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Theorem chtppilimlem1 20622
Description: Lemma for chtppilim 20624. (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 10390 . . . . . 6  |-  ( ph  ->  A  e.  RR )
32recnd 8861 . . . . 5  |-  ( ph  ->  A  e.  CC )
43sqvald 11242 . . . 4  |-  ( ph  ->  ( A ^ 2 )  =  ( A  x.  A ) )
54oveq1d 5873 . . 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 9815 . . . . . . . . . 10  |-  2  e.  RR
8 elicopnf 10739 . . . . . . . . . 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 20369 . . . . . . 7  |-  ( N  e.  RR  ->  (π `  N )  e.  NN0 )
1311, 12syl 15 . . . . . 6  |-  ( ph  ->  (π `  N )  e. 
NN0 )
1413nn0red 10019 . . . . 5  |-  ( ph  ->  (π `  N )  e.  RR )
1514recnd 8861 . . . 4  |-  ( ph  ->  (π `  N )  e.  CC )
16 0re 8838 . . . . . . . . 9  |-  0  e.  RR
1716a1i 10 . . . . . . . 8  |-  ( ph  ->  0  e.  RR )
187a1i 10 . . . . . . . 8  |-  ( ph  ->  2  e.  RR )
19 2pos 9828 . . . . . . . . 9  |-  0  <  2
2019a1i 10 . . . . . . . 8  |-  ( ph  ->  0  <  2 )
2110simprd 449 . . . . . . . 8  |-  ( ph  ->  2  <_  N )
2217, 18, 11, 20, 21ltletrd 8976 . . . . . . 7  |-  ( ph  ->  0  <  N )
2311, 22elrpd 10388 . . . . . 6  |-  ( ph  ->  N  e.  RR+ )
2423relogcld 19974 . . . . 5  |-  ( ph  ->  ( log `  N
)  e.  RR )
2524recnd 8861 . . . 4  |-  ( ph  ->  ( log `  N
)  e.  CC )
263, 3, 15, 25mul4d 9024 . . 3  |-  ( ph  ->  ( ( A  x.  A )  x.  (
(π `  N )  x.  ( log `  N
) ) )  =  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) ) )
275, 26eqtrd 2315 . 2  |-  ( ph  ->  ( ( A ^
2 )  x.  (
(π `  N )  x.  ( log `  N
) ) )  =  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) ) )
282, 14remulcld 8863 . . . 4  |-  ( ph  ->  ( A  x.  (π `  N ) )  e.  RR )
292, 24remulcld 8863 . . . 4  |-  ( ph  ->  ( A  x.  ( log `  N ) )  e.  RR )
3028, 29remulcld 8863 . . 3  |-  ( ph  ->  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) )  e.  RR )
3123, 2rpcxpcld 20077 . . . . . . . 8  |-  ( ph  ->  ( N  ^ c  A )  e.  RR+ )
3231rpred 10390 . . . . . . 7  |-  ( ph  ->  ( N  ^ c  A )  e.  RR )
33 ppicl 20369 . . . . . . 7  |-  ( ( N  ^ c  A
)  e.  RR  ->  (π `  ( N  ^ c  A ) )  e. 
NN0 )
3432, 33syl 15 . . . . . 6  |-  ( ph  ->  (π `  ( N  ^ c  A ) )  e. 
