Theorem chtnprm 16274

Description: The Chebyshev function at a non-prime. (Contributed by Mario Carneiro, 19-Sep-2014.)

Assertion

Ref	Expression
chtnprm	|- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( theta ` ( A + 1 ) ) = ( theta ` A ) )

Proof of Theorem chtnprm

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

Step	Hyp	Ref	Expression
1		simprr 537	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) )
2	1	elin2d 3419	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> x e. Prime )
3		simprl 535	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> -. ( A + 1 ) e. Prime )
4		nelne2 2511	. . . . . . . . . . . 12 |- ( ( x e. Prime /\ -. ( A + 1 ) e. Prime ) -> x =/= ( A + 1 ) )
5	2, 3, 4	syl2anc 415	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> x =/= ( A + 1 ) )
6		velsn 3726	. . . . . . . . . . . 12 |- ( x e. { ( A + 1 ) } <-> x = ( A + 1 ) )
7	6	necon3bbii 2457	. . . . . . . . . . 11 |- ( -. x e. { ( A + 1 ) } <-> x =/= ( A + 1 ) )
8	5, 7	sylibr 134	. . . . . . . . . 10 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> -. x e. { ( A + 1 ) } )
9	1	elin1d 3418	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> x e. ( 2 ... ( A + 1 ) ) )
10		2z 9684	. . . . . . . . . . . . 13 |- 2 e. ZZ
11		zcn 9661	. . . . . . . . . . . . . . . . 17 |- ( A e. ZZ -> A e. CC )
12	11	adantr 276	. . . . . . . . . . . . . . . 16 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> A e. CC )
13		ax-1cn 8279	. . . . . . . . . . . . . . . 16 |- 1 e. CC
14		pncan 8540	. . . . . . . . . . . . . . . 16 |- ( ( A e. CC /\ 1 e. CC ) -> ( ( A + 1 ) - 1 ) = A )
15	12, 13, 14	sylancl 417	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> ( ( A + 1 ) - 1 ) = A )
16		elfzuz2 10451	. . . . . . . . . . . . . . . 16 |- ( x e. ( 2 ... ( A + 1 ) ) -> ( A + 1 ) e. ( ZZ>= ` 2 ) )
17		uz2m1nn 10022	. . . . . . . . . . . . . . . 16 |- ( ( A + 1 ) e. ( ZZ>= ` 2 ) -> ( ( A + 1 ) - 1 ) e. NN )
18	9, 16, 17	3syl 17	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> ( ( A + 1 ) - 1 ) e. NN )
19	15, 18	eqeltrrd 2316	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> A e. NN )
20		nnuz 9975	. . . . . . . . . . . . . . 15 |- NN = ( ZZ>= ` 1 )
21		2m1e1 9432	. . . . . . . . . . . . . . . 16 |- ( 2 - 1 ) = 1
22	21	fveq2i 5701	. . . . . . . . . . . . . . 15 |- ( ZZ>= ` ( 2 - 1 ) ) = ( ZZ>= ` 1 )
23	20, 22	eqtr4i 2262	. . . . . . . . . . . . . 14 |- NN = ( ZZ>= ` ( 2 - 1 ) )
24	19, 23	eleqtrdi 2331	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> A e. ( ZZ>= ` ( 2 - 1 ) ) )
25		fzsuc2 10504	. . . . . . . . . . . . 13 |- ( ( 2 e. ZZ /\ A e. ( ZZ>= ` ( 2 - 1 ) ) ) -> ( 2 ... ( A + 1 ) ) = ( ( 2 ... A ) u. { ( A + 1 ) } ) )
26	10, 24, 25	sylancr 418	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> ( 2 ... ( A + 1 ) ) = ( ( 2 ... A ) u. { ( A + 1 ) } ) )
27	9, 26	eleqtrd 2317	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> x e. ( ( 2 ... A ) u. { ( A + 1 ) } ) )
28		elun 3370	. . . . . . . . . . 11 |- ( x e. ( ( 2 ... A ) u. { ( A + 1 ) } ) <-> ( x e. ( 2 ... A ) \/ x e. { ( A + 1 ) } ) )
29	27, 28	sylib 122	. . . . . . . . . 10 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> ( x e. ( 2 ... A ) \/ x e. { ( A + 1 ) } ) )
30	8, 29	ecased 1390	. . . . . . . . 9 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> x e. ( 2 ... A ) )
31	30, 2	elind 3414	. . . . . . . 8 |- ( ( A e. ZZ /\ ( -. ( A + 1 ) e. Prime /\ x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) ) ) -> x e. ( ( 2 ... A ) i^i Prime ) )
32	31	expr 375	. . . . . . 7 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( x e. ( ( 2 ... ( A + 1 ) ) i^i Prime ) -> x e. ( ( 2 ... A ) i^i Prime ) ) )
33	32	ssrdv 3254	. . . . . 6 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( ( 2 ... ( A + 1 ) ) i^i Prime ) C_ ( ( 2 ... A ) i^i Prime ) )
34		uzid 9953	. . . . . . . 8 |- ( A e. ZZ -> A e. ( ZZ>= ` A ) )
35	34	adantr 276	. . . . . . 7 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> A e. ( ZZ>= ` A ) )
36		peano2uz 10000	. . . . . . 7 |- ( A e. ( ZZ>= ` A ) -> ( A + 1 ) e. ( ZZ>= ` A ) )
