Theorem chtvalz 30707

Description: Value of the Chebyshev function for integers. (Contributed by Thierry Arnoux, 28-Dec-2021.)

Assertion

Ref	Expression
chtvalz	|- ( N e. ZZ -> ( theta ` N ) = sum_ n e. ( ( 1 ... N ) i^i Prime ) ( log ` n ) )

Distinct variable group:   n, N

Proof of Theorem chtvalz

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zre 11381	. . 3 |- ( N e. ZZ -> N e. RR )
2		chtval 24836	. . 3 |- ( N e. RR -> ( theta ` N ) = sum_ n e. ( ( 0 [,] N ) i^i Prime ) ( log ` n ) )
3	1, 2	syl 17	. 2 |- ( N e. ZZ -> ( theta ` N ) = sum_ n e. ( ( 0 [,] N ) i^i Prime ) ( log ` n ) )
4		nnz 11399	. . . . . . 7 |- ( N e. NN -> N e. ZZ )
5		ppisval 24830	. . . . . . . . 9 |- ( N e. RR -> ( ( 0 [,] N ) i^i Prime ) = ( ( 2 ... ( |_ ` N ) ) i^i Prime ) )
6	1, 5	syl 17	. . . . . . . 8 |- ( N e. ZZ -> ( ( 0 [,] N ) i^i Prime ) = ( ( 2 ... ( |_ ` N ) ) i^i Prime ) )
7		flid 12609	. . . . . . . . . 10 |- ( N e. ZZ -> ( |_ ` N ) = N )
8	7	oveq2d 6666	. . . . . . . . 9 |- ( N e. ZZ -> ( 2 ... ( |_ ` N ) ) = ( 2 ... N ) )
9	8	ineq1d 3813	. . . . . . . 8 |- ( N e. ZZ -> ( ( 2 ... ( |_ ` N ) ) i^i Prime ) = ( ( 2 ... N ) i^i Prime ) )
10	6, 9	eqtrd 2656	. . . . . . 7 |- ( N e. ZZ -> ( ( 0 [,] N ) i^i Prime ) = ( ( 2 ... N ) i^i Prime ) )
11	4, 10	syl 17	. . . . . 6 |- ( N e. NN -> ( ( 0 [,] N ) i^i Prime ) = ( ( 2 ... N ) i^i Prime ) )
12		2nn 11185	. . . . . . . . . . . . 13 |- 2 e. NN
13		nnuz 11723	. . . . . . . . . . . . 13 |- NN = ( ZZ>= ` 1 )
14	12, 13	eleqtri 2699	. . . . . . . . . . . 12 |- 2 e. ( ZZ>= ` 1 )
15		fzss1 12380	. . . . . . . . . . . 12 |- ( 2 e. ( ZZ>= ` 1 ) -> ( 2 ... N ) C_ ( 1 ... N ) )
16	14, 15	ax-mp 5	. . . . . . . . . . 11 |- ( 2 ... N ) C_ ( 1 ... N )
17		ssdif0 3942	. . . . . . . . . . 11 |- ( ( 2 ... N ) C_ ( 1 ... N ) <-> ( ( 2 ... N ) \ ( 1 ... N ) ) = (/) )
18	16, 17	mpbi 220	. . . . . . . . . 10 |- ( ( 2 ... N ) \ ( 1 ... N ) ) = (/)
19	18	ineq1i 3810	. . . . . . . . 9 |- ( ( ( 2 ... N ) \ ( 1 ... N ) ) i^i Prime ) = ( (/) i^i Prime )
20		0in 3969	. . . . . . . . 9 |- ( (/) i^i Prime ) = (/)
21	19, 20	eqtri 2644	. . . . . . . 8 |- ( ( ( 2 ... N ) \ ( 1 ... N ) ) i^i Prime ) = (/)
22	21	a1i 11	. . . . . . 7 |- ( N e. NN -> ( ( ( 2 ... N ) \ ( 1 ... N ) ) i^i Prime ) = (/) )
23	13	eleq2i 2693	. . . . . . . . . . . . 13 |- ( N e. NN <-> N e. ( ZZ>= ` 1 ) )
24		fzpred 12389	. . . . . . . . . . . . 13 |- ( N e. ( ZZ>= ` 1 ) -> ( 1 ... N ) = ( { 1 } u. ( ( 1 + 1 ) ... N ) ) )
25	23, 24	sylbi 207	. . . . . . . . . . . 12 |- ( N e. NN -> ( 1 ... N ) = ( { 1 } u. ( ( 1 + 1 ) ... N ) ) )
