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Theorem chtppilimlem1 21120
Description: Lemma for chtppilim 21122. (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 10604 . . . . . 6  |-  ( ph  ->  A  e.  RR )
32recnd 9070 . . . . 5  |-  ( ph  ->  A  e.  CC )
43sqvald 11475 . . . 4  |-  ( ph  ->  ( A ^ 2 )  =  ( A  x.  A ) )
54oveq1d 6055 . . 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 10025 . . . . . . . . . 10  |-  2  e.  RR
8 elicopnf 10956 . . . . . . . . . 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 20867 . . . . . . 7  |-  ( N  e.  RR  ->  (π `  N )  e.  NN0 )
1311, 12syl 16 . . . . . 6  |-  ( ph  ->  (π `  N )  e. 
NN0 )
1413nn0red 10231 . . . . 5  |-  ( ph  ->  (π `  N )  e.  RR )
1514recnd 9070 . . . 4  |-  ( ph  ->  (π `  N )  e.  CC )
16 0re 9047 . . . . . . . . 9  |-  0  e.  RR
1716a1i 11 . . . . . . . 8  |-  ( ph  ->  0  e.  RR )
187a1i 11 . . . . . . . 8  |-  ( ph  ->  2  e.  RR )
19 2pos 10038 . . . . . . . . 9  |-  0  <  2
2019a1i 11 . . . . . . . 8  |-  ( ph  ->  0  <  2 )
2110simprd 450 . . . . . . . 8  |-  ( ph  ->  2  <_  N )
2217, 18, 11, 20, 21ltletrd 9186 . . . . . . 7  |-  ( ph  ->  0  <  N )
2311, 22elrpd 10602 . . . . . 6  |-  ( ph  ->  N  e.  RR+ )
2423relogcld 20471 . . . . 5  |-  ( ph  ->  ( log `  N
)  e.  RR )
2524recnd 9070 . . . 4  |-  ( ph  ->  ( log `  N
)  e.  CC )
263, 3, 15, 25mul4d 9234 . . 3  |-  ( ph  ->  ( ( A  x.  A )  x.  (
(π `  N )  x.  ( log `  N
) ) )  =  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) ) )
275, 26eqtrd 2436 . 2  |-  ( ph  ->  ( ( A ^
2 )  x.  (
(π `  N )  x.  ( log `  N
) ) )  =  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) ) )
282, 14remulcld 9072 . . . 4  |-  ( ph  ->  ( A  x.  (π `  N ) )  e.  RR )
292, 24remulcld 9072 . . . 4  |-  ( ph  ->  ( A  x.  ( log `  N ) )  e.  RR )
3028, 29remulcld 9072 . . 3  |-  ( ph  ->  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) )  e.  RR )
3123, 2rpcxpcld 20574 . . . . . . . 8  |-  ( ph  ->  ( N  ^ c  A )  e.  RR+ )
3231rpred 10604 . . . . . . 7  |-  ( ph  ->  ( N  ^ c  A )  e.  RR )
33 ppicl 20867 . . . . . . 7  |-  ( ( N  ^ c  A
)  e.  RR  ->  (π `  ( N  ^ c  A ) )  e. 
NN0 )
3432, 33syl 16 . . . . . 6  |-  ( ph  ->  (π `  ( N  ^ c  A ) )  e. 
NN0 )
3534nn0red 10231 . . . . 5  |-  ( ph  ->  (π `  ( N  ^ c  A ) )  e.  RR )
3614, 35resubcld 9421 . . . 4  |-  ( ph  ->  ( (π `  N )  -  (π `
 ( N  ^ c  A ) ) )  e.  RR )
3736, 29remulcld 9072 . . 3  |-  ( ph  ->  ( ( (π `  N
)  -  (π `  ( N  ^ c  A ) ) )  x.  ( A  x.  ( log `  N ) ) )  e.  RR )
38 chtcl 20845 . . . 4  |-  ( N  e.  RR  ->  ( theta `  N )  e.  RR )
3911, 38syl 16 . . 3  |-  ( ph  ->  ( theta `  N )  e.  RR )
40 1re 9046 . . . . . . . 8  |-  1  e.  RR
4140a1i 11 . . . . . . 7  |-  ( ph  ->  1  e.  RR )
42 1lt2 10098 . . . . . . . 8  |-  1  <  2
4342a1i 11 . . . . . . 7  |-  ( ph  ->  1  <  2 )
4441, 18, 11, 43, 21ltletrd 9186 . . . . . 6  |-  ( ph  ->  1  <  N )
4511, 44rplogcld 20477 . . . . 5  |-  ( ph  ->  ( log `  N
)  e.  RR+ )
461, 45rpmulcld 10620 . . . 4  |-  ( ph  ->  ( A  x.  ( log `  N ) )  e.  RR+ )
4714, 32resubcld 9421 . . . . 5  |-  ( ph  ->  ( (π `  N )  -  ( N  ^ c  A ) )  e.  RR )
48 ppinncl 20910 . . . . . . . . . 10  |-  ( ( N  e.  RR  /\  2  <_  N )  -> 
(π `  N )  e.  NN )
4910, 48syl 16 . . . . . . . . 9  |-  ( ph  ->  (π `  N )  e.  NN )
5032, 49nndivred 10004 . . . . . . . 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 9588 . . . . . . 7  |-  ( ph  ->  A  <  ( 1  -  ( ( N  ^ c  A )  /  (π `  N ) ) ) )
5332recnd 9070 . . . . . . . . 9  |-  ( ph  ->  ( N  ^ c  A )  e.  CC )
5449nnrpd 10603 . . . . . . . . . 10  |-  ( ph  ->  (π `  N )  e.  RR+ )
5554rpcnne0d 10613 . . . . . . . . 9  |-  ( ph  ->  ( (π `  N )  e.  CC  /\  (π `  N
)  =/=  0 ) )
56 divsubdir 9666 . . . . . . . . 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 9661 . . . . . . . . . 10  |-  ( ( (π `  N )  e.  CC  /\  (π `  N
)  =/=  0 )  ->  ( (π `  N
)  /  (π `  N
) )  =  1 )
5955, 58syl 16 . . . . . . . . 9  |-  ( ph  ->  ( (π `  N )  / 
(π `  N ) )  =  1 )
6059oveq1d 6055 . . . . . . . 8  |-  ( ph  ->  ( ( (π `  N
)  /  (π `  N
) )  -  (