37		fzss2 10488	. . . . . . 7 |- ( ( A + 1 ) e. ( ZZ>= ` A ) -> ( 2 ... A ) C_ ( 2 ... ( A + 1 ) ) )
38		ssrin 3456	. . . . . . 7 |- ( ( 2 ... A ) C_ ( 2 ... ( A + 1 ) ) -> ( ( 2 ... A ) i^i Prime ) C_ ( ( 2 ... ( A + 1 ) ) i^i Prime ) )
39	35, 36, 37, 38	4syl 18	. . . . . 6 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( ( 2 ... A ) i^i Prime ) C_ ( ( 2 ... ( A + 1 ) ) i^i Prime ) )
40	33, 39	eqssd 3265	. . . . 5 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( ( 2 ... ( A + 1 ) ) i^i Prime ) = ( ( 2 ... A ) i^i Prime ) )
41		peano2z 9692	. . . . . . . . 9 |- ( A e. ZZ -> ( A + 1 ) e. ZZ )
42	41	adantr 276	. . . . . . . 8 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( A + 1 ) e. ZZ )
43		flid 10741	. . . . . . . 8 |- ( ( A + 1 ) e. ZZ -> ( |_ ` ( A + 1 ) ) = ( A + 1 ) )
44	42, 43	syl 14	. . . . . . 7 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( |_ ` ( A + 1 ) ) = ( A + 1 ) )
45	44	oveq2d 6104	. . . . . 6 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( 2 ... ( |_ ` ( A + 1 ) ) ) = ( 2 ... ( A + 1 ) ) )
46	45	ineq1d 3431	. . . . 5 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( ( 2 ... ( |_ ` ( A + 1 ) ) ) i^i Prime ) = ( ( 2 ... ( A + 1 ) ) i^i Prime ) )
47		flid 10741	. . . . . . . 8 |- ( A e. ZZ -> ( |_ ` A ) = A )
48	47	adantr 276	. . . . . . 7 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( |_ ` A ) = A )
49	48	oveq2d 6104	. . . . . 6 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( 2 ... ( |_ ` A ) ) = ( 2 ... A ) )
50	49	ineq1d 3431	. . . . 5 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( ( 2 ... ( |_ ` A ) ) i^i Prime ) = ( ( 2 ... A ) i^i Prime ) )
51	40, 46, 50	3eqtr4d 2281	. . . 4 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( ( 2 ... ( |_ ` ( A + 1 ) ) ) i^i Prime ) = ( ( 2 ... ( |_ ` A ) ) i^i Prime ) )
52		zq 10043	. . . . . 6 |- ( ( A + 1 ) e. ZZ -> ( A + 1 ) e. QQ )
53	42, 52	syl 14	. . . . 5 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( A + 1 ) e. QQ )
54		ppiqsval 16252	. . . . 5 |- ( ( A + 1 ) e. QQ -> ( ( 0 [,] ( A + 1 ) ) i^i Prime ) = ( ( 2 ... ( |_ ` ( A + 1 ) ) ) i^i Prime ) )
55	53, 54	syl 14	. . . 4 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( ( 0 [,] ( A + 1 ) ) i^i Prime ) = ( ( 2 ... ( |_ ` ( A + 1 ) ) ) i^i Prime ) )
56		zq 10043	. . . . . 6 |- ( A e. ZZ -> A e. QQ )
57	56	adantr 276	. . . . 5 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> A e. QQ )
58		ppiqsval 16252	. . . . 5 |- ( A e. QQ -> ( ( 0 [,] A ) i^i Prime ) = ( ( 2 ... ( |_ ` A ) ) i^i Prime ) )
59	57, 58	syl 14	. . . 4 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( ( 0 [,] A ) i^i Prime ) = ( ( 2 ... ( |_ ` A ) ) i^i Prime ) )
60	51, 55, 59	3eqtr4d 2281	. . 3 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( ( 0 [,] ( A + 1 ) ) i^i Prime ) = ( ( 0 [,] A ) i^i Prime ) )
61	60	sumeq1d 12158	. 2 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> sum_ p e. ( ( 0 [,] ( A + 1 ) ) i^i Prime ) ( log ` p ) = sum_ p e. ( ( 0 [,] A ) i^i Prime ) ( log ` p ) )
62		chtqval 16257	. . 3 |- ( ( A + 1 ) e. QQ -> ( theta ` ( A + 1 ) ) = sum_ p e. ( ( 0 [,] ( A + 1 ) ) i^i Prime ) ( log ` p ) )
63	53, 62	syl 14	. 2 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( theta ` ( A + 1 ) ) = sum_ p e. ( ( 0 [,] ( A + 1 ) ) i^i Prime ) ( log ` p ) )
64		chtqval 16257	. . 3 |- ( A e. QQ -> ( theta ` A ) = sum_ p e. ( ( 0 [,] A ) i^i Prime ) ( log ` p ) )
65	57, 64	syl 14	. 2 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( theta ` A ) = sum_ p e. ( ( 0 [,] A ) i^i Prime ) ( log ` p ) )
66	61, 63, 65	3eqtr4d 2281	1 |- ( ( A e. ZZ /\ -. ( A + 1 ) e. Prime ) -> ( theta ` ( A + 1 ) ) = ( theta ` A ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   =/= wne 2420   u. cun 3218   i^i cin 3219   C_ wss 3220   {csn 3709   ` cfv 5380  (class class class)co 6088   CCcc 8184   0cc0 8186   1c1 8187   + caddc 8189   - cmin 8505   NNcn 9314   2c2 9365   ZZcz 9656   ZZ>=cuz 9938   QQcq 10036   [,]cicc 10311   ...cfz 10429   |_cfl 10721   sum_csu 12145   Primecprime 12911   logclog 16097   thetaccht 16245

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

This theorem is used by:  chtqub  16308