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Theorem chtppilimlem1 20638
Description: Lemma for chtppilim 20640. (Contributed by Mario Carneiro, 22-Sep-2014.)
Hypotheses
Ref Expression
chtppilim.1  |-  ( ph  ->  A  e.  RR+ )
chtppilim.2  |-  ( ph  ->  A  <  1 )
chtppilim.3  |-  ( ph  ->  N  e.  ( 2 [,)  +oo ) )
chtppilim.4  |-  ( ph  ->  ( ( N  ^ c  A )  /  (π `  N ) )  < 
( 1  -  A
) )
Assertion
Ref Expression
chtppilimlem1  |-  ( ph  ->  ( ( A ^
2 )  x.  (
(π `  N )  x.  ( log `  N
) ) )  < 
( theta `  N )
)

Proof of Theorem chtppilimlem1
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 chtppilim.1 . . . . . . 7  |-  ( ph  ->  A  e.  RR+ )
21rpred 10406 . . . . . 6  |-  ( ph  ->  A  e.  RR )
32recnd 8877 . . . . 5  |-  ( ph  ->  A  e.  CC )
43sqvald 11258 . . . 4  |-  ( ph  ->  ( A ^ 2 )  =  ( A  x.  A ) )
54oveq1d 5889 . . 3  |-  ( ph  ->  ( ( A ^
2 )  x.  (
(π `  N )  x.  ( log `  N
) ) )  =  ( ( A  x.  A )  x.  (
(π `  N )  x.  ( log `  N
) ) ) )
6 chtppilim.3 . . . . . . . . 9  |-  ( ph  ->  N  e.  ( 2 [,)  +oo ) )
7 2re 9831 . . . . . . . . . 10  |-  2  e.  RR
8 elicopnf 10755 . . . . . . . . . 10  |-  ( 2  e.  RR  ->  ( N  e.  ( 2 [,)  +oo )  <->  ( N  e.  RR  /\  2  <_  N ) ) )
97, 8ax-mp 8 . . . . . . . . 9  |-  ( N  e.  ( 2 [,) 
+oo )  <->  ( N  e.  RR  /\  2  <_  N ) )
106, 9sylib 188 . . . . . . . 8  |-  ( ph  ->  ( N  e.  RR  /\  2  <_  N )
)
1110simpld 445 . . . . . . 7  |-  ( ph  ->  N  e.  RR )
12 ppicl 20385 . . . . . . 7  |-  ( N  e.  RR  ->  (π `  N )  e.  NN0 )
1311, 12syl 15 . . . . . 6  |-  ( ph  ->  (π `  N )  e. 
NN0 )
1413nn0red 10035 . . . . 5  |-  ( ph  ->  (π `  N )  e.  RR )
1514recnd 8877 . . . 4  |-  ( ph  ->  (π `  N )  e.  CC )
16 0re 8854 . . . . . . . . 9  |-  0  e.  RR
1716a1i 10 . . . . . . . 8  |-  ( ph  ->  0  e.  RR )
187a1i 10 . . . . . . . 8  |-  ( ph  ->  2  e.  RR )
19 2pos 9844 . . . . . . . . 9  |-  0  <  2
2019a1i 10 . . . . . . . 8  |-  ( ph  ->  0  <  2 )
2110simprd 449 . . . . . . . 8  |-  ( ph  ->  2  <_  N )
2217, 18, 11, 20, 21ltletrd 8992 . . . . . . 7  |-  ( ph  ->  0  <  N )
2311, 22elrpd 10404 . . . . . 6  |-  ( ph  ->  N  e.  RR+ )
2423relogcld 19990 . . . . 5  |-  ( ph  ->  ( log `  N
)  e.  RR )
2524recnd 8877 . . . 4  |-  ( ph  ->  ( log `  N
)  e.  CC )
263, 3, 15, 25mul4d 9040 . . 3  |-  ( ph  ->  ( ( A  x.  A )  x.  (
(π `  N )  x.  ( log `  N
) ) )  =  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) ) )
275, 26eqtrd 2328 . 2  |-  ( ph  ->  ( ( A ^
2 )  x.  (
(π `  N )  x.  ( log `  N
) ) )  =  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) ) )
282, 14remulcld 8879 . . . 4  |-  ( ph  ->  ( A  x.  (π `  N ) )  e.  RR )
292, 24remulcld 8879 . . . 4  |-  ( ph  ->  ( A  x.  ( log `  N ) )  e.  RR )
3028, 29remulcld 8879 . . 3  |-  ( ph  ->  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) )  e.  RR )
3123, 2rpcxpcld 20093 . . . . . . . 8  |-  ( ph  ->  ( N  ^ c  A )  e.  RR+ )
3231rpred 10406 . . . . . . 7  |-  ( ph  ->  ( N  ^ c  A )  e.  RR )
33 ppicl 20385 . . . . . . 7  |-  ( ( N  ^ c  A
)  e.  RR  ->  (π `  ( N  ^ c  A ) )  e. 
