Step |
Hyp |
Ref |
Expression |
1 |
|
chtppilim.1 |
|- ( ph -> A e. RR+ ) |
2 |
|
chtppilim.2 |
|- ( ph -> A < 1 ) |
3 |
|
chtppilim.3 |
|- ( ph -> N e. ( 2 [,) +oo ) ) |
4 |
|
chtppilim.4 |
|- ( ph -> ( ( N ^c A ) / ( ppi ` N ) ) < ( 1 - A ) ) |
5 |
1
|
rpred |
|- ( ph -> A e. RR ) |
6 |
5
|
recnd |
|- ( ph -> A e. CC ) |
7 |
6
|
sqvald |
|- ( ph -> ( A ^ 2 ) = ( A x. A ) ) |
8 |
7
|
oveq1d |
|- ( ph -> ( ( A ^ 2 ) x. ( ( ppi ` N ) x. ( log ` N ) ) ) = ( ( A x. A ) x. ( ( ppi ` N ) x. ( log ` N ) ) ) ) |
9 |
|
2re |
|- 2 e. RR |
10 |
|
elicopnf |
|- ( 2 e. RR -> ( N e. ( 2 [,) +oo ) <-> ( N e. RR /\ 2 <_ N ) ) ) |
11 |
9 10
|
ax-mp |
|- ( N e. ( 2 [,) +oo ) <-> ( N e. RR /\ 2 <_ N ) ) |
12 |
3 11
|
sylib |
|- ( ph -> ( N e. RR /\ 2 <_ N ) ) |
13 |
12
|
simpld |
|- ( ph -> N e. RR ) |
14 |
|
ppicl |
|- ( N e. RR -> ( ppi ` N ) e. NN0 ) |
15 |
13 14
|
syl |
|- ( ph -> ( ppi ` N ) e. NN0 ) |
16 |
15
|
nn0red |
|- ( ph -> ( ppi ` N ) e. RR ) |
17 |
16
|
recnd |
|- ( ph -> ( ppi ` N ) e. CC ) |
18 |
|
0red |
|- ( ph -> 0 e. RR ) |
19 |
9
|
a1i |
|- ( ph -> 2 e. RR ) |
20 |
|
2pos |
|- 0 < 2 |
21 |
20
|
a1i |
|- ( ph -> 0 < 2 ) |
22 |
12
|
simprd |
|- ( ph -> 2 <_ N ) |
23 |
18 19 13 21 22
|
ltletrd |
|- ( ph -> 0 < N ) |
24 |
13 23
|
elrpd |
|- ( ph -> N e. RR+ ) |
25 |
24
|
relogcld |
|- ( ph -> ( log ` N ) e. RR ) |
26 |
25
|
recnd |
|- ( ph -> ( log ` N ) e. CC ) |
27 |
6 6 17 26
|
mul4d |
|- ( ph -> ( ( A x. A ) x. ( ( ppi ` N ) x. ( log ` N ) ) ) = ( ( A x. ( ppi ` N ) ) x. ( A x. ( log ` N ) ) ) ) |
28 |
8 27
|
eqtrd |
|- ( ph -> ( ( A ^ 2 ) x. ( ( ppi ` N ) x. ( log ` N ) ) ) = ( ( A x. ( ppi ` N ) ) x. ( A x. ( log ` N ) ) ) ) |
29 |
5 16
|
remulcld |
|- ( ph -> ( A x. ( ppi ` N ) ) e. RR ) |
30 |
5 25
|
remulcld |
|- ( ph -> ( A x. ( log ` N ) ) e. RR ) |
31 |
29 30
|
remulcld |
|- ( ph -> ( ( A x. ( ppi ` N ) ) x. ( A x. ( log ` N ) ) ) e. RR ) |
32 |
24 5
|
rpcxpcld |
|- ( ph -> ( N ^c A ) e. RR+ ) |
33 |
32
|
rpred |
|- ( ph -> ( N ^c A ) e. RR ) |
34 |
|
ppicl |
|- ( ( N ^c A ) e. RR -> ( ppi ` ( N ^c A ) ) e. NN0 ) |
35 |
33 34
|
syl |
|- ( ph -> ( ppi ` ( N ^c A ) ) e. NN0 ) |
36 |
35
|
nn0red |
|- ( ph -> ( ppi ` ( N ^c A ) ) e. RR ) |
37 |
16 36
|
resubcld |
|- ( ph -> ( ( ppi ` N ) - ( ppi ` ( N ^c A ) ) ) e. RR ) |
38 |
37 30
|
remulcld |
|- ( ph -> ( ( ( ppi ` N ) - ( ppi ` ( N ^c A ) ) ) x. ( A x. ( log ` N ) ) ) e. RR ) |
39 |
|
chtcl |
|- ( N e. RR -> ( theta ` N ) e. RR ) |
40 |
13 39
|
syl |
|- ( ph -> ( theta ` N ) e. RR ) |
41 |
|
1red |
|- ( ph -> 1 e. RR ) |
42 |
|
1lt2 |
|- 1 < 2 |
43 |
42
|
a1i |
|- ( ph -> 1 < 2 ) |
44 |
41 19 13 43 22
|
ltletrd |
