Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dya2iocival Structured version   Unicode version

Theorem dya2iocival 24615
Description: The function  I returns closed below opened above dyadic rational intervals covering the the real line. This is the same construction as in dyadmbl 19484. (Contributed by Thierry Arnoux, 24-Sep-2017.)
Hypotheses
Ref Expression
sxbrsiga.0  |-  J  =  ( topGen `  ran  (,) )
dya2ioc.1  |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
Assertion
Ref Expression
dya2iocival  |-  ( ( N  e.  ZZ  /\  X  e.  ZZ )  ->  ( X I N )  =  ( ( X  /  ( 2 ^ N ) ) [,) ( ( X  +  1 )  / 
( 2 ^ N
) ) ) )
Distinct variable group:    x, n
Allowed substitution hints:    I( x, n)    J( x, n)    N( x, n)    X( x, n)

Proof of Theorem dya2iocival
Dummy variables  m  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6080 . . . 4  |-  ( u  =  X  ->  (
u  /  ( 2 ^ m ) )  =  ( X  / 
( 2 ^ m
) ) )
2 oveq1 6080 . . . . 5  |-  ( u  =  X  ->  (
u  +  1 )  =  ( X  + 
1 ) )
32oveq1d 6088 . . . 4  |-  ( u  =  X  ->  (
( u  +  1 )  /  ( 2 ^ m ) )  =  ( ( X  +  1 )  / 
( 2 ^ m
) ) )
41, 3oveq12d 6091 . . 3  |-  ( u  =  X  ->  (
( u  /  (
2 ^ m ) ) [,) ( ( u  +  1 )  /  ( 2 ^ m ) ) )  =  ( ( X  /  ( 2 ^ m ) ) [,) ( ( X  + 
1 )  /  (
2 ^ m ) ) ) )
5 oveq2 6081 . . . . 5  |-  ( m  =  N  ->  (
2 ^ m )  =  ( 2 ^ N ) )
65oveq2d 6089 . . . 4  |-  ( m  =  N  ->  ( X  /  ( 2 ^ m ) )  =  ( X  /  (
2 ^ N ) ) )
75oveq2d 6089 . . . 4  |-  ( m  =  N  ->  (
( X  +  1 )  /  ( 2 ^ m ) )  =  ( ( X  +  1 )  / 
( 2 ^ N
) ) )
86, 7oveq12d 6091 . . 3  |-  ( m  =  N  ->  (
( X  /  (
2 ^ m ) ) [,) ( ( X  +  1 )  /  ( 2 ^ m ) ) )  =  ( ( X  /  ( 2 ^ N ) ) [,) ( ( X  + 
1 )  /  (
2 ^ N ) ) ) )
9 dya2ioc.1 . . . 4  |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
10 oveq1 6080 . . . . . 6  |-  ( u  =  x  ->  (
u  /  ( 2 ^ m ) )  =  ( x  / 
( 2 ^ m
) ) )
11 oveq1 6080 . . . . . . 7  |-  ( u  =  x  ->  (
u  +  1 )  =  ( x  + 
1 ) )
1211oveq1d 6088 . . . . . 6  |-  ( u  =  x  ->  (
( u  +  1 )  /  ( 2 ^ m ) )  =  ( ( x  +  1 )  / 
( 2 ^ m
) ) )
1310, 12oveq12d 6091 . . . . 5  |-  ( u  =  x  ->  (
( u  /  (
2 ^ m ) ) [,) ( ( u  +  1 )  /  ( 2 ^ m ) ) )  =  ( ( x  /  ( 2 ^ m ) ) [,) ( ( x  + 
1 )  /  (
2 ^ m ) ) ) )
14 oveq2 6081 . . . . . . 7  |-  ( m  =  n  ->  (
2 ^ m )  =  ( 2 ^ n ) )
1514oveq2d 6089 . . . . . 6  |-  ( m  =  n  ->  (
x  /  ( 2 ^ m ) )  =  ( x  / 
( 2 ^ n
) ) )
1614oveq2d 6089 . . . . . 6  |-  ( m  =  n  ->  (
( x  +  1 )  /  ( 2 ^ m ) )  =  ( ( x  +  1 )  / 
( 2 ^ n
) ) )
1715, 16oveq12d 6091 . . . . 5  |-  ( m  =  n  ->  (
( x  /  (
2 ^ m ) ) [,) ( ( x  +  1 )  /  ( 2 ^ m ) ) )  =  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  + 
1 )  /  (
2 ^ n ) ) ) )
1813, 17cbvmpt2v 6144 . . . 4  |-  ( u  e.  ZZ ,  m  e.  ZZ  |->  ( ( u  /  ( 2 ^ m ) ) [,) ( ( u  + 
1 )  /  (
2 ^ m ) ) ) )  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
199, 18eqtr4i 2458 . . 3  |-  I  =  ( u  e.  ZZ ,  m  e.  ZZ  |->  ( ( u  / 
( 2 ^ m
) ) [,) (
( u  +  1 )  /  ( 2 ^ m ) ) ) )
20 ovex 6098 . . 3  |-  ( ( X  /  ( 2 ^ N ) ) [,) ( ( X  +  1 )  / 
( 2 ^ N
) ) )  e. 
_V
214, 8, 19, 20ovmpt2 6201 . 2  |-  ( ( X  e.  ZZ  /\  N  e.  ZZ )  ->  ( X I N )  =  ( ( X  /  ( 2 ^ N ) ) [,) ( ( X  +  1 )  / 
( 2 ^ N
) ) ) )
2221ancoms 440 1  |-  ( ( N  e.  ZZ  /\  X  e.  ZZ )  ->  ( X I N )  =  ( ( X  /  ( 2 ^ N ) ) [,) ( ( X  +  1 )  / 
( 2 ^ N
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ran crn 4871   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1c1 8983    + caddc 8985    / cdiv 9669   2c2 10041   ZZcz 10274   (,)cioo 10908   [,)cico 10910   ^cexp 11374   topGenctg 13657
This theorem is referenced by:  dya2iocress  24616  dya2iocbrsiga  24617  dya2icoseg  24619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078
  Copyright terms: Public domain W3C validator