Metamath Proof Explorer


Theorem aks4d1p1p3

Description: Bound of a ceiling of the binary logarithm to the fifth power. (Contributed by metakunt, 19-Aug-2024)

Ref Expression
Hypotheses aks4d1p1p3.1 φ N
aks4d1p1p3.2 B = log 2 N 5
aks4d1p1p3.3 φ 3 N
Assertion aks4d1p1p3 φ N log 2 B < N log 2 log 2 N 5 + 1

Proof

Step Hyp Ref Expression
1 aks4d1p1p3.1 φ N
2 aks4d1p1p3.2 B = log 2 N 5
3 aks4d1p1p3.3 φ 3 N
4 2re 2
5 4 a1i φ 2
6 2pos 0 < 2
7 6 a1i φ 0 < 2
8 1 nnred φ N
9 1 nngt0d φ 0 < N
10 1red φ 1
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 log 2 N
16 5nn0 5 0
17 16 a1i φ 5 0
18 15 17 reexpcld φ log 2 N 5
19 ceilcl log 2 N 5 log 2 N 5
20 18 19 syl φ log 2 N 5
21 20 zred φ log 2 N 5
22 2 a1i φ B = log 2 N 5
23 22 eleq1d φ B log 2 N 5
24 21 23 mpbird φ B
25 0red φ 0
26 7re 7
27 26 a1i φ 7
28 7pos 0 < 7
29 28 a1i φ 0 < 7
30 8 3 3lexlogpow5ineq3 φ 7 < log 2 N 5
31 ceilge log 2 N 5 log 2 N 5 log 2 N 5
32 18 31 syl φ log 2 N 5 log 2 N 5
33 27 18 21 30 32 ltletrd φ 7 < log 2 N 5
34 22 eqcomd φ log 2 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 log 2 B
38 37 flcld φ log 2 B
39 38 zred φ log 2 B
40 18 10 readdcld φ log 2 N 5 + 1
41 18 ltp1d φ log 2 N 5 < log 2 N 5 + 1
42 27 18 40 30 41 lttrd φ 7 < log 2 N 5 + 1
43 25 27 40 29 42 lttrd φ 0 < log 2 N 5 + 1
44 5 7 40 43 14 relogbcld φ log 2 log 2 N 5 + 1
45 flle log 2 B log 2 B log 2 B
46 37 45 syl φ log 2 B log 2 B
47 ceilm1lt log 2 N 5 log 2 N 5 1 < log 2 N 5
48 18 47 syl φ log 2 N 5 1 < log 2 N 5
49 21 10 18 ltsubaddd φ log 2 N 5 1 < log 2 N 5 log 2 N 5 < log 2 N 5 + 1
50 48 49 mpbid φ log 2 N 5 < log 2 N 5 + 1
51 22 50 eqbrtrd φ B < log 2 N 5 + 1
52 2z 2
53 52 a1i φ 2
54 53 uzidd φ 2 2
55 24 36 elrpd φ B +
56 40 43 elrpd φ log 2 N 5 + 1 +
57 logblt 2 2 B + log 2 N 5 + 1 + B < log 2 N 5 + 1 log 2 B < log 2 log 2 N 5 + 1
58 54 55 56 57 syl3anc φ B < log 2 N 5 + 1 log 2 B < log 2 log 2 N 5 + 1
59 51 58 mpbid φ log 2 B < log 2 log 2 N 5 + 1
60 39 37 44 46 59 lelttrd φ log 2 B < log 2 log 2 N 5 + 1
61 3re 3
62 61 a1i φ 3
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 log 2 B < log 2 log 2 N 5 + 1 N log 2 B < N log 2 log 2 N 5 + 1
67 60 66 mpbid φ N log 2 B < N log 2 log 2 N 5 + 1