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Theorem chtleppi 20862
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 10551 . . . 4  |-  ( A  e.  RR+  ->  A  e.  RR )
2 ppifi 20756 . . . 4  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  e. 
Fin )
31, 2syl 16 . . 3  |-  ( A  e.  RR+  ->  ( ( 0 [,] A )  i^i  Prime )  e.  Fin )
4 inss2 3506 . . . . . . 7  |-  ( ( 0 [,] A )  i^i  Prime )  C_  Prime
5 simpr 448 . . . . . . 7  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  p  e.  ( ( 0 [,] A )  i^i  Prime ) )
64, 5sseldi 3290 . . . . . 6  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  p  e.  Prime )
7 prmnn 13010 . . . . . 6  |-  ( p  e.  Prime  ->  p  e.  NN )
86, 7syl 16 . . . . 5  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  p  e.  NN )
98nnrpd 10580 . . . 4  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  p  e.  RR+ )
109relogcld 20386 . . 3  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  ( log `  p )  e.  RR )
11 relogcl 20341 . . . 4  |-  ( A  e.  RR+  ->  ( log `  A )  e.  RR )
1211adantr 452 . . 3  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  ( log `  A )  e.  RR )
13 inss1 3505 . . . . . . . 8  |-  ( ( 0 [,] A )  i^i  Prime )  C_  (
0 [,] A )
1413, 5sseldi 3290 . . . . . . 7  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  p  e.  ( 0 [,] A
) )
15 0re 9025 . . . . . . . . 9  |-  0  e.  RR
16 elicc2 10908 . . . . . . . . 9  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( p  e.  ( 0 [,] A )  <-> 
( p  e.  RR  /\  0  <_  p  /\  p  <_  A ) ) )
1715, 1, 16sylancr 645 . . . . . . . 8  |-  ( A  e.  RR+  ->  ( p  e.  ( 0 [,] A )  <->  ( p  e.  RR  /\  0  <_  p  /\  p  <_  A
) ) )
1817biimpa 471 . . . . . . 7  |-  ( ( A  e.  RR+  /\  p  e.  ( 0 [,] A
) )  ->  (
p  e.  RR  /\  0  <_  p  /\  p  <_  A ) )
1914, 18syldan 457 . . . . . 6  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  ( p  e.  RR  /\  0  <_  p  /\  p  <_  A
) )
2019simp3d 971 . . . . 5  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  p  <_  A )
219reeflogd 20387 . . . . 5  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  ( exp `  ( log `  p
) )  =  p )
22 reeflog 20343 . . . . . 6  |-  ( A  e.  RR+  ->  ( exp `  ( log `  A
) )  =  A )
2322adantr 452 . . . . 5  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  ( exp `  ( log `  A
) )  =  A )
2420, 21, 233brtr4d 4184 . . . 4  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  ( exp `  ( log `  p
) )  <_  ( exp `  ( log `  A
) ) )
25 efle 12647 . . . . 5  |-  ( ( ( log `  p
)  e.  RR  /\  ( log `  A )  e.  RR )  -> 
( ( log `  p
)  <_  ( log `  A )  <->  ( exp `  ( log `  p
) )  <_  ( exp `  ( log `  A
) ) ) )
2610, 12, 25syl2anc 643 . . . 4  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  ( ( log `  p )  <_ 
( log `  A
)  <->  ( exp `  ( log `  p ) )  <_  ( exp `  ( log `  A ) ) ) )
2724, 26mpbird 224 . . 3  |-  ( ( A  e.  RR+  /\  p  e.  ( ( 0 [,] A )  i^i  Prime ) )  ->  ( log `  p )  <_  ( log `  A ) )
283, 10, 12, 27fsumle 12506 . 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 20761 . . 3  |-  ( A  e.  RR  ->  ( theta `  A )  = 
