Theorem chtqleppi 16306

Description: Upper bound on the theta function. (Contributed by Mario Carneiro, 22-Sep-2014.)

Assertion

Ref	Expression
chtqleppi	|- ( ( A e. QQ /\ 0 < A ) -> ( theta ` A ) <_ ( (π ` A ) x. ( log ` A ) ) )

Proof of Theorem chtqleppi

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ppiqfi 16254	. . . 4 |- ( A e. QQ -> ( ( 0 [,] A ) i^i Prime ) e. Fin )
2	1	adantr 276	. . 3 |- ( ( A e. QQ /\ 0 < A ) -> ( ( 0 [,] A ) i^i Prime ) e. Fin )
3		simpr 110	. . . . . . 7 |- ( ( ( A e. QQ /\ 0 < A ) /\ p e. ( ( 0 [,] A ) i^i Prime ) ) -> p e. ( ( 0 [,] A ) i^i Prime ) )
4	3	elin2d 3419	. . . . . 6 |- ( ( ( A e. QQ /\ 0 < A ) /\ p e. ( ( 0 [,] A ) i^i Prime ) ) -> p e. Prime )
5		prmnn 12914	. . . . . 6 |- ( p e. Prime -> p e. NN )
6	4, 5	syl 14	. . . . 5 |- ( ( ( A e. QQ /\ 0 < A ) /\ p e. ( ( 0 [,] A ) i^i Prime ) ) -> p e. NN )
7	6	nnrpd 10113	. . . 4 |- ( ( ( A e. QQ /\ 0 < A ) /\ p e. ( ( 0 [,] A ) i^i Prime ) ) -> p e. RR+ )
8	7	relogcld 16124	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ p e. ( ( 0 [,] A ) i^i Prime ) ) -> ( log ` p ) e. RR )
9		qre 10042	. . . . . . 7 |- ( A e. QQ -> A e. RR )
10	9	adantr 276	. . . . . 6 |- ( ( A e. QQ /\ 0 < A ) -> A e. RR )
11		simpr 110	. . . . . 6 |- ( ( A e. QQ /\ 0 < A ) -> 0 < A )
12	10, 11	elrpd 10112	. . . . 5 |- ( ( A e. QQ /\ 0 < A ) -> A e. RR+ )
13	12	relogcld 16124	. . . 4 |- ( ( A e. QQ /\ 0 < A ) -> ( log ` A ) e. RR )
14	13	adantr 276	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ p e. ( ( 0 [,] A ) i^i Prime ) ) -> ( log ` A ) e. RR )
15	3	elin1d 3418	. . . . . . 7 |- ( ( ( A e. QQ /\ 0 < A ) /\ p e. ( ( 0 [,] A ) i^i Prime ) ) -> p e. ( 0 [,] A ) )
16		0re 8333	. . . . . . . . 9 |- 0 e. RR
17		elicc2 10358	. . . . . . . . 9 |- ( ( 0 e. RR /\ A e. RR ) -> ( p e. ( 0 [,] A ) <-> ( p e. RR /\ 0 <_ p /\ p <_ A ) ) )
18	16, 10, 17	sylancr 418	. . . . . . . 8 |- ( ( A e. QQ /\ 0 < A ) -> ( p e. ( 0 [,] A ) <-> ( p e. RR /\ 0 <_ p /\ p <_ A ) ) )
19	18	biimpa 296	. . . . . . 7 |- ( ( ( A e. QQ /\ 0 < A ) /\ p e. ( 0 [,] A ) ) -> ( p e. RR /\ 0 <_ p /\ p <_ A ) )
20	15, 19	syldan 282	. . . . . 6 |- ( ( ( A e. QQ /\ 0 < A ) /\ p e. ( ( 0 [,] A ) i^i Prime ) ) -> ( p e. RR /\ 0 <_ p /\ p <_ A ) )
21	20	simp3d 1042	. . . . 5 |- ( ( ( A e. QQ /\ 0 < A ) /\ p e. ( ( 0 [,] A ) i^i Prime ) ) -> p <_ A )
22	7	reeflogd 16125	. . . . 5 |- ( ( ( A e. QQ /\ 0 < A ) /\ p e. ( ( 0 [,] A ) i^i Prime ) ) -> ( exp ` ( log ` p ) ) = p )
23	12	reeflogd 16125	. . . . . 6 |- ( ( A e. QQ /\ 0 < A ) -> ( exp ` ( log ` A ) ) = A )
24	23	adantr 276	. . . . 5 |- ( ( ( A e. QQ /\ 0 < A ) /\ p e. ( ( 0 [,] A ) i^i Prime ) ) -> ( exp ` ( log ` A ) ) = A )
25	21, 22, 24	3brtr4d 4163	. . . 4 |- ( ( ( A e. QQ /\ 0 < A ) /\ p e. ( ( 0 [,] A ) i^i Prime ) ) -> ( exp ` ( log ` p ) ) <_ ( exp ` ( log ` A ) ) )
26		efle 16014	. . . . 5 |- ( ( ( log ` p ) e. RR /\ ( log ` A ) e. RR ) -> ( ( log ` p ) <_ ( log ` A ) <-> ( exp ` ( log ` p ) ) <_ ( exp ` ( log ` A ) ) ) )
27	8, 14, 26	syl2anc 415	. . . 4 |- ( ( ( A e. QQ /\ 0 < A ) /\ p e. ( ( 0 [,] A ) i^i Prime ) ) -> ( ( log ` p ) <_ ( log ` A ) <-> ( exp ` ( log ` p ) ) <_ ( exp ` ( log ` A ) ) ) )
28	25, 27	mpbird 167	. . 3 |- ( ( ( A e. QQ /\ 0 < A ) /\ p e. ( ( 0 [,] A ) i^i Prime ) ) -> ( log ` p ) <_ ( log ` A ) )
29	2, 8, 14, 28	fsumle 12256	. 2 |- ( ( A e. QQ /\ 0 < A ) -> sum_ p e. ( ( 0 [,] A ) i^i Prime ) ( log ` p ) <_ sum_ p e. ( ( 0 [,] A ) i^i Prime ) ( log ` A ) )
30		chtqval 16257	. . 3 |- ( A e. QQ -> ( theta ` A ) = sum_ p e. ( ( 0 [,] A ) i^i Prime ) ( log ` p ) )
31	30	adantr 276	. 2 |- ( ( A e. QQ /\ 0 < A ) -> ( theta ` A ) = sum_ p e. ( ( 0 [,] A ) i^i Prime ) ( log ` p ) )
32		ppiqval 16260	. . . . 5 |- ( A e. QQ -> (π ` A ) = (♯ ` ( ( 0 [,] A ) i^i Prime ) ) )
33	32	adantr 276	. . . 4 |- ( ( A e. QQ /\ 0 < A ) -> (π ` A ) = (♯ ` ( ( 0 [,] A ) i^i Prime ) ) )
34	33	oveq1d 6103	. . 3 |- ( ( A e. QQ /\ 0 < A ) -> ( (π ` A ) x. ( log ` A ) ) = ( (♯ ` ( ( 0 [,] A ) i^i Prime ) ) x. ( log ` A ) ) )
35		relogcl 16103	. . . . . 6 |- ( A e. RR+ -> ( log ` A ) e. RR )
36	35	recnd 8361	. . . . 5 |- ( A e. RR+ -> ( log ` A ) e. CC )
37	12, 36	syl 14	. . . 4 |- ( ( A e. QQ /\ 0 < A ) -> ( log ` A ) e. CC )
38		fsumconst 12247	. . . 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 ) ) )
39	2, 37, 38	syl2anc 415	. . 3 |- ( ( A e. QQ /\ 0 < A ) -> sum_ p e. ( ( 0 [,] A ) i^i Prime ) ( log ` A ) = ( (♯ ` ( ( 0 [,] A ) i^i Prime ) ) x. ( log ` A ) ) )
40	34, 39	eqtr4d 2274	. 2 |- ( ( A e. QQ /\ 0 < A ) -> ( (π ` A ) x. ( log ` A ) ) = sum_ p e. ( ( 0 [,] A ) i^i Prime ) ( log ` A ) )
41	29, 31, 40	3brtr4d 4163	1 |- ( ( A e. QQ /\ 0 < A ) -> ( theta ` A ) <_ ( (π ` A ) x. ( log ` A ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   i^i cin 3219   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   Fincfn 7027   CCcc 8184   RRcr 8185   0cc0 8186   x. cmul 8191   < clt 8367   <_ cle 8368   NNcn 9314   QQcq 10036   RR+crp 10072   [,]cicc 10311  ♯chash 11237   sum_csu 12145   expce 12435   Primecprime 12911   logclog 16097   thetaccht 16245  πcppi 16246

