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Theorem chtleppi 24217
Description: Upper bound on the  theta function. (Contributed by Mario Carneiro, 22-Sep-2014.)
Assertion
Ref Expression
chtleppi  |-  ( A  e.  RR+  ->  ( theta `  A )  <_  (
(π `  A )  x.  ( log `  A
) ) )

Proof of Theorem chtleppi
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 rpre 11331 . . . 4  |-  ( A  e.  RR+  ->  A  e.  RR )
2 ppifi 24111 . . . 4  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  e. 
Fin )
31, 2syl 17 . . 3  |-  ( A  e.  RR+  ->  ( ( 0 [,] A )  i^i  Prime )  e.  Fin )
4 inss2 3644 . . . . . . 7  |-  ( ( 0 [,] A )  i^i  Prime )  C_  Prime
5 simpr 468 . . . . . . 7  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  p  e.  ( ( 0 [,] A )  i^i  Prime ) )
64, 5sseldi 3416 . . . . . 6  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  p  e.  Prime )
7 prmnn 14704 . . . . . 6  |-  ( p  e.  Prime  ->  p  e.  NN )
86, 7syl 17 . . . . 5  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  p  e.  NN )
98nnrpd 11362 . . . 4  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  p  e.  RR+ )
109relogcld 23651 . . 3  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  ( log `  p )  e.  RR )
11 relogcl 23604 . . . 4  |-  ( A  e.  RR+  ->  ( log `  A )  e.  RR )
1211adantr 472 . . 3  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  ( log `  A )  e.  RR )
13 inss1 3643 . . . . . . . 8  |-  ( ( 0 [,] A )  i^i  Prime )  C_  (
0 [,] A )
1413, 5sseldi 3416 . . . . . . 7  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  p  e.  ( 0 [,] A
) )
15 0re 9661 . . . . . . . . 9  |-  0  e.  RR
16 elicc2 11724 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) ) )
1715, 1, 16sylancr 676 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( p  e.  ( 0 [,] A )  <->  ( p  e.  RR  /\  0  <_  p  /\  p  <_  A
) ) )
1817biimpa 492 . . . . . . 7  |-  ( ( A  e.  RR+  /\  p  e.  ( 0 [,] A
) )  ->  (
p  e.  RR  /\  0  <_  p  /\  p  <_  A ) )
1914, 18syldan 478 . . . . . 6  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  ( p  e.  RR  /\  0  <_  p  /\  p  <_  A
) )
2019simp3d 1044 . . . . 5  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  p  <_  A )
219reeflogd 23652 . . . . 5  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  ( exp `  ( log `  p
) )  =  p )
22 reeflog 23609 . . . . . 6  |-  ( A  e.  RR+  ->  ( exp `  ( log `  A
) )  =  A )
2322adantr 472 . . . . 5  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  ( exp `  ( log `  A
) )  =  A )
2420, 21, 233brtr4d 4426 . . . 4  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  ( exp `  ( log `  p
) )  <_  ( exp `  ( log `  A
) ) )
25 efle 14249 . . . . 5  |-  ( ( ( log `  p
)  e.  RR  /\  ( log `  A )  e.  RR )  -> 
( ( log `  p
)  <_  ( log `  A )  <->  ( exp `  ( log `  p
) )  <_  ( exp `  ( log `  A
) ) ) )
2610, 12, 25syl2anc 673 . . . 4  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  ( ( log `  p )  <_ 
( log `  A
)  <->  ( exp `  ( log `  p ) )  <_  ( exp `  ( log `  A ) ) ) )
2724, 26mpbird 240 . . 3  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  ( log `  p )  <_  ( log `  A ) )
283, 10, 12, 27fsumle 13936 . 2  |-  ( A  e.  RR+  ->  sum_ p  e.  ( ( 0 [,] A )  i^i  Prime ) ( log `  p
)  <_  sum_ p  e.  ( ( 0 [,] A )  i^i  Prime ) ( log `  A
) )
29 chtval 24116 . . 3  |-  ( A  e.  RR  ->  ( theta `  A )  = 
sum_ p  e.  (
( 0 [,] A
)  i^i  Prime ) ( log `  p ) )
301, 29syl 17 . 2  |-  ( A  e.  RR+  ->  ( theta `  A )  =  sum_ p  e.  ( ( 0 [,] A )  i^i 
Prime ) ( log `  p
) )
31 ppival 24133 . . . . 5  |-  ( A  e.  RR  ->  (π `  A )  =  (
# `  ( (
0 [,] A )  i^i  Prime ) ) )
321, 31syl 17 . . . 4  |-  ( A  e.  RR+  ->  (π `  A
)  =  ( # `  ( ( 0 [,] A )  i^i  Prime ) ) )
3332oveq1d 6323 . . 3  |-  ( A  e.  RR+  ->  ( (π `  A )  x.  ( log `  A ) )  =  ( ( # `  ( ( 0 [,] A )  i^i  Prime ) )  x.  ( log `  A ) ) )
3411recnd 9687 . . . 4  |-  ( A  e.  RR+  ->  ( log `  A )  e.  CC )
35 fsumconst 13928 . . . 4  |-  ( ( ( ( 0 [,] A )  i^i  Prime )  e.  Fin  /\  ( log `  A )  e.  CC )  ->  sum_ p  e.  ( ( 0 [,] A )  i^i  Prime ) ( log `  A
)  =  ( (
# `  ( (
0 [,] A )  i^i  Prime ) )  x.  ( log `  A
) ) )
363, 34, 35syl2anc 673 . . 3  |-  ( A  e.  RR+  ->  sum_ p  e.  ( ( 0 [,] A )  i^i  Prime ) ( log `  A
)  =  ( (
# `  ( (
0 [,] A )  i^i  Prime ) )  x.  ( log `  A
) ) )
3733, 36eqtr4d 2508 . 2  |-  ( A  e.  RR+  ->  ( (π `  A )  x.  ( log `  A ) )  =  sum_ p  e.  ( ( 0 [,] A
)  i^i  Prime ) ( log `  A ) )
3828, 30, 373brtr4d 4426 1  |-  ( A  e.  RR+  ->  ( theta `  A )  <_  (
(π `  A )  x.  ( log `  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    i^i cin 3389   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   Fincfn 7587   CCcc 9555   RRcr 9556   0cc0 9557    x. cmul 9562    <_ cle 9694   NNcn 10631   RR+crp 11325   [,]cicc 11663   #chash 12553   sum_csu 13829   expce 14191   Primecprime 14701   logclog 23583   thetaccht 24096  πcppi 24099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-pi 14203  df-dvds 14383  df-prm 14702  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-log 23585  df-cht 24102  df-ppi 24105
This theorem is referenced by:  chtppilim  24392
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