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Theorem chtppilimlem1 21159
Description: Lemma for chtppilim 21161. (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 10640 . . . . . 6  |-  ( ph  ->  A  e.  RR )
32recnd 9106 . . . . 5  |-  ( ph  ->  A  e.  CC )
43sqvald 11512 . . . 4  |-  ( ph  ->  ( A ^ 2 )  =  ( A  x.  A ) )
54oveq1d 6088 . . 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 10061 . . . . . . . . . 10  |-  2  e.  RR
8 elicopnf 10992 . . . . . . . . . 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 20906 . . . . . . 7  |-  ( N  e.  RR  ->  (π `  N )  e.  NN0 )
1311, 12syl 16 . . . . . 6  |-  ( ph  ->  (π `  N )  e. 
NN0 )
1413nn0red 10267 . . . . 5  |-  ( ph  ->  (π `  N )  e.  RR )
1514recnd 9106 . . . 4  |-  ( ph  ->  (π `  N )  e.  CC )
16 0re 9083 . . . . . . . . 9  |-  0  e.  RR
1716a1i 11 . . . . . . . 8  |-  ( ph  ->  0  e.  RR )
187a1i 11 . . . . . . . 8  |-  ( ph  ->  2  e.  RR )
19 2pos 10074 . . . . . . . . 9  |-  0  <  2
2019a1i 11 . . . . . . . 8  |-  ( ph  ->  0  <  2 )
2110simprd 450 . . . . . . . 8  |-  ( ph  ->  2  <_  N )
2217, 18, 11, 20, 21ltletrd 9222 . . . . . . 7  |-  ( ph  ->  0  <  N )
2311, 22elrpd 10638 . . . . . 6  |-  ( ph  ->  N  e.  RR+ )
2423relogcld 20510 . . . . 5  |-  ( ph  ->  ( log `  N
)  e.  RR )
2524recnd 9106 . . . 4  |-  ( ph  ->  ( log `  N
)  e.  CC )
263, 3, 15, 25mul4d 9270 . . 3  |-  ( ph  ->  ( ( A  x.  A )  x.  (
(π `  N )  x.  ( log `  N
) ) )  =  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) ) )
275, 26eqtrd 2467 . 2  |-  ( ph  ->  ( ( A ^
2 )  x.  (
(π `  N )  x.  ( log `  N
) ) )  =  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) ) )
282, 14remulcld 9108 . . . 4  |-  ( ph  ->  ( A  x.  (π `  N ) )  e.  RR )
292, 24remulcld 9108 . . . 4  |-  ( ph  ->  ( A  x.  ( log `  N ) )  e.  RR )
3028, 29remulcld 9108 . . 3  |-  ( ph  ->  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) )  e.  RR )
3123, 2rpcxpcld 20613 . . . . . . . 8  |-  ( ph  ->  ( N  ^ c  A )  e.  RR+ )
3231rpred 10640 . . . . . . 7  |-  ( ph  ->  ( N  ^ c  A )  e.  RR )
33 ppicl 20906 . . . . . . 7  |-  ( ( N  ^ c  A
)  e.  RR  ->  (π `  ( N  ^ c  A ) )  e. 
NN0 )
3432, 33syl 16 . . . . . 6  |-  ( ph  ->  (π `  ( N  ^ c  A ) )  e. 
