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Theorem dya2iocival 26833
Description: The function  I returns closed-below open-above dyadic rational intervals covering the real line. This is the same construction as in dyadmbl 21214. (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 6208 . . . 4  |-  ( u  =  X  ->  (
u  /  ( 2 ^ m ) )  =  ( X  / 
( 2 ^ m
) ) )
2 oveq1 6208 . . . . 5  |-  ( u  =  X  ->  (
u  +  1 )  =  ( X  + 
1 ) )
32oveq1d 6216 . . . 4  |-  ( u  =  X  ->  (
( u  +  1 )  /  ( 2 ^ m ) )  =  ( ( X  +  1 )  / 
( 2 ^ m
) ) )
41, 3oveq12d 6219 . . 3  |-  ( u  =  X  ->  (
( u  /  (
2 ^ m ) ) [,) ( ( u  +  1 )  /  ( 2 ^ m ) ) )  =  ( ( X  /  ( 2 ^ m ) ) [,) ( ( X  + 
1 )  /  (
2 ^ m ) ) ) )
5 oveq2 6209 . . . . 5  |-  ( m  =  N  ->  (
2 ^ m )  =  ( 2 ^ N ) )
65oveq2d 6217 . . . 4  |-  ( m  =  N  ->  ( X  /  ( 2 ^ m ) )  =  ( X  /  (
2 ^ N ) ) )
75oveq2d 6217 . . . 4  |-  ( m  =  N  ->  (
( X  +  1 )  /  ( 2 ^ m ) )  =  ( ( X  +  1 )  / 
( 2 ^ N
) ) )
86, 7oveq12d 6219 . . 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 6208 . . . . . 6  |-  ( u  =  x  ->  (
u  /  ( 2 ^ m ) )  =  ( x  / 
( 2 ^ m
) ) )
11 oveq1 6208 . . . . . . 7  |-  ( u  =  x  ->  (
u  +  1 )  =  ( x  + 
1 ) )
1211oveq1d 6216 . . . . . 6  |-  ( u  =  x  ->  (
( u  +  1 )  /  ( 2 ^ m ) )  =  ( ( x  +  1 )  / 
( 2 ^ m
) ) )
1310, 12oveq12d 6219 . . . . 5  |-  ( u  =  x  ->  (
( u  /  (
2 ^ m ) ) [,) ( ( u  +  1 )  /  ( 2 ^ m ) ) )  =  ( ( x  /  ( 2 ^ m ) ) [,) ( ( x  + 
1 )  /  (
2 ^ m ) ) ) )
14 oveq2 6209 . . . . . . 7  |-  ( m  =  n  ->  (
2 ^ m )  =  ( 2 ^ n ) )
1514oveq2d 6217 . . . . . 6  |-  ( m  =  n  ->  (
x  /  ( 2 ^ m ) )  =  ( x  / 
( 2 ^ n
) ) )
1614oveq2d 6217 . . . . . 6  |-  ( m  =  n  ->  (
( x  +  1 )  /  ( 2 ^ m ) )  =  ( ( x  +  1 )  / 
( 2 ^ n
) ) )
1715, 16oveq12d 6219 . . . . 5  |-  ( m  =  n  ->  (
( x  /  (
2 ^ m ) ) [,) ( ( x  +  1 )  /  ( 2 ^ m ) ) )  =  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  + 
1 )  /  (
2 ^ n ) ) ) )
1813, 17cbvmpt2v 6276 . . . 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 2486 . . 3  |-  I  =  ( u  e.  ZZ ,  m  e.  ZZ  |->  ( ( u  / 
( 2 ^ m
) ) [,) (
( u  +  1 )  /  ( 2 ^ m ) ) ) )
20 ovex 6226 . . 3  |-  ( ( X  /  ( 2 ^ N ) ) [,) ( ( X  +  1 )  / 
( 2 ^ N
) ) )  e. 
_V
214, 8, 19, 20ovmpt2 6337 . 2  |-  ( ( X  e.  ZZ  /\  N  e.  ZZ )  ->  ( X I N )  =  ( ( X  /  ( 2 ^ N ) ) [,) ( ( X  +  1 )  / 
( 2 ^ N
) ) ) )
2221ancoms 453 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 1370    e. wcel 1758   ran crn 4950   ` cfv 5527  (class class class)co 6201    |-> cmpt2 6203   1c1 9395    + caddc 9397    / cdiv 10105   2c2 10483   ZZcz 10758   (,)cioo 11412   [,)cico 11414   ^cexp 11983   topGenctg 14496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-iota 5490  df-fun 5529  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206
This theorem is referenced by:  dya2iocress  26834  dya2iocbrsiga  26835  dya2icoseg  26837
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