Theorem prmorcht 24904

Description: Relate the primorial (product of the first n primes) to the Chebyshev function. (Contributed by Mario Carneiro, 22-Sep-2014.)

Hypothesis

Ref	Expression
prmorcht.1	|- F = ( n e. NN |-> if ( n e. Prime , n , 1 ) )

Assertion

Ref	Expression
prmorcht	|- ( A e. NN -> ( exp ` ( theta ` A ) ) = ( seq 1 ( x. , F ) ` A ) )

Proof of Theorem prmorcht

Dummy variables k p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnre 11027	. . . . . . 7 |- ( A e. NN -> A e. RR )
2		chtval 24836	. . . . . . 7 |- ( A e. RR -> ( theta ` A ) = sum_ k e. ( ( 0 [,] A ) i^i Prime ) ( log ` k ) )
3	1, 2	syl 17	. . . . . 6 |- ( A e. NN -> ( theta ` A ) = sum_ k e. ( ( 0 [,] A ) i^i Prime ) ( log ` k ) )
4		2eluzge1 11734	. . . . . . . . . 10 |- 2 e. ( ZZ>= ` 1 )
5		ppisval2 24831	. . . . . . . . . 10 |- ( ( A e. RR /\ 2 e. ( ZZ>= ` 1 ) ) -> ( ( 0 [,] A ) i^i Prime ) = ( ( 1 ... ( |_ ` A ) ) i^i Prime ) )
6	1, 4, 5	sylancl 694	. . . . . . . . 9 |- ( A e. NN -> ( ( 0 [,] A ) i^i Prime ) = ( ( 1 ... ( |_ ` A ) ) i^i Prime ) )
7		nnz 11399	. . . . . . . . . . . 12 |- ( A e. NN -> A e. ZZ )
8		flid 12609	. . . . . . . . . . . 12 |- ( A e. ZZ -> ( |_ ` A ) = A )
9	7, 8	syl 17	. . . . . . . . . . 11 |- ( A e. NN -> ( |_ ` A ) = A )
10	9	oveq2d 6666	. . . . . . . . . 10 |- ( A e. NN -> ( 1 ... ( |_ ` A ) ) = ( 1 ... A ) )
11	10	ineq1d 3813	. . . . . . . . 9 |- ( A e. NN -> ( ( 1 ... ( |_ ` A ) ) i^i Prime ) = ( ( 1 ... A ) i^i Prime ) )
12	6, 11	eqtrd 2656	. . . . . . . 8 |- ( A e. NN -> ( ( 0 [,] A ) i^i Prime ) = ( ( 1 ... A ) i^i Prime ) )
13	12	sumeq1d 14431	. . . . . . 7 |- ( A e. NN -> sum_ k e. ( ( 0 [,] A ) i^i Prime ) ( log ` k ) = sum_ k e. ( ( 1 ... A ) i^i Prime ) ( log ` k ) )
14		inss1 3833	. . . . . . . 8 |- ( ( 1 ... A ) i^i Prime ) C_ ( 1 ... A )
15	14	sseli 3599	. . . . . . . . . 10 |- ( k e. ( ( 1 ... A ) i^i Prime ) -> k e. ( 1 ... A ) )
16		elfznn 12370	. . . . . . . . . . . . . 14 |- ( k e. ( 1 ... A ) -> k e. NN )
17	16	adantl 482	. . . . . . . . . . . . 13 |- ( ( A e. NN /\ k e. ( 1 ... A ) ) -> k e. NN )
18	17	nnrpd 11870	. . . . . . . . . . . 12 |- ( ( A e. NN /\ k e. ( 1 ... A ) ) -> k e. RR+ )
19	18	relogcld 24369	. . . . . . . . . . 11 |- ( ( A e. NN /\ k e. ( 1 ... A ) ) -> ( log ` k ) e. RR )
20	19	recnd 10068	. . . . . . . . . 10 |- ( ( A e. NN /\ k e. ( 1 ... A ) ) -> ( log ` k ) e. CC )
21	15, 20	sylan2 491	. . . . . . . . 9 |- ( ( A e. NN /\ k e. ( ( 1 ... A ) i^i Prime ) ) -> ( log ` k ) e. CC )