sum_ p  e.  (
( 0 [,] A
)  i^i  Prime ) ( log `  p ) )
301, 29syl 16 . 2  |-  ( A  e.  RR+  ->  ( theta `  A )  =  sum_ p  e.  ( ( 0 [,] A )  i^i 
Prime ) ( log `  p
) )
31 ppival 20778 . . . . 5  |-  ( A  e.  RR  ->  (π `  A )  =  (
# `  ( (
0 [,] A )  i^i  Prime ) ) )
321, 31syl 16 . . . 4  |-  ( A  e.  RR+  ->  (π `  A
)  =  ( # `  ( ( 0 [,] A )  i^i  Prime ) ) )
3332oveq1d 6036 . . 3  |-  ( A  e.  RR+  ->  ( (π `  A )  x.  ( log `  A ) )  =  ( ( # `  ( ( 0 [,] A )  i^i  Prime ) )  x.  ( log `  A ) ) )
3411recnd 9048 . . . 4  |-  ( A  e.  RR+  ->  ( log `  A )  e.  CC )
35 fsumconst 12501 . . . 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 643 . . 3  |-  ( A  e.  RR+  ->  sum_ p  e.  ( ( 0 [,] A )  i^i  Prime ) ( log `  A
)  =  ( (
# `  ( (
0 [,] A )  i^i  Prime ) )  x.  ( log `  A
) ) )
3733, 36eqtr4d 2423 . 2  |-  ( A  e.  RR+  ->  ( (π `  A )  x.  ( log `  A ) )  =  sum_ p  e.  ( ( 0 [,] A
)  i^i  Prime ) ( log `  A ) )
3828, 30, 373brtr4d 4184 1  |-  ( A  e.  RR+  ->  ( theta `  A )  <_  (
(π `  A )  x.  ( log `  A
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    i^i cin 3263   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   Fincfn 7046   CCcc 8922   RRcr 8923   0cc0 8924    x. cmul 8929    <_ cle 9055   NNcn 9933   RR+crp 10545   [,]cicc 10852   #chash 11546   sum_csu 12407   expce 12592   Primecprime 13007   logclog 20320   thetaccht 20741  πcppi 20744
This theorem is referenced by:  chtppilim  21037
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003  ax-mulf 9004
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-ixp 7001  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-sup 7382  df-oi 7413  df-card 7760  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-ioo 10853  df-ioc 10854  df-ico 10855  df-icc 10856  df-fz 10977  df-fzo 11067  df-fl 11130  df-mod 11179  df-seq 11252  df-exp 11311  df-fac 11495  df-bc 11522  df-hash 11547  df-shft 11810  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-limsup 12193  df-clim 12210  df-rlim 12211  df-sum 12408  df-ef 12598  df-sin 12600  df-cos 12601  df-pi 12603  df-dvds 12781  df-prm 13008  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-starv 13472  df-sca 13473  df-vsca 13474  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-hom 13481  df-cco 13482  df-rest 13578  df-topn 13579  df-topgen 13595  df-pt 13596  df-prds 13599  df-xrs 13654  df-0g 13655  df-gsum 13656  df-qtop 13661  df-imas 13662  df-xps 13664  df-mre 13739  df-mrc 13740  df-acs 13742  df-mnd 14618  df-submnd 14667  df-mulg 14743  df-cntz 15044  df-cmn 15342  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-fbas 16624  df-fg 16625  df-cnfld 16628  df-top 16887  df-bases 16889  df-topon 16890  df-topsp 16891  df-cld 17007  df-ntr 17008  df-cls 17009  df-nei 17086  df-lp 17124  df-perf 17125  df-cn 17214  df-cnp 17215  df-haus 17302  df-tx 17516  df-hmeo 17709  df-fil 17800  df-fm 17892  df-flim 17893  df-flf 17894  df-xms 18260  df-ms 18261  df-tms 18262  df-cncf 18780  df-limc 19621  df-dv 19622  df-log 20322  df-cht 20747  df-ppi 20750
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