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Theorem dya2iocseg 23581
Description: For any point and any closed below, opened above interval of  RR centered on that point, there is a closed below opened above dyadic rational interval which contains that point and is included in the original interval. (Contributed by Thierry Arnoux, 19-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 ) ) ) )
dya2ioc.2  |-  R  =  ( n  e.  ZZ  |->  ran  ( x  e.  ZZ ,  y  e.  ZZ  |->  ( ( x I n )  X.  (
y I n ) ) ) )
dya2iocseg.1  |-  N  =  ( |_ `  (
1  -  ( 2logb D ) ) )
dya2iocseg.2  |-  G  =  ( I  o.  `' ( 1st  |`  ( _V  X.  { N } ) ) )
Assertion
Ref Expression
dya2iocseg  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  E. b  e.  ( G " ZZ ) ( X  e.  b  /\  b  C_  ( ( X  -  D ) [,) ( X  +  D
) ) ) )
Distinct variable groups:    x, y, n    D, b    x, b, G    I, b, x    N, b, x    X, b, x
Allowed substitution hints:    D( x, y, n)    R( x, y, n, b)    G( y, n)    I(
y, n)    J( x, y, n, b)    N( y, n)    X( y, n)

Proof of Theorem dya2iocseg
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 simpl 443 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  X  e.  RR )
2 dya2iocseg.1 . . . . . . . . . 10  |-  N  =  ( |_ `  (
1  -  ( 2logb D ) ) )
3 1re 8839 . . . . . . . . . . . . 13  |-  1  e.  RR
43a1i 10 . . . . . . . . . . . 12  |-  ( D  e.  RR+  ->  1  e.  RR )
5 2z 10056 . . . . . . . . . . . . . 14  |-  2  e.  ZZ
6 uzid 10244 . . . . . . . . . . . . . 14  |-  ( 2  e.  ZZ  ->  2  e.  ( ZZ>= `  2 )
)
75, 6ax-mp 8 . . . . . . . . . . . . 13  |-  2  e.  ( ZZ>= `  2 )
8 rnlogbcl 23405 . . . . . . . . . . . . 13  |-  ( ( 2  e.  ( ZZ>= ` 
2 )  /\  D  e.  RR+ )  ->  (
2logb D )  e.  RR )
97, 8mpan 651 . . . . . . . . . . . 12  |-  ( D  e.  RR+  ->  ( 2logb D )  e.  RR )
104, 9resubcld 9213 . . . . . . . . . . 11  |-  ( D  e.  RR+  ->  ( 1  -  ( 2logb D
) )  e.  RR )
11 flcl 10929 . . . . . . . . . . 11  |-  ( ( 1  -  ( 2logb D ) )  e.  RR  ->  ( |_ `  ( 1  -  (
2logb D ) ) )  e.  ZZ )
1210, 11syl 15 . . . . . . . . . 10  |-  ( D  e.  RR+  ->  ( |_
`  ( 1  -  ( 2logb D ) ) )  e.  ZZ )
132, 12syl5eqel 2369 . . . . . . . . 9  |-  ( D  e.  RR+  ->  N  e.  ZZ )
14 2rp 10361 . . . . . . . . . 10  |-  2  e.  RR+
15 rpexpcl 11124 . . . . . . . . . . 11  |-  ( ( 2  e.  RR+  /\  N  e.  ZZ )  ->  (
2 ^ N )  e.  RR+ )
1615rpred 10392 . . . . . . . . . 10  |-  ( ( 2  e.  RR+  /\  N  e.  ZZ )  ->  (
2 ^ N )  e.  RR )
1714, 16mpan 651 . . . . . . . . 9  |-  ( N  e.  ZZ  ->  (