|- ( ph -> 1 < N ) |
45 |
13 44
|
rplogcld |
|- ( ph -> ( log ` N ) e. RR+ ) |
46 |
1 45
|
rpmulcld |
|- ( ph -> ( A x. ( log ` N ) ) e. RR+ ) |
47 |
16 33
|
resubcld |
|- ( ph -> ( ( ppi ` N ) - ( N ^c A ) ) e. RR ) |
48 |
|
ppinncl |
|- ( ( N e. RR /\ 2 <_ N ) -> ( ppi ` N ) e. NN ) |
49 |
12 48
|
syl |
|- ( ph -> ( ppi ` N ) e. NN ) |
50 |
33 49
|
nndivred |
|- ( ph -> ( ( N ^c A ) / ( ppi ` N ) ) e. RR ) |
51 |
50 41 5 4
|
ltsub13d |
|- ( ph -> A < ( 1 - ( ( N ^c A ) / ( ppi ` N ) ) ) ) |
52 |
33
|
recnd |
|- ( ph -> ( N ^c A ) e. CC ) |
53 |
49
|
nnrpd |
|- ( ph -> ( ppi ` N ) e. RR+ ) |
54 |
53
|
rpcnne0d |
|- ( ph -> ( ( ppi ` N ) e. CC /\ ( ppi ` N ) =/= 0 ) ) |
55 |
|
divsubdir |
|- ( ( ( ppi ` N ) e. CC /\ ( N ^c A ) e. CC /\ ( ( ppi ` N ) e. CC /\ ( ppi ` N ) =/= 0 ) ) -> ( ( ( ppi ` N ) - ( N ^c A ) ) / ( ppi ` N ) ) = ( ( ( ppi ` N ) / ( ppi ` N ) ) - ( ( N ^c A ) / ( ppi ` N ) ) ) ) |
56 |
17 52 54 55
|
syl3anc |
|- ( ph -> ( ( ( ppi ` N ) - ( N ^c A ) ) / ( ppi ` N ) ) = ( ( ( ppi ` N ) / ( ppi ` N ) ) - ( ( N ^c A ) / ( ppi ` N ) ) ) ) |
57 |
|
divid |
|- ( ( ( ppi ` N ) e. CC /\ ( ppi ` N ) =/= 0 ) -> ( ( ppi ` N ) / ( ppi ` N ) ) = 1 ) |
58 |
54 57
|
syl |
|- ( ph -> ( ( ppi ` N ) / ( ppi ` N ) ) = 1 ) |
59 |
58
|
oveq1d |
|- ( ph -> ( ( ( ppi ` N ) / ( ppi ` N ) ) - ( ( N ^c A ) / ( ppi ` N ) ) ) = ( 1 - ( ( N ^c A ) / ( ppi ` N ) ) ) ) |
60 |
56 59
|
eqtrd |
|- ( ph -> ( ( ( ppi ` N ) - ( N ^c A ) ) / ( ppi ` N ) ) = ( 1 - ( ( N ^c A ) / ( ppi ` N ) ) ) ) |
61 |
51 60
|
breqtrrd |
|- ( ph -> A < ( ( ( ppi ` N ) - ( N ^c A ) ) / ( ppi ` N ) ) ) |
62 |
5 47 53
|
ltmuldivd |
|- ( ph -> ( ( A x. ( ppi ` N ) ) < ( ( ppi ` N ) - ( N ^c A ) ) <-> A < ( ( ( ppi ` N ) - ( N ^c A ) ) / ( ppi ` N ) ) ) ) |
63 |
61 62
|
mpbird |
|- ( ph -> ( A x. ( ppi ` N ) ) < ( ( ppi ` N ) - ( N ^c A ) ) ) |
64 |
|
ppiltx |
|- ( ( N ^c A ) e. RR+ -> ( ppi ` ( N ^c A ) ) < ( N ^c A ) ) |
65 |
32 64
|
syl |
|- ( ph -> ( ppi ` ( N ^c A ) ) < ( N ^c A ) ) |
66 |
36 33 16 65
|
ltsub2dd |
|- ( ph -> ( ( ppi ` N ) - ( N ^c A ) ) < ( ( ppi ` N ) - ( ppi ` ( N ^c A ) ) ) ) |
67 |
29 47 37 63 66
|
lttrd |
|- ( ph -> ( A x. ( ppi ` N ) ) < ( ( ppi ` N ) - ( ppi ` ( N ^c A ) ) ) ) |
68 |
29 37 46 67
|
ltmul1dd |
|- ( ph -> ( ( A x. ( ppi ` N ) ) x. ( A x. ( log ` N ) ) ) < ( ( ( ppi ` N ) - ( ppi ` ( N ^c A ) ) ) x. ( A x. ( log ` N ) ) ) ) |
69 |
|
fzfid |
|- ( ph -> ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) e. Fin ) |
70 |
|
inss1 |
|- ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) C_ ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) |
71 |
|
ssfi |
|- ( ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) e. Fin /\ ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) C_ ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) ) -> ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) e. Fin ) |
72 |
69 70 71
|
sylancl |
|- ( ph -> ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) e. Fin ) |
73 |
|
simpr |
|- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) |
74 |
73
|
elin2d |
|- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> p e. Prime ) |
75 |
|
prmnn |
|- ( p e. Prime -> p e. NN ) |
76 |
75
|
nnrpd |
|- ( p e. Prime -> p e. RR+ ) |
77 |
74 76
|
syl |
|- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> p e. RR+ ) |
78 |
77
|
relogcld |
|- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( log ` p ) e. RR ) |
79 |
72 78
|
fsumrecl |
|- ( ph -> sum_ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ( log ` p ) e. RR ) |
80 |
30
|
recnd |
|- ( ph -> ( A x. ( log ` N ) ) e. CC ) |
81 |
|
fsumconst |
|- ( ( ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) e. Fin /\ ( A x. ( log ` N ) ) e. CC ) -> sum_ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ( A x. ( log ` N ) ) = ( ( # ` ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) x. ( A x. ( log ` N ) ) ) ) |
82 |
72 80 81
|
syl2anc |
|- ( ph -> sum_ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ( A x. ( log ` N ) ) = ( ( # ` ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) x. ( A x. ( log ` N ) ) ) ) |
83 |
|
ppifl |
|- ( N e. RR -> ( ppi ` ( |_ ` N ) ) = ( ppi ` N ) ) |
84 |
13 83
|
syl |
|- ( ph -> ( ppi ` ( |_ ` N ) ) = ( ppi ` N ) ) |
85 |
|
ppifl |
|- ( ( N ^c A ) e. RR -> ( ppi ` ( |_ ` ( N ^c A ) ) ) = ( ppi ` ( N ^c A ) ) ) |
86 |
33 85
|
syl |
|- ( ph -> ( ppi ` ( |_ ` ( N ^c A ) ) ) = ( ppi ` ( N ^c A ) ) ) |
87 |
84 86
|
oveq12d |
|- ( ph -> ( ( ppi ` ( |_ ` N ) ) - ( ppi ` ( |_ ` ( N ^c A ) ) ) ) = ( ( ppi ` N ) - ( ppi ` ( N ^c A ) ) ) ) |
88 |
41 13 44
|
ltled |
|- ( ph -> 1 <_ N ) |
89 |
|
1re |
|- 1 e. RR |
90 |
|
ltle |
|- ( ( A e. RR /\ 1 e. RR ) -> ( A < 1 -> A <_ 1 ) ) |
91 |
5 89 90
|
sylancl |
|- ( ph -> ( A < 1 -> A <_ 1 ) ) |
92 |
2 91
|
mpd |
|- ( ph -> A <_ 1 ) |
93 |
13 88 5 41 92
|
cxplead |
|- ( ph -> ( N ^c A ) <_ ( N ^c 1 ) ) |
94 |
13
|
recnd |
|- ( ph -> N e. CC ) |
95 |
94
|
cxp1d |
|- ( ph -> ( N ^c 1 ) = N ) |
96 |
93 95
|
breqtrd |
|- ( ph -> ( N ^c A ) <_ N ) |
97 |
|
flword2 |
|- ( ( ( N ^c A ) e. RR /\ N e. RR /\ ( N ^c A ) <_ N ) -> ( |_ ` N ) e. ( ZZ>= ` ( |_ ` ( N ^c A ) ) ) ) |
98 |
33 13 96 97
|
syl3anc |
|- ( ph -> ( |_ ` N ) e. ( ZZ>= ` ( |_ ` ( N ^c A ) ) ) ) |
99 |
|
ppidif |