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305  ax-caucvg 8306  ax-pre-suploc 8307  ax-addf 8308  ax-mulf 8309

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-disj 4107  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-of 6305  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6811  df-map 6928  df-pm 6929  df-en 7028  df-dom 7029  df-fin 7030  df-sup 7331  df-inf 7332  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-n0 9576  df-z 9657  df-uz 9939  df-q 10037  df-rp 10073  df-xneg 10192  df-xadd 10193  df-ioo 10312  df-ico 10314  df-icc 10315  df-fz 10430  df-fzo 10568  df-fl 10723  df-mod 10782  df-seqfrec 10907  df-exp 10998  df-fac 11187  df-bc 11209  df-ihash 11238  df-shft 11603  df-cj 11630  df-re 11631  df-im 11632  df-rsqrt 11787  df-abs 11788  df-clim 12071  df-sumdc 12146  df-ef 12441  df-e 12442  df-dvds 12581  df-prm 12912  df-rest 13655  df-topgen 13674  df-psmet 14990  df-xmet 14991  df-met 14992  df-bl 14993  df-mopn 14994  df-top 15236  df-topon 15249  df-bases 15281  df-ntr 15334  df-cn 15426  df-cnp 15427  df-tx 15491  df-cncf 15809  df-limced 15894  df-dvap 15895  df-relog 16099  df-cht 16248  df-ppi 16249

This theorem is used by: (None)