22	21	ralrimiva 2966	. . . . . . . 8 |- ( A e. NN -> A. k e. ( ( 1 ... A ) i^i Prime ) ( log ` k ) e. CC )
23		fzfi 12771	. . . . . . . . . 10 |- ( 1 ... A ) e. Fin
24	23	olci 406	. . . . . . . . 9 |- ( ( 1 ... A ) C_ ( ZZ>= ` 1 ) \/ ( 1 ... A ) e. Fin )
25		sumss2 14457	. . . . . . . . 9 |- ( ( ( ( ( 1 ... A ) i^i Prime ) C_ ( 1 ... A ) /\ A. k e. ( ( 1 ... A ) i^i Prime ) ( log ` k ) e. CC ) /\ ( ( 1 ... A ) C_ ( ZZ>= ` 1 ) \/ ( 1 ... A ) e. Fin ) ) -> sum_ k e. ( ( 1 ... A ) i^i Prime ) ( log ` k ) = sum_ k e. ( 1 ... A ) if ( k e. ( ( 1 ... A ) i^i Prime ) , ( log ` k ) , 0 ) )
26	24, 25	mpan2 707	. . . . . . . 8 |- ( ( ( ( 1 ... A ) i^i Prime ) C_ ( 1 ... A ) /\ A. k e. ( ( 1 ... A ) i^i Prime ) ( log ` k ) e. CC ) -> sum_ k e. ( ( 1 ... A ) i^i Prime ) ( log ` k ) = sum_ k e. ( 1 ... A ) if ( k e. ( ( 1 ... A ) i^i Prime ) , ( log ` k ) , 0 ) )
27	14, 22, 26	sylancr 695	. . . . . . 7 |- ( A e. NN -> sum_ k e. ( ( 1 ... A ) i^i Prime ) ( log ` k ) = sum_ k e. ( 1 ... A ) if ( k e. ( ( 1 ... A ) i^i Prime ) , ( log ` k ) , 0 ) )
28	13, 27	eqtrd 2656	. . . . . 6 |- ( A e. NN -> sum_ k e. ( ( 0 [,] A ) i^i Prime ) ( log ` k ) = sum_ k e. ( 1 ... A ) if ( k e. ( ( 1 ... A ) i^i Prime ) , ( log ` k ) , 0 ) )
29	3, 28	eqtrd 2656	. . . . 5 |- ( A e. NN -> ( theta ` A ) = sum_ k e. ( 1 ... A ) if ( k e. ( ( 1 ... A ) i^i Prime ) , ( log ` k ) , 0 ) )
30		elin 3796	. . . . . . . 8 |- ( k e. ( ( 1 ... A ) i^i Prime ) <-> ( k e. ( 1 ... A ) /\ k e. Prime ) )
31	30	baibr 945	. . . . . . 7 |- ( k e. ( 1 ... A ) -> ( k e. Prime <-> k e. ( ( 1 ... A ) i^i Prime ) ) )
32	31	ifbid 4108	. . . . . 6 |- ( k e. ( 1 ... A ) -> if ( k e. Prime , ( log ` k ) , 0 ) = if ( k e. ( ( 1 ... A ) i^i Prime ) , ( log ` k ) , 0 ) )
33	32	sumeq2i 14429	. . . . 5 |- sum_ k e. ( 1 ... A ) if ( k e. Prime , ( log ` k ) , 0 ) = sum_ k e. ( 1 ... A ) if ( k e. ( ( 1 ... A ) i^i Prime ) , ( log ` k ) , 0 )
34	29, 33	syl6eqr 2674	. . . 4 |- ( A e. NN -> ( theta ` A ) = sum_ k e. ( 1 ... A ) if ( k e. Prime , ( log ` k ) , 0 ) )
35		eleq1 2689	. . . . . . . 8 |- ( n = k -> ( n e. Prime <-> k e. Prime ) )
36		fveq2 6191	. . . . . . . 8 |- ( n = k -> ( log ` n ) = ( log ` k ) )
37	35, 36	ifbieq1d 4109	. . . . . . 7 |- ( n = k -> if ( n e. Prime , ( log ` n ) , 0 ) = if ( k e. Prime , ( log ` k ) , 0 ) )
38		eqid 2622	. . . . . . 7 |- ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) = ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) )