26	25	eqcomd 2628	. . . . . . . . . . 11 |- ( N e. NN -> ( { 1 } u. ( ( 1 + 1 ) ... N ) ) = ( 1 ... N ) )
27		1p1e2 11134	. . . . . . . . . . . . 13 |- ( 1 + 1 ) = 2
28	27	oveq1i 6660	. . . . . . . . . . . 12 |- ( ( 1 + 1 ) ... N ) = ( 2 ... N )
29	28	a1i 11	. . . . . . . . . . 11 |- ( N e. NN -> ( ( 1 + 1 ) ... N ) = ( 2 ... N ) )
30	26, 29	difeq12d 3729	. . . . . . . . . 10 |- ( N e. NN -> ( ( { 1 } u. ( ( 1 + 1 ) ... N ) ) \ ( ( 1 + 1 ) ... N ) ) = ( ( 1 ... N ) \ ( 2 ... N ) ) )
31		difun2 4048	. . . . . . . . . . 11 |- ( ( { 1 } u. ( ( 1 + 1 ) ... N ) ) \ ( ( 1 + 1 ) ... N ) ) = ( { 1 } \ ( ( 1 + 1 ) ... N ) )
32		fzpreddisj 12390	. . . . . . . . . . . . 13 |- ( N e. ( ZZ>= ` 1 ) -> ( { 1 } i^i ( ( 1 + 1 ) ... N ) ) = (/) )
33	23, 32	sylbi 207	. . . . . . . . . . . 12 |- ( N e. NN -> ( { 1 } i^i ( ( 1 + 1 ) ... N ) ) = (/) )
34		disjdif2 4047	. . . . . . . . . . . 12 |- ( ( { 1 } i^i ( ( 1 + 1 ) ... N ) ) = (/) -> ( { 1 } \ ( ( 1 + 1 ) ... N ) ) = { 1 } )
35	33, 34	syl 17	. . . . . . . . . . 11 |- ( N e. NN -> ( { 1 } \ ( ( 1 + 1 ) ... N ) ) = { 1 } )
36	31, 35	syl5eq 2668	. . . . . . . . . 10 |- ( N e. NN -> ( ( { 1 } u. ( ( 1 + 1 ) ... N ) ) \ ( ( 1 + 1 ) ... N ) ) = { 1 } )
37	30, 36	eqtr3d 2658	. . . . . . . . 9 |- ( N e. NN -> ( ( 1 ... N ) \ ( 2 ... N ) ) = { 1 } )
38	37	ineq1d 3813	. . . . . . . 8 |- ( N e. NN -> ( ( ( 1 ... N ) \ ( 2 ... N ) ) i^i Prime ) = ( { 1 } i^i Prime ) )
39		incom 3805	. . . . . . . . 9 |- ( Prime i^i { 1 } ) = ( { 1 } i^i Prime )
40		1nprm 15392	. . . . . . . . . 10 |- -. 1 e. Prime
41		disjsn 4246	. . . . . . . . . 10 |- ( ( Prime i^i { 1 } ) = (/) <-> -. 1 e. Prime )
42	40, 41	mpbir 221	. . . . . . . . 9 |- ( Prime i^i { 1 } ) = (/)
43	39, 42	eqtr3i 2646	. . . . . . . 8 |- ( { 1 } i^i Prime ) = (/)
44	38, 43	syl6eq 2672	. . . . . . 7 |- ( N e. NN -> ( ( ( 1 ... N ) \ ( 2 ... N ) ) i^i Prime ) = (/) )
45		difininv 29354	. . . . . . 7 |- ( ( ( ( ( 2 ... N ) \ ( 1 ... N ) ) i^i Prime ) = (/) /\ ( ( ( 1 ... N ) \ ( 2 ... N ) ) i^i Prime ) = (/) ) -> ( ( 2 ... N ) i^i Prime ) = ( ( 1 ... N ) i^i Prime ) )
46	22, 44, 45	syl2anc 693	. . . . . 6 |- ( N e. NN -> ( ( 2 ... N ) i^i Prime ) = ( ( 1 ... N ) i^i Prime ) )
47	11, 46	eqtrd 2656	. . . . 5 |- ( N e. NN -> ( ( 0 [,] N ) i^i Prime ) = ( ( 1 ... N ) i^i Prime ) )
48	47	adantl 482	. . . 4 |- ( ( N e. ZZ /\ N e. NN ) -> ( ( 0 [,] N ) i^i Prime ) = ( ( 1 ... N ) i^i Prime ) )
49		znnnlt1 11404	. . . . . 6 |- ( N e. ZZ -> ( -. N e. NN <-> N < 1 ) )
