Step |
Hyp |
Ref |
Expression |
1 |
|
aks4d1p1p4.1 |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
2 |
|
aks4d1p1p4.2 |
⊢ 𝐴 = ( ( 𝑁 ↑ ( ⌊ ‘ ( 2 logb 𝐵 ) ) ) · ∏ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( 2 logb 𝑁 ) ↑ 2 ) ) ) ( ( 𝑁 ↑ 𝑘 ) − 1 ) ) |
3 |
|
aks4d1p1p4.3 |
⊢ 𝐵 = ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) |
4 |
|
aks4d1p1p4.4 |
⊢ ( 𝜑 → 3 ≤ 𝑁 ) |
5 |
|
aks4d1p1p4.5 |
⊢ 𝐶 = ( 2 logb ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) |
6 |
|
aks4d1p1p4.6 |
⊢ 𝐷 = ( ( 2 logb 𝑁 ) ↑ 2 ) |
7 |
|
aks4d1p1p4.7 |
⊢ 𝐸 = ( ( 2 logb 𝑁 ) ↑ 4 ) |
8 |
|
aks4d1p1p4.8 |
⊢ ( 𝜑 → ( ( 2 · 𝐶 ) + 𝐷 ) ≤ 𝐸 ) |
9 |
1
|
nnred |
⊢ ( 𝜑 → 𝑁 ∈ ℝ ) |
10 |
|
2re |
⊢ 2 ∈ ℝ |
11 |
10
|
a1i |
⊢ ( 𝜑 → 2 ∈ ℝ ) |
12 |
|
2pos |
⊢ 0 < 2 |
13 |
12
|
a1i |
⊢ ( 𝜑 → 0 < 2 ) |
14 |
1
|
nngt0d |
⊢ ( 𝜑 → 0 < 𝑁 ) |
15 |
|
1red |
⊢ ( 𝜑 → 1 ∈ ℝ ) |
16 |
|
1lt2 |
⊢ 1 < 2 |
17 |
16
|
a1i |
⊢ ( 𝜑 → 1 < 2 ) |
18 |
15 17
|
ltned |
⊢ ( 𝜑 → 1 ≠ 2 ) |
19 |
18
|
necomd |
⊢ ( 𝜑 → 2 ≠ 1 ) |
20 |
11 13 9 14 19
|
relogbcld |
⊢ ( 𝜑 → ( 2 logb 𝑁 ) ∈ ℝ ) |
21 |
|
5nn0 |
⊢ 5 ∈ ℕ0 |
22 |
21
|
a1i |
⊢ ( 𝜑 → 5 ∈ ℕ0 ) |
23 |
20 22
|
reexpcld |
⊢ ( 𝜑 → ( ( 2 logb 𝑁 ) ↑ 5 ) ∈ ℝ ) |
24 |
|
ceilcl |
⊢ ( ( ( 2 logb 𝑁 ) ↑ 5 ) ∈ ℝ → ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) ∈ ℤ ) |
25 |
23 24
|
syl |
⊢ ( 𝜑 → ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) ∈ ℤ ) |
26 |
25
|
zred |
⊢ ( 𝜑 → ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) ∈ ℝ ) |
27 |
3
|
a1i |
⊢ ( 𝜑 → 𝐵 = ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) ) |
28 |
27
|
eleq1d |
⊢ ( 𝜑 → ( 𝐵 ∈ ℝ ↔ ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) ∈ ℝ ) ) |
29 |
26 28
|
mpbird |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
30 |
|
0red |
⊢ ( 𝜑 → 0 ∈ ℝ ) |
31 |
|
7re |
⊢ 7 ∈ ℝ |
32 |
31
|
a1i |
⊢ ( 𝜑 → 7 ∈ ℝ ) |
33 |
|
7pos |
⊢ 0 < 7 |
34 |
33
|
a1i |
⊢ ( 𝜑 → 0 < 7 ) |
35 |
9 4
|
3lexlogpow5ineq3 |
⊢ ( 𝜑 → 7 < ( ( 2 logb 𝑁 ) ↑ 5 ) ) |
36 |
32 23 35
|
ltled |
⊢ ( 𝜑 → 7 ≤ ( ( 2 logb 𝑁 ) ↑ 5 ) ) |
37 |
|
ceilge |
⊢ ( ( ( 2 logb 𝑁 ) ↑ 5 ) ∈ ℝ → ( ( 2 logb 𝑁 ) ↑ 5 ) ≤ ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) ) |
38 |
23 37
|
syl |
⊢ ( 𝜑 → ( ( 2 logb 𝑁 ) ↑ 5 ) ≤ ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) ) |
39 |
