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Theorem chtppilimlem1 21027
Description: Lemma for chtppilim 21029. (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 10573 . . . . . 6  |-  ( ph  ->  A  e.  RR )
32recnd 9040 . . . . 5  |-  ( ph  ->  A  e.  CC )
43sqvald 11440 . . . 4  |-  ( ph  ->  ( A ^ 2 )  =  ( A  x.  A ) )
54oveq1d 6028 . . 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 9994 . . . . . . . . . 10  |-  2  e.  RR
8 elicopnf 10925 . . . . . . . . . 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 189 . . . . . . . 8  |-  ( ph  ->  ( N  e.  RR  /\  2  <_  N )
)
1110simpld 446 . . . . . . 7  |-  ( ph  ->  N  e.  RR )
12 ppicl 20774 . . . . . . 7  |-  ( N  e.  RR  ->  (π `  N )  e.  NN0 )
1311, 12syl 16 . . . . . 6  |-  ( ph  ->  (π `  N )  e. 
NN0 )
1413nn0red 10200 . . . . 5  |-  ( ph  ->  (π `  N )  e.  RR )
1514recnd 9040 . . . 4  |-  ( ph  ->  (π `  N )  e.  CC )
16 0re 9017 . . . . . . . . 9  |-  0  e.  RR
1716a1i 11 . . . . . . . 8  |-  ( ph  ->  0  e.  RR )
187a1i 11 . . . . . . . 8  |-  ( ph  ->  2  e.  RR )
19 2pos 10007 . . . . . . . . 9  |-  0  <  2
2019a1i 11 . . . . . . . 8  |-  ( ph  ->  0  <  2 )
2110simprd 450 . . . . . . . 8  |-  ( ph  ->  2  <_  N )
2217, 18, 11, 20, 21ltletrd 9155 . . . . . . 7  |-  ( ph  ->  0  <  N )
2311, 22elrpd 10571 . . . . . 6  |-  ( ph  ->  N  e.  RR+ )
2423relogcld 20378 . . . . 5  |-  ( ph  ->  ( log `  N
)  e.  RR )
2524recnd 9040 . . . 4  |-  ( ph  ->  ( log `  N
)  e.  CC )
263, 3, 15, 25mul4d 9203 . . 3  |-  ( ph  ->  ( ( A  x.  A )  x.  (
(π `  N )  x.  ( log `  N
) ) )  =  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) ) )
275, 26eqtrd 2412 . 2  |-  ( ph  ->  ( ( A ^
2 )  x.  (
(π `  N )  x.  ( log `  N
) ) )  =  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) ) )
282, 14remulcld 9042 . . . 4  |-  ( ph  ->  ( A  x.  (π `  N ) )  e.  RR )
292, 24remulcld 9042 . . . 4  |-  ( ph  ->  ( A  x.  ( log `  N ) )  e.  RR )
3028, 29remulcld 9042 . . 3  |-  ( ph  ->  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) )  e.  RR )
3123, 2rpcxpcld 20481 . . . . . . . 8  |-  ( ph  ->  ( N  ^ c  A )  e.  RR+ )
3231rpred 10573 . . . . . . 7  |-  ( ph  ->  ( N  ^ c  A )  e.  RR )
33 ppicl 20774 . . . . . . 7  |-  ( ( N  ^ c  A
)  e.  RR  ->  (π `  ( N  ^ c  A ) )  e. 
NN0 )
3432, 33syl 16 . . . . . 6  |-  ( ph  ->  (π `  ( N  ^ c  A ) )  e. 
