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Theorem pntrval 23475
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 5864 . . 3  |-  ( a  =  A  ->  (ψ `  a )  =  (ψ `  A ) )
2 id 22 . . 3  |-  ( a  =  A  ->  a  =  A )
31, 2oveq12d 6300 . 2  |-  ( a  =  A  ->  (
(ψ `  a )  -  a )  =  ( (ψ `  A
)  -  A ) )
4 pntrval.r . 2  |-  R  =  ( a  e.  RR+  |->  ( (ψ `  a )  -  a ) )
5 ovex 6307 . 2  |-  ( (ψ `  A )  -  A
)  e.  _V
63, 4, 5fvmpt 5948 1  |-  ( A  e.  RR+  ->  ( R `
 A )  =  ( (ψ `  A
)  -  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    |-> cmpt 4505   ` cfv 5586  (class class class)co 6282    - cmin 9801   RR+crp 11216  ψcchp 23094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285
This theorem is referenced by:  pntrmax  23477  pntrsumo1  23478  selbergr  23481  selberg3r  23482  selberg4r  23483  pntrlog2bndlem2  23491  pntrlog2bndlem4  23493  pntrlog2bnd  23497  pntpbnd1a  23498  pntibndlem2  23504  pntlem3  23522
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