38 27
|
breqtrrd |
⊢ ( 𝜑 → ( ( 2 logb 𝑁 ) ↑ 5 ) ≤ 𝐵 ) |
40 |
32 23 29 36 39
|
letrd |
⊢ ( 𝜑 → 7 ≤ 𝐵 ) |
41 |
30 32 29 34 40
|
ltletrd |
⊢ ( 𝜑 → 0 < 𝐵 ) |
42 |
11 13 29 41 19
|
relogbcld |
⊢ ( 𝜑 → ( 2 logb 𝐵 ) ∈ ℝ ) |
43 |
42
|
flcld |
⊢ ( 𝜑 → ( ⌊ ‘ ( 2 logb 𝐵 ) ) ∈ ℤ ) |
44 |
30 15
|
readdcld |
⊢ ( 𝜑 → ( 0 + 1 ) ∈ ℝ ) |
45 |
43
|
zred |
⊢ ( 𝜑 → ( ⌊ ‘ ( 2 logb 𝐵 ) ) ∈ ℝ ) |
46 |
45 15
|
readdcld |
⊢ ( 𝜑 → ( ( ⌊ ‘ ( 2 logb 𝐵 ) ) + 1 ) ∈ ℝ ) |
47 |
11 13 11 13 19
|
relogbcld |
⊢ ( 𝜑 → ( 2 logb 2 ) ∈ ℝ ) |
48 |
15
|
leidd |
⊢ ( 𝜑 → 1 ≤ 1 ) |
49 |
|
1cnd |
⊢ ( 𝜑 → 1 ∈ ℂ ) |
50 |
49
|
addid2d |
⊢ ( 𝜑 → ( 0 + 1 ) = 1 ) |
51 |
11
|
recnd |
⊢ ( 𝜑 → 2 ∈ ℂ ) |
52 |
30 13
|
gtned |
⊢ ( 𝜑 → 2 ≠ 0 ) |
53 |
|
logbid1 |
⊢ ( ( 2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1 ) → ( 2 logb 2 ) = 1 ) |
54 |
51 52 19 53
|
syl3anc |
⊢ ( 𝜑 → ( 2 logb 2 ) = 1 ) |
55 |
54
|
eqcomd |
⊢ ( 𝜑 → 1 = ( 2 logb 2 ) ) |
56 |
55
|
eqcomd |
⊢ ( 𝜑 → ( 2 logb 2 ) = 1 ) |
57 |
50 56
|
breq12d |
⊢ ( 𝜑 → ( ( 0 + 1 ) ≤ ( 2 logb 2 ) ↔ 1 ≤ 1 ) ) |
58 |
48 57
|
mpbird |
⊢ ( 𝜑 → ( 0 + 1 ) ≤ ( 2 logb 2 ) ) |
59 |
|
5re |
⊢ 5 ∈ ℝ |
60 |
59
|
a1i |
⊢ ( 𝜑 → 5 ∈ ℝ ) |
61 |
11 60
|
readdcld |
⊢ ( 𝜑 → ( 2 + 5 ) ∈ ℝ ) |
62 |
10 21
|
nn0addge1i |
⊢ 2 ≤ ( 2 + 5 ) |
63 |
62
|
a1i |
⊢ ( 𝜑 → 2 ≤ ( 2 + 5 ) ) |
64 |
10
|
recni |
⊢ 2 ∈ ℂ |
65 |
|
5cn |
⊢ 5 ∈ ℂ |
66 |
64 65
|
addcomi |
⊢ ( 2 + 5 ) = ( 5 + 2 ) |
67 |
|
5p2e7 |
⊢ ( 5 + 2 ) = 7 |
68 |
66 67
|
eqtri |
⊢ ( 2 + 5 ) = 7 |
69 |
68
|
a1i |
⊢ ( 𝜑 → ( 2 + 5 ) = 7 ) |
70 |
32
|
leidd |
⊢ ( 𝜑 → 7 ≤ 7 ) |
71 |
69 70
|
eqbrtrd |
⊢ ( 𝜑 → ( 2 + 5 ) ≤ 7 ) |
72 |
11 61 32 63 71
|
letrd |
⊢ ( 𝜑 → 2 ≤ 7 ) |
73 |
11 32 29 72 40
|
letrd |
⊢ ( 𝜑 → 2 ≤ 𝐵 ) |
74 |
|
2z |
⊢ 2 ∈ ℤ |
75 |
74
|
a1i |
⊢ ( 𝜑 → 2 ∈ ℤ ) |
76 |
75
|
uzidd |
⊢ ( 𝜑 → 2 ∈ ( ℤ≥ ‘ 2 ) ) |
77 |
|
2rp |
⊢ 2 ∈ ℝ+ |
78 |
77
|
a1i |
⊢ ( 𝜑 → 2 ∈ ℝ+ ) |
79 |
29 41
|
elrpd |
⊢ ( 𝜑 → 𝐵 ∈ ℝ+ ) |
80 |
|
logbleb |
⊢ ( ( 2 ∈ ( ℤ≥ ‘ 2 ) ∧ 2 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ) → ( 2 ≤ 𝐵 ↔ ( 2 logb 2 ) ≤ ( 2 logb 𝐵 ) ) ) |
81 |
76 78 79 80
|
syl3anc |
⊢ ( 𝜑 → ( 2 ≤ 𝐵 ↔ ( 2 logb 2 ) ≤ ( 2 logb 𝐵 ) ) ) |
82 |
73 81
|
mpbid |
⊢ ( 𝜑 → ( 2 logb 2 ) ≤ ( 2 logb 𝐵 ) ) |
83 |
44 47 42 58 82
|
letrd |
⊢ ( 𝜑 → ( 0 + 1 ) ≤ ( 2 logb 𝐵 ) ) |
84 |
|
fllep1 |