( N  ^ c  A )  /  (π `  N ) ) )  =  ( 1  -  ( ( N  ^ c  A )  /  (π `  N ) ) ) )
6157, 60eqtrd 2436 . . . . . . 7  |-  ( ph  ->  ( ( (π `  N
)  -  ( N  ^ c  A ) )  /  (π `  N
) )  =  ( 1  -  ( ( N  ^ c  A
)  /  (π `  N
) ) ) )
6252, 61breqtrrd 4198 . . . . . 6  |-  ( ph  ->  A  <  ( ( (π `  N )  -  ( N  ^ c  A ) )  / 
(π `  N ) ) )
632, 47, 54ltmuldivd 10647 . . . . . 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 20913 . . . . . . 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 9595 . . . . 5  |-  ( ph  ->  ( (π `  N )  -  ( N  ^ c  A ) )  < 
( (π `  N )  -  (π `
 ( N  ^ c  A ) ) ) )
6828, 47, 36, 64, 67lttrd 9187 . . . 4  |-  ( ph  ->  ( A  x.  (π `  N ) )  < 
( (π `  N )  -  (π `
 ( N  ^ c  A ) ) ) )
6928, 36, 46, 68ltmul1dd 10655 . . 3  |-  ( ph  ->  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) )  <  (
( (π `  N )  -  (π `
 ( N  ^ c  A ) ) )  x.  ( A  x.  ( log `  N ) ) ) )
70 fzfid 11267 . . . . . 6  |-  ( ph  ->  ( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  e.  Fin )
71 inss1 3521 . . . . . 6  |-  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime )  C_  (
( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )
72 ssfi 7288 . . . . . 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 3522 . . . . . . . 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 3306 . . . . . . 7  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  p  e.  Prime )
77 prmnn 13037 . . . . . . . 8  |-  ( p  e.  Prime  ->  p  e.  NN )
7877nnrpd 10603 . . . . . . 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 20471 . . . . 5  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( log `  p )  e.  RR )
8173, 80fsumrecl 12483 . . . 4  |-  ( ph  -> 
sum_ p  e.  (
( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime ) ( log `  p )  e.  RR )
8229recnd 9070 . . . . . . 7  |-  ( ph  ->  ( A  x.  ( log `  N ) )  e.  CC )
83 fsumconst 12528 . . . . . . 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 20896 . . . . . . . . . 10  |-  ( N  e.  RR  ->  (π `  ( |_ `  N
) )  =  (π `  N ) )
8611, 85syl 16 . . . . . . . . 9  |-  ( ph  ->  (π `  ( |_ `  N ) )  =  (π `  N ) )
87 ppifl 20896 . . . . . . . . . 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 6058 . . . . . . . 8  |-  ( ph  ->  ( (π `  ( |_ `  N ) )  -  (π `
 ( |_ `  ( N  ^ c  A ) ) ) )  =  ( (π `  N )  -  (π `  ( N  ^ c  A ) ) ) )
9041, 11, 44ltled 9177 . . . . . . . . . . . 12  |-  ( ph  ->  1  <_  N )
91 chtppilim.2 . . . . . . . . . . . . 13  |-  ( ph  ->  A  <  1 )
92 ltle 9119 . . . . . . . . . . . . . 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 20565 . . . . . . . . . . 11  |-  ( ph  ->  ( N  ^ c  A )  <_  ( N  ^ c  1 ) )
9611recnd 9070 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  CC )
9796cxp1d 20550 . . . . . . . . . . 11  |-  ( ph  ->  ( N  ^ c 
1 )  =  N )
9895, 97breqtrd 4196 . . . . . . . . . 10  |-  ( ph  ->  ( N  ^ c  A )  <_  N
)
99 flword2 11175 . . . . . . . . . 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 20899 . . . . . . . . 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 2438 . . . . . . 7  |-  ( ph  ->  ( (π `  N )  -  (π `
 ( N  ^ c  A ) ) )  =  ( # `  (
( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime ) ) )
104103oveq1d 6055 . . . . . 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 2439 . . . . 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 11160 . . . . . . . . . . 11  |-  ( ( N  ^ c  A
)  e.  RR  ->  ( |_ `  ( N  ^ c  A ) )  e.  RR )
109 peano2re 9195 . . . . . . . . . . 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 10604 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  p  e.  RR )
113 fllep1 11165 . . . . . . . . . . 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 3306 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  p  e.  ( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) ) )
117 elfzle1 11016 . . . . . . . . . 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 9183 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( N  ^ c  A )  <_  p )
12023rpne0d 10609 . . . . . . . . . . 11  |-  ( ph  ->  N  =/=  0 )
12196, 120, 3cxpefd 20556 . . . . . . . . . 10  |-  ( ph  ->  ( N  ^ c  A )  =  ( exp `  ( A  x.  ( log `  N