2 ^ N )  e.  RR )
1813, 17syl 15 . . . . . . . 8  |-  ( D  e.  RR+  ->  ( 2 ^ N )  e.  RR )
1918adantl 452 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 2 ^ N
)  e.  RR )
201, 19remulcld 8865 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  x.  (
2 ^ N ) )  e.  RR )
21 flcl 10929 . . . . . 6  |-  ( ( X  x.  ( 2 ^ N ) )  e.  RR  ->  ( |_ `  ( X  x.  ( 2 ^ N
) ) )  e.  ZZ )
2220, 21syl 15 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( X  x.  ( 2 ^ N ) ) )  e.  ZZ )
23 eqid 2285 . . . . 5  |-  ( ( |_ `  ( X  x.  ( 2 ^ N ) ) ) I N )  =  ( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N )
24 oveq1 5867 . . . . . . 7  |-  ( x  =  ( |_ `  ( X  x.  (
2 ^ N ) ) )  ->  (
x I N )  =  ( ( |_
`  ( X  x.  ( 2 ^ N
) ) ) I N ) )
2524eqeq2d 2296 . . . . . 6  |-  ( x  =  ( |_ `  ( X  x.  (
2 ^ N ) ) )  ->  (
( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N )  =  ( x I N )  <->  ( ( |_ `  ( X  x.  ( 2 ^ N
) ) ) I N )  =  ( ( |_ `  ( X  x.  ( 2 ^ N ) ) ) I N ) ) )
2625rspcev 2886 . . . . 5  |-  ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  e.  ZZ  /\  ( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N )  =  ( ( |_ `  ( X  x.  ( 2 ^ N ) ) ) I N ) )  ->  E. x  e.  ZZ  ( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N )  =  ( x I N ) )
2722, 23, 26sylancl 643 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  E. x  e.  ZZ  ( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N )  =  ( x I N ) )
28 nfv 1607 . . . . 5  |-  F/ u E. x  e.  ZZ  ( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N )  =  ( x I N )
29 ovex 5885 . . . . 5  |-  ( ( |_ `  ( X  x.  ( 2 ^ N ) ) ) I N )  e. 
_V
30 eqeq1 2291 . . . . . 6  |-  ( u  =  ( ( |_
`  ( X  x.  ( 2 ^ N
) ) ) I N )  ->  (
u  =  ( x I N )  <->  ( ( |_ `  ( X  x.  ( 2 ^ N
) ) ) I N )  =  ( x I N ) ) )
3130rexbidv 2566 . . . . 5  |-  ( u  =  ( ( |_
`  ( X  x.  ( 2 ^ N
) ) ) I N )  ->  ( E. x  e.  ZZ  u  =  ( x I N )  <->  E. x  e.  ZZ  ( ( |_
`  ( X  x.  ( 2 ^ N
) ) ) I N )  =  ( x I N ) ) )
3228, 29, 31elabf 2915 . . . 4  |-  ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) ) I N )  e.  { u  |  E. x  e.  ZZ  u  =  ( x I N ) }  <->  E. x  e.  ZZ  ( ( |_
`  ( X  x.  ( 2 ^ N
) ) ) I N )  =  ( x I N ) )
3327, 32sylibr 203 . . 3  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N )  e.  { u  |  E. x  e.  ZZ  u  =  ( x I N ) } )
34 ovex 5885 . . . . . . . 8  |-  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) )  e. 