⊢ ( ( 2 logb 𝐵 ) ∈ ℝ → ( 2 logb 𝐵 ) ≤ ( ( ⌊ ‘ ( 2 logb 𝐵 ) ) + 1 ) ) |
85 |
42 84
|
syl |
⊢ ( 𝜑 → ( 2 logb 𝐵 ) ≤ ( ( ⌊ ‘ ( 2 logb 𝐵 ) ) + 1 ) ) |
86 |
44 42 46 83 85
|
letrd |
⊢ ( 𝜑 → ( 0 + 1 ) ≤ ( ( ⌊ ‘ ( 2 logb 𝐵 ) ) + 1 ) ) |
87 |
30 45 15
|
leadd1d |
⊢ ( 𝜑 → ( 0 ≤ ( ⌊ ‘ ( 2 logb 𝐵 ) ) ↔ ( 0 + 1 ) ≤ ( ( ⌊ ‘ ( 2 logb 𝐵 ) ) + 1 ) ) ) |
88 |
86 87
|
mpbird |
⊢ ( 𝜑 → 0 ≤ ( ⌊ ‘ ( 2 logb 𝐵 ) ) ) |
89 |
43 88
|
jca |
⊢ ( 𝜑 → ( ( ⌊ ‘ ( 2 logb 𝐵 ) ) ∈ ℤ ∧ 0 ≤ ( ⌊ ‘ ( 2 logb 𝐵 ) ) ) ) |
90 |
|
elnn0z |
⊢ ( ( ⌊ ‘ ( 2 logb 𝐵 ) ) ∈ ℕ0 ↔ ( ( ⌊ ‘ ( 2 logb 𝐵 ) ) ∈ ℤ ∧ 0 ≤ ( ⌊ ‘ ( 2 logb 𝐵 ) ) ) ) |
91 |
89 90
|
sylibr |
⊢ ( 𝜑 → ( ⌊ ‘ ( 2 logb 𝐵 ) ) ∈ ℕ0 ) |
92 |
9 91
|
reexpcld |
⊢ ( 𝜑 → ( 𝑁 ↑ ( ⌊ ‘ ( 2 logb 𝐵 ) ) ) ∈ ℝ ) |
93 |
|
fzfid |
⊢ ( 𝜑 → ( 1 ... ( ⌊ ‘ ( ( 2 logb 𝑁 ) ↑ 2 ) ) ) ∈ Fin ) |
94 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( 2 logb 𝑁 ) ↑ 2 ) ) ) ) → 𝑁 ∈ ℝ ) |
95 |
|
elfznn |
⊢ ( 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( 2 logb 𝑁 ) ↑ 2 ) ) ) → 𝑘 ∈ ℕ ) |
96 |
95
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( 2 logb 𝑁 ) ↑ 2 ) ) ) ) → 𝑘 ∈ ℕ ) |
97 |
96
|
nnnn0d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( 2 logb 𝑁 ) ↑ 2 ) ) ) ) → 𝑘 ∈ ℕ0 ) |
98 |
94 97
|
reexpcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( 2 logb 𝑁 ) ↑ 2 ) ) ) ) → ( 𝑁 ↑ 𝑘 ) ∈ ℝ ) |
99 |
|
1red |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( 2 logb 𝑁 ) ↑ 2 ) ) ) ) → 1 ∈ ℝ ) |
100 |
98 99
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( 2 logb 𝑁 ) ↑ 2 ) ) ) ) → ( ( 𝑁 ↑ 𝑘 ) − 1 ) ∈ ℝ ) |
101 |
93 100
|
fprodrecl |
⊢ ( 𝜑 → ∏ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( 2 logb 𝑁 ) ↑ 2 ) ) ) ( ( 𝑁 ↑ 𝑘 ) − 1 ) ∈ ℝ ) |
102 |
92 101
|
remulcld |
⊢ ( 𝜑 → ( ( 𝑁 ↑ ( ⌊ ‘ ( 2 logb 𝐵 ) ) ) · ∏ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( 2 logb 𝑁 ) ↑ 2 ) ) ) ( ( 𝑁 ↑ 𝑘 ) − 1 ) ) ∈ ℝ ) |
103 |
2
|
a1i |
⊢ ( 𝜑 → 𝐴 = ( ( 𝑁 ↑ ( ⌊ ‘ ( 2 logb 𝐵 ) ) ) · ∏ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( 2 logb 𝑁 ) ↑ 2 ) ) ) ( ( 𝑁 ↑ 𝑘 ) − 1 ) ) ) |
104 |
103
|
eleq1d |
⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ↔ ( ( 𝑁 ↑ ( ⌊ ‘ ( 2 logb 𝐵 ) ) ) · ∏ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( ( 2 logb 𝑁 ) ↑ 2 ) ) ) ( ( 𝑁 ↑ 𝑘 ) − 1 ) ) ∈ ℝ ) ) |
105 |
102 104
|