|- ( ( |_ ` N ) e. ( ZZ>= ` ( |_ ` ( N ^c A ) ) ) -> ( ( ppi ` ( |_ ` N ) ) - ( ppi ` ( |_ ` ( N ^c A ) ) ) ) = ( # ` ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) ) |
100 |
98 99
|
syl |
|- ( ph -> ( ( ppi ` ( |_ ` N ) ) - ( ppi ` ( |_ ` ( N ^c A ) ) ) ) = ( # ` ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) ) |
101 |
87 100
|
eqtr3d |
|- ( ph -> ( ( ppi ` N ) - ( ppi ` ( N ^c A ) ) ) = ( # ` ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) ) |
102 |
101
|
oveq1d |
|- ( ph -> ( ( ( ppi ` N ) - ( ppi ` ( N ^c A ) ) ) x. ( A x. ( log ` N ) ) ) = ( ( # ` ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) x. ( A x. ( log ` N ) ) ) ) |
103 |
82 102
|
eqtr4d |
|- ( ph -> sum_ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ( A x. ( log ` N ) ) = ( ( ( ppi ` N ) - ( ppi ` ( N ^c A ) ) ) x. ( A x. ( log ` N ) ) ) ) |
104 |
30
|
adantr |
|- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( A x. ( log ` N ) ) e. RR ) |
105 |
33
|
adantr |
|- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( N ^c A ) e. RR ) |
106 |
|
reflcl |
|- ( ( N ^c A ) e. RR -> ( |_ ` ( N ^c A ) ) e. RR ) |
107 |
|
peano2re |
|- ( ( |_ ` ( N ^c A ) ) e. RR -> ( ( |_ ` ( N ^c A ) ) + 1 ) e. RR ) |
108 |
33 106 107
|
3syl |
|- ( ph -> ( ( |_ ` ( N ^c A ) ) + 1 ) e. RR ) |
109 |
108
|
adantr |
|- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( ( |_ ` ( N ^c A ) ) + 1 ) e. RR ) |
110 |
77
|
rpred |
|- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> p e. RR ) |
111 |
|
fllep1 |
|- ( ( N ^c A ) e. RR -> ( N ^c A ) <_ ( ( |_ ` ( N ^c A ) ) + 1 ) ) |
112 |
33 111
|
syl |
|- ( ph -> ( N ^c A ) <_ ( ( |_ ` ( N ^c A ) ) + 1 ) ) |
113 |
112
|
adantr |
|- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( N ^c A ) <_ ( ( |_ ` ( N ^c A ) ) + 1 ) ) |
114 |
73
|
elin1d |
|- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> p e. ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) ) |
115 |
|
elfzle1 |
|- ( p e. ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) -> ( ( |_ ` ( N ^c A ) ) + 1 ) <_ p ) |
116 |
114 115
|
syl |
|- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( ( |_ ` ( N ^c A ) ) + 1 ) <_ p ) |
117 |
105 109 110 113 116
|
letrd |
|- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( N ^c A ) <_ p ) |
118 |
24
|
rpne0d |
|- ( ph -> N =/= 0 ) |
119 |
94 118 6
|
cxpefd |
|- ( ph -> ( N ^c A ) = ( exp ` ( A x. ( log ` N ) ) ) ) |
120 |
119
|
eqcomd |
|- ( ph -> ( exp ` ( A x. ( log ` N ) ) ) = ( N ^c A ) ) |
121 |
120
|
adantr |
|- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( exp ` ( A x. ( log ` N ) ) ) = ( N ^c A ) ) |
122 |
77
|
reeflogd |
|- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( exp ` ( log ` p ) ) = p ) |
123 |
117 121 122
|
3brtr4d |
|- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( exp ` ( A x. ( log ` N ) ) ) <_ ( exp ` ( log ` p ) ) ) |
124 |
|
efle |
|- ( ( ( A x. ( log ` N ) ) e. RR /\ ( log ` p ) e. RR ) -> ( ( A x. ( log ` N ) ) <_ ( log ` p ) <-> ( exp ` ( A x. ( log ` N ) ) ) <_ ( exp ` ( log ` p ) ) ) ) |
125 |
104 78 124
|
syl2anc |
|- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( ( A x. ( log ` N ) ) <_ ( log ` p ) <-> ( exp ` ( A x. ( log ` N ) ) ) <_ ( exp ` ( log ` p ) ) ) ) |
126 |
123 125
|
mpbird |
|- ( ( ph /\ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ) -> ( A x. ( log ` N ) ) <_ ( log ` p ) ) |
127 |
72 104 78 126
|
fsumle |
|- ( ph -> sum_ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ( A x. ( log ` N ) ) <_ sum_ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ( log ` p ) ) |
128 |
103 127
|
eqbrtrrd |
|- ( ph -> ( ( ( ppi ` N ) - ( ppi ` ( N ^c A ) ) ) x. ( A x. ( log ` N ) ) ) <_ sum_ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ( log ` p ) ) |
129 |
|
fzfid |
|- ( ph -> ( 1 ... ( |_ ` N ) ) e. Fin ) |
130 |
|
inss1 |
|- ( ( 1 ... ( |_ ` N ) ) i^i Prime ) C_ ( 1 ... ( |_ ` N ) ) |
131 |
|
ssfi |
|- ( ( ( 1 ... ( |_ ` N ) ) e. Fin /\ ( ( 1 ... ( |_ ` N ) ) i^i Prime ) C_ ( 1 ... ( |_ ` N ) ) ) -> ( ( 1 ... ( |_ ` N ) ) i^i Prime ) e. Fin ) |
132 |
129 130 131
|
sylancl |
|- ( ph -> ( ( 1 ... ( |_ ` N ) ) i^i Prime ) e. Fin ) |
133 |
|
simpr |
|- ( ( ph /\ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) -> p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) |
134 |
133
|
elin2d |
|- ( ( ph /\ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) -> p e. Prime ) |
135 |
|
prmuz2 |
|- ( p e. Prime -> p e. ( ZZ>= ` 2 ) ) |
136 |
134 135
|
syl |
|- ( ( ph /\ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) -> p e. ( ZZ>= ` 2 ) ) |
137 |
|
eluz2b2 |
|- ( p e. ( ZZ>= ` 2 ) <-> ( p e. NN /\ 1 < p ) ) |
138 |
136 137
|
sylib |
|- ( ( ph /\ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) -> ( p e. NN /\ 1 < p ) ) |
139 |
138
|
simpld |
|- ( ( ph /\ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) -> p e. NN ) |
140 |
139
|
nnred |
|- ( ( ph /\ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) -> p e. RR ) |
141 |
138
|
simprd |
|- ( ( ph /\ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) -> 1 < p ) |
142 |
140 141
|
rplogcld |
|- ( ( ph /\ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) -> ( log ` p ) e. RR+ ) |
143 |
142
|
rpred |
|- ( ( ph /\ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) -> ( log ` p ) e. RR ) |
