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Theorem dya2iocival 26542
Description: The function  I returns closed-below open-above dyadic rational intervals covering the real line. This is the same construction as in dyadmbl 20922. (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 6087 . . . 4  |-  ( u  =  X  ->  (
u  /  ( 2 ^ m ) )  =  ( X  / 
( 2 ^ m
) ) )
2 oveq1 6087 . . . . 5  |-  ( u  =  X  ->  (
u  +  1 )  =  ( X  + 
1 ) )
32oveq1d 6095 . . . 4  |-  ( u  =  X  ->  (
( u  +  1 )  /  ( 2 ^ m ) )  =  ( ( X  +  1 )  / 
( 2 ^ m
) ) )
41, 3oveq12d 6098 . . 3  |-  ( u  =  X  ->  (
( u  /  (
2 ^ m ) ) [,) ( ( u  +  1 )  /  ( 2 ^ m ) ) )  =  ( ( X  /  ( 2 ^ m ) ) [,) ( ( X  + 
1 )  /  (
2 ^ m ) ) ) )
5 oveq2 6088 . . . . 5  |-  ( m  =  N  ->  (
2 ^ m )  =  ( 2 ^ N ) )
65oveq2d 6096 . . . 4  |-  ( m  =  N  ->  ( X  /  ( 2 ^ m ) )  =  ( X  /  (
2 ^ N ) ) )
75oveq2d 6096 . . . 4  |-  ( m  =  N  ->  (
( X  +  1 )  /  ( 2 ^ m ) )  =  ( ( X  +  1 )  / 
( 2 ^ N
) ) )
86, 7oveq12d 6098 . . 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 6087 . . . . . 6  |-  ( u  =  x  ->  (
u  /  ( 2 ^ m ) )  =  ( x  / 
( 2 ^ m
) ) )
11 oveq1 6087 . . . . . . 7  |-  ( u  =  x  ->  (
u  +  1 )  =  ( x  + 
1 ) )
1211oveq1d 6095 . . . . . 6  |-  ( u  =  x  ->  (
( u  +  1 )  /  ( 2 ^ m ) )  =  ( ( x  +  1 )  / 
( 2 ^ m
) ) )
1310, 12oveq12d 6098 . . . . 5  |-  ( u  =  x  ->  (
( u  /  (
2 ^ m ) ) [,) ( ( u  +  1 )  /  ( 2 ^ m ) ) )  =  ( ( x  /  ( 2 ^ m ) ) [,) ( ( x  + 
1 )  /  (
2 ^ m ) ) ) )
14 oveq2 6088 . . . . . . 7  |-  ( m  =  n  ->  (
2 ^ m )  =  ( 2 ^ n ) )
1514oveq2d 6096 . . . . . 6  |-  ( m  =  n  ->  (
x  /  ( 2 ^ m ) )  =  ( x  / 
( 2 ^ n
) ) )
1614oveq2d 6096 . . . . . 6  |-  ( m  =  n  ->  (
( x  +  1 )  /  ( 2 ^ m ) )  =  ( ( x  +  1 )  / 
( 2 ^ n
) ) )
1715, 16oveq12d 6098 . . . . 5  |-  ( m  =  n  ->  (
( x  /  (
2 ^ m ) ) [,) ( ( x  +  1 )  /  ( 2 ^ m ) ) )  =  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  + 
1 )  /  (
2 ^ n ) ) ) )
1813, 17cbvmpt2v 6155 . . . 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 2456 . . 3  |-  I  =  ( u  e.  ZZ ,  m  e.  ZZ  |->  ( ( u  / 
( 2 ^ m
) ) [,) (
( u  +  1 )  /  ( 2 ^ m ) ) ) )
20 ovex 6105 . . 3  |-  ( ( X  /  ( 2 ^ N ) ) [,) ( ( X  +  1 )  / 
( 2 ^ N
) ) )  e. 
_V
214, 8, 19, 20ovmpt2 6215 . 2  |-  ( ( X  e.  ZZ  /\  N  e.  ZZ )  ->  ( X I N )  =  ( ( X  /  ( 2 ^ N ) ) [,) ( ( X  +  1 )  / 
( 2 ^ N
) ) ) )
2221ancoms 450 1  |-  ( ( N  e.  ZZ  /\  X  e.  ZZ )  ->  ( X I N )  =  ( ( X  /  ( 2 ^ N ) ) [,) ( ( X  +  1 )  / 
( 2 ^ N
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1362    e. wcel 1755   ran crn 4828   ` cfv 5406  (class class class)co 6080    e. cmpt2 6082   1c1 9271    + caddc 9273    / cdiv 9981   2c2 10359   ZZcz 10634   (,)cioo 11288   [,)cico 11290   ^cexp 11849   topGenctg 14359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-iota 5369  df-fun 5408  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085
This theorem is referenced by:  dya2iocress  26543  dya2iocbrsiga  26544  dya2icoseg  26546
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