NN0 )
3432, 33syl 15 . . . . . 6  |-  ( ph  ->  (π `  ( N  ^ c  A ) )  e. 
NN0 )
3534nn0red 10035 . . . . 5  |-  ( ph  ->  (π `  ( N  ^ c  A ) )  e.  RR )
3614, 35resubcld 9227 . . . 4  |-  ( ph  ->  ( (π `  N )  -  (π `
 ( N  ^ c  A ) ) )  e.  RR )
3736, 29remulcld 8879 . . 3  |-  ( ph  ->  ( ( (π `  N
)  -  (π `  ( N  ^ c  A ) ) )  x.  ( A  x.  ( log `  N ) ) )  e.  RR )
38 chtcl 20363 . . . 4  |-  ( N  e.  RR  ->  ( theta `  N )  e.  RR )
3911, 38syl 15 . . 3  |-  ( ph  ->  ( theta `  N )  e.  RR )
40 1re 8853 . . . . . . . 8  |-  1  e.  RR
4140a1i 10 . . . . . . 7  |-  ( ph  ->  1  e.  RR )
42 1lt2 9902 . . . . . . . 8  |-  1  <  2
4342a1i 10 . . . . . . 7  |-  ( ph  ->  1  <  2 )
4441, 18, 11, 43, 21ltletrd 8992 . . . . . 6  |-  ( ph  ->  1  <  N )
4511, 44rplogcld 19996 . . . . 5  |-  ( ph  ->  ( log `  N
)  e.  RR+ )
461, 45rpmulcld 10422 . . . 4  |-  ( ph  ->  ( A  x.  ( log `  N ) )  e.  RR+ )
4714, 32resubcld 9227 . . . . 5  |-  ( ph  ->  ( (π `  N )  -  ( N  ^ c  A ) )  e.  RR )
48 ppinncl 20428 . . . . . . . . . 10  |-  ( ( N  e.  RR  /\  2  <_  N )  -> 
(π `  N )  e.  NN )
4910, 48syl 15 . . . . . . . . 9  |-  ( ph  ->  (π `  N )  e.  NN )
5032, 49nndivred 9810 . . . . . . . 8  |-  ( ph  ->  ( ( N  ^ c  A )  /  (π `  N ) )  e.  RR )
51 chtppilim.4 . . . . . . . 8  |-  ( ph  ->  ( ( N  ^ c  A )  /  (π `  N ) )  < 
( 1  -  A
) )
5250, 41, 2, 51ltsub13d 9394 . . . . . . 7  |-  ( ph  ->  A  <  ( 1  -  ( ( N  ^ c  A )  /  (π `  N ) ) ) )
5332recnd 8877 . . . . . . . . 9  |-  ( ph  ->  ( N  ^ c  A )  e.  CC )
5449nnrpd 10405 . . . . . . . . . 10  |-  ( ph  ->  (π `  N )  e.  RR+ )
5554rpcnne0d 10415 . . . . . . . . 9  |-  ( ph  ->  ( (π `  N )  e.  CC  /\  (π `  N
)  =/=  0 ) )
56 divsubdir 9472 . . . . . . . . 9  |-  ( ( (π `  N )  e.  CC  /\  ( N  ^ c  A )  e.  CC  /\  (
(π `  N )  e.  CC  /\  (π `  N
)  =/=  0 ) )  ->  ( (
(π `  N )  -  ( N  ^ c  A ) )  / 
(π `  N ) )  =  ( ( (π `  N )  /  (π `  N ) )  -  ( ( N  ^ c  A )  /  (π `  N ) ) ) )
5715, 53, 55, 56syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  ( ( (π `  N
)  -  ( N  ^ c  A ) )  /  (π `  N
) )  =  ( ( (π `  N )  / 
(π `  N ) )  -  ( ( N  ^ c  A )  /  (π `  N ) ) ) )
58 divid 9467 . . . . . . . . . 10  |-  ( ( (π `  N )  e.  CC  /\  (π `  N
)  =/=  0 )  ->  ( (π `  N
)  /  (π `  N
) )  =  1 )
5955, 58syl 15 . . . . . . . . 9  |-  ( ph  ->  ( (π `  N )  / 
(π `  N ) )  =  1 )
6059oveq1d 5889 . . . . . . . 8  |-  ( ph  ->  ( ( (π `  N
)  /  (π `  N
) )  -  (
( N  ^ c  A )  /  (π `  N ) ) )  =  ( 1  -  ( ( N  ^ c  A )  /  (π `  N ) ) ) )