mpbird |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
106 |
7
|
a1i |
⊢ ( 𝜑 → 𝐸 = ( ( 2 logb 𝑁 ) ↑ 4 ) ) |
107 |
106
|
oveq2d |
⊢ ( 𝜑 → ( 𝑁 ↑𝑐 𝐸 ) = ( 𝑁 ↑𝑐 ( ( 2 logb 𝑁 ) ↑ 4 ) ) ) |
108 |
|
2cnd |
⊢ ( 𝜑 → 2 ∈ ℂ ) |
109 |
78
|
rpne0d |
⊢ ( 𝜑 → 2 ≠ 0 ) |
110 |
109 19
|
nelprd |
⊢ ( 𝜑 → ¬ 2 ∈ { 0 , 1 } ) |
111 |
108 110
|
eldifd |
⊢ ( 𝜑 → 2 ∈ ( ℂ ∖ { 0 , 1 } ) ) |
112 |
9
|
recnd |
⊢ ( 𝜑 → 𝑁 ∈ ℂ ) |
113 |
30 14
|
ltned |
⊢ ( 𝜑 → 0 ≠ 𝑁 ) |
114 |
|
necom |
⊢ ( 0 ≠ 𝑁 ↔ 𝑁 ≠ 0 ) |
115 |
114
|
imbi2i |
⊢ ( ( 𝜑 → 0 ≠ 𝑁 ) ↔ ( 𝜑 → 𝑁 ≠ 0 ) ) |
116 |
113 115
|
mpbi |
⊢ ( 𝜑 → 𝑁 ≠ 0 ) |
117 |
116
|
neneqd |
⊢ ( 𝜑 → ¬ 𝑁 = 0 ) |
118 |
|
c0ex |
⊢ 0 ∈ V |
119 |
118
|
elsn2 |
⊢ ( 𝑁 ∈ { 0 } ↔ 𝑁 = 0 ) |
120 |
117 119
|
sylnibr |
⊢ ( 𝜑 → ¬ 𝑁 ∈ { 0 } ) |
121 |
112 120
|
eldifd |
⊢ ( 𝜑 → 𝑁 ∈ ( ℂ ∖ { 0 } ) ) |
122 |
|
cxplogb |
⊢ ( ( 2 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝑁 ∈ ( ℂ ∖ { 0 } ) ) → ( 2 ↑𝑐 ( 2 logb 𝑁 ) ) = 𝑁 ) |
123 |
111 121 122
|
syl2anc |
⊢ ( 𝜑 → ( 2 ↑𝑐 ( 2 logb 𝑁 ) ) = 𝑁 ) |
124 |
123
|
eqcomd |
⊢ ( 𝜑 → 𝑁 = ( 2 ↑𝑐 ( 2 logb 𝑁 ) ) ) |
125 |
124
|
oveq1d |
⊢ ( 𝜑 → ( 𝑁 ↑𝑐 ( ( 2 logb 𝑁 ) ↑ 4 ) ) = ( ( 2 ↑𝑐 ( 2 logb 𝑁 ) ) ↑𝑐 ( ( 2 logb 𝑁 ) ↑ 4 ) ) ) |
126 |
|
eqidd |
⊢ ( 𝜑 → ( ( 2 ↑𝑐 ( 2 logb 𝑁 ) ) ↑𝑐 ( ( 2 logb 𝑁 ) ↑ 4 ) ) = ( ( 2 ↑𝑐 ( 2 logb 𝑁 ) ) ↑𝑐 ( ( 2 logb 𝑁 ) ↑ 4 ) ) ) |
127 |
125 126
|
eqtrd |
⊢ ( 𝜑 → ( 𝑁 ↑𝑐 ( ( 2 logb 𝑁 ) ↑ 4 ) ) = ( ( 2 ↑𝑐 ( 2 logb 𝑁 ) ) ↑𝑐 ( ( 2 logb 𝑁 ) ↑ 4 ) ) ) |
128 |
107 127
|
eqtrd |
⊢ ( 𝜑 → ( 𝑁 ↑𝑐 𝐸 ) = ( ( 2 ↑𝑐 ( 2 logb 𝑁 ) ) ↑𝑐 ( ( 2 logb 𝑁 ) ↑ 4 ) ) ) |
129 |
106
|
eqcomd |
⊢ ( 𝜑 → ( ( 2 logb 𝑁 ) ↑ 4 ) = 𝐸 ) |
130 |
|
4nn0 |
⊢ 4 ∈ ℕ0 |
131 |
130
|
a1i |
⊢ ( 𝜑 → 4 ∈ ℕ0 ) |
132 |
20 131
|
reexpcld |
⊢ ( 𝜑 → ( ( 2 logb 𝑁 ) ↑ 4 ) ∈ ℝ ) |
133 |
106
|
eleq1d |
⊢ ( 𝜑 → ( 𝐸 ∈ ℝ ↔ ( ( 2 logb 𝑁 ) ↑ 4 ) ∈ ℝ ) ) |
134 |
132 133
|
mpbird |
⊢ ( 𝜑 → 𝐸 ∈ ℝ ) |
135 |
134
|
recnd |
⊢ ( 𝜑 → 𝐸 ∈ ℂ ) |
136 |
129 135
|
eqeltrd |
⊢ ( 𝜑 → ( ( 2 logb 𝑁 ) ↑ 4 ) ∈ ℂ ) |
137 |
78 20 136
|
cxpmuld |
⊢ ( 𝜑 → ( 2 ↑𝑐 ( ( 2 logb 𝑁 ) · ( ( 2 logb 𝑁 ) ↑ 4 ) ) ) = ( ( 2 ↑𝑐 ( 2 logb 𝑁 ) ) ↑𝑐 ( ( 2 logb 𝑁 ) ↑ 4 ) ) ) |
138 |
137
|
eqcomd |
⊢ ( 𝜑 → ( ( 2 ↑𝑐 ( 2 logb 𝑁 ) ) ↑𝑐 ( ( 2 logb 𝑁 ) ↑ 4 ) ) = ( 2 ↑𝑐 ( ( 2 logb 𝑁 ) · ( ( 2 logb 𝑁 ) ↑ 4 ) ) ) ) |
139 |
128 138
|
eqtrd |