NN0 )
3534nn0red 10019 . . . . 5  |-  ( ph  ->  (π `  ( N  ^ c  A ) )  e.  RR )
3614, 35resubcld 9211 . . . 4  |-  ( ph  ->  ( (π `  N )  -  (π `
 ( N  ^ c  A ) ) )  e.  RR )
3736, 29remulcld 8863 . . 3  |-  ( ph  ->  ( ( (π `  N
)  -  (π `  ( N  ^ c  A ) ) )  x.  ( A  x.  ( log `  N ) ) )  e.  RR )
38 chtcl 20347 . . . 4  |-  ( N  e.  RR  ->  ( theta `  N )  e.  RR )
3911, 38syl 15 . . 3  |-  ( ph  ->  ( theta `  N )  e.  RR )
40 1re 8837 . . . . . . . 8  |-  1  e.  RR
4140a1i 10 . . . . . . 7  |-  ( ph  ->  1  e.  RR )
42 1lt2 9886 . . . . . . . 8  |-  1  <  2
4342a1i 10 . . . . . . 7  |-  ( ph  ->  1  <  2 )
4441, 18, 11, 43, 21ltletrd 8976 . . . . . 6  |-  ( ph  ->  1  <  N )
4511, 44rplogcld 19980 . . . . 5  |-  ( ph  ->  ( log `  N
)  e.  RR+ )
461, 45rpmulcld 10406 . . . 4  |-  ( ph  ->  ( A  x.  ( log `  N ) )  e.  RR+ )
4714, 32resubcld 9211 . . . . 5  |-  ( ph  ->  ( (π `  N )  -  ( N  ^ c  A ) )  e.  RR )
48 ppinncl 20412 . . . . . . . . . 10  |-  ( ( N  e.  RR  /\  2  <_  N )  -> 
(π `  N )  e.  NN )
4910, 48syl 15 . . . . . . . . 9  |-  ( ph  ->  (π `  N )  e.  NN )
5032, 49nndivred 9794 . . . . . . . 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 9378 . . . . . . 7  |-  ( ph  ->  A  <  ( 1  -  ( ( N  ^ c  A )  /  (π `  N ) ) ) )
5332recnd 8861 . . . . . . . . 9  |-  ( ph  ->  ( N  ^ c  A )  e.  CC )
5449nnrpd 10389 . . . . . . . . . 10  |-  ( ph  ->  (π `  N )  e.  RR+ )
5554rpcnne0d 10399 . . . . . . . . 9  |-  ( ph  ->  ( (π `  N )  e.  CC  /\  (π `  N
)  =/=  0 ) )
56 divsubdir 9456 . . . . . . . . 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 9451 . . . . . . . . . 10  |-  ( ( (π `  N )  e.  CC  /\  (π `  N
)  =/=  0 )  ->  ( (π `  N
)  /  (π `  N
) )  =  1 )
5955, 58syl 15 . . . . . . . . 9  |-  ( ph  ->  ( (π `  N )  / 
(π `  N ) )  =  1 )
6059oveq1d 5873 . . . . . . . 8  |-  ( ph  ->  ( ( (π `  N
)  /  (π `  N
) )  -  (
( N  ^ c  A )  /  (π `  N ) ) )  =  ( 1  -  ( ( N  ^ c  A )  /  (π `  N ) ) ) )
6157, 60eqtrd 2315 . . . . . . 7  |-  ( ph  ->  ( ( (π `  N
)  -  ( N  ^ c  A ) )  /  (π `  N
) )  =  ( 1  -  ( ( N  ^ c  A
)  /  (π `  N
) ) ) )
6252, 61breqtrrd 4049 . . . . . 6  |-  ( ph  ->  A  <  ( ( (π `  N )  -  ( N  ^ c  A ) )  / 
(π `  N ) ) )
632, 47, 54ltmuldivd 10433 . . . . . 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 20415 . . . . . . 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 9385 . . . . 5  |-  ( ph  ->  ( (π `  N )  -  ( N  ^ c  A ) )  < 
( (π `  N )  -  (π `
 ( N  ^ c  A ) ) ) )
6828, 47, 36, 64, 67lttrd 8977 . . . 4  |-  ( ph  ->  ( A  x.  (π `  N ) )  < 
( (π `  N )  -  (π `
 ( N  ^ c  A ) ) ) )
6928, 36, 46, 68ltmul1dd 10441 . . 3  |-  ( ph  ->  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) )  <  (
( (π `  N )  -  (π `
 ( N  ^ c  A ) ) )  x.  ( A  x.  ( log `  N ) ) ) )
70 fzfid 11035 . . . . . 6  |-  ( ph  ->  ( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  e.  Fin )
71 inss1 3389 . . . . . 6  |-  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime )  C_  (
( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )
72 ssfi 7083 . . . . . 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 3390 . . . . . . . 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 3178 . . . . . . 7  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  p  e.  Prime )
77 prmnn 12761 . . . . . . . 8  |-  ( p  e.  Prime  ->  p  e.  NN )