NN0 )
3534nn0red 10200 . . . . 5  |-  ( ph  ->  (π `  ( N  ^ c  A ) )  e.  RR )
3614, 35resubcld 9390 . . . 4  |-  ( ph  ->  ( (π `  N )  -  (π `
 ( N  ^ c  A ) ) )  e.  RR )
3736, 29remulcld 9042 . . 3  |-  ( ph  ->  ( ( (π `  N
)  -  (π `  ( N  ^ c  A ) ) )  x.  ( A  x.  ( log `  N ) ) )  e.  RR )
38 chtcl 20752 . . . 4  |-  ( N  e.  RR  ->  ( theta `  N )  e.  RR )
3911, 38syl 16 . . 3  |-  ( ph  ->  ( theta `  N )  e.  RR )
40 1re 9016 . . . . . . . 8  |-  1  e.  RR
4140a1i 11 . . . . . . 7  |-  ( ph  ->  1  e.  RR )
42 1lt2 10067 . . . . . . . 8  |-  1  <  2
4342a1i 11 . . . . . . 7  |-  ( ph  ->  1  <  2 )
4441, 18, 11, 43, 21ltletrd 9155 . . . . . 6  |-  ( ph  ->  1  <  N )
4511, 44rplogcld 20384 . . . . 5  |-  ( ph  ->  ( log `  N
)  e.  RR+ )
461, 45rpmulcld 10589 . . . 4  |-  ( ph  ->  ( A  x.  ( log `  N ) )  e.  RR+ )
4714, 32resubcld 9390 . . . . 5  |-  ( ph  ->  ( (π `  N )  -  ( N  ^ c  A ) )  e.  RR )
48 ppinncl 20817 . . . . . . . . . 10  |-  ( ( N  e.  RR  /\  2  <_  N )  -> 
(π `  N )  e.  NN )
4910, 48syl 16 . . . . . . . . 9  |-  ( ph  ->  (π `  N )  e.  NN )
5032, 49nndivred 9973 . . . . . . . 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 9557 . . . . . . 7  |-  ( ph  ->  A  <  ( 1  -  ( ( N  ^ c  A )  /  (π `  N ) ) ) )
5332recnd 9040 . . . . . . . . 9  |-  ( ph  ->  ( N  ^ c  A )  e.  CC )
5449nnrpd 10572 . . . . . . . . . 10  |-  ( ph  ->  (π `  N )  e.  RR+ )
5554rpcnne0d 10582 . . . . . . . . 9  |-  ( ph  ->  ( (π `  N )  e.  CC  /\  (π `  N
)  =/=  0 ) )
56 divsubdir 9635 . . . . . . . . 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 1184 . . . . . . . 8  |-  ( ph  ->  ( ( (π `  N
)  -  ( N  ^ c  A ) )  /  (π `  N
) )  =  ( ( (π `  N )  / 
(π `  N ) )  -  ( ( N  ^ c  A )  /  (π `  N ) ) ) )
58 divid 9630 . . . . . . . . . 10  |-  ( ( (π `  N )  e.  CC  /\  (π `  N
)  =/=  0 )  ->  ( (π `  N
)  /  (π `  N
) )  =  1 )
5955, 58syl 16 . . . . . . . . 9  |-  ( ph  ->  ( (π `  N )  / 
(π `  N ) )  =  1 )
6059oveq1d 6028 . . . . . . . 8  |-  ( ph  ->  ( ( (π `  N
)  /  (π `  N
) )  -  (
( N  ^ c  A )  /  (π `  N ) ) )  =  ( 1  -  ( ( N  ^ c  A )  /  (π `  N ) ) ) )
6157, 60eqtrd 2412 . . . . . . 7  |-  ( ph  ->  ( ( (π `  N
)  -  ( N  ^ c  A ) )  /  (π `  N
) )  =  ( 1  -  ( ( N  ^ c  A
)  /  (π `  N
) ) ) )
6252, 61breqtrrd 4172 . . . . . 6  |-  ( ph  ->  A  <  ( ( (π `  N )  -  ( N  ^ c  A ) )  / 
(π `  N ) ) )
632, 47, 54ltmuldivd 10616 . . . . . 6  |-  ( ph  ->  ( ( A  x.  (π `
 N ) )  <  ( (π `  N
)  -  ( N  ^ c  A ) )  <->  A  <  ( ( (π `  N )  -  ( N  ^ c  A ) )  / 
(π `  N ) ) ) )
6462, 63mpbird 224 . . . . 5  |-  ( ph  ->  ( A  x.  (π `  N ) )  < 
( (π `  N )  -  ( N  ^ c  A ) ) )
65 ppiltx 20820 . . . . . . 7  |-  ( ( N  ^ c  A
)  e.  RR+  ->  (π `  ( N  ^ c  A ) )  < 
( N  ^ c  A ) )
6631, 65syl 16 . . . . . 6  |-  ( ph  ->  (π `  ( N  ^ c  A ) )  < 
( N  ^ c  A ) )
6735, 32, 14, 66ltsub2dd 9564 . . . . 5  |-  ( ph  ->  ( (π `  N )  -  ( N  ^ c  A ) )  < 
( (π `  N )  -  (π `
 ( N  ^ c  A ) ) ) )
6828, 47, 36, 64, 67lttrd 9156 . . . 4  |-  ( ph  ->  ( A  x.  (π `  N ) )  < 
( (π `  N )  -  (π `
 ( N  ^ c  A ) ) ) )
6928, 36, 46, 68ltmul1dd 10624 . . 3  |-  ( ph  ->  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) )  <  (
( (π `  N )  -  (π `
 ( N  ^ c  A ) ) )  x.  ( A  x.  ( log `  N ) ) ) )