6157, 60eqtrd 2328 . . . . . . 7  |-  ( ph  ->  ( ( (π `  N
)  -  ( N  ^ c  A ) )  /  (π `  N
) )  =  ( 1  -  ( ( N  ^ c  A
)  /  (π `  N
) ) ) )
6252, 61breqtrrd 4065 . . . . . 6  |-  ( ph  ->  A  <  ( ( (π `  N )  -  ( N  ^ c  A ) )  / 
(π `  N ) ) )
632, 47, 54ltmuldivd 10449 . . . . . 6  |-  ( ph  ->  ( ( A  x.  (π `
 N ) )  <  ( (π `  N
)  -  ( N  ^ c  A ) )  <->  A  <  ( ( (π `  N )  -  ( N  ^ c  A ) )  / 
(π `  N ) ) ) )
6462, 63mpbird 223 . . . . 5  |-  ( ph  ->  ( A  x.  (π `  N ) )  < 
( (π `  N )  -  ( N  ^ c  A ) ) )
65 ppiltx 20431 . . . . . . 7  |-  ( ( N  ^ c  A
)  e.  RR+  ->  (π `  ( N  ^ c  A ) )  < 
( N  ^ c  A ) )
6631, 65syl 15 . . . . . 6  |-  ( ph  ->  (π `  ( N  ^ c  A ) )  < 
( N  ^ c  A ) )
6735, 32, 14, 66ltsub2dd 9401 . . . . 5  |-  ( ph  ->  ( (π `  N )  -  ( N  ^ c  A ) )  < 
( (π `  N )  -  (π `
 ( N  ^ c  A ) ) ) )
6828, 47, 36, 64, 67lttrd 8993 . . . 4  |-  ( ph  ->  ( A  x.  (π `  N ) )  < 
( (π `  N )  -  (π `
 ( N  ^ c  A ) ) ) )
6928, 36, 46, 68ltmul1dd 10457 . . 3  |-  ( ph  ->  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) )  <  (
( (π `  N )  -  (π `
 ( N  ^ c  A ) ) )  x.  ( A  x.  ( log `  N ) ) ) )
70 fzfid 11051 . . . . . 6  |-  ( ph  ->  ( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  e.  Fin )
71 inss1 3402 . . . . . 6  |-  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime )  C_  (
( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )
72 ssfi 7099 . . . . . 6  |-  ( ( ( ( ( |_
`  ( 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 )
7370, 71, 72sylancl 643 . . . . 5  |-  ( ph  ->  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime )  e.  Fin )
74 inss2 3403 . . . . . . . 8  |-  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime )  C_  Prime
75 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )
7674, 75sseldi 3191 . . . . . . 7  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  p  e.  Prime )
77 prmnn 12777 . . . . . . . 8  |-  ( p  e.  Prime  ->  p  e.  NN )
7877nnrpd 10405 . . . . . . 7  |-  ( p  e.  Prime  ->  p  e.  RR+ )
7976, 78syl 15 . . . . . 6  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  p  e.  RR+ )
8079relogcld 19990 . . . . 5  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( log `  p )  e.  RR )
8173, 80fsumrecl 12223 . . . 4  |-  ( ph  -> 
sum_ p  e.  (
( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime ) ( log `  p )  e.  RR )
8229recnd 8877 . . . . . . 7  |-  ( ph  ->  ( A  x.  ( log `  N ) )  e.  CC )
83 fsumconst 12268 . . . . . . 7  |-  ( ( ( ( ( ( |_ `  ( 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 ) ) ) )