39		fvex 6201	. . . . . . . 8 |- ( log ` k ) e. _V
40		0cn 10032	. . . . . . . . 9 |- 0 e. CC
41	40	elexi 3213	. . . . . . . 8 |- 0 e. _V
42	39, 41	ifex 4156	. . . . . . 7 |- if ( k e. Prime , ( log ` k ) , 0 ) e. _V
43	37, 38, 42	fvmpt 6282	. . . . . 6 |- ( k e. NN -> ( ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ` k ) = if ( k e. Prime , ( log ` k ) , 0 ) )
44	17, 43	syl 17	. . . . 5 |- ( ( A e. NN /\ k e. ( 1 ... A ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ` k ) = if ( k e. Prime , ( log ` k ) , 0 ) )
45		elnnuz 11724	. . . . . 6 |- ( A e. NN <-> A e. ( ZZ>= ` 1 ) )
46	45	biimpi 206	. . . . 5 |- ( A e. NN -> A e. ( ZZ>= ` 1 ) )
47		ifcl 4130	. . . . . 6 |- ( ( ( log ` k ) e. CC /\ 0 e. CC ) -> if ( k e. Prime , ( log ` k ) , 0 ) e. CC )
48	20, 40, 47	sylancl 694	. . . . 5 |- ( ( A e. NN /\ k e. ( 1 ... A ) ) -> if ( k e. Prime , ( log ` k ) , 0 ) e. CC )
49	44, 46, 48	fsumser 14461	. . . 4 |- ( A e. NN -> sum_ k e. ( 1 ... A ) if ( k e. Prime , ( log ` k ) , 0 ) = ( seq 1 ( + , ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ) ` A ) )
50	34, 49	eqtrd 2656	. . 3 |- ( A e. NN -> ( theta ` A ) = ( seq 1 ( + , ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ) ` A ) )
51	50	fveq2d 6195	. 2 |- ( A e. NN -> ( exp ` ( theta ` A ) ) = ( exp ` ( seq 1 ( + , ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ) ` A ) ) )
52		addcl 10018	. . . 4 |- ( ( k e. CC /\ p e. CC ) -> ( k + p ) e. CC )
53	52	adantl 482	. . 3 |- ( ( A e. NN /\ ( k e. CC /\ p e. CC ) ) -> ( k + p ) e. CC )
54	44, 48	eqeltrd 2701	. . 3 |- ( ( A e. NN /\ k e. ( 1 ... A ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ` k ) e. CC )
55		efadd 14824	. . . 4 |- ( ( k e. CC /\ p e. CC ) -> ( exp ` ( k + p ) ) = ( ( exp ` k ) x. ( exp ` p ) ) )
56	55	adantl 482	. . 3 |- ( ( A e. NN /\ ( k e. CC /\ p e. CC ) ) -> ( exp ` ( k + p ) ) = ( ( exp ` k ) x. ( exp ` p ) ) )
57		1nn 11031	. . . . . . 7 |- 1 e. NN
58		ifcl 4130	. . . . . . 7 |- ( ( k e. NN /\ 1 e. NN ) -> if ( k e. Prime , k , 1 ) e. NN )
59	17, 57, 58	sylancl 694	. . . . . 6 |- ( ( A e. NN /\ k e. ( 1 ... A ) ) -> if ( k e. Prime , k , 1 ) e. NN )
60	59	nnrpd 11870	. . . . 5 |- ( ( A e. NN /\ k e. ( 1 ... A ) ) -> if ( k e. Prime , k , 1 ) e. RR+ )
61	60	reeflogd 24370	. . . 4 |- ( ( A e. NN /\ k e. ( 1 ... A ) ) -> ( exp ` ( log ` if ( k e. Prime , k , 1 ) ) ) = if ( k e. Prime , k , 1 ) )