_V
3534rgen2w 2613 . . . . . . 7  |-  A. x  e.  ZZ  A. n  e.  ZZ  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  e.  _V
36 dya2ioc.1 . . . . . . . 8  |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
3736fnmpt2 6194 . . . . . . 7  |-  ( A. x  e.  ZZ  A. n  e.  ZZ  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  + 
1 )  /  (
2 ^ n ) ) )  e.  _V  ->  I  Fn  ( ZZ 
X.  ZZ ) )
3835, 37ax-mp 8 . . . . . 6  |-  I  Fn  ( ZZ  X.  ZZ )
3938a1i 10 . . . . 5  |-  ( D  e.  RR+  ->  I  Fn  ( ZZ  X.  ZZ ) )
40 ssid 3199 . . . . . 6  |-  ZZ  C_  ZZ
4140a1i 10 . . . . 5  |-  ( D  e.  RR+  ->  ZZ  C_  ZZ )
42 dya2iocseg.2 . . . . . 6  |-  G  =  ( I  o.  `' ( 1st  |`  ( _V  X.  { N } ) ) )
4342curry2ima 23249 . . . . 5  |-  ( ( I  Fn  ( ZZ 
X.  ZZ )  /\  N  e.  ZZ  /\  ZZ  C_  ZZ )  ->  ( G " ZZ )  =  { u  |  E. x  e.  ZZ  u  =  ( x I N ) } )
4439, 13, 41, 43syl3anc 1182 . . . 4  |-  ( D  e.  RR+  ->  ( G
" ZZ )  =  { u  |  E. x  e.  ZZ  u  =  ( x I N ) } )
4544adantl 452 . . 3  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( G " ZZ )  =  { u  |  E. x  e.  ZZ  u  =  ( x I N ) } )
4633, 45eleqtrrd 2362 . 2  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N )  e.  ( G
" ZZ ) )
47 fllelt 10931 . . . . . . . . . 10  |-  ( ( X  x.  ( 2 ^ N ) )  e.  RR  ->  (
( |_ `  ( X  x.  ( 2 ^ N ) ) )  <_  ( X  x.  ( 2 ^ N
) )  /\  ( X  x.  ( 2 ^ N ) )  <  ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 ) ) )
4820, 47syl 15 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  <_  ( X  x.  ( 2 ^ N ) )  /\  ( X  x.  ( 2 ^ N
) )  <  (
( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 ) ) )
4948simpld 445 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( X  x.  ( 2 ^ N ) ) )  <_  ( X  x.  ( 2 ^ N
) ) )
5022zred 10119 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( X  x.  ( 2 ^ N ) ) )  e.  RR )
5113adantl 452 . . . . . . . . . 10  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  N  e.  ZZ )
5214, 51, 15sylancr 644 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 2 ^ N
)  e.  RR+ )
5350, 20, 52lediv1d 10434 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  <_  ( X  x.  ( 2 ^ N ) )  <-> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  /  (
2 ^ N ) )  <_  ( ( X  x.  ( 2 ^ N ) )  /  ( 2 ^ N ) ) ) )
5449, 53mpbid 201 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  /  (
2 ^ N ) )  <_  ( ( X  x.  ( 2 ^ N ) )  /  ( 2 ^ N ) ) )
551recnd 8863 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  X  e.  CC )
5619recnd 8863 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 2 ^ N
)  e.  CC )
57 2cn 9818 . . . . . . . . . 10  |-  2  e.  CC
5857a1i 10 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
2  e.  CC )
59 2ne0 9831 . . . . . . . . . 10  |-  2  =/=  0
6059a1i 10 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
2  =/=  0 )
6158, 60, 51expne0d 11253 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 2 ^ N
)  =/=  0 )
62 divcan4 9451 . . . . . . . 8  |-  ( ( X  e.  CC  /\  ( 2 ^ N
)  e.  CC  /\  ( 2 ^ N
)  =/=  0 )  ->  ( ( X  x.  ( 2 ^ N ) )  / 
( 2 ^ N
) )  =  X )
6355, 56, 61, 62syl3anc 1182 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  /  (
2 ^ N ) )  =  X )
6454, 63breqtrd 4049 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  /  (
2 ^ N ) )  <_  X )
6548simprd 449 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  x.  (
2 ^ N ) )  <  ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 ) )