NN0 )
3534nn0red 10267 . . . . 5  |-  ( ph  ->  (π `  ( N  ^ c  A ) )  e.  RR )
3614, 35resubcld 9457 . . . 4  |-  ( ph  ->  ( (π `  N )  -  (π `
 ( N  ^ c  A ) ) )  e.  RR )
3736, 29remulcld 9108 . . 3  |-  ( ph  ->  ( ( (π `  N
)  -  (π `  ( N  ^ c  A ) ) )  x.  ( A  x.  ( log `  N ) ) )  e.  RR )
38 chtcl 20884 . . . 4  |-  ( N  e.  RR  ->  ( theta `  N )  e.  RR )
3911, 38syl 16 . . 3  |-  ( ph  ->  ( theta `  N )  e.  RR )
40 1re 9082 . . . . . . . 8  |-  1  e.  RR
4140a1i 11 . . . . . . 7  |-  ( ph  ->  1  e.  RR )
42 1lt2 10134 . . . . . . . 8  |-  1  <  2
4342a1i 11 . . . . . . 7  |-  ( ph  ->  1  <  2 )
4441, 18, 11, 43, 21ltletrd 9222 . . . . . 6  |-  ( ph  ->  1  <  N )
4511, 44rplogcld 20516 . . . . 5  |-  ( ph  ->  ( log `  N
)  e.  RR+ )
461, 45rpmulcld 10656 . . . 4  |-  ( ph  ->  ( A  x.  ( log `  N ) )  e.  RR+ )
4714, 32resubcld 9457 . . . . 5  |-  ( ph  ->  ( (π `  N )  -  ( N  ^ c  A ) )  e.  RR )
48 ppinncl 20949 . . . . . . . . . 10  |-  ( ( N  e.  RR  /\  2  <_  N )  -> 
(π `  N )  e.  NN )
4910, 48syl 16 . . . . . . . . 9  |-  ( ph  ->  (π `  N )  e.  NN )
5032, 49nndivred 10040 . . . . . . . 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 9624 . . . . . . 7  |-  ( ph  ->  A  <  ( 1  -  ( ( N  ^ c  A )  /  (π `  N ) ) ) )
5332recnd 9106 . . . . . . . . 9  |-  ( ph  ->  ( N  ^ c  A )  e.  CC )
5449nnrpd 10639 . . . . . . . . . 10  |-  ( ph  ->  (π `  N )  e.  RR+ )
5554rpcnne0d 10649 . . . . . . . . 9  |-  ( ph  ->  ( (π `  N )  e.  CC  /\  (π `  N
)  =/=  0 ) )
56 divsubdir 9702 . . . . . . . . 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 9697 . . . . . . . . . 10  |-  ( ( (π `  N )  e.  CC  /\  (π `  N
)  =/=  0 )  ->  ( (π `  N
)  /  (π `  N
) )  =  1 )
5955, 58syl 16 . . . . . . . . 9  |-  ( ph  ->  ( (π `  N )  / 
(π `  N ) )  =  1 )
6059oveq1d 6088 . . . . . . . 8  |-  ( ph  ->  ( ( (π `  N
)  /  (π `  N
) )  -  (
( N  ^ c  A )  /  (π `  N ) ) )  =  ( 1  -  ( ( N  ^ c  A )  /  (π `  N ) ) ) )
6157, 60eqtrd 2467 . . . . . . 7  |-  ( ph  ->  ( ( (π `  N
)  -  ( N  ^ c  A ) )  /  (π `  N
) )  =  ( 1  -  ( ( N  ^ c  A
)  /  (π `  N
) ) ) )
6252, 61breqtrrd 4230 . . . . . 6  |-  ( ph  ->  A  <  ( ( (π `  N )  -  ( N  ^ c  A ) )  / 
(π `  N ) ) )
632, 47, 54ltmuldivd 10683 . . . . . 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 20952 . . . . . . 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 9631 . . . . 5  |-  ( ph  ->  ( (π `  N )  -  ( N  ^ c  A ) )  < 
( (π `  N )  -  (π `
 ( N  ^ c  A ) ) ) )
6828, 47, 36, 64, 67lttrd 9223 . . . 4  |-  ( ph  ->  ( A  x.  (π `  N ) )  < 
( (π `  N )  -  (π `
 ( N  ^ c  A ) ) ) )
6928, 36, 46, 68ltmul1dd 10691 . . 3  |-  ( ph  ->  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) )  <  (
( (π `  N )  -  (π `
 ( N  ^ c  A ) ) )  x.  ( A  x.  ( log `  N ) ) ) )
70 fzfid 11304 . . . . . 6  |-  ( ph  ->  ( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  e.  Fin )
71 inss1 3553 . . . . . 6  |-  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime )  C_  (
( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )
72 ssfi 7321 . . . . . 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 3554 . . . . . . . 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 3338 . . . . . . 7  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  p  e.  Prime )
77 prmnn 13074 . . . . . . . 8  |-  ( p  e.  Prime  ->  p  e.  NN )
7877nnrpd 10639 . . . . . . 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 20510 . . . . 5  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( log `  p )  e.  RR )
8173, 80fsumrecl 12520 . . . 4  |-  ( ph  -> 
sum_ p  e.  (
( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime ) ( log `  p )  e.  RR )
8229recnd 9106 . . . . . . 7  |-  ( ph  ->  ( A  x.  ( log `  N ) )  e.  CC )
83 fsumconst 12565 . . . . . . 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 20935 . . . . . . . . . 10  |-  ( N  e.  RR  ->  (π `  ( |_ `  N
) )  =  (π `  N ) )
8611, 85syl 16 . . . . . . . . 9  |-  ( ph  ->  (π `  ( |_ `  N ) )  =  (π `  N ) )
87 ppifl 20935 . . . . . . . . . 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 6091 . . . . . . . 8  |-  ( ph  ->  ( (π `  ( |_ `  N ) )  -  (π `
 ( |_ `  ( N  ^ c  A ) ) ) )  =  ( (π `  N )  -  (π `  ( N  ^ c  A ) ) ) )
9041, 11, 44ltled 9213 . . . . . . . . . . . 12  |-  ( ph  ->  1  <_  N )
91 chtppilim.2 . . . . . . . . . . . . 13  |-  ( ph  ->  A  <  1 )
92 ltle 9155 . . . . . . . . . . . . . 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 20604 . . . . . . . . . . 11  |-  ( ph  ->  ( N  ^ c  A )  <_  ( N  ^ c  1 ) )
9611recnd 9106 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  CC )
9796cxp1d 20589 . . . . . . . . . . 11  |-  ( ph  ->  ( N  ^ c 
1 )  =  N )
9895, 97breqtrd 4228 . . . . . . . . . 10  |-  ( ph  ->  ( N  ^ c  A )  <_  N
)
99 flword2 11212 . . . . . . . . . 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 20938 . . . . . . . . 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 2469 . . . . . . 7  |-  ( ph  ->  ( (π `  N )  -  (π `
 ( N  ^ c  A ) ) )  =  ( # `  (
( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime ) ) )
104103oveq1d 6088 . . . . . 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 2470 . . . . 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 11197 . . . . . . . . . . 11  |-  ( ( N  ^ c  A
)  e.  RR  ->  ( |_ `  ( N  ^ c  A ) )  e.  RR )
109 peano2re 9231 . . . . . . . . . . 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 10640 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  p  e.  RR )
113 fllep1 11202 . . . . . . . . . . 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 3338 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  p  e.  ( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) ) )
117 elfzle1 11052 . . . . . . . . . 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 9219 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( N  ^ c  A )  <_  p )
12023rpne0d 10645 . . . . . . . . . . 11  |-  ( ph  ->  N  =/=  0 )
12196, 120, 3cxpefd 20595 . . . . . . . . . 10  |-  ( ph  ->  ( N  ^ c  A )  =  ( exp `  ( A  x.  ( log `  N
) ) ) )
122121eqcomd 2440 . . . . . . . . 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 20511 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( exp `  ( log `  p
) )  =  p )
125119, 123, 1243brtr4d 4234 . . . . . . 7  |-  ( (
ph  /\  p  e.  ( ( ( ( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i 
Prime ) )  ->  ( exp `  ( A  x.  ( log `  N ) ) )  <_  ( exp `  ( log `  p
) ) )
126 efle 12711 . . . . . . . 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 12570 . . . . 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 4226 . . . 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 11304 . . . . . . 7  |-  ( ph  ->  ( 1 ... ( |_ `  N ) )  e.  Fin )
132 inss1 3553 . . . . . . 7  |-  ( ( 1 ... ( |_
`  N ) )  i^i  Prime )  C_  (
1 ... ( |_ `  N ) )
133 ssfi 7321 . . . . . . 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 3554 . . . . . . . . . . . . 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 3338 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  p  e.  Prime )