50	49	biimpa 501	. . . . 5 |- ( ( N e. ZZ /\ -. N e. NN ) -> N < 1 )
51		incom 3805	. . . . . . 7 |- ( ( 0 [,] N ) i^i Prime ) = ( Prime i^i ( 0 [,] N ) )
52		isprm3 15396	. . . . . . . . . . 11 |- ( n e. Prime <-> ( n e. ( ZZ>= ` 2 ) /\ A. i e. ( 2 ... ( n - 1 ) ) -. i || n ) )
53	52	simplbi 476	. . . . . . . . . 10 |- ( n e. Prime -> n e. ( ZZ>= ` 2 ) )
54	53	ssriv 3607	. . . . . . . . 9 |- Prime C_ ( ZZ>= ` 2 )
55	12	nnzi 11401	. . . . . . . . . 10 |- 2 e. ZZ
56		uzssico 29546	. . . . . . . . . 10 |- ( 2 e. ZZ -> ( ZZ>= ` 2 ) C_ ( 2 [,) +oo ) )
57	55, 56	ax-mp 5	. . . . . . . . 9 |- ( ZZ>= ` 2 ) C_ ( 2 [,) +oo )
58	54, 57	sstri 3612	. . . . . . . 8 |- Prime C_ ( 2 [,) +oo )
59		incom 3805	. . . . . . . . 9 |- ( ( 0 [,] N ) i^i ( 2 [,) +oo ) ) = ( ( 2 [,) +oo ) i^i ( 0 [,] N ) )
60		0xr 10086	. . . . . . . . . . . 12 |- 0 e. RR*
61	60	a1i 11	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ N < 1 ) -> 0 e. RR* )
62	12	nnrei 11029	. . . . . . . . . . . . 13 |- 2 e. RR
63	62	rexri 10097	. . . . . . . . . . . 12 |- 2 e. RR*
64	63	a1i 11	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ N < 1 ) -> 2 e. RR* )
65		0le0 11110	. . . . . . . . . . . 12 |- 0 <_ 0
66	65	a1i 11	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ N < 1 ) -> 0 <_ 0 )
67	1	adantr 481	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ N < 1 ) -> N e. RR )
68		1red 10055	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ N < 1 ) -> 1 e. RR )
69	62	a1i 11	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ N < 1 ) -> 2 e. RR )
70		simpr 477	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ N < 1 ) -> N < 1 )
71		1lt2 11194	. . . . . . . . . . . . 13 |- 1 < 2
72	71	a1i 11	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ N < 1 ) -> 1 < 2 )
73	67, 68, 69, 70, 72	lttrd 10198	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ N < 1 ) -> N < 2 )
74		iccssico 12245	. . . . . . . . . . 11 |- ( ( ( 0 e. RR* /\ 2 e. RR* ) /\ ( 0 <_ 0 /\ N < 2 ) ) -> ( 0 [,] N ) C_ ( 0 [,) 2 ) )
75	61, 64, 66, 73, 74	syl22anc 1327	. . . . . . . . . 10 |- ( ( N e. ZZ /\ N < 1 ) -> ( 0 [,] N ) C_ ( 0 [,) 2 ) )
76		pnfxr 10092	. . . . . . . . . . 11 |- +oo e. RR*
77		icodisj 12297	. . . . . . . . . . 11 |- ( ( 0 e. RR* /\ 2 e. RR* /\ +oo e. RR* ) -> ( ( 0 [,) 2 ) i^i ( 2 [,) +oo ) ) = (/) )
78	60, 63, 76, 77	mp3an 1424	. . . . . . . . . 10 |- ( ( 0 [,) 2 ) i^i ( 2 [,) +oo ) ) = (/)
79		ssdisj 4026	. . . . . . . . . 10 |- ( ( ( 0 [,] N ) C_ ( 0 [,) 2 ) /\ ( ( 0 [,) 2 ) i^i ( 2 [,) +oo ) ) = (/) ) -> ( ( 0 [,] N ) i^i ( 2 [,) +oo ) ) = (/) )