70 fzfid 11232 . . . . . 6  |-  ( ph  ->  ( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  e.  Fin )
71 inss1 3497 . . . . . 6  |-  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime )  C_  (
( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )
72 ssfi 7258 . . . . . 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 644 . . . . 5  |-  ( ph  ->  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime )  e.  Fin )
74 inss2 3498 . . . . . . . 8  |-  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime )  C_  Prime
75 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )
7674, 75sseldi 3282 . . . . . . 7  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  p  e.  Prime )
77 prmnn 13002 . . . . . . . 8  |-  ( p  e.  Prime  ->  p  e.  NN )
7877nnrpd 10572 . . . . . . 7  |-  ( p  e.  Prime  ->  p  e.  RR+ )
7976, 78syl 16 . . . . . 6  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  p  e.  RR+ )
8079relogcld 20378 . . . . 5  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( log `  p )  e.  RR )
8173, 80fsumrecl 12448 . . . 4  |-  ( ph  -> 
sum_ p  e.  (
( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime ) ( log `  p )  e.  RR )
8229recnd 9040 . . . . . . 7  |-  ( ph  ->  ( A  x.  ( log `  N ) )  e.  CC )
83 fsumconst 12493 . . . . . . 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 643 . . . . . 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 20803 . . . . . . . . . 10  |-  ( N  e.  RR  ->  (π `  ( |_ `  N
) )  =  (π `  N ) )
8611, 85syl 16 . . . . . . . . 9  |-  ( ph  ->  (π `  ( |_ `  N ) )  =  (π `  N ) )
87 ppifl 20803 . . . . . . . . . 10  |-  ( ( N  ^ c  A
)  e.  RR  ->  (π `  ( |_ `  ( N  ^ c  A ) ) )  =  (π `  ( N  ^ c  A ) ) )
8832, 87syl 16 . . . . . . . . 9  |-  ( ph  ->  (π `  ( |_ `  ( N  ^ c  A ) ) )  =  (π `  ( N  ^ c  A ) ) )
8986, 88oveq12d 6031 . . . . . . . 8  |-  ( ph  ->  ( (π `  ( |_ `  N ) )  -  (π `
 ( |_ `  ( N  ^ c  A ) ) ) )  =  ( (π `  N )  -  (π `  ( N  ^ c  A ) ) ) )
9041, 11, 44ltled 9146 . . . . . . . . . . . 12  |-  ( ph  ->  1  <_  N )
91 chtppilim.2 . . . . . . . . . . . . 13  |-  ( ph  ->  A  <  1 )
92 ltle 9089 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  <  1  ->  A  <_  1 ) )
932, 40, 92sylancl 644 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  <  1  ->  A  <_  1 ) )
9491, 93mpd 15 . . . . . . . . . . . 12  |-  ( ph  ->  A  <_  1 )
9511, 90, 2, 41, 94cxplead 20472 . . . . . . . . . . 11  |-  ( ph  ->  ( N  ^ c  A )  <_  ( N  ^ c  1 ) )
9611recnd 9040 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  CC )
9796cxp1d 20457 . . . . . . . . . . 11  |-  ( ph  ->  ( N  ^ c 
1 )  =  N )
9895, 97breqtrd 4170 . . . . . . . . . 10  |-  ( ph  ->  ( N  ^ c  A )  <_  N
)
99 flword2 11140 . . . . . . . . . 10  |-  ( ( ( N  ^ c  A )  e.  RR  /\  N  e.  RR  /\  ( N  ^ c  A )  <_  N
)  ->  ( |_ `  N )  e.  (
ZZ>= `  ( |_ `  ( N  ^ c  A ) ) ) )
10032, 11, 98, 99syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  ( |_ `  N
)  e.  ( ZZ>= `  ( |_ `  ( N  ^ c  A ) ) ) )
101 ppidif 20806 . . . . . . . . 9  |-  ( ( |_ `  N )  e.  ( ZZ>= `  ( |_ `  ( N  ^ c  A ) ) )  ->  ( (π `  ( |_ `  N ) )  -  (π `  ( |_ `  ( N  ^ c  A ) ) ) )  =  ( # `  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) ) )
102100, 101syl 16 . . . . . . . 8  |-  ( ph  ->  ( (π `  ( |_ `  N ) )  -  (π `
 ( |_ `  ( N  ^ c  A ) ) ) )  =  ( # `  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) ) )