7877nnrpd 10389 . . . . . . 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 19974 . . . . 5  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( log `  p )  e.  RR )
8173, 80fsumrecl 12207 . . . 4  |-  ( ph  -> 
sum_ p  e.  (
( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime ) ( log `  p )  e.  RR )
8229recnd 8861 . . . . . . 7  |-  ( ph  ->  ( A  x.  ( log `  N ) )  e.  CC )
83 fsumconst 12252 . . . . . . 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 20398 . . . . . . . . . 10  |-  ( N  e.  RR  ->  (π `  ( |_ `  N
) )  =  (π `  N ) )
8611, 85syl 15 . . . . . . . . 9  |-  ( ph  ->  (π `  ( |_ `  N ) )  =  (π `  N ) )
87 ppifl 20398 . . . . . . . . . 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 5876 . . . . . . . 8  |-  ( ph  ->  ( (π `  ( |_ `  N ) )  -  (π `
 ( |_ `  ( N  ^ c  A ) ) ) )  =  ( (π `  N )  -  (π `  ( N  ^ c  A ) ) ) )
9041, 11, 44ltled 8967 . . . . . . . . . . . 12  |-  ( ph  ->  1  <_  N )
91 chtppilim.2 . . . . . . . . . . . . 13  |-  ( ph  ->  A  <  1 )
92 ltle 8910 . . . . . . . . . . . . . 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 20068 . . . . . . . . . . 11  |-  ( ph  ->  ( N  ^ c  A )  <_  ( N  ^ c  1 ) )
9611recnd 8861 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  CC )
9796cxp1d 20053 . . . . . . . . . . 11  |-  ( ph  ->  ( N  ^ c 
1 )  =  N )
9895, 97breqtrd 4047 . . . . . . . . . 10  |-  ( ph  ->  ( N  ^ c  A )  <_  N
)
99 flword2 10943 . . . . . . . . . 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 20401 . . . . . . . . 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 2317 . . . . . . 7  |-  ( ph  ->  ( (π `  N )  -  (π `
 ( N  ^ c  A ) ) )  =  ( # `  (
( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime ) ) )
104103oveq1d 5873 . . . . . 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 2318 . . . . 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 10928 . . . . . . . . . . 11  |-  ( ( N  ^ c  A
)  e.  RR  ->  ( |_ `  ( N  ^ c  A ) )  e.  RR )
109 peano2re 8985 . . . . . . . . . . 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 10390 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  p  e.  RR )
113 fllep1 10933 . . . . . . . . . . 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 3178 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  p  e.  ( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) ) )
117 elfzle1 10799 . . . . . . . . . 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 8973 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( N  ^ c  A )  <_  p )
12023rpne0d 10395 . . . . . . . . . . 11  |-  ( ph  ->  N  =/=  0 )
12196, 120, 3cxpefd 20059 . . . . . . . . . 10  |-  ( ph  ->  ( N  ^ c  A )  =  ( exp `  ( A  x.  ( log `  N
) ) ) )
122121eqcomd 2288 . . . . . . . . 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 19975 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( exp `  ( log `  p
) )  =  p )
125119, 123, 1243brtr4d 4053 . . . . . . 7  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( exp `  ( A  x.  ( log `  N ) ) )  <_  ( exp `  ( log `  p
) ) )
126 efle 12398 . . . . . . . 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 12257 . . . . 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 4045 . . . 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 11035 . . . . . . 7  |-  ( ph  ->  ( 1 ... ( |_ `  N ) )  e.  Fin )
132 inss1 3389 . . . . . . 7  |-  ( ( 1 ... ( |_
`  N ) )  i^i  Prime )  C_  (
1 ... ( |_ `  N ) )