663a1i 10 . . . . . . . . . 10  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
1  e.  RR )
6750, 66readdcld 8864 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  +  1 )  e.  RR )
6820, 67, 52ltdiv1d 10433 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  <  (
( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  <-> 
( ( X  x.  ( 2 ^ N
) )  /  (
2 ^ N ) )  <  ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  /  ( 2 ^ N ) ) ) )
6965, 68mpbid 201 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  /  (
2 ^ N ) )  <  ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  /  ( 2 ^ N ) ) )
7063, 69eqbrtrrd 4047 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  X  <  ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  / 
( 2 ^ N
) ) )
711, 64, 703jca 1132 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  e.  RR  /\  ( ( |_ `  ( X  x.  (
2 ^ N ) ) )  /  (
2 ^ N ) )  <_  X  /\  X  <  ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  / 
( 2 ^ N
) ) ) )
7250, 19, 61redivcld 9590 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  /  (
2 ^ N ) )  e.  RR )
7367, 19, 61redivcld 9590 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  e.  RR )
7473rexrd 8883 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  e.  RR* )
75 elico2 10716 . . . . . 6  |-  ( ( ( ( |_ `  ( X  x.  (
2 ^ N ) ) )  /  (
2 ^ N ) )  e.  RR  /\  ( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  e.  RR* )  ->  ( X  e.  ( ( ( |_ `  ( X  x.  (
2 ^ N ) ) )  /  (
2 ^ N ) ) [,) ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  /  ( 2 ^ N ) ) )  <-> 
( X  e.  RR  /\  ( ( |_ `  ( X  x.  (
2 ^ N ) ) )  /  (
2 ^ N ) )  <_  X  /\  X  <  ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  / 
( 2 ^ N
) ) ) ) )
7672, 74, 75syl2anc 642 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  e.  ( ( ( |_ `  ( X  x.  (
2 ^ N ) ) )  /  (
2 ^ N ) ) [,) ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  /  ( 2 ^ N ) ) )  <-> 
( X  e.  RR  /\  ( ( |_ `  ( X  x.  (
2 ^ N ) ) )  /  (
2 ^ N ) )  <_  X  /\  X  <  ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  / 
( 2 ^ N
) ) ) ) )
7771, 76mpbird 223 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  X  e.  ( (
( |_ `  ( X  x.  ( 2 ^ N ) ) )  /  ( 2 ^ N ) ) [,) ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  / 
( 2 ^ N
) ) ) )
78 sxbrsiga.0 . . . . . 6  |-  J  =  ( topGen `  ran  (,) )
79 dya2ioc.2 . . . . . 6  |-  R  =  ( n  e.  ZZ  |->  ran  ( x  e.  ZZ ,  y  e.  ZZ  |->  ( ( x I n )  X.  (
y I n ) ) ) )
8078, 36, 79dya2iocival 23578 . . . . 5  |-  ( ( N  e.  ZZ  /\  ( |_ `  ( X  x.  ( 2 ^ N ) ) )  e.  ZZ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N )  =  ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  /  ( 2 ^ N ) ) [,) ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  / 
( 2 ^ N
) ) ) )
8151, 22, 80syl2anc 642 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N )  =  ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  /  ( 2 ^ N ) ) [,) ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  / 
( 2 ^ N
) ) ) )
8277, 81eleqtrrd 2362 . . 3  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  X  e.  ( ( |_ `  ( X  x.  ( 2 ^ N
) ) ) I N ) )
83 simpr 447 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  D  e.  RR+ )
8483rpred 10392 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  D  e.  RR )
851, 84resubcld 9213 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  D