8473, 82, 83syl2anc 642 . . . . . 6  |-  ( 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 ) ) ) )
85 ppifl 20414 . . . . . . . . . 10  |-  ( N  e.  RR  ->  (π `  ( |_ `  N
) )  =  (π `  N ) )
8611, 85syl 15 . . . . . . . . 9  |-  ( ph  ->  (π `  ( |_ `  N ) )  =  (π `  N ) )
87 ppifl 20414 . . . . . . . . . 10  |-  ( ( N  ^ c  A
)  e.  RR  ->  (π `  ( |_ `  ( N  ^ c  A ) ) )  =  (π `  ( N  ^ c  A ) ) )
8832, 87syl 15 . . . . . . . . 9  |-  ( ph  ->  (π `  ( |_ `  ( N  ^ c  A ) ) )  =  (π `  ( N  ^ c  A ) ) )
8986, 88oveq12d 5892 . . . . . . . 8  |-  ( ph  ->  ( (π `  ( |_ `  N ) )  -  (π `
 ( |_ `  ( N  ^ c  A ) ) ) )  =  ( (π `  N )  -  (π `  ( N  ^ c  A ) ) ) )
9041, 11, 44ltled 8983 . . . . . . . . . . . 12  |-  ( ph  ->  1  <_  N )
91 chtppilim.2 . . . . . . . . . . . . 13  |-  ( ph  ->  A  <  1 )
92 ltle 8926 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  <  1  ->  A  <_  1 ) )
932, 40, 92sylancl 643 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  <  1  ->  A  <_  1 ) )
9491, 93mpd 14 . . . . . . . . . . . 12  |-  ( ph  ->  A  <_  1 )
9511, 90, 2, 41, 94cxplead 20084 . . . . . . . . . . 11  |-  ( ph  ->  ( N  ^ c  A )  <_  ( N  ^ c  1 ) )
9611recnd 8877 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  CC )
9796cxp1d 20069 . . . . . . . . . . 11  |-  ( ph  ->  ( N  ^ c 
1 )  =  N )
9895, 97breqtrd 4063 . . . . . . . . . 10  |-  ( ph  ->  ( N  ^ c  A )  <_  N
)
99 flword2 10959 . . . . . . . . . 10  |-  ( ( ( N  ^ c  A )  e.  RR  /\  N  e.  RR  /\  ( N  ^ c  A )  <_  N
)  ->  ( |_ `  N )  e.  (
ZZ>= `  ( |_ `  ( N  ^ c  A ) ) ) )
10032, 11, 98, 99syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( |_ `  N
)  e.  ( ZZ>= `  ( |_ `  ( N  ^ c  A ) ) ) )
101 ppidif 20417 . . . . . . . . 9  |-  ( ( |_ `  N )  e.  ( ZZ>= `  ( |_ `  ( N  ^ c  A ) ) )  ->  ( (π `  ( |_ `  N ) )  -  (π `  ( |_ `  ( N  ^ c  A ) ) ) )  =  ( # `  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) ) )
102100, 101syl 15 . . . . . . . 8  |-  ( ph  ->  ( (π `  ( |_ `  N ) )  -  (π `
 ( |_ `  ( N  ^ c  A ) ) ) )  =  ( # `  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) ) )
10389, 102eqtr3d 2330 . . . . . . 7  |-  ( ph  ->  ( (π `  N )  -  (π `
 ( N  ^ c  A ) ) )  =  ( # `  (
( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime ) ) )
104103oveq1d 5889 . . . . . 6  |-  ( ph  ->  ( ( (π `  N
)  -  (π `  ( N  ^ c  A ) ) )  x.  ( A  x.  ( log `  N ) ) )  =  ( ( # `  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  x.  ( A  x.  ( log `  N ) ) ) )
10584, 104eqtr4d 2331 . . . . 5  |-  ( ph  -> 
sum_ p  e.  (
( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime ) ( A  x.  ( log `  N
) )  =  ( ( (π `  N )  -  (π `
 ( N  ^ c  A ) ) )  x.  ( A  x.  ( log `  N ) ) ) )
10629adantr 451 . . . . . 6  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( A  x.  ( log `  N ) )  e.  RR )
10732adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( N  ^ c  A )  e.  RR )
108 reflcl 10944 . . . . . . . . . . 11  |-  ( ( N  ^ c  A
)  e.  RR  ->  ( |_ `  ( N  ^ c  A ) )  e.  RR )
109 peano2re 9001 . . . . . . . . . . 11  |-  ( ( |_ `  ( N  ^ c  A ) )  e.  RR  ->  ( ( |_ `  ( N  ^ c  A ) )  +  1 )  e.  RR )
11032, 108, 1093syl 18 . . . . . . . . . 10  |-  ( ph  ->  ( ( |_ `  ( N  ^ c  A ) )  +  1 )  e.  RR )
111110adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  (
( |_ `  ( N  ^ c  A ) )  +  1 )  e.  RR )
11279rpred 10406 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  p  e.  RR )
113 fllep1 10949 . . . . . . . . . . 11  |-  ( ( N  ^ c  A
)  e.  RR  ->  ( N  ^ c  A
)  <_  ( ( |_ `  ( N  ^ c  A ) )  +  1 ) )
11432, 113syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( N  ^ c  A )  <_  (
( |_ `  ( N  ^ c  A ) )  +  1 ) )
115114adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( N  ^ c  A )  <_  ( ( |_
`  ( N  ^ c  A ) )  +  1 ) )
11671, 75sseldi 3191 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  p  e.  ( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) ) )
117 elfzle1 10815 . . . . . . . . . 10  |-  ( p  e.  ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  -> 
( ( |_ `  ( N  ^ c  A ) )  +  1 )  <_  p
)
118116, 117syl 15 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  (
( |_ `  ( N  ^ c  A ) )  +  1 )  <_  p )
119107, 111, 112, 115, 118letrd 8989 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( N  ^ c  A )  <_  p )
12023rpne0d 10411 . . . . . . . . . . 11  |-  ( ph  ->  N  =/=  0 )
12196, 120, 3cxpefd 20075 . . . . . . . . . 10  |-  ( ph  ->  ( N  ^ c  A )  =  ( exp `  ( A  x.  ( log `  N
) ) ) )
122121eqcomd 2301 . . . . . . . . 9  |-  ( ph  ->  ( exp `  ( A  x.  ( log `  N ) ) )  =  ( N  ^ c  A ) )
123122adantr 451 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( exp `  ( A  x.  ( log `  N ) ) )  =  ( N  ^ c  A
) )
12479reeflogd 19991 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( exp `  ( log `  p
) )  =  p )
125119, 123, 1243brtr4d 4069 . . . . . . 7  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( exp `  ( A  x.  ( log `  N ) ) )  <_  ( exp `  ( log `  p
) ) )
126 efle 12414 . . . . . . . 8  |-  ( ( ( A  x.  ( log `  N ) )  e.  RR  /\  ( log `  p )  e.  RR )  ->  (
( A  x.  ( log `  N ) )  <_  ( log `  p
