Theorem pntrval 25251

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

Step	Hyp	Ref	Expression
1		fveq2 6191	. . 3 |- ( a = A -> (ψ ` a ) = (ψ ` A ) )
2		id 22	. . 3 |- ( a = A -> a = A )
3	1, 2	oveq12d 6668	. 2 |- ( a = A -> ( (ψ ` a ) - a ) = ( (ψ ` A ) - A ) )
4		pntrval.r	. 2 |- R = ( a e. RR+ |-> ( (ψ ` a ) - a ) )
5		ovex 6678	. 2 |- ( (ψ ` A ) - A ) e. _V
6	3, 4, 5	fvmpt 6282	1 |- ( A e. RR+ -> ( R ` A ) = ( (ψ ` A ) - A ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   - cmin 10266   RR+crp 11832  ψcchp 24819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653

This theorem is referenced by:  pntrmax  25253  pntrsumo1  25254  selbergr  25257  selberg3r  25258  selberg4r  25259  pntrlog2bndlem2  25267  pntrlog2bndlem4  25269  pntrlog2bnd  25273  pntpbnd1a  25274  pntibndlem2  25280  pntlem3  25298