)  e.  RR )
8685rexrd 8883 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  D
)  e.  RR* )
871, 84readdcld 8864 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  +  D
)  e.  RR )
8887rexrd 8883 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  +  D
)  e.  RR* )
8919, 61rereccld 9589 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 1  /  (
2 ^ N ) )  e.  RR )
901, 89resubcld 9213 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  (
1  /  ( 2 ^ N ) ) )  e.  RR )
91 dya2ub 23577 . . . . . . . . . 10  |-  ( D  e.  RR+  ->  ( 1  /  ( 2 ^ ( |_ `  (
1  -  ( 2logb D ) ) ) ) )  <  D
)
9283, 91syl 15 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 1  /  (
2 ^ ( |_
`  ( 1  -  ( 2logb D ) ) ) ) )  <  D )
932oveq2i 5871 . . . . . . . . . . 11  |-  ( 2 ^ N )  =  ( 2 ^ ( |_ `  ( 1  -  ( 2logb D ) ) ) )
9493oveq2i 5871 . . . . . . . . . 10  |-  ( 1  /  ( 2 ^ N ) )  =  ( 1  /  (
2 ^ ( |_
`  ( 1  -  ( 2logb D ) ) ) ) )
9594breq1i 4032 . . . . . . . . 9  |-  ( ( 1  /  ( 2 ^ N ) )  <  D  <->  ( 1  /  ( 2 ^ ( |_ `  (
1  -  ( 2logb D ) ) ) ) )  <  D
)
9692, 95sylibr 203 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 1  /  (
2 ^ N ) )  <  D )
9789, 84, 1, 96ltsub2dd 9387 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  D
)  <  ( X  -  ( 1  / 
( 2 ^ N
) ) ) )
9855, 56mulcld 8857 . . . . . . . . . 10  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  x.  (
2 ^ N ) )  e.  CC )
99 ax-1cn 8797 . . . . . . . . . . 11  |-  1  e.  CC
10099a1i 10 . . . . . . . . . 10  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
1  e.  CC )
10198, 100, 56, 61divsubdird 9577 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  - 
1 )  /  (
2 ^ N ) )  =  ( ( ( X  x.  (
2 ^ N ) )  /  ( 2 ^ N ) )  -  ( 1  / 
( 2 ^ N
) ) ) )
10263oveq1d 5875 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  / 
( 2 ^ N
) )  -  (
1  /  ( 2 ^ N ) ) )  =  ( X  -  ( 1  / 
( 2 ^ N
) ) ) )
103101, 102eqtrd 2317 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  - 
1 )  /  (
2 ^ N ) )  =  ( X  -  ( 1  / 
( 2 ^ N
) ) ) )
10420, 66resubcld 9213 . . . . . . . . . 10  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  -  1 )  e.  RR )
10520, 67, 66, 65ltsub1dd 9386 . . . . . . . . . . 11  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  -  1 )  <  ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  -  1 ) )
10650recnd 8863 . . . . . . . . . . . 12  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( X  x.  ( 2 ^ N ) ) )  e.  CC )
107106, 100pncand 9160 . . . . . . . . . . 11  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  -  1 )  =  ( |_
`  ( X  x.  ( 2 ^ N
) ) ) )
108105, 107breqtrd 4049 . . . . . . . . . 10  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  -  1 )  <  ( |_
`  ( X  x.  ( 2 ^ N
) ) ) )
109104, 50, 108ltled 8969 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  -  1 )  <_  ( |_ `  ( X  x.  (
2 ^ N ) ) ) )
110104, 50, 52lediv1d 10434 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  - 
1 )  <_  ( |_ `  ( X  x.  ( 2 ^ N
) ) )  <->  ( (
( X  x.  (
2 ^ N ) )  -  1 )  /  ( 2 ^ N ) )  <_ 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  /  (
2 ^ N ) ) ) )
111109, 110mpbid 201 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  - 
1 )  /  (
2 ^ N ) )  <_  ( ( |_ `  ( X  x.  ( 2 ^ N
) ) )  / 
( 2 ^ N
) ) )
112103, 111eqbrtrrd 4047 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  (
1  /  ( 2 ^ N ) ) )  <_  ( ( |_ `  ( X  x.  ( 2 ^ N
