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Theorem chtnprm 20890
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.
StepHypRef Expression
1 inss2 3522 . . . . . . . . . . . . 13  |-  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  C_  Prime
2 simprr 734 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime ) )
31, 2sseldi 3306 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  x  e.  Prime )
4 simprl 733 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  -.  ( A  +  1
)  e.  Prime )
5 nelne2 2657 . . . . . . . . . . . 12  |-  ( ( x  e.  Prime  /\  -.  ( A  +  1
)  e.  Prime )  ->  x  =/=  ( A  +  1 ) )
63, 4, 5syl2anc 643 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  x  =/=  ( A  +  1 ) )
7 elsn 3789 . . . . . . . . . . . 12  |-  ( x  e.  { ( A  +  1 ) }  <-> 
x  =  ( A  +  1 ) )
87necon3bbii 2598 . . . . . . . . . . 11  |-  ( -.  x  e.  { ( A  +  1 ) }  <->  x  =/=  ( A  +  1 ) )
96, 8sylibr 204 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  -.  x  e.  { ( A  +  1 ) } )
10 inss1 3521 . . . . . . . . . . . . . 14  |-  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  C_  (
2 ... ( A  + 
1 ) )
1110, 2sseldi 3306 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  x  e.  ( 2 ... ( A  +  1 ) ) )
12 2z 10268 . . . . . . . . . . . . . 14  |-  2  e.  ZZ
13 zcn 10243 . . . . . . . . . . . . . . . . . 18  |-  ( A  e.  ZZ  ->  A  e.  CC )
1413adantr 452 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  A  e.  CC )
15 ax-1cn 9004 . . . . . . . . . . . . . . . . 17  |-  1  e.  CC
16 pncan 9267 . . . . . . . . . . . . . . . . 17  |-  ( ( A  e.  CC  /\  1  e.  CC )  ->  ( ( A  + 
1 )  -  1 )  =  A )
1714, 15, 16sylancl 644 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  (
( A  +  1 )  -  1 )  =  A )
18 elfzuz2 11018 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ( 2 ... ( A  +  1 ) )  ->  ( A  +  1 )  e.  ( ZZ>= `  2
) )
19 uz2m1nn 10506 . . . . . . . . . . . . . . . . 17  |-  ( ( A  +  1 )  e.  ( ZZ>= `  2
)  ->  ( ( A  +  1 )  -  1 )  e.  NN )
2011, 18, 193syl 19 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  (
( A  +  1 )  -  1 )  e.  NN )
2117, 20eqeltrrd 2479 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  A  e.  NN )
22 nnuz 10477 . . . . . . . . . . . . . . . 16  |-  NN  =  ( ZZ>= `  1 )
23 2m1e1 10051 . . . . . . . . . . . . . . . . 17  |-  ( 2  -  1 )  =  1
2423fveq2i 5690 . . . . . . . . . . . . . . . 16  |-  ( ZZ>= `  ( 2  -  1 ) )  =  (
ZZ>= `  1 )
2522, 24eqtr4i 2427 . . . . . . . . . . . . . . 15  |-  NN  =  ( ZZ>= `  ( 2  -  1 ) )
2621, 25syl6eleq 2494 . . . . . . . . . . . . . 14  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  A  e.  ( ZZ>= `  ( 2  -  1 ) ) )
27 fzsuc2 11060 . . . . . . . . . . . . . 14  |-  ( ( 2  e.  ZZ  /\  A  e.  ( ZZ>= `  ( 2  -  1 ) ) )  -> 
( 2 ... ( A  +  1 ) )  =  ( ( 2 ... A )  u.  { ( A  +  1 ) } ) )
2812, 26, 27sylancr 645 . . . . . . . . . . . . 13  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  (
2 ... ( A  + 
1 ) )  =  ( ( 2 ... A )  u.  {
( A  +  1 ) } ) )
2911, 28eleqtrd 2480 . . . . . . . . . . . 12  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  x  e.  ( ( 2 ... A )  u.  {
( A  +  1 ) } ) )
30 elun 3448 . . . . . . . . . . . 12  |-  ( x  e.  ( ( 2 ... A )  u. 
{ ( A  + 
1 ) } )  <-> 
( x  e.  ( 2 ... A )  \/  x  e.  {
( A  +  1 ) } ) )
3129, 30sylib 189 . . . . . . . . . . 11  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  (
x  e.  ( 2 ... A )  \/  x  e.  { ( A  +  1 ) } ) )
3231ord 367 . . . . . . . . . 10  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  ( -.  x  e.  (
2 ... A )  ->  x  e.  { ( A  +  1 ) } ) )
339, 32mt3d 119 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  x  e.  ( 2 ... A
) )
34 elin 3490 . . . . . . . . 9  |-  ( x  e.  ( ( 2 ... A )  i^i 
Prime )  <->  ( x  e.  ( 2 ... A
)  /\  x  e.  Prime ) )
3533, 3, 34sylanbrc 646 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  ( -.  ( A  +  1 )  e. 
