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Theorem dya2iocival 29168
Description: The function  I returns closed-below open-above dyadic rational intervals covering the real line. This is the same construction as in dyadmbl 22637. (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 6315 . . . 4  |-  ( u  =  X  ->  (
u  /  ( 2 ^ m ) )  =  ( X  / 
( 2 ^ m
) ) )
2 oveq1 6315 . . . . 5  |-  ( u  =  X  ->  (
u  +  1 )  =  ( X  + 
1 ) )
32oveq1d 6323 . . . 4  |-  ( u  =  X  ->  (
( u  +  1 )  /  ( 2 ^ m ) )  =  ( ( X  +  1 )  / 
( 2 ^ m
) ) )
41, 3oveq12d 6326 . . 3  |-  ( u  =  X  ->  (
( u  /  (
2 ^ m ) ) [,) ( ( u  +  1 )  /  ( 2 ^ m ) ) )  =  ( ( X  /  ( 2 ^ m ) ) [,) ( ( X  + 
1 )  /  (
2 ^ m ) ) ) )
5 oveq2 6316 . . . . 5  |-  ( m  =  N  ->  (
2 ^ m )  =  ( 2 ^ N ) )
65oveq2d 6324 . . . 4  |-  ( m  =  N  ->  ( X  /  ( 2 ^ m ) )  =  ( X  /  (
2 ^ N ) ) )
75oveq2d 6324 . . . 4  |-  ( m  =  N  ->  (
( X  +  1 )  /  ( 2 ^ m ) )  =  ( ( X  +  1 )  / 
( 2 ^ N
) ) )
86, 7oveq12d 6326 . . 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 6315 . . . . . 6  |-  ( u  =  x  ->  (
u  /  ( 2 ^ m ) )  =  ( x  / 
( 2 ^ m
) ) )
11 oveq1 6315 . . . . . . 7  |-  ( u  =  x  ->  (
u  +  1 )  =  ( x  + 
1 ) )
1211oveq1d 6323 . . . . . 6  |-  ( u  =  x  ->  (
( u  +  1 )  /  ( 2 ^ m ) )  =  ( ( x  +  1 )  / 
( 2 ^ m
) ) )
1310, 12oveq12d 6326 . . . . 5  |-  ( u  =  x  ->  (
( u  /  (
2 ^ m ) ) [,) ( ( u  +  1 )  /  ( 2 ^ m ) ) )  =  ( ( x  /  ( 2 ^ m ) ) [,) ( ( x  + 
1 )  /  (
2 ^ m ) ) ) )
14 oveq2 6316 . . . . . . 7  |-  ( m  =  n  ->  (
2 ^ m )  =  ( 2 ^ n ) )
1514oveq2d 6324 . . . . . 6  |-  ( m  =  n  ->  (
x  /  ( 2 ^ m ) )  =  ( x  / 
( 2 ^ n
) ) )
1614oveq2d 6324 . . . . . 6  |-  ( m  =  n  ->  (
( x  +  1 )  /  ( 2 ^ m ) )  =  ( ( x  +  1 )  / 
( 2 ^ n
) ) )
1715, 16oveq12d 6326 . . . . 5  |-  ( m  =  n  ->  (
( x  /  (
2 ^ m ) ) [,) ( ( x  +  1 )  /  ( 2 ^ m ) ) )  =  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  + 
1 )  /  (
2 ^ n ) ) ) )
1813, 17cbvmpt2v 6390 . . . 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 2496 . . 3  |-  I  =  ( u  e.  ZZ ,  m  e.  ZZ  |->  ( ( u  / 
( 2 ^ m
) ) [,) (
( u  +  1 )  /  ( 2 ^ m ) ) ) )
20 ovex 6336 . . 3  |-  ( ( X  /  ( 2 ^ N ) ) [,) ( ( X  +  1 )  / 
( 2 ^ N
) ) )  e. 
_V
214, 8, 19, 20ovmpt2 6451 . 2  |-  ( ( X  e.  ZZ  /\  N  e.  ZZ )  ->  ( X I N )  =  ( ( X  /  ( 2 ^ N ) ) [,) ( ( X  +  1 )  / 
( 2 ^ N
) ) ) )
2221ancoms 460 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 376    = wceq 1452    e. wcel 1904   ran crn 4840   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   1c1 9558    + caddc 9560    / cdiv 10291   2c2 10681   ZZcz 10961   (,)cioo 11660   [,)cico 11662   ^cexp 12310   topGenctg 15414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313
This theorem is referenced by:  dya2iocress  29169  dya2iocbrsiga  29170  dya2icoseg  29172
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