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Theorem pntrval 24128
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 5849 . . 3  |-  ( a  =  A  ->  (ψ `  a )  =  (ψ `  A ) )
2 id 22 . . 3  |-  ( a  =  A  ->  a  =  A )
31, 2oveq12d 6296 . 2  |-  ( a  =  A  ->  (
(ψ `  a )  -  a )  =  ( (ψ `  A
)  -  A ) )
4 pntrval.r . 2  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
5 ovex 6306 . 2  |-  ( (ψ `  A )  -  A
)  e.  _V
63, 4, 5fvmpt 5932 1  |-  ( A  e.  RR+  ->  ( R `
 A )  =  ( (ψ `  A
)  -  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842    |-> cmpt 4453   ` cfv 5569  (class class class)co 6278    - cmin 9841   RR+crp 11265  ψcchp 23747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-ov 6281
This theorem is referenced by:  pntrmax  24130  pntrsumo1  24131  selbergr  24134  selberg3r  24135  selberg4r  24136  pntrlog2bndlem2  24144  pntrlog2bndlem4  24146  pntrlog2bnd  24150  pntpbnd1a  24151  pntibndlem2  24157  pntlem3  24175
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