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Theorem dya2iocival 24576
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 19445. (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 6047 . . . 4  |-  ( u  =  X  ->  (
u  /  ( 2 ^ m ) )  =  ( X  / 
( 2 ^ m
) ) )
2 oveq1 6047 . . . . 5  |-  ( u  =  X  ->  (
u  +  1 )  =  ( X  + 
1 ) )
32oveq1d 6055 . . . 4  |-  ( u  =  X  ->  (
( u  +  1 )  /  ( 2 ^ m ) )  =  ( ( X  +  1 )  / 
( 2 ^ m
) ) )
41, 3oveq12d 6058 . . 3  |-  ( u  =  X  ->  (
( u  /  (
2 ^ m ) ) [,) ( ( u  +  1 )  /  ( 2 ^ m ) ) )  =  ( ( X  /  ( 2 ^ m ) ) [,) ( ( X  + 
1 )  /  (
2 ^ m ) ) ) )
5 oveq2 6048 . . . . 5  |-  ( m  =  N  ->  (
2 ^ m )  =  ( 2 ^ N ) )
65oveq2d 6056 . . . 4  |-  ( m  =  N  ->  ( X  /  ( 2 ^ m ) )  =  ( X  /  (
2 ^ N ) ) )
75oveq2d 6056 . . . 4  |-  ( m  =  N  ->  (
( X  +  1 )  /  ( 2 ^ m ) )  =  ( ( X  +  1 )  / 
( 2 ^ N
) ) )
86, 7oveq12d 6058 . . 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 6047 . . . . . 6  |-  ( u  =  x  ->  (
u  /  ( 2 ^ m ) )  =  ( x  / 
( 2 ^ m
) ) )
11 oveq1 6047 . . . . . . 7  |-  ( u  =  x  ->  (
u  +  1 )  =  ( x  + 
1 ) )
1211oveq1d 6055 . . . . . 6  |-  ( u  =  x  ->  (
( u  +  1 )  /  ( 2 ^ m ) )  =  ( ( x  +  1 )  / 
( 2 ^ m
) ) )
1310, 12oveq12d 6058 . . . . 5  |-  ( u  =  x  ->  (
( u  /  (
2 ^ m ) ) [,) ( ( u  +  1 )  /  ( 2 ^ m ) ) )  =  ( ( x  /  ( 2 ^ m ) ) [,) ( ( x  + 
1 )  /  (
2 ^ m ) ) ) )
14 oveq2 6048 . . . . . . 7  |-  ( m  =  n  ->  (
2 ^ m )  =  ( 2 ^ n ) )
1514oveq2d 6056 . . . . . 6  |-  ( m  =  n  ->  (
x  /  ( 2 ^ m ) )  =  ( x  / 
( 2 ^ n
) ) )
1614oveq2d 6056 . . . . . 6  |-  ( m  =  n  ->  (
( x  +  1 )  /  ( 2 ^ m ) )  =  ( ( x  +  1 )  / 
( 2 ^ n
) ) )
1715, 16oveq12d 6058 . . . . 5  |-  ( m  =  n  ->  (
( x  /  (
2 ^ m ) ) [,) ( ( x  +  1 )  /  ( 2 ^ m ) ) )  =  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  + 
1 )  /  (
2 ^ n ) ) ) )
1813, 17cbvmpt2v 6111 . . . 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 2427 . . 3  |-  I  =  ( u  e.  ZZ ,  m  e.  ZZ  |->  ( ( u  / 
( 2 ^ m
) ) [,) (
( u  +  1 )  /  ( 2 ^ m ) ) ) )
20 ovex 6065 . . 3  |-  ( ( X  /  ( 2 ^ N ) ) [,) ( ( X  +  1 )  / 
( 2 ^ N
) ) )  e. 
_V
214, 8, 19, 20ovmpt2 6168 . 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 1649    e. wcel 1721   ran crn 4838   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1c1 8947    + caddc 8949    / cdiv 9633   2c2 10005   ZZcz 10238   (,)cioo 10872   [,)cico 10874   ^cexp 11337   topGenctg 13620
This theorem is referenced by:  dya2iocress  24577  dya2iocbrsiga  24578  dya2icoseg  24580
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045
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