) ) )  / 
( 2 ^ N
) ) )
11385, 90, 72, 97, 112ltletrd 8978 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  D
)  <  ( ( |_ `  ( X  x.  ( 2 ^ N
) ) )  / 
( 2 ^ N
) ) )
11485, 72, 113ltled 8969 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  D
)  <_  ( ( |_ `  ( X  x.  ( 2 ^ N
) ) )  / 
( 2 ^ N
) ) )
1151, 89readdcld 8864 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  +  ( 1  /  ( 2 ^ N ) ) )  e.  RR )
11650, 20, 66, 49leadd1dd 9388 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  +  1 )  <_  ( ( X  x.  ( 2 ^ N ) )  +  1 ) )
11720, 66readdcld 8864 . . . . . . . . . 10  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  +  1 )  e.  RR )
11867, 117, 52lediv1d 10434 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  <_  (
( X  x.  (
2 ^ N ) )  +  1 )  <-> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  <_  ( (
( X  x.  (
2 ^ N ) )  +  1 )  /  ( 2 ^ N ) ) ) )
119116, 118mpbid 201 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  <_  ( (
( X  x.  (
2 ^ N ) )  +  1 )  /  ( 2 ^ N ) ) )
120 divdir 9449 . . . . . . . . . 10  |-  ( ( ( X  x.  (
2 ^ N ) )  e.  CC  /\  1  e.  CC  /\  (
( 2 ^ N
)  e.  CC  /\  ( 2 ^ N
)  =/=  0 ) )  ->  ( (
( X  x.  (
2 ^ N ) )  +  1 )  /  ( 2 ^ N ) )  =  ( ( ( X  x.  ( 2 ^ N ) )  / 
( 2 ^ N
) )  +  ( 1  /  ( 2 ^ N ) ) ) )
12198, 100, 56, 61, 120syl112anc 1186 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  +  1 )  /  (
2 ^ N ) )  =  ( ( ( X  x.  (
2 ^ N ) )  /  ( 2 ^ N ) )  +  ( 1  / 
( 2 ^ N
) ) ) )
12263oveq1d 5875 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  / 
( 2 ^ N
) )  +  ( 1  /  ( 2 ^ N ) ) )  =  ( X  +  ( 1  / 
( 2 ^ N
) ) ) )
123121, 122eqtrd 2317 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  +  1 )  /  (
2 ^ N ) )  =  ( X  +  ( 1  / 
( 2 ^ N
) ) ) )
124119, 123breqtrd 4049 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  <_  ( X  +  ( 1  / 
( 2 ^ N
) ) ) )
12589, 84, 1, 96ltadd2dd 8977 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  +  ( 1  /  ( 2 ^ N ) ) )  <  ( X  +  D ) )
12673, 115, 87, 124, 125lelttrd 8976 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  <  ( X  +  D ) )
12773, 87, 126ltled 8969 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  <_  ( X  +  D ) )
128 icossico 23265 . . . . 5  |-  ( ( ( ( X  -  D )  e.  RR*  /\  ( X  +  D
)  e.  RR* )  /\  ( ( X  -  D )  <_  (
( |_ `  ( X  x.  ( 2 ^ N ) ) )  /  ( 2 ^ N ) )  /\  ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  / 
( 2 ^ N
) )  <_  ( X  +  D )
) )  ->  (
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  /  (
2 ^ N ) ) [,) ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  /  ( 2 ^ N ) ) ) 
C_  ( ( X  -  D ) [,) ( X  +  D
) ) )
12986, 88, 114, 127, 128syl22anc 1183 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  / 
( 2 ^ N
) ) [,) (
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  +  1 )  /  ( 2 ^ N ) ) )  C_  ( ( X  -  D ) [,) ( X  +  D
) ) )
13081, 129eqsstrd 3214 . . 3  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N )  C_  ( ( X  -  D ) [,) ( X  +  D
) ) )
13182, 130jca 518 . 2  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  e.  ( ( |_ `  ( X  x.  ( 2 ^ N ) ) ) I N )  /\  ( ( |_
`  ( X  x.  ( 2 ^ N
) ) ) I N )  C_  (
( X  -  D
) [,) ( X  +  D ) ) ) )
132 eleq2 2346 . . . 4  |-  ( b  =  ( ( |_
`  ( X  x.  ( 2 ^ N
) ) ) I N )  ->  ( X  e.  b  <->  X  e.  ( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N ) ) )
133 sseq1 3201 . . . 4  |-  ( b  =  ( ( |_