)  <->  ( exp `  ( A  x.  ( log `  N ) ) )  <_  ( exp `  ( log `  p ) ) ) )
127106, 80, 126syl2anc 642 . . . . . . 7  |-  ( (
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 ) ) ) )
128125, 127mpbird 223 . . . . . 6  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( A  x.  ( log `  N ) )  <_ 
( log `  p
) )
12973, 106, 80, 128fsumle 12273 . . . . 5  |-  ( 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
) )
130105, 129eqbrtrrd 4061 . . . 4  |-  ( ph  ->  ( ( (π `  N
)  -  (π `  ( N  ^ c  A ) ) )  x.  ( A  x.  ( log `  N ) ) )  <_  sum_ p  e.  ( ( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime ) ( log `  p ) )
131 fzfid 11051 . . . . . . 7  |-  ( ph  ->  ( 1 ... ( |_ `  N ) )  e.  Fin )
132 inss1 3402 . . . . . . 7  |-  ( ( 1 ... ( |_
`  N ) )  i^i  Prime )  C_  (
1 ... ( |_ `  N ) )
133 ssfi 7099 . . . . . . 7  |-  ( ( ( 1 ... ( |_ `  N ) )  e.  Fin  /\  (
( 1 ... ( |_ `  N ) )  i^i  Prime )  C_  (
1 ... ( |_ `  N ) ) )  ->  ( ( 1 ... ( |_ `  N ) )  i^i 
Prime )  e.  Fin )
134131, 132, 133sylancl 643 . . . . . 6  |-  ( ph  ->  ( ( 1 ... ( |_ `  N
) )  i^i  Prime )  e.  Fin )
135 inss2 3403 . . . . . . . . . . . . 13  |-  ( ( 1 ... ( |_
`  N ) )  i^i  Prime )  C_  Prime
136 simpr 447 . . . . . . . . . . . . 13  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )
137135, 136sseldi 3191 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  p  e.  Prime )
138 prmuz2 12792 . . . . . . . . . . . 12  |-  ( p  e.  Prime  ->  p  e.  ( ZZ>= `  2 )
)
139137, 138syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  p  e.  ( ZZ>= `  2 )
)
140 eluz2b2 10306 . . . . . . . . . . 11  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
141139, 140sylib 188 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  ( p  e.  NN  /\  1  < 
p ) )
142141simpld 445 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  p  e.  NN )
143142nnred 9777 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  p  e.  RR )
144141simprd 449 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  1  <  p )
145143, 144rplogcld 19996 . . . . . . 7  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  ( log `  p )  e.  RR+ )
146145rpred 10406 . . . . . 6  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  ( log `  p )  e.  RR )
147145rpge0d 10410 . . . . . 6  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  0  <_  ( log `  p ) )
14831rpge0d 10410 . . . . . . . . . 10  |-  ( ph  ->  0  <_  ( N  ^ c  A )
)
149 flge0nn0 10964 . . . . . . . . . 10  |-  ( ( ( N  ^ c  A )  e.  RR  /\  0  <_  ( N  ^ c  A )
)  ->  ( |_ `  ( N  ^ c  A ) )  e. 
