ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) ∈ ℝ ) |
22 |
2
|
a1i |
⊢ ( 𝜑 → 𝐵 = ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) ) |
23 |
22
|
eleq1d |
⊢ ( 𝜑 → ( 𝐵 ∈ ℝ ↔ ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) ∈ ℝ ) ) |
24 |
21 23
|
mpbird |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
25 |
|
0red |
⊢ ( 𝜑 → 0 ∈ ℝ ) |
26 |
|
7re |
⊢ 7 ∈ ℝ |
27 |
26
|
a1i |
⊢ ( 𝜑 → 7 ∈ ℝ ) |
28 |
|
7pos |
⊢ 0 < 7 |
29 |
28
|
a1i |
⊢ ( 𝜑 → 0 < 7 ) |
30 |
8 3
|
3lexlogpow5ineq3 |
⊢ ( 𝜑 → 7 < ( ( 2 logb 𝑁 ) ↑ 5 ) ) |
31 |
|
ceilge |
⊢ ( ( ( 2 logb 𝑁 ) ↑ 5 ) ∈ ℝ → ( ( 2 logb 𝑁 ) ↑ 5 ) ≤ ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) ) |
32 |
18 31
|
syl |
⊢ ( 𝜑 → ( ( 2 logb 𝑁 ) ↑ 5 ) ≤ ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) ) |
33 |
27 18 21 30 32
|
ltletrd |
⊢ ( 𝜑 → 7 < ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) ) |
34 |
22
|
eqcomd |
⊢ ( 𝜑 → ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) = 𝐵 ) |
35 |
33 34
|
breqtrd |
⊢ ( 𝜑 → 7 < 𝐵 ) |
36 |
25 27 24 29 35
|
lttrd |
⊢ ( 𝜑 → 0 < 𝐵 ) |
37 |
5 7 24 36 14
|
relogbcld |
⊢ ( 𝜑 → ( 2 logb 𝐵 ) ∈ ℝ ) |
38 |
37
|
flcld |
⊢ ( 𝜑 → ( ⌊ ‘ ( 2 logb 𝐵 ) ) ∈ ℤ ) |
39 |
38
|
zred |
⊢ ( 𝜑 → ( ⌊ ‘ ( 2 logb 𝐵 ) ) ∈ ℝ ) |
40 |
18 10
|
readdcld |
⊢ ( 𝜑 → ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ∈ ℝ ) |
41 |
18
|
ltp1d |
⊢ ( 𝜑 → ( ( 2 logb 𝑁 ) ↑ 5 ) < ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) |
42 |
27 18 40 30 41
|
lttrd |
⊢ ( 𝜑 → 7 < ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) |
43 |
25 27 40 29 42
|
lttrd |
⊢ ( 𝜑 → 0 < ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) |
44 |
5 7 40 43 14
|
relogbcld |
⊢ ( 𝜑 → ( 2 logb ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) ∈ ℝ ) |
45 |
|
flle |
⊢ ( ( 2 logb 𝐵 ) ∈ ℝ → ( ⌊ ‘ ( 2 logb 𝐵 ) ) ≤ ( 2 logb 𝐵 ) ) |
46 |
37 45
|
syl |
⊢ ( 𝜑 → ( ⌊ ‘ ( 2 logb 𝐵 ) ) ≤ ( 2 logb 𝐵 ) ) |
47 |
|
ceilm1lt |
⊢ ( ( ( 2 logb 𝑁 ) ↑ 5 ) ∈ ℝ → ( ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) − 1 ) < ( ( 2 logb 𝑁 ) ↑ 5 ) ) |
48 |
18 47
|
syl |
⊢ ( 𝜑 → ( ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) − 1 ) < ( ( 2 logb 𝑁 ) ↑ 5 ) ) |
49 |
21 10 18
|
ltsubaddd |
⊢ ( 𝜑 → ( ( ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) − 1 ) < ( ( 2 logb 𝑁 ) ↑ 5 ) ↔ ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) < ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) ) |
50 |
48 49
|
mpbid |
⊢ ( 𝜑 → ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) < ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) |
51 |
22 50
|
eqbrtrd |
⊢ ( 𝜑 → 𝐵 < ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) |
52 |
|
2z |
⊢ 2 ∈ ℤ |
53 |
52
|
a1i |
⊢ ( 𝜑 → 2 ∈ ℤ ) |
54 |
53
|
uzidd |
⊢ ( 𝜑 → 2 ∈ ( ℤ≥ ‘ 2 ) ) |
55 |
24 36
|
elrpd |
⊢ ( 𝜑 → 𝐵 ∈ ℝ+ ) |
56 |
40 43
|
elrpd |
⊢ ( 𝜑 → ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ∈ ℝ+ ) |
57 |
|
logblt |
⊢ ( ( 2 ∈ ( ℤ≥ ‘ 2 ) ∧ 𝐵 ∈ ℝ+ ∧ ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ∈ ℝ+ ) → ( 𝐵 < ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ↔ ( 2 logb 𝐵 ) < ( 2 logb ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) ) ) |
58 |
54 55 56 57
|
syl3anc |
⊢ ( 𝜑 → ( 𝐵 < ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ↔ ( 2 logb 𝐵 ) < ( 2 logb ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) ) ) |
59 |
51 58
|
mpbid |
⊢ ( 𝜑 → ( 2 logb 𝐵 ) < ( 2 logb ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) ) |
60 |
39 37 44 46 59
|
lelttrd |
⊢ ( 𝜑 → ( ⌊ ‘ ( 2 logb 𝐵 ) ) < ( 2 logb ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) ) |
61 |
|
3re |
⊢ 3 ∈ ℝ |
62 |
61
|
a1i |
⊢ ( 𝜑 → 3 ∈ ℝ ) |
63 |
|
1lt3 |
⊢ 1 < 3 |
64 |
63
|
a1i |
⊢ ( 𝜑 → 1 < 3 ) |
65 |
10 62 8 64 3
|
ltletrd |
⊢ ( 𝜑 → 1 < 𝑁 ) |
66 |
8 65 39 44
|
cxpltd |
⊢ ( 𝜑 → ( ( ⌊ ‘ ( 2 logb 𝐵 ) ) < ( 2 logb ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) ↔ ( 𝑁 ↑𝑐 ( ⌊ ‘ ( 2 logb 𝐵 ) ) ) < ( 𝑁 ↑𝑐 ( 2 logb ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) ) ) ) |
67 |
60 66
|
mpbid |
⊢ ( 𝜑 → ( 𝑁 ↑𝑐 ( ⌊ ‘ ( 2 logb 𝐵 ) ) ) < ( 𝑁 ↑𝑐 ( 2 logb ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) ) ) |