⊢ ( 𝜑 → ( 𝑁 ↑𝑐 𝐸 ) = ( 2 ↑𝑐 ( ( 2 logb 𝑁 ) · ( ( 2 logb 𝑁 ) ↑ 4 ) ) ) ) |
140 |
20
|
recnd |
⊢ ( 𝜑 → ( 2 logb 𝑁 ) ∈ ℂ ) |
141 |
140
|
exp1d |
⊢ ( 𝜑 → ( ( 2 logb 𝑁 ) ↑ 1 ) = ( 2 logb 𝑁 ) ) |
142 |
141
|
eqcomd |
⊢ ( 𝜑 → ( 2 logb 𝑁 ) = ( ( 2 logb 𝑁 ) ↑ 1 ) ) |
143 |
142
|
oveq1d |
⊢ ( 𝜑 → ( ( 2 logb 𝑁 ) · ( ( 2 logb 𝑁 ) ↑ 4 ) ) = ( ( ( 2 logb 𝑁 ) ↑ 1 ) · ( ( 2 logb 𝑁 ) ↑ 4 ) ) ) |
144 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
145 |
144
|
a1i |
⊢ ( 𝜑 → 1 ∈ ℕ0 ) |
146 |
140 131 145
|
expaddd |
⊢ ( 𝜑 → ( ( 2 logb 𝑁 ) ↑ ( 1 + 4 ) ) = ( ( ( 2 logb 𝑁 ) ↑ 1 ) · ( ( 2 logb 𝑁 ) ↑ 4 ) ) ) |
147 |
146
|
eqcomd |
⊢ ( 𝜑 → ( ( ( 2 logb 𝑁 ) ↑ 1 ) · ( ( 2 logb 𝑁 ) ↑ 4 ) ) = ( ( 2 logb 𝑁 ) ↑ ( 1 + 4 ) ) ) |
148 |
143 147
|
eqtrd |
⊢ ( 𝜑 → ( ( 2 logb 𝑁 ) · ( ( 2 logb 𝑁 ) ↑ 4 ) ) = ( ( 2 logb 𝑁 ) ↑ ( 1 + 4 ) ) ) |
149 |
148
|
oveq2d |
⊢ ( 𝜑 → ( 2 ↑𝑐 ( ( 2 logb 𝑁 ) · ( ( 2 logb 𝑁 ) ↑ 4 ) ) ) = ( 2 ↑𝑐 ( ( 2 logb 𝑁 ) ↑ ( 1 + 4 ) ) ) ) |
150 |
139 149
|
eqtrd |
⊢ ( 𝜑 → ( 𝑁 ↑𝑐 𝐸 ) = ( 2 ↑𝑐 ( ( 2 logb 𝑁 ) ↑ ( 1 + 4 ) ) ) ) |
151 |
|
4cn |
⊢ 4 ∈ ℂ |
152 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
153 |
|
4p1e5 |
⊢ ( 4 + 1 ) = 5 |
154 |
151 152 153
|
addcomli |
⊢ ( 1 + 4 ) = 5 |
155 |
154
|
a1i |
⊢ ( 𝜑 → ( 1 + 4 ) = 5 ) |
156 |
155
|
oveq2d |
⊢ ( 𝜑 → ( ( 2 logb 𝑁 ) ↑ ( 1 + 4 ) ) = ( ( 2 logb 𝑁 ) ↑ 5 ) ) |
157 |
156
|
oveq2d |
⊢ ( 𝜑 → ( 2 ↑𝑐 ( ( 2 logb 𝑁 ) ↑ ( 1 + 4 ) ) ) = ( 2 ↑𝑐 ( ( 2 logb 𝑁 ) ↑ 5 ) ) ) |
158 |
150 157
|
eqtrd |
⊢ ( 𝜑 → ( 𝑁 ↑𝑐 𝐸 ) = ( 2 ↑𝑐 ( ( 2 logb 𝑁 ) ↑ 5 ) ) ) |
159 |
|
3re |
⊢ 3 ∈ ℝ |
160 |
159
|
a1i |
⊢ ( 𝜑 → 3 ∈ ℝ ) |
161 |
|
0le1 |
⊢ 0 ≤ 1 |
162 |
161
|
a1i |
⊢ ( 𝜑 → 0 ≤ 1 ) |
163 |
|
1lt3 |
⊢ 1 < 3 |
164 |
163
|
a1i |
⊢ ( 𝜑 → 1 < 3 ) |
165 |
15 160 164
|
ltled |
⊢ ( 𝜑 → 1 ≤ 3 ) |
166 |
30 15 160 162 165
|
letrd |
⊢ ( 𝜑 → 0 ≤ 3 ) |
167 |
30 160 9 166 4
|
letrd |
⊢ ( 𝜑 → 0 ≤ 𝑁 ) |
168 |
9 167 134
|
recxpcld |
⊢ ( 𝜑 → ( 𝑁 ↑𝑐 𝐸 ) ∈ ℝ ) |
169 |
158 168
|
eqeltrrd |
⊢ ( 𝜑 → ( 2 ↑𝑐 ( ( 2 logb 𝑁 ) ↑ 5 ) ) ∈ ℝ ) |
170 |
27
|
eleq1d |
⊢ ( 𝜑 → ( 𝐵 ∈ ℤ ↔ ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) ∈ ℤ ) ) |
171 |
25 170
|
mpbird |
⊢ ( 𝜑 → 𝐵 ∈ ℤ ) |
172 |
30 29 41
|
ltled |
⊢ ( 𝜑 → 0 ≤ 𝐵 ) |
173 |
171 172
|
jca |
⊢ ( 𝜑 → ( 𝐵 ∈ ℤ ∧ 0 ≤ 𝐵 ) ) |
174 |
|
elnn0z |
⊢ ( 𝐵 ∈ ℕ0 ↔ ( 𝐵 ∈ ℤ ∧ 0 ≤ 𝐵 ) ) |
175 |
173 174