NN0 )
15032, 148, 149syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( |_ `  ( N  ^ c  A ) )  e.  NN0 )
151 nn0p1nn 10019 . . . . . . . . 9  |-  ( ( |_ `  ( N  ^ c  A ) )  e.  NN0  ->  ( ( |_ `  ( N  ^ c  A ) )  +  1 )  e.  NN )
152150, 151syl 15 . . . . . . . 8  |-  ( ph  ->  ( ( |_ `  ( N  ^ c  A ) )  +  1 )  e.  NN )
153 nnuz 10279 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
154152, 153syl6eleq 2386 . . . . . . 7  |-  ( ph  ->  ( ( |_ `  ( N  ^ c  A ) )  +  1 )  e.  (
ZZ>= `  1 ) )
155 fzss1 10846 . . . . . . 7  |-  ( ( ( |_ `  ( N  ^ c  A ) )  +  1 )  e.  ( ZZ>= `  1
)  ->  ( (
( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  C_  ( 1 ... ( |_ `  N ) ) )
156 ssrin 3407 . . . . . . 7  |-  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) ) 
C_  ( 1 ... ( |_ `  N
) )  ->  (
( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime )  C_  (
( 1 ... ( |_ `  N ) )  i^i  Prime ) )
157154, 155, 1563syl 18 . . . . . 6  |-  ( ph  ->  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime )  C_  ( ( 1 ... ( |_
`  N ) )  i^i  Prime ) )
158134, 146, 147, 157fsumless 12270 . . . . 5  |-  ( ph  -> 
sum_ p  e.  (
( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime ) ( log `  p )  <_  sum_ p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) ( log `  p
) )
159 chtval 20364 . . . . . . 7  |-  ( N  e.  RR  ->  ( theta `  N )  = 
sum_ p  e.  (
( 0 [,] N
)  i^i  Prime ) ( log `  p ) )
16011, 159syl 15 . . . . . 6  |-  ( ph  ->  ( theta `  N )  =  sum_ p  e.  ( ( 0 [,] N
)  i^i  Prime ) ( log `  p ) )
161 2nn 9893 . . . . . . . . 9  |-  2  e.  NN
162161, 153eleqtri 2368 . . . . . . . 8  |-  2  e.  ( ZZ>= `  1 )
163 ppisval2 20358 . . . . . . . 8  |-  ( ( N  e.  RR  /\  2  e.  ( ZZ>= ` 
1 ) )  -> 
( ( 0 [,] N )  i^i  Prime )  =  ( ( 1 ... ( |_ `  N ) )  i^i 
Prime ) )
16411, 162, 163sylancl 643 . . . . . . 7  |-  ( ph  ->  ( ( 0 [,] N )  i^i  Prime )  =  ( ( 1 ... ( |_ `  N ) )  i^i 
Prime ) )
165164sumeq1d 12190 . . . . . 6  |-  ( ph  -> 
sum_ p  e.  (
( 0 [,] N
)  i^i  Prime ) ( log `  p )  =  sum_ p  e.  ( ( 1 ... ( |_ `  N ) )  i^i  Prime ) ( log `  p ) )
166160, 165eqtrd 2328 . . . . 5  |-  ( ph  ->  ( theta `  N )  =  sum_ p  e.  ( ( 1 ... ( |_ `  N ) )  i^i  Prime ) ( log `  p ) )
167158, 166breqtrrd 4065 . . . 4  |-  ( ph  -> 
sum_ p  e.  (
( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime ) ( log `  p )  <_  ( theta `  N ) )
16837, 81, 39, 130, 167letrd 8989 . . 3  |-  ( ph  ->  ( ( (π `  N
)  -  (π `  ( N  ^ c  A ) ) )  x.  ( A  x.  ( log `  N ) ) )  <_  ( theta `  N
) )
16930, 37, 39, 69, 168ltletrd 8992 . 2  |-  ( ph  ->  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) )  <  ( theta `  N ) )
17027, 169eqbrtrd 4059 1  |-  ( ph  ->  ( ( A ^
2 )  x.  (
(π `  N )  x.  ( log `  N
) ) )  < 
( theta `  N )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    i^i cin 3164    C_ wss 3165   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Fincfn 6879   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    +oocpnf 8880    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981   ZZ>=cuz 10246   RR+crp 10370   [,)cico 10674   [,]cicc 10675   ...cfz 10798   |_cfl 10940   ^cexp 11120   #chash 11353   sum_csu 12174   expce 12359   Primecprime 12774   logclog 19928    ^ c ccxp 19929   thetaccht 20344  πcppi 20347
This theorem is referenced by:  chtppilimlem2  20639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-dvds 12548  df-prm 12775  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930  df-cxp 19931  df-cht 20350  df-ppi 20353
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