10389, 102eqtr3d 2414 . . . . . . 7  |-  ( ph  ->  ( (π `  N )  -  (π `
 ( N  ^ c  A ) ) )  =  ( # `  (
( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime ) ) )
104103oveq1d 6028 . . . . . 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 2415 . . . . 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 452 . . . . . 6  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( A  x.  ( log `  N ) )  e.  RR )
10732adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( N  ^ c  A )  e.  RR )
108 reflcl 11125 . . . . . . . . . . 11  |-  ( ( N  ^ c  A
)  e.  RR  ->  ( |_ `  ( N  ^ c  A ) )  e.  RR )
109 peano2re 9164 . . . . . . . . . . 11  |-  ( ( |_ `  ( N  ^ c  A ) )  e.  RR  ->  ( ( |_ `  ( N  ^ c  A ) )  +  1 )  e.  RR )
11032, 108, 1093syl 19 . . . . . . . . . 10  |-  ( ph  ->  ( ( |_ `  ( N  ^ c  A ) )  +  1 )  e.  RR )
111110adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  (
( |_ `  ( N  ^ c  A ) )  +  1 )  e.  RR )
11279rpred 10573 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  p  e.  RR )
113 fllep1 11130 . . . . . . . . . . 11  |-  ( ( N  ^ c  A
)  e.  RR  ->  ( N  ^ c  A
)  <_  ( ( |_ `  ( N  ^ c  A ) )  +  1 ) )
11432, 113syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( N  ^ c  A )  <_  (
( |_ `  ( N  ^ c  A ) )  +  1 ) )
115114adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( N  ^ c  A )  <_  ( ( |_
`  ( N  ^ c  A ) )  +  1 ) )
11671, 75sseldi 3282 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  p  e.  ( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) ) )
117 elfzle1 10985 . . . . . . . . . 10  |-  ( p  e.  ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  -> 
( ( |_ `  ( N  ^ c  A ) )  +  1 )  <_  p
)
118116, 117syl 16 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  (
( |_ `  ( N  ^ c  A ) )  +  1 )  <_  p )
119107, 111, 112, 115, 118letrd 9152 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( N  ^ c  A )  <_  p )
12023rpne0d 10578 . . . . . . . . . . 11  |-  ( ph  ->  N  =/=  0 )
12196, 120, 3cxpefd 20463 . . . . . . . . . 10  |-  ( ph  ->  ( N  ^ c  A )  =  ( exp `  ( A  x.  ( log `  N
) ) ) )
122121eqcomd 2385 . . . . . . . . 9  |-  ( ph  ->  ( exp `  ( A  x.  ( log `  N ) ) )  =  ( N  ^ c  A ) )
123122adantr 452 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( exp `  ( A  x.  ( log `  N ) ) )  =  ( N  ^ c  A
) )
12479reeflogd 20379 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( exp `  ( log `  p
) )  =  p )
125119, 123, 1243brtr4d 4176 . . . . . . 7  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( exp `  ( A  x.  ( log `  N ) ) )  <_  ( exp `  ( log `  p
) ) )
126 efle 12639 . . . . . . . 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 643 . . . . . . 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 224 . . . . . 6  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( A  x.  ( log `  N ) )  <_ 
( log `  p
) )
12973, 106, 80, 128fsumle 12498 . . . . 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 4168 . . . 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 11232 . . . . . . 7  |-  ( ph  ->  ( 1 ... ( |_ `  N ) )  e.  Fin )
132 inss1 3497 . . . . . . 7  |-  ( ( 1 ... ( |_
`  N ) )  i^i  Prime )  C_  (
1 ... ( |_ `  N ) )
133 ssfi 7258 . . . . . . 7  |-  ( ( ( 1 ... ( |_ `  N ) )  e.  Fin  /\  (
( 1 ... ( |_ `  N ) )  i^i  Prime )  C_  (