133 ssfi 7083 . . . . . . 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 3390 . . . . . . . . . . . . 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 3178 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  p  e.  Prime )
138 prmuz2 12776 . . . . . . . . . . . 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 10290 . . . . . . . . . . 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 9761 . . . . . . . 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 19980 . . . . . . 7  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  ( log `  p )  e.  RR+ )
146145rpred 10390 . . . . . 6  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  ( log `  p )  e.  RR )
147145rpge0d 10394 . . . . . 6  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  0  <_  ( log `  p ) )
14831rpge0d 10394 . . . . . . . . . 10  |-  ( ph  ->  0  <_  ( N  ^ c  A )
)
149 flge0nn0 10948 . . . . . . . . . 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 10003 . . . . . . . . 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 10263 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
154152, 153syl6eleq 2373 . . . . . . 7  |-  ( ph  ->  ( ( |_ `  ( N  ^ c  A ) )  +  1 )  e.  (
ZZ>= `  1 ) )
155 fzss1 10830 . . . . . . 7  |-  ( ( ( |_ `  ( N  ^ c  A ) )  +  1 )  e.  ( ZZ>= `  1
)  ->  ( (
( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  C_  ( 1 ... ( |_ `  N ) ) )
156 ssrin 3394 . . . . . . 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 12254 . . . . 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 20348 . . . . . . 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 9877 . . . . . . . . 9  |-  2  e.  NN
162161, 153eleqtri 2355 . . . . . . . 8  |-  2  e.  ( ZZ>= `  1 )
163 ppisval2 20342 . . . . . . . 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 12174 . . . . . 6  |-  ( ph  -> 
sum_ p  e.  (
( 0 [,] N
)  i^i  Prime ) ( log `  p )  =  sum_ p  e.  ( ( 1 ... ( |_ `  N ) )  i^i  Prime ) ( log `  p ) )
166160, 165eqtrd 2315 . . . . 5  |-  ( ph  ->  ( theta `  N )  =  sum_ p  e.  ( ( 1 ... ( |_ `  N ) )  i^i  Prime ) ( log `  p ) )
167158, 166breqtrrd 4049 . . . 4  |-  ( ph  -> 
sum_ p  e.  (
( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime ) ( log `  p )  <_  ( theta `  N ) )
16837, 81, 39, 130, 167letrd 8973 . . 3  |-  ( ph  ->  ( ( (π `  N
)  -  (π `  ( N  ^ c  A ) ) )  x.  ( A  x.  ( log `  N ) ) )  <_  ( theta `  N
) )
16930, 37, 39, 69, 168ltletrd 8976 . 2  |-  ( ph  ->  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) )  <  ( theta `  N ) )
17027, 169eqbrtrd 4043 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 1623    e. wcel 1684    =/= wne 2446    i^i cin 3151    C_ wss 3152   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Fincfn 6863   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    +oocpnf 8864    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   2c2 9795   NN0cn0 9965   ZZ>=cuz 10230   RR+crp 10354   [,)cico 10658   [,]cicc 10659   ...cfz 10782   |_cfl 10924   ^cexp 11104   #chash 11337   sum_csu 12158   expce 12343   Primecprime 12758   logclog 19912    ^ c ccxp 19913   thetaccht 20328  πcppi 20331
This theorem is referenced by:  chtppilimlem2  20623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ioc 10661  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351  df-cos 12352  df-pi 12354  df-dvds 12532  df-prm 12759  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217  df-log 19914  df-cxp 19915  df-cht 20334  df-ppi 20337
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