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Theorem chtfl 20932
Description: The Chebyshev function does not change off the integers. (Contributed by Mario Carneiro, 22-Sep-2014.)
Assertion
Ref Expression
chtfl  |-  ( A  e.  RR  ->  ( theta `  ( |_ `  A ) )  =  ( theta `  A )
)

Proof of Theorem chtfl
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 flidm 11217 . . . . . 6  |-  ( A  e.  RR  ->  ( |_ `  ( |_ `  A ) )  =  ( |_ `  A
) )
21oveq2d 6097 . . . . 5  |-  ( A  e.  RR  ->  (
2 ... ( |_ `  ( |_ `  A ) ) )  =  ( 2 ... ( |_
`  A ) ) )
32ineq1d 3541 . . . 4  |-  ( A  e.  RR  ->  (
( 2 ... ( |_ `  ( |_ `  A ) ) )  i^i  Prime )  =  ( ( 2 ... ( |_ `  A ) )  i^i  Prime ) )
4 reflcl 11205 . . . . 5  |-  ( A  e.  RR  ->  ( |_ `  A )  e.  RR )
5 ppisval 20886 . . . . 5  |-  ( ( |_ `  A )  e.  RR  ->  (
( 0 [,] ( |_ `  A ) )  i^i  Prime )  =  ( ( 2 ... ( |_ `  ( |_ `  A ) ) )  i^i  Prime ) )
64, 5syl 16 . . . 4  |-  ( A  e.  RR  ->  (
( 0 [,] ( |_ `  A ) )  i^i  Prime )  =  ( ( 2 ... ( |_ `  ( |_ `  A ) ) )  i^i  Prime ) )
7 ppisval 20886 . . . 4  |-  ( A  e.  RR  ->  (
( 0 [,] A
)  i^i  Prime )  =  ( ( 2 ... ( |_ `  A
) )  i^i  Prime ) )
83, 6, 73eqtr4d 2478 . . 3  |-  ( A  e.  RR  ->  (
( 0 [,] ( |_ `  A ) )  i^i  Prime )  =  ( ( 0 [,] A
)  i^i  Prime ) )
98sumeq1d 12495 . 2  |-  ( A  e.  RR  ->  sum_ p  e.  ( ( 0 [,] ( |_ `  A
) )  i^i  Prime ) ( log `  p
)  =  sum_ p  e.  ( ( 0 [,] A )  i^i  Prime ) ( log `  p
) )
10 chtval 20893 . . 3  |-  ( ( |_ `  A )  e.  RR  ->  ( theta `  ( |_ `  A ) )  = 
sum_ p  e.  (
( 0 [,] ( |_ `  A ) )  i^i  Prime ) ( log `  p ) )
114, 10syl 16 . 2  |-  ( A  e.  RR  ->  ( theta `  ( |_ `  A ) )  = 
sum_ p  e.  (
( 0 [,] ( |_ `  A ) )  i^i  Prime ) ( log `  p ) )
12 chtval 20893 . 2  |-  ( A  e.  RR  ->  ( theta `  A )  = 
sum_ p  e.  (
( 0 [,] A
)  i^i  Prime ) ( log `  p ) )
139, 11, 123eqtr4d 2478 1  |-  ( A  e.  RR  ->  ( theta `  ( |_ `  A ) )  =  ( theta `  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    i^i cin 3319   ` cfv 5454  (class class class)co 6081   RRcr 8989   0cc0 8990   2c2 10049   [,]cicc 10919   ...cfz 11043   |_cfl 11201   sum_csu 12479   Primecprime 13079   logclog 20452   thetaccht 20873
This theorem is referenced by:  efchtdvds  20942  chtub  20996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-icc 10923  df-fz 11044  df-fl 11202  df-seq 11324  df-sum 12480  df-dvds 12853  df-prm 13080  df-cht 20879
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