|
sylibr |
⊢ ( 𝜑 → 𝐵 ∈ ℕ0 ) |
176 |
11 175
|
reexpcld |
⊢ ( 𝜑 → ( 2 ↑ 𝐵 ) ∈ ℝ ) |
177 |
9 14
|
elrpd |
⊢ ( 𝜑 → 𝑁 ∈ ℝ+ ) |
178 |
23 15
|
readdcld |
⊢ ( 𝜑 → ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ∈ ℝ ) |
179 |
22
|
nn0zd |
⊢ ( 𝜑 → 5 ∈ ℤ ) |
180 |
|
logb1 |
⊢ ( ( 2 ∈ ℂ ∧ 2 ≠ 0 ∧ 2 ≠ 1 ) → ( 2 logb 1 ) = 0 ) |
181 |
51 52 19 180
|
syl3anc |
⊢ ( 𝜑 → ( 2 logb 1 ) = 0 ) |
182 |
181 30
|
eqeltrd |
⊢ ( 𝜑 → ( 2 logb 1 ) ∈ ℝ ) |
183 |
30
|
leidd |
⊢ ( 𝜑 → 0 ≤ 0 ) |
184 |
181
|
eqcomd |
⊢ ( 𝜑 → 0 = ( 2 logb 1 ) ) |
185 |
183 184
|
breqtrd |
⊢ ( 𝜑 → 0 ≤ ( 2 logb 1 ) ) |
186 |
15 160 9 164 4
|
ltletrd |
⊢ ( 𝜑 → 1 < 𝑁 ) |
187 |
|
1rp |
⊢ 1 ∈ ℝ+ |
188 |
187
|
a1i |
⊢ ( 𝜑 → 1 ∈ ℝ+ ) |
189 |
|
logblt |
⊢ ( ( 2 ∈ ( ℤ≥ ‘ 2 ) ∧ 1 ∈ ℝ+ ∧ 𝑁 ∈ ℝ+ ) → ( 1 < 𝑁 ↔ ( 2 logb 1 ) < ( 2 logb 𝑁 ) ) ) |
190 |
76 188 177 189
|
syl3anc |
⊢ ( 𝜑 → ( 1 < 𝑁 ↔ ( 2 logb 1 ) < ( 2 logb 𝑁 ) ) ) |
191 |
186 190
|
mpbid |
⊢ ( 𝜑 → ( 2 logb 1 ) < ( 2 logb 𝑁 ) ) |
192 |
30 182 20 185 191
|
lelttrd |
⊢ ( 𝜑 → 0 < ( 2 logb 𝑁 ) ) |
193 |
20 179 192
|
3jca |
⊢ ( 𝜑 → ( ( 2 logb 𝑁 ) ∈ ℝ ∧ 5 ∈ ℤ ∧ 0 < ( 2 logb 𝑁 ) ) ) |
194 |
|
expgt0 |
⊢ ( ( ( 2 logb 𝑁 ) ∈ ℝ ∧ 5 ∈ ℤ ∧ 0 < ( 2 logb 𝑁 ) ) → 0 < ( ( 2 logb 𝑁 ) ↑ 5 ) ) |
195 |
193 194
|
syl |
⊢ ( 𝜑 → 0 < ( ( 2 logb 𝑁 ) ↑ 5 ) ) |
196 |
|
ltp1 |
⊢ ( ( ( 2 logb 𝑁 ) ↑ 5 ) ∈ ℝ → ( ( 2 logb 𝑁 ) ↑ 5 ) < ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) |
197 |
23 196
|
syl |
⊢ ( 𝜑 → ( ( 2 logb 𝑁 ) ↑ 5 ) < ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) |
198 |
30 23 178 195 197
|
lttrd |
⊢ ( 𝜑 → 0 < ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) |
199 |
11 13 178 198 19
|
relogbcld |
⊢ ( 𝜑 → ( 2 logb ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) ∈ ℝ ) |
200 |
5
|
a1i |
⊢ ( 𝜑 → 𝐶 = ( 2 logb ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) ) |
201 |
200
|
eleq1d |
⊢ ( 𝜑 → ( 𝐶 ∈ ℝ ↔ ( 2 logb ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) ∈ ℝ ) ) |
202 |
199 201
|
mpbird |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
203 |
20
|
resqcld |
⊢ ( 𝜑 → ( ( 2 logb 𝑁 ) ↑ 2 ) ∈ ℝ ) |
204 |
6
|
a1i |
⊢ ( 𝜑 → 𝐷 = ( ( 2 logb 𝑁 ) ↑ 2 ) ) |
205 |
204
|
eleq1d |
⊢ ( 𝜑 → ( 𝐷 ∈ ℝ ↔ ( ( 2 logb 𝑁 ) ↑ 2 ) ∈ ℝ ) ) |
206 |
203 205
|
mpbird |
⊢ ( 𝜑 → 𝐷 ∈ ℝ ) |
207 |
206
|
rehalfcld |