144 |
142
|
rpge0d |
|- ( ( ph /\ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) -> 0 <_ ( log ` p ) ) |
145 |
32
|
rpge0d |
|- ( ph -> 0 <_ ( N ^c A ) ) |
146 |
|
flge0nn0 |
|- ( ( ( N ^c A ) e. RR /\ 0 <_ ( N ^c A ) ) -> ( |_ ` ( N ^c A ) ) e. NN0 ) |
147 |
33 145 146
|
syl2anc |
|- ( ph -> ( |_ ` ( N ^c A ) ) e. NN0 ) |
148 |
|
nn0p1nn |
|- ( ( |_ ` ( N ^c A ) ) e. NN0 -> ( ( |_ ` ( N ^c A ) ) + 1 ) e. NN ) |
149 |
147 148
|
syl |
|- ( ph -> ( ( |_ ` ( N ^c A ) ) + 1 ) e. NN ) |
150 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
151 |
149 150
|
eleqtrdi |
|- ( ph -> ( ( |_ ` ( N ^c A ) ) + 1 ) e. ( ZZ>= ` 1 ) ) |
152 |
|
fzss1 |
|- ( ( ( |_ ` ( N ^c A ) ) + 1 ) e. ( ZZ>= ` 1 ) -> ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) C_ ( 1 ... ( |_ ` N ) ) ) |
153 |
|
ssrin |
|- ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) C_ ( 1 ... ( |_ ` N ) ) -> ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) C_ ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) |
154 |
151 152 153
|
3syl |
|- ( ph -> ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) C_ ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) |
155 |
132 143 144 154
|
fsumless |
|- ( ph -> sum_ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ( log ` p ) <_ sum_ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ( log ` p ) ) |
156 |
|
chtval |
|- ( N e. RR -> ( theta ` N ) = sum_ p e. ( ( 0 [,] N ) i^i Prime ) ( log ` p ) ) |
157 |
13 156
|
syl |
|- ( ph -> ( theta ` N ) = sum_ p e. ( ( 0 [,] N ) i^i Prime ) ( log ` p ) ) |
158 |
|
2eluzge1 |
|- 2 e. ( ZZ>= ` 1 ) |
159 |
|
ppisval2 |
|- ( ( N e. RR /\ 2 e. ( ZZ>= ` 1 ) ) -> ( ( 0 [,] N ) i^i Prime ) = ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) |
160 |
13 158 159
|
sylancl |
|- ( ph -> ( ( 0 [,] N ) i^i Prime ) = ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ) |
161 |
160
|
sumeq1d |
|- ( ph -> sum_ p e. ( ( 0 [,] N ) i^i Prime ) ( log ` p ) = sum_ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ( log ` p ) ) |
162 |
157 161
|
eqtrd |
|- ( ph -> ( theta ` N ) = sum_ p e. ( ( 1 ... ( |_ ` N ) ) i^i Prime ) ( log ` p ) ) |
163 |
155 162
|
breqtrrd |
|- ( ph -> sum_ p e. ( ( ( ( |_ ` ( N ^c A ) ) + 1 ) ... ( |_ ` N ) ) i^i Prime ) ( log ` p ) <_ ( theta ` N ) ) |
164 |
38 79 40 128 163
|
letrd |
|- ( ph -> ( ( ( ppi ` N ) - ( ppi ` ( N ^c A ) ) ) x. ( A x. ( log ` N ) ) ) <_ ( theta ` N ) ) |
165 |
31 38 40 68 164
|
ltletrd |
|- ( ph -> ( ( A x. ( ppi ` N ) ) x. ( A x. ( log ` N ) ) ) < ( theta ` N ) ) |
166 |
28 165
|
eqbrtrd |
|- ( ph -> ( ( A ^ 2 ) x. ( ( ppi ` N ) x. ( log ` N ) ) ) < ( theta ` N ) ) |