62		fvif 6204	. . . . . . 7 |- ( log ` if ( k e. Prime , k , 1 ) ) = if ( k e. Prime , ( log ` k ) , ( log ` 1 ) )
63		log1 24332	. . . . . . . 8 |- ( log ` 1 ) = 0
64		ifeq2 4091	. . . . . . . 8 |- ( ( log ` 1 ) = 0 -> if ( k e. Prime , ( log ` k ) , ( log ` 1 ) ) = if ( k e. Prime , ( log ` k ) , 0 ) )
65	63, 64	ax-mp 5	. . . . . . 7 |- if ( k e. Prime , ( log ` k ) , ( log ` 1 ) ) = if ( k e. Prime , ( log ` k ) , 0 )
66	62, 65	eqtri 2644	. . . . . 6 |- ( log ` if ( k e. Prime , k , 1 ) ) = if ( k e. Prime , ( log ` k ) , 0 )
67	44, 66	syl6eqr 2674	. . . . 5 |- ( ( A e. NN /\ k e. ( 1 ... A ) ) -> ( ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ` k ) = ( log ` if ( k e. Prime , k , 1 ) ) )
68	67	fveq2d 6195	. . . 4 |- ( ( A e. NN /\ k e. ( 1 ... A ) ) -> ( exp ` ( ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ` k ) ) = ( exp ` ( log ` if ( k e. Prime , k , 1 ) ) ) )
69		id 22	. . . . . . 7 |- ( n = k -> n = k )
70	35, 69	ifbieq1d 4109	. . . . . 6 |- ( n = k -> if ( n e. Prime , n , 1 ) = if ( k e. Prime , k , 1 ) )
71		prmorcht.1	. . . . . 6 |- F = ( n e. NN |-> if ( n e. Prime , n , 1 ) )
72		vex 3203	. . . . . . 7 |- k e. _V
73	57	elexi 3213	. . . . . . 7 |- 1 e. _V
74	72, 73	ifex 4156	. . . . . 6 |- if ( k e. Prime , k , 1 ) e. _V
75	70, 71, 74	fvmpt 6282	. . . . 5 |- ( k e. NN -> ( F ` k ) = if ( k e. Prime , k , 1 ) )
76	17, 75	syl 17	. . . 4 |- ( ( A e. NN /\ k e. ( 1 ... A ) ) -> ( F ` k ) = if ( k e. Prime , k , 1 ) )
77	61, 68, 76	3eqtr4d 2666	. . 3 |- ( ( A e. NN /\ k e. ( 1 ... A ) ) -> ( exp ` ( ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ` k ) ) = ( F ` k ) )
78	53, 54, 46, 56, 77	seqhomo 12848	. 2 |- ( A e. NN -> ( exp ` ( seq 1 ( + , ( n e. NN |-> if ( n e. Prime , ( log ` n ) , 0 ) ) ) ` A ) ) = ( seq 1 ( x. , F ) ` A ) )
79	51, 78	eqtrd 2656	1 |- ( A e. NN -> ( exp ` ( theta ` A ) ) = ( seq 1 ( x. , F ) ` A ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   i^i cin 3573   C_ wss 3574   ifcif 4086   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   NNcn 11020   2c2 11070   ZZcz 11377   ZZ>=cuz 11687   [,]cicc 12178   ...cfz 12326   |_cfl 12591   seqcseq 12801   sum_csu 14416   expce 14792   Primecprime 15385   logclog 24301   thetaccht 24817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-dvds 14984  df-prm 15386  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-cht 24823

This theorem is referenced by:  chtublem  24936  bposlem6  25014