⊢ ( 𝜑 → ( 𝐷 / 2 ) ∈ ℝ ) |
208 |
202 207
|
readdcld |
⊢ ( 𝜑 → ( 𝐶 + ( 𝐷 / 2 ) ) ∈ ℝ ) |
209 |
134 11 109
|
redivcld |
⊢ ( 𝜑 → ( 𝐸 / 2 ) ∈ ℝ ) |
210 |
208 209
|
readdcld |
⊢ ( 𝜑 → ( ( 𝐶 + ( 𝐷 / 2 ) ) + ( 𝐸 / 2 ) ) ∈ ℝ ) |
211 |
177 210
|
rpcxpcld |
⊢ ( 𝜑 → ( 𝑁 ↑𝑐 ( ( 𝐶 + ( 𝐷 / 2 ) ) + ( 𝐸 / 2 ) ) ) ∈ ℝ+ ) |
212 |
211
|
rpred |
⊢ ( 𝜑 → ( 𝑁 ↑𝑐 ( ( 𝐶 + ( 𝐷 / 2 ) ) + ( 𝐸 / 2 ) ) ) ∈ ℝ ) |
213 |
1 2 3 4
|
aks4d1p1p2 |
⊢ ( 𝜑 → 𝐴 < ( 𝑁 ↑𝑐 ( ( ( 2 logb ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) + ( ( ( 2 logb 𝑁 ) ↑ 2 ) / 2 ) ) + ( ( ( 2 logb 𝑁 ) ↑ 4 ) / 2 ) ) ) ) |
214 |
129
|
oveq1d |
⊢ ( 𝜑 → ( ( ( 2 logb 𝑁 ) ↑ 4 ) / 2 ) = ( 𝐸 / 2 ) ) |
215 |
214
|
oveq2d |
⊢ ( 𝜑 → ( ( ( 2 logb ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) + ( ( ( 2 logb 𝑁 ) ↑ 2 ) / 2 ) ) + ( ( ( 2 logb 𝑁 ) ↑ 4 ) / 2 ) ) = ( ( ( 2 logb ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) + ( ( ( 2 logb 𝑁 ) ↑ 2 ) / 2 ) ) + ( 𝐸 / 2 ) ) ) |
216 |
200
|
eqcomd |
⊢ ( 𝜑 → ( 2 logb ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) = 𝐶 ) |
217 |
216
|
oveq1d |
⊢ ( 𝜑 → ( ( 2 logb ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) + ( ( ( 2 logb 𝑁 ) ↑ 2 ) / 2 ) ) = ( 𝐶 + ( ( ( 2 logb 𝑁 ) ↑ 2 ) / 2 ) ) ) |
218 |
204
|
eqcomd |
⊢ ( 𝜑 → ( ( 2 logb 𝑁 ) ↑ 2 ) = 𝐷 ) |
219 |
218
|
oveq1d |
⊢ ( 𝜑 → ( ( ( 2 logb 𝑁 ) ↑ 2 ) / 2 ) = ( 𝐷 / 2 ) ) |
220 |
219
|
oveq2d |
⊢ ( 𝜑 → ( 𝐶 + ( ( ( 2 logb 𝑁 ) ↑ 2 ) / 2 ) ) = ( 𝐶 + ( 𝐷 / 2 ) ) ) |
221 |
217 220
|
eqtrd |
⊢ ( 𝜑 → ( ( 2 logb ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) + ( ( ( 2 logb 𝑁 ) ↑ 2 ) / 2 ) ) = ( 𝐶 + ( 𝐷 / 2 ) ) ) |
222 |
221
|
oveq1d |
⊢ ( 𝜑 → ( ( ( 2 logb ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) + ( ( ( 2 logb 𝑁 ) ↑ 2 ) / 2 ) ) + ( 𝐸 / 2 ) ) = ( ( 𝐶 + ( 𝐷 / 2 ) ) + ( 𝐸 / 2 ) ) ) |
223 |
215 222
|
eqtrd |
⊢ ( 𝜑 → ( ( ( 2 logb ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) + ( ( ( 2 logb 𝑁 ) ↑ 2 ) / 2 ) ) + ( ( ( 2 logb 𝑁 ) ↑ 4 ) / 2 ) ) = ( ( 𝐶 + ( 𝐷 / 2 ) ) + ( 𝐸 / 2 ) ) ) |
224 |
223
|
oveq2d |
⊢ ( 𝜑 → ( 𝑁 ↑𝑐 ( ( ( 2 logb ( ( ( 2 logb 𝑁 ) ↑ 5 ) + 1 ) ) + ( ( ( 2 logb 𝑁 ) ↑ 2 ) / 2 ) ) + ( ( ( 2 logb 𝑁 ) ↑ 4 ) / 2 ) ) ) = ( 𝑁 ↑𝑐 ( ( 𝐶 + ( 𝐷 / 2 ) ) + ( 𝐸 / 2 ) ) ) ) |
225 |
213 224
|
breqtrd |
⊢ ( 𝜑 → 𝐴 < ( 𝑁 ↑𝑐 ( ( 𝐶 + ( 𝐷 / 2 ) ) + ( 𝐸 / 2 ) ) ) ) |