138 prmuz2 13089 . . . . . . . . . . . 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 10540 . . . . . . . . . . 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 10007 . . . . . . . 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 20516 . . . . . . 7  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  ( log `  p )  e.  RR+ )
146145rpred 10640 . . . . . 6  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  ( log `  p )  e.  RR )
147145rpge0d 10644 . . . . . 6  |-  ( (
ph  /\  p  e.  ( ( 1 ... ( |_ `  N
) )  i^i  Prime ) )  ->  0  <_  ( log `  p ) )
14831rpge0d 10644 . . . . . . . . . 10  |-  ( ph  ->  0  <_  ( N  ^ c  A )
)
149 flge0nn0 11217 . . . . . . . . . 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 10251 . . . . . . . . 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 10513 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
154152, 153syl6eleq 2525 . . . . . . 7  |-  ( ph  ->  ( ( |_ `  ( N  ^ c  A ) )  +  1 )  e.  (
ZZ>= `  1 ) )
155 fzss1 11083 . . . . . . 7  |-  ( ( ( |_ `  ( N  ^ c  A ) )  +  1 )  e.  ( ZZ>= `  1
)  ->  ( (
( |_ `  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  C_  ( 1 ... ( |_ `  N ) ) )
156 ssrin 3558 . . . . . . 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 12567 . . . . 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 20885 . . . . . . 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 10125 . . . . . . . . 9  |-  2  e.  NN
162161, 153eleqtri 2507 . . . . . . . 8  |-  2  e.  ( ZZ>= `  1 )
163 ppisval2 20879 . . . . . . . 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 12487 . . . . . 6  |-  ( ph  -> 
sum_ p  e.  (
( 0 [,] N
)  i^i  Prime ) ( log `  p )  =  sum_ p  e.  ( ( 1 ... ( |_ `  N ) )  i^i  Prime ) ( log `  p ) )
166160, 165eqtrd 2467 . . . . 5  |-  ( ph  ->  ( theta `  N )  =  sum_ p  e.  ( ( 1 ... ( |_ `  N ) )  i^i  Prime ) ( log `  p ) )
167158, 166breqtrrd 4230 . . . 4  |-  ( ph  -> 
sum_ p  e.  (
( ( ( |_
`  ( N  ^ c  A ) )  +  1 ) ... ( |_ `  N ) )  i^i  Prime ) ( log `  p )  <_  ( theta `  N ) )
16837, 81, 39, 130, 167letrd 9219 . . 3  |-  ( ph  ->  ( ( (π `  N
)  -  (π `  ( N  ^ c  A ) ) )  x.  ( A  x.  ( log `  N ) ) )  <_  ( theta `  N
) )
16930, 37, 39, 69, 168ltletrd 9222 . 2  |-  ( ph  ->  ( ( A  x.  (π `
 N ) )  x.  ( A  x.  ( log `  N ) ) )  <  ( theta `  N ) )
17027, 169eqbrtrd 4224 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 1652    e. wcel 1725    =/= wne 2598    i^i cin 3311    C_ wss 3312   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Fincfn 7101   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    +oocpnf 9109    < clt 9112    <_ cle 9113    - cmin 9283    / cdiv 9669   NNcn 9992   2c2 10041   NN0cn0 10213   ZZ>=cuz 10480   RR+crp 10604   [,)cico 10910   [,]cicc 10911   ...cfz 11035   |_cfl 11193   ^cexp 11374   #chash 11610   sum_csu 12471   expce 12656   Primecprime 13071   logclog 20444    ^ c ccxp 20445   thetaccht 20865  πcppi 20868
This theorem is referenced by:  chtppilimlem2  21160
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ioc 10913  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-shft 11874  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-sum 12472  df-ef 12662  df-sin 12664  df-cos 12665  df-pi 12667  df-dvds 12845  df-prm 13072  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-fbas 16691  df-fg 16692  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-lp 17192  df-perf 17193  df-cn 17283  df-cnp 17284  df-haus 17371  df-tx 17586  df-hmeo 17779  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-xms 18342  df-ms 18343  df-tms 18344  df-cncf 18900  df-limc 19745  df-dv 19746  df-log 20446  df-cxp 20447  df-cht 20871  df-ppi 20874
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