1 ... ( |_ `  N ) ) )  ->  ( ( 1 ... ( |_ `  N ) )  i^i 
Prime )  e.  Fin )
134131, 132, 133sylancl 644 . . . . . 6  |-  ( ph  ->  ( ( 1 ... ( |_ `  N
) )  i^i  Prime )  e.  Fin )
135 inss2 3498 . . . . . . . . . . . . 13  |-  ( ( 1 ... ( |_
`  N ) )  i^i  Prime )  C_  Prime
136 simpr 448 . . . . . . . . . . . . 13  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )
137135, 136sseldi 3282 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  p  e.  Prime )
138 prmuz2 13017 . . . . . . . . . . . 12  |-  ( p  e.  Prime  ->  p  e.  ( ZZ>= `  2 )
)
139137, 138syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  p  e.  ( ZZ>= `  2 )
)
140 eluz2b2 10473 . . . . . . . . . . 11  |-  ( p  e.  ( ZZ>= `  2
)  <->  ( p  e.  NN  /\  1  < 
p ) )
141139, 140sylib 189 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  ( p  e.  NN  /\  1  < 
p ) )
142141simpld 446 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  p  e.  NN )
143142nnred 9940 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  p  e.  RR )
144141simprd 450 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  1  <  p )
145143, 144rplogcld 20384 . . . . . . 7  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  ( log `  p )  e.  RR+ )
146145rpred 10573 . . . . . 6  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  ( log `  p )  e.  RR )
147145rpge0d 10577 . . . . . 6  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  0  <_  ( log `  p ) )
14831rpge0d 10577 . . . . . . . . . 10  |-  ( ph  ->  0  <_  ( N  ^ c  A )
)
149 flge0nn0 11145 . . . . . . . . . 10  |-  ( ( ( N  ^ c  A )  e.  RR  /\  0  <_  ( N  ^ c  A )
)  ->  ( |_ `  ( N  ^ c  A ) )  e. 
NN0 )
15032, 148, 149syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( |_ `  ( N  ^ c  A ) )  e.  NN0 )
151 nn0p1nn 10184 . . . . . . . . 9  |-  ( ( |_ `  ( N  ^ c  A ) )  e.  NN0  ->  ( ( |_ `  ( N  ^ c  A ) )  +  1 )  e.  NN )
152150, 151syl 16 . . . . . . . 8  |-  ( ph  ->  ( ( |_ `  ( N  ^ c  A ) )  +  1 )  e.  NN )
153 nnuz 10446 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
154152, 153syl6eleq 2470 . . . . . . 7  |-  ( ph  ->  ( ( |_ `  ( N  ^ c  A ) )  +  1 )  e.  (
ZZ>= `  1 ) )
155 fzss1 11016 . . . . . . 7  |-  ( ( ( |_ `  ( N  ^ c  A ) )  +  1 )  e.  ( ZZ>= `  1
)  ->  ( (
( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  C_  ( 1 ... ( |_ `  N ) ) )
156 ssrin 3502 . . . . . . 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 19 . . . . . 6  |-  ( ph  ->  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime )  C_  ( ( 1 ... ( |_
`  N ) )  i^i  Prime ) )
158134, 146, 147, 157fsumless 12495 . . . . 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 20753 . . . . . . 7  |-  ( N  e.  RR  ->  ( theta `  N )  = 
sum_ p  e.  (
( 0 [,] N
)  i^i  Prime ) ( log `  p ) )
16011, 159syl 16 . . . . . 6  |-  ( ph  ->  ( theta `  N )  =  sum_ p  e.  ( ( 0 [,] N
)  i^i  Prime ) ( log `  p ) )
161 2nn 10058 . . . . . . . . 9  |-  2  e.  NN
162161, 153eleqtri 2452 . . . . . . . 8  |-  2  e.  ( ZZ>= `  1 )
163 ppisval2 20747 . . . . . . . 8  |-  ( ( N  e.  RR  /\  2  e.  ( ZZ>= ` 
1 ) )  -> 
( ( 0 [,] N )  i^i  Prime )  =  ( ( 1 ... ( |_ `  N ) )  i^i 
Prime ) )
16411, 162, 163sylancl 644 . . . . . . 7  |-  ( ph  ->  ( ( 0 [,] N )  i^i  Prime )  =  ( ( 1 ... ( |_ `  N ) )  i^i 
Prime ) )
165164sumeq1d 12415 . . . . . 6  |-  ( ph  -> 
sum_ p  e.  (
( 0 [,] N