226 |
202
|
recnd |
⊢ ( 𝜑 → 𝐶 ∈ ℂ ) |
227 |
226 108 109
|
divcan3d |
⊢ ( 𝜑 → ( ( 2 · 𝐶 ) / 2 ) = 𝐶 ) |
228 |
227
|
eqcomd |
⊢ ( 𝜑 → 𝐶 = ( ( 2 · 𝐶 ) / 2 ) ) |
229 |
228
|
oveq1d |
⊢ ( 𝜑 → ( 𝐶 + ( 𝐷 / 2 ) ) = ( ( ( 2 · 𝐶 ) / 2 ) + ( 𝐷 / 2 ) ) ) |
230 |
11 202
|
remulcld |
⊢ ( 𝜑 → ( 2 · 𝐶 ) ∈ ℝ ) |
231 |
230
|
recnd |
⊢ ( 𝜑 → ( 2 · 𝐶 ) ∈ ℂ ) |
232 |
206
|
recnd |
⊢ ( 𝜑 → 𝐷 ∈ ℂ ) |
233 |
231 232 51 109
|
divdird |
⊢ ( 𝜑 → ( ( ( 2 · 𝐶 ) + 𝐷 ) / 2 ) = ( ( ( 2 · 𝐶 ) / 2 ) + ( 𝐷 / 2 ) ) ) |
234 |
233
|
eqcomd |
⊢ ( 𝜑 → ( ( ( 2 · 𝐶 ) / 2 ) + ( 𝐷 / 2 ) ) = ( ( ( 2 · 𝐶 ) + 𝐷 ) / 2 ) ) |
235 |
229 234
|
eqtrd |
⊢ ( 𝜑 → ( 𝐶 + ( 𝐷 / 2 ) ) = ( ( ( 2 · 𝐶 ) + 𝐷 ) / 2 ) ) |
236 |
230 206
|
readdcld |
⊢ ( 𝜑 → ( ( 2 · 𝐶 ) + 𝐷 ) ∈ ℝ ) |
237 |
236 134 78
|
lediv1d |
⊢ ( 𝜑 → ( ( ( 2 · 𝐶 ) + 𝐷 ) ≤ 𝐸 ↔ ( ( ( 2 · 𝐶 ) + 𝐷 ) / 2 ) ≤ ( 𝐸 / 2 ) ) ) |
238 |
8 237
|
mpbid |
⊢ ( 𝜑 → ( ( ( 2 · 𝐶 ) + 𝐷 ) / 2 ) ≤ ( 𝐸 / 2 ) ) |
239 |
235 238
|
eqbrtrd |
⊢ ( 𝜑 → ( 𝐶 + ( 𝐷 / 2 ) ) ≤ ( 𝐸 / 2 ) ) |
240 |
208 209 209 239
|
leadd1dd |
⊢ ( 𝜑 → ( ( 𝐶 + ( 𝐷 / 2 ) ) + ( 𝐸 / 2 ) ) ≤ ( ( 𝐸 / 2 ) + ( 𝐸 / 2 ) ) ) |
241 |
135
|
2halvesd |
⊢ ( 𝜑 → ( ( 𝐸 / 2 ) + ( 𝐸 / 2 ) ) = 𝐸 ) |
242 |
240 241
|
breqtrd |
⊢ ( 𝜑 → ( ( 𝐶 + ( 𝐷 / 2 ) ) + ( 𝐸 / 2 ) ) ≤ 𝐸 ) |
243 |
9 186 210 134
|
cxpled |
⊢ ( 𝜑 → ( ( ( 𝐶 + ( 𝐷 / 2 ) ) + ( 𝐸 / 2 ) ) ≤ 𝐸 ↔ ( 𝑁 ↑𝑐 ( ( 𝐶 + ( 𝐷 / 2 ) ) + ( 𝐸 / 2 ) ) ) ≤ ( 𝑁 ↑𝑐 𝐸 ) ) ) |
244 |
242 243
|
mpbid |
⊢ ( 𝜑 → ( 𝑁 ↑𝑐 ( ( 𝐶 + ( 𝐷 / 2 ) ) + ( 𝐸 / 2 ) ) ) ≤ ( 𝑁 ↑𝑐 𝐸 ) ) |
245 |
105 212 168 225 244
|
ltletrd |
⊢ ( 𝜑 → 𝐴 < ( 𝑁 ↑𝑐 𝐸 ) ) |
246 |
245 158
|
breqtrd |
⊢ ( 𝜑 → 𝐴 < ( 2 ↑𝑐 ( ( 2 logb 𝑁 ) ↑ 5 ) ) ) |
247 |
|
1le2 |
⊢ 1 ≤ 2 |
248 |
247
|
a1i |
⊢ ( 𝜑 → 1 ≤ 2 ) |
249 |
175
|
nn0red |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
250 |
27
|
eqcomd |
⊢ ( 𝜑 → ( ⌈ ‘ ( ( 2 logb 𝑁 ) ↑ 5 ) ) = 𝐵 ) |
251 |
38 250
|
breqtrd |
⊢ ( 𝜑 → ( ( 2 logb 𝑁 ) ↑ 5 ) ≤ 𝐵 ) |
252 |
11 248 23 249 251
|
cxplead |
⊢ ( 𝜑 → ( 2 ↑𝑐 ( ( 2 logb 𝑁 ) ↑ 5 ) ) ≤ ( 2 ↑𝑐 𝐵 ) ) |
253 |
|
cxpexp |
⊢ ( ( 2 ∈ ℂ ∧ 𝐵 ∈ ℕ0 ) → ( 2 ↑𝑐 𝐵 ) = ( 2 ↑ 𝐵 ) ) |
254 |
108 175 253
|
syl2anc |
⊢ ( 𝜑 → ( 2 ↑𝑐 𝐵 ) = ( 2 ↑ 𝐵 ) ) |
255 |
252 254
|
breqtrd |
⊢ ( 𝜑 → ( 2 ↑𝑐 ( ( 2 logb 𝑁 ) ↑ 5 ) ) ≤ ( 2 ↑ 𝐵 ) ) |
256 |
105 169 176 246 255
|
ltletrd |
⊢ ( 𝜑 → 𝐴 < ( 2 ↑ 𝐵 ) ) |