Prime  /\  x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )
) )  ->  x  e.  ( ( 2 ... A )  i^i  Prime ) )
3635expr 599 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( x  e.  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  ->  x  e.  ( ( 2 ... A )  i^i  Prime ) ) )
3736ssrdv 3314 . . . . . 6  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime ) 
C_  ( ( 2 ... A )  i^i 
Prime ) )
38 uzid 10456 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  A  e.  ( ZZ>= `  A )
)
3938adantr 452 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  A  e.  ( ZZ>= `  A ) )
40 peano2uz 10486 . . . . . . . 8  |-  ( A  e.  ( ZZ>= `  A
)  ->  ( A  +  1 )  e.  ( ZZ>= `  A )
)
4139, 40syl 16 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( A  +  1 )  e.  ( ZZ>= `  A ) )
42 fzss2 11048 . . . . . . 7  |-  ( ( A  +  1 )  e.  ( ZZ>= `  A
)  ->  ( 2 ... A )  C_  ( 2 ... ( A  +  1 ) ) )
43 ssrin 3526 . . . . . . 7  |-  ( ( 2 ... A ) 
C_  ( 2 ... ( A  +  1 ) )  ->  (
( 2 ... A
)  i^i  Prime )  C_  ( ( 2 ... ( A  +  1 ) )  i^i  Prime ) )
4441, 42, 433syl 19 . . . . . 6  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( ( 2 ... A )  i^i  Prime ) 
C_  ( ( 2 ... ( A  + 
1 ) )  i^i 
Prime ) )
4537, 44eqssd 3325 . . . . 5  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( ( 2 ... ( A  +  1 ) )  i^i  Prime )  =  ( ( 2 ... A )  i^i 
Prime ) )
46 peano2z 10274 . . . . . . . . 9  |-  ( A  e.  ZZ  ->  ( A  +  1 )  e.  ZZ )
4746adantr 452 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( A  +  1 )  e.  ZZ )
48 flid 11171 . . . . . . . 8  |-  ( ( A  +  1 )  e.  ZZ  ->  ( |_ `  ( A  + 
1 ) )  =  ( A  +  1 ) )
4947, 48syl 16 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( |_ `  ( A  +  1 ) )  =  ( A  +  1 ) )
5049oveq2d 6056 . . . . . 6  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( 2 ... ( |_ `  ( A  + 
1 ) ) )  =  ( 2 ... ( A  +  1 ) ) )
5150ineq1d 3501 . . . . 5  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( ( 2 ... ( |_ `  ( A  +  1 ) ) )  i^i  Prime )  =  ( ( 2 ... ( A  + 
1 ) )  i^i 
Prime ) )
52 flid 11171 . . . . . . . 8  |-  ( A  e.  ZZ  ->  ( |_ `  A )  =  A )
5352adantr 452 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( |_ `  A
)  =  A )
5453oveq2d 6056 . . . . . 6  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( 2 ... ( |_ `  A ) )  =  ( 2 ... A ) )
5554ineq1d 3501 . . . . 5  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( ( 2 ... ( |_ `  A
) )  i^i  Prime )  =  ( ( 2 ... A )  i^i 
Prime ) )
5645, 51, 553eqtr4d 2446 . . . 4  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( ( 2 ... ( |_ `  ( A  +  1 ) ) )  i^i  Prime )  =  ( ( 2 ... ( |_ `  A ) )  i^i 
Prime ) )
57 zre 10242 . . . . . 6  |-  ( A  e.  ZZ  ->  A  e.  RR )
5857adantr 452 . . . . 5  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  A  e.  RR )
59 peano2re 9195 . . . . 5  |-  ( A  e.  RR  ->  ( A  +  1 )  e.  RR )
60 ppisval 20839 . . . . 5  |-  ( ( A  +  1 )  e.  RR  ->  (
( 0 [,] ( A  +  1 ) )  i^i  Prime )  =  ( ( 2 ... ( |_ `  ( A  +  1
) ) )  i^i 
Prime ) )
6158, 59, 603syl 19 . . . 4  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( ( 0 [,] ( A  +  1 ) )  i^i  Prime )  =  ( ( 2 ... ( |_ `  ( A  +  1
) ) )  i^i 
Prime ) )
62 ppisval 20839 . . . . 5  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  =  ( ( 2 ... ( |_ `  A
) )  i^i  Prime ) )
6358, 62syl 16 . . . 4  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( ( 0 [,] A )  i^i  Prime )  =  ( ( 2 ... ( |_ `  A ) )  i^i 
Prime ) )
6456, 61, 633eqtr4d 2446 . . 3  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( ( 0 [,] ( A  +  1 ) )  i^i  Prime )  =  ( ( 0 [,] A )  i^i 
Prime ) )
6564sumeq1d 12450 . 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
) )
66 chtval 20846 . . 3  |-  ( ( A  +  1 )  e.  RR  ->  ( theta `  ( A  + 
1 ) )  = 
sum_ p  e.  (
( 0 [,] ( A  +  1 ) )  i^i  Prime )
( log `  p
) )
6758, 59, 663syl 19 . 2  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( theta `  ( A  +  1 ) )  =  sum_ p  e.  ( ( 0 [,] ( A  +  1 ) )  i^i  Prime )
( log `  p
) )
68 chtval 20846 . . 3  |-  ( A  e.  RR  ->  ( theta `  A )  = 
sum_ p  e.  (
( 0 [,] A
)  i^i  Prime ) ( log `  p ) )
6958, 68syl 16 . 2  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( theta `  A )  =  sum_ p  e.  ( ( 0 [,] A
)  i^i  Prime ) ( log `  p ) )
7065, 67, 693eqtr4d 2446 1  |-  ( ( A  e.  ZZ  /\  -.  ( A  +  1 )  e.  Prime )  ->  ( theta `  ( A  +  1 ) )  =  ( theta `  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567    u. cun 3278    i^i cin 3279    C_ wss 3280   {csn 3774   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    - cmin 9247   NNcn 9956   2c2 10005   ZZcz 10238   ZZ>=cuz 10444   [,]cicc 10875   ...cfz 10999   |_cfl 11156   sum_csu 12434   Primecprime 13034   logclog 20405   thetaccht 20826
This theorem is referenced by:  chtub  20949
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-icc 10879  df-fz 11000  df-fl 11157  df-seq 11279  df-sum 12435  df-dvds 12808  df-prm 13035  df-cht 20832
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