)  i^i  Prime ) ( log `  p )  =  sum_ p  e.  ( ( 1 ... ( |_ `  N ) )  i^i  Prime ) ( log `  p ) )
166160, 165eqtrd 2412 . . . . 5  |-  ( ph  ->  ( theta `  N )  =  sum_ p  e.  ( ( 1 ... ( |_ `  N ) )  i^i  Prime ) ( log `  p ) )
167158, 166breqtrrd 4172 . . . 4  |-  ( ph  -> 
sum_ p  e.  (
( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime ) ( log `  p )  <_  ( theta `  N ) )
16837, 81, 39, 130, 167letrd 9152 . . 3  |-  ( ph  ->  ( ( (π `  N
)  -  (π `  ( N  ^ c  A ) ) )  x.  ( A  x.  ( log `  N ) ) )  <_  ( theta `  N
) )
16930, 37, 39, 69, 168ltletrd 9155 . 2  |-  ( ph  ->  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) )  <  ( theta `  N ) )
17027, 169eqbrtrd 4166 1  |-  ( ph  ->  ( ( A ^
2 )  x.  (
(π `  N )  x.  ( log `  N
) ) )  < 
( theta `  N )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2543    i^i cin 3255    C_ wss 3256   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   Fincfn 7038   CCcc 8914   RRcr 8915   0cc0 8916   1c1 8917    + caddc 8919    x. cmul 8921    +oocpnf 9043    < clt 9046    <_ cle 9047    - cmin 9216    / cdiv 9602   NNcn 9925   2c2 9974   NN0cn0 10146   ZZ>=cuz 10413   RR+crp 10537   [,)cico 10843   [,]cicc 10844   ...cfz 10968   |_cfl 11121   ^cexp 11302   #chash 11538   sum_csu 12399   expce 12584   Primecprime 12999   logclog 20312    ^ c ccxp 20313   thetaccht 20733  πcppi 20736
This theorem is referenced by:  chtppilimlem2  21028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994  ax-addf 8995  ax-mulf 8996
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-2o 6654  df-oadd 6657  df-er 6834  df-map 6949  df-pm 6950  df-ixp 6993  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-fi 7344  df-sup 7374  df-oi 7405  df-card 7752  df-cda 7974  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-10 9991  df-n0 10147  df-z 10208  df-dec 10308  df-uz 10414  df-q 10500  df-rp 10538  df-xneg 10635  df-xadd 10636  df-xmul 10637  df-ioo 10845  df-ioc 10846  df-ico 10847  df-icc 10848  df-fz 10969  df-fzo 11059  df-fl 11122  df-mod 11171  df-seq 11244  df-exp 11303  df-fac 11487  df-bc 11514  df-hash 11539  df-shft 11802  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-limsup 12185  df-clim 12202  df-rlim 12203  df-sum 12400  df-ef 12590  df-sin 12592  df-cos 12593  df-pi 12595  df-dvds 12773  df-prm 13000  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-mulr 13463  df-starv 13464  df-sca 13465  df-vsca 13466  df-tset 13468  df-ple 13469  df-ds 13471  df-unif 13472  df-hom 13473  df-cco 13474  df-rest 13570  df-topn 13571  df-topgen 13587  df-pt 13588  df-prds 13591  df-xrs 13646  df-0g 13647  df-gsum 13648  df-qtop 13653  df-imas 13654  df-xps 13656  df-mre 13731  df-mrc 13732  df-acs 13734  df-mnd 14610  df-submnd 14659  df-mulg 14735  df-cntz 15036  df-cmn 15334  df-xmet 16612  df-met 16613  df-bl 16614  df-mopn 16615  df-fbas 16616  df-fg 16617  df-cnfld 16620  df-top 16879  df-bases 16881  df-topon 16882  df-topsp 16883  df-cld 16999  df-ntr 17000  df-cls 17001  df-nei 17078  df-lp 17116  df-perf 17117  df-cn 17206  df-cnp 17207  df-haus 17294  df-tx 17508  df-hmeo 17701  df-fil 17792  df-fm 17884  df-flim 17885  df-flf 17886  df-xms 18252  df-ms 18253  df-tms 18254  df-cncf 18772  df-limc 19613  df-dv 19614  df-log 20314  df-cxp 20315  df-cht 20739  df-ppi 20742
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