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Theorem pntrval 22816
Description: Define the residual of the second Chebyshev function. The goal is to have  R ( x )  e.  o ( x ), or  R ( x )  /  x  ~~> r  0. (Contributed by Mario Carneiro, 8-Apr-2016.)
Hypothesis
Ref Expression
pntrval.r  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
Assertion
Ref Expression
pntrval  |-  ( A  e.  RR+  ->  ( R `
 A )  =  ( (ψ `  A
)  -  A ) )
Distinct variable group:    A, a
Allowed substitution hint:    R( a)

Proof of Theorem pntrval
StepHypRef Expression
1 fveq2 5696 . . 3  |-  ( a  =  A  ->  (ψ `  a )  =  (ψ `  A ) )
2 id 22 . . 3  |-  ( a  =  A  ->  a  =  A )
31, 2oveq12d 6114 . 2  |-  ( a  =  A  ->  (
(ψ `  a )  -  a )  =  ( (ψ `  A
)  -  A ) )
4 pntrval.r . 2  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
5 ovex 6121 . 2  |-  ( (ψ `  A )  -  A
)  e.  _V
63, 4, 5fvmpt 5779 1  |-  ( A  e.  RR+  ->  ( R `
 A )  =  ( (ψ `  A
)  -  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756    e. cmpt 4355   ` cfv 5423  (class class class)co 6096    - cmin 9600   RR+crp 10996  ψcchp 22435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5386  df-fun 5425  df-fv 5431  df-ov 6099
This theorem is referenced by:  pntrmax  22818  pntrsumo1  22819  selbergr  22822  selberg3r  22823  selberg4r  22824  pntrlog2bndlem2  22832  pntrlog2bndlem4  22834  pntrlog2bnd  22838  pntpbnd1a  22839  pntibndlem2  22845  pntlem3  22863
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