`  ( X  x.  ( 2 ^ N
) ) ) I N )  ->  (
b  C_  ( ( X  -  D ) [,) ( X  +  D
) )  <->  ( ( |_ `  ( X  x.  ( 2 ^ N
) ) ) I N )  C_  (
( X  -  D
) [,) ( X  +  D ) ) ) )
134132, 133anbi12d 691 . . 3  |-  ( b  =  ( ( |_
`  ( X  x.  ( 2 ^ N
) ) ) I N )  ->  (
( X  e.  b  /\  b  C_  (
( X  -  D
) [,) ( X  +  D ) ) )  <->  ( X  e.  ( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N )  /\  ( ( |_ `  ( X  x.  ( 2 ^ N ) ) ) I N )  C_  ( ( X  -  D ) [,) ( X  +  D )
) ) ) )
135134rspcev 2886 . 2  |-  ( ( ( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N )  e.  ( G
" ZZ )  /\  ( X  e.  (
( |_ `  ( X  x.  ( 2 ^ N ) ) ) I N )  /\  ( ( |_
`  ( X  x.  ( 2 ^ N
) ) ) I N )  C_  (
( X  -  D
) [,) ( X  +  D ) ) ) )  ->  E. b  e.  ( G " ZZ ) ( X  e.  b  /\  b  C_  ( ( X  -  D ) [,) ( X  +  D )
) ) )
13646, 131, 135syl2anc 642 1  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  E. b  e.  ( G " ZZ ) ( X  e.  b  /\  b  C_  ( ( X  -  D ) [,) ( X  +  D
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686   {cab 2271    =/= wne 2448   A.wral 2545   E.wrex 2546   _Vcvv 2790    C_ wss 3154   {csn 3642   class class class wbr 4025    e. cmpt 4079    X. cxp 4689   `'ccnv 4690   ran crn 4692    |` cres 4693   "cima 4694    o. ccom 4695    Fn wfn 5252   ` cfv 5257  (class class class)co 5860    e. cmpt2 5862   1stc1st 6122   CCcc 8737   RRcr 8738   0cc0 8739   1c1 8740    + caddc 8742    x. cmul 8744   RR*cxr 8868    < clt 8869    <_ cle 8870    - cmin 9039    / cdiv 9425   2c2 9797   ZZcz 10026   ZZ>=cuz 10232   RR+crp 10356   (,)cioo 10658   [,)cico 10660   |_cfl 10926   ^cexp 11106   topGenctg 13344  logbclogb 23392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-addf 8818  ax-mulf 8819
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-map 6776  df-pm 6777  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-fi 7167  df-sup 7196  df-oi 7227  df-card 7574  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-div 9426  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-q 10319  df-rp 10357  df-xneg 10454  df-xadd 10455  df-xmul 10456  df-ioo 10662  df-ioc 10663  df-ico 10664  df-icc 10665  df-fz 10785  df-fzo 10873  df-fl 10927  df-mod 10976  df-seq 11049  df-exp 11107  df-fac 11291  df-bc 11318  df-hash 11340  df-shft 11564  df-cj 11586  df-re 11587  df-im 11588  df-sqr 11722  df-abs 11723  df-limsup 11947  df-clim 11964  df-rlim 11965  df-sum 12161  df-ef 12351  df-sin 12353  df-cos 12354  df-pi 12356  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-starv 13225  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-hom 13234  df-cco 13235  df-rest 13329  df-topn 13330  df-topgen 13346  df-pt 13347  df-prds 13350  df-xrs 13405  df-0g 13406  df-gsum 13407  df-qtop 13412  df-imas 13413  df-xps 13415  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-submnd 14418  df-mulg 14494  df-cntz 14795  df-cmn 15093  df-xmet 16375  df-met 16376  df-bl 16377  df-mopn 16378  df-cnfld 16380  df-top 16638  df-bases 16640  df-topon 16641  df-topsp 16642  df-cld 16758  df-ntr 16759  df-cls 16760  df-nei 16837  df-lp 16870  df-perf 16871  df-cn 16959  df-cnp 16960  df-haus 17045  df-tx 17259  df-hmeo 17448  df-fbas 17522  df-fg 17523  df-fil 17543  df-fm 17635  df-flim 17636  df-flf 17637  df-xms 17887  df-ms 17888  df-tms 17889  df-cncf 18384  df-limc 19218  df-dv 19219  df-log 19916  df-cxp 19917  df-logb 23393
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