80	75, 78, 79	sylancl 694	. . . . . . . . 9 |- ( ( N e. ZZ /\ N < 1 ) -> ( ( 0 [,] N ) i^i ( 2 [,) +oo ) ) = (/) )
81	59, 80	syl5eqr 2670	. . . . . . . 8 |- ( ( N e. ZZ /\ N < 1 ) -> ( ( 2 [,) +oo ) i^i ( 0 [,] N ) ) = (/) )
82		ssdisj 4026	. . . . . . . 8 |- ( ( Prime C_ ( 2 [,) +oo ) /\ ( ( 2 [,) +oo ) i^i ( 0 [,] N ) ) = (/) ) -> ( Prime i^i ( 0 [,] N ) ) = (/) )
83	58, 81, 82	sylancr 695	. . . . . . 7 |- ( ( N e. ZZ /\ N < 1 ) -> ( Prime i^i ( 0 [,] N ) ) = (/) )
84	51, 83	syl5eq 2668	. . . . . 6 |- ( ( N e. ZZ /\ N < 1 ) -> ( ( 0 [,] N ) i^i Prime ) = (/) )
85		1zzd 11408	. . . . . . . . 9 |- ( ( N e. ZZ /\ N < 1 ) -> 1 e. ZZ )
86		simpl 473	. . . . . . . . 9 |- ( ( N e. ZZ /\ N < 1 ) -> N e. ZZ )
87		fzn 12357	. . . . . . . . . 10 |- ( ( 1 e. ZZ /\ N e. ZZ ) -> ( N < 1 <-> ( 1 ... N ) = (/) ) )
88	87	biimpa 501	. . . . . . . . 9 |- ( ( ( 1 e. ZZ /\ N e. ZZ ) /\ N < 1 ) -> ( 1 ... N ) = (/) )
89	85, 86, 70, 88	syl21anc 1325	. . . . . . . 8 |- ( ( N e. ZZ /\ N < 1 ) -> ( 1 ... N ) = (/) )
90	89	ineq1d 3813	. . . . . . 7 |- ( ( N e. ZZ /\ N < 1 ) -> ( ( 1 ... N ) i^i Prime ) = ( (/) i^i Prime ) )
91	90, 20	syl6eq 2672	. . . . . 6 |- ( ( N e. ZZ /\ N < 1 ) -> ( ( 1 ... N ) i^i Prime ) = (/) )
92	84, 91	eqtr4d 2659	. . . . 5 |- ( ( N e. ZZ /\ N < 1 ) -> ( ( 0 [,] N ) i^i Prime ) = ( ( 1 ... N ) i^i Prime ) )
93	50, 92	syldan 487	. . . 4 |- ( ( N e. ZZ /\ -. N e. NN ) -> ( ( 0 [,] N ) i^i Prime ) = ( ( 1 ... N ) i^i Prime ) )
94		exmidd 432	. . . 4 |- ( N e. ZZ -> ( N e. NN \/ -. N e. NN ) )
95	48, 93, 94	mpjaodan 827	. . 3 |- ( N e. ZZ -> ( ( 0 [,] N ) i^i Prime ) = ( ( 1 ... N ) i^i Prime ) )
96	95	sumeq1d 14431	. 2 |- ( N e. ZZ -> sum_ n e. ( ( 0 [,] N ) i^i Prime ) ( log ` n ) = sum_ n e. ( ( 1 ... N ) i^i Prime ) ( log ` n ) )
97	3, 96	eqtrd 2656	1 |- ( N e. ZZ -> ( theta ` N ) = sum_ n e. ( ( 1 ... N ) i^i Prime ) ( log ` n ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   +oocpnf 10071   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   2c2 11070   ZZcz 11377   ZZ>=cuz 11687   [,)cico 12177   [,]cicc 12178   ...cfz 12326   |_cfl 12591   sum_csu 14416   || cdvds 14983   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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-iun 4522  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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-icc 12182  df-fz 12327  df-fl 12593  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-sum 14417  df-dvds 14984  df-prm 15386  df-cht 24823

This theorem is referenced by:  hgt750lemd  30726