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Theorem pntrval 20727
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 5541 . . 3  |-  ( a  =  A  ->  (ψ `  a )  =  (ψ `  A ) )
2 id 19 . . 3  |-  ( a  =  A  ->  a  =  A )
31, 2oveq12d 5892 . 2  |-  ( a  =  A  ->  (
(ψ `  a )  -  a )  =  ( (ψ `  A
)  -  A ) )
4 pntrval.r . 2  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
5 ovex 5899 . 2  |-  ( (ψ `  A )  -  A
)  e.  _V
63, 4, 5fvmpt 5618 1  |-  ( A  e.  RR+  ->  ( R `
 A )  =  ( (ψ `  A
)  -  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    e. cmpt 4093   ` cfv 5271  (class class class)co 5874    - cmin 9053   RR+crp 10370  ψcchp 20346
This theorem is referenced by:  pntrmax  20729  pntrsumo1  20730  selbergr  20733  selberg3r  20734  selberg4r  20735  pntrlog2bndlem2  20743  pntrlog2bndlem4  20745  pntrlog2bnd  20749  pntpbnd1a  20750  pntibndlem2  20756  pntlem3  20774
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877
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