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Theorem pntrval 22770
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 5688 . . 3  |-  ( a  =  A  ->  (ψ `  a )  =  (ψ `  A ) )
2 id 22 . . 3  |-  ( a  =  A  ->  a  =  A )
31, 2oveq12d 6108 . 2  |-  ( a  =  A  ->  (
(ψ `  a )  -  a )  =  ( (ψ `  A
)  -  A ) )
4 pntrval.r . 2  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
5 ovex 6115 . 2  |-  ( (ψ `  A )  -  A
)  e.  _V
63, 4, 5fvmpt 5771 1  |-  ( A  e.  RR+  ->  ( R `
 A )  =  ( (ψ `  A
)  -  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 1761    e. cmpt 4347   ` cfv 5415  (class class class)co 6090    - cmin 9591   RR+crp 10987  ψcchp 22389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-iota 5378  df-fun 5417  df-fv 5423  df-ov 6093
This theorem is referenced by:  pntrmax  22772  pntrsumo1  22773  selbergr  22776  selberg3r  22777  selberg4r  22778  pntrlog2bndlem2  22786  pntrlog2bndlem4  22788  pntrlog2bnd  22792  pntpbnd1a  22793  pntibndlem2  22799  pntlem3  22817
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