) ) ) )
122121eqcomd 2409 . . . . . . . . 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 20472 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( exp `  ( log `  p
) )  =  p )
125119, 123, 1243brtr4d 4202 . . . . . . 7  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( exp `  ( A  x.  ( log `  N ) ) )  <_  ( exp `  ( log `  p
) ) )
126 efle 12674 . . . . . . . 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 12533 . . . . 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 4194 . . . 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 11267 . . . . . . 7  |-  ( ph  ->  ( 1 ... ( |_ `  N ) )  e.  Fin )
132 inss1 3521 . . . . . . 7  |-  ( ( 1 ... ( |_
`  N ) )  i^i  Prime )  C_  (
1 ... ( |_ `  N ) )
133 ssfi 7288 . . . . . . 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 3522 . . . . . . . . . . . . 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 3306 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  p  e.  Prime )
138 prmuz2 13052 . . . . . . . . . . . 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 10504 . . . . . . . . . . 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 9971 . . . . . . . 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 20477 . . . . . . 7  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  ( log `  p )  e.  RR+ )
146145rpred 10604 . . . . . 6  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  ( log `  p )  e.  RR )
147145rpge0d 10608 . . . . . 6  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  0  <_  ( log `  p ) )
14831rpge0d 10608 . . . . . . . . . 10  |-  ( ph  ->  0  <_  ( N  ^ c  A )
)
149 flge0nn0 11180 . . . . . . . . . 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 10215 . . . . . . . . 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 10477 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
154152, 153syl6eleq 2494 . . . . . . 7  |-  ( ph  ->  ( ( |_ `  ( N  ^ c  A ) )  +  1 )  e.  (
ZZ>= `  1 ) )
155 fzss1 11047 . . . . . . 7  |-  ( ( ( |_ `  ( N  ^ c  A ) )  +  1 )  e.  ( ZZ>= `  1
)  ->  ( (
( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  C_  ( 1 ... ( |_ `  N ) ) )
156 ssrin 3526 . . . . . . 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 12530 . . . . 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 20846 . . . . . . 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 10089 . . . . . . . . 9  |-  2  e.  NN
162161, 153eleqtri 2476 . . . . . . . 8  |-  2  e.  ( ZZ>= `  1 )
163 ppisval2 20840 . . . . . . . 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 12450 . . . . . 6  |-  ( ph  -> 
sum_ p  e.  (
( 0 [,] N
)  i^i  Prime ) ( log `  p )  =  sum_ p  e.  ( ( 1 ... ( |_ `  N ) )  i^i  Prime ) ( log `  p ) )
166160, 165eqtrd 2436 . . . . 5  |-  ( ph  ->  ( theta `  N )  =  sum_ p  e.  ( ( 1 ... ( |_ `  N ) )  i^i  Prime ) ( log `  p ) )
167158, 166breqtrrd 4198 . . . 4  |-  ( ph  -> 
sum_ p  e.  (
( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime ) ( log `  p )  <_  ( theta `  N ) )
16837, 81, 39, 130, 167letrd 9183 . . 3  |-  ( ph  ->  ( ( (π `  N
)  -  (π `  ( N  ^ c  A ) ) )  x.  ( A  x.  ( log `  N ) ) )  <_  ( theta `  N
) )
16930, 37, 39, 69, 168ltletrd 9186 . 2  |-  ( ph  ->  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) )  <  ( theta `  N ) )
17027, 169eqbrtrd 4192 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 1721    =/= wne 2567    i^i cin 3279    C_ wss 3280   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Fincfn 7068   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    +oocpnf 9073    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   NNcn 9956   2c2 10005   NN0cn0 10177   ZZ>=cuz 10444   RR+crp 10568   [,)cico 10874   [,]cicc 10875   ...cfz 10999   |_cfl 11156   ^cexp 11337   #chash 11573   sum_csu 12434   expce 12619   Primecprime 13034   logclog 20405    ^ c ccxp 20406   thetaccht 20826  πcppi 20829
This theorem is referenced by:  chtppilimlem2  21121
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-dvds 12808  df-prm 13035  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408  df-cht 20832  df-ppi 20835
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