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Theorem dya2icoseg 24580
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 ) ) ) )
dya2icoseg.1  |-  N  =  ( |_ `  (
1  -  ( 2logb D ) ) )
Assertion
Ref Expression
dya2icoseg  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  E. b  e.  ran  I ( X  e.  b  /\  b  C_  ( ( X  -  D ) (,) ( X  +  D )
) ) )
Distinct variable groups:    x, n    x, I    D, b    x, b, I    N, b, x    X, b, x
Allowed substitution hints:    D( x, n)    I( n)    J( x, n, b)    N( n)    X( n)

Proof of Theorem dya2icoseg
StepHypRef Expression
1 dya2ioc.1 . . . . 5  |-  I  =  ( x  e.  ZZ ,  n  e.  ZZ  |->  ( ( x  / 
( 2 ^ n
) ) [,) (
( x  +  1 )  /  ( 2 ^ n ) ) ) )
2 ovex 6065 . . . . 5  |-  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) )  e. 
_V
31, 2fnmpt2i 6379 . . . 4  |-  I  Fn  ( ZZ  X.  ZZ )
43a1i 11 . . 3  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  I  Fn  ( ZZ  X.  ZZ ) )
5 simpl 444 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  X  e.  RR )
6 2rp 10573 . . . . . . 7  |-  2  e.  RR+
7 dya2icoseg.1 . . . . . . . 8  |-  N  =  ( |_ `  (
1  -  ( 2logb D ) ) )
8 1re 9046 . . . . . . . . . . 11  |-  1  e.  RR
98a1i 11 . . . . . . . . . 10  |-  ( D  e.  RR+  ->  1  e.  RR )
10 2z 10268 . . . . . . . . . . . 12  |-  2  e.  ZZ
11 uzid 10456 . . . . . . . . . . . 12  |-  ( 2  e.  ZZ  ->  2  e.  ( ZZ>= `  2 )
)
1210, 11ax-mp 8 . . . . . . . . . . 11  |-  2  e.  ( ZZ>= `  2 )
13 rnlogbcl 24354 . . . . . . . . . . 11  |-  ( ( 2  e.  ( ZZ>= ` 
2 )  /\  D  e.  RR+ )  ->  (
2logb D )  e.  RR )
1412, 13mpan 652 . . . . . . . . . 10  |-  ( D  e.  RR+  ->  ( 2logb D )  e.  RR )
159, 14resubcld 9421 . . . . . . . . 9  |-  ( D  e.  RR+  ->  ( 1  -  ( 2logb D
) )  e.  RR )
1615flcld 11162 . . . . . . . 8  |-  ( D  e.  RR+  ->  ( |_
`  ( 1  -  ( 2logb D ) ) )  e.  ZZ )
177, 16syl5eqel 2488 . . . . . . 7  |-  ( D  e.  RR+  ->  N  e.  ZZ )
18 rpexpcl 11355 . . . . . . . 8  |-  ( ( 2  e.  RR+  /\  N  e.  ZZ )  ->  (
2 ^ N )  e.  RR+ )
1918rpred 10604 . . . . . . 7  |-  ( ( 2  e.  RR+  /\  N  e.  ZZ )  ->  (
2 ^ N )  e.  RR )
206, 17, 19sylancr 645 . . . . . 6  |-  ( D  e.  RR+  ->  ( 2 ^ N )  e.  RR )
2120adantl 453 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 2 ^ N
)  e.  RR )
225, 21remulcld 9072 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  x.  (
2 ^ N ) )  e.  RR )
2322flcld 11162 . . 3  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( X  x.  ( 2 ^ N ) ) )  e.  ZZ )
2417adantl 453 . . 3  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  N  e.  ZZ )
25 fnovrn 6180 . . 3  |-  ( ( I  Fn  ( ZZ 
X.  ZZ )  /\  ( |_ `  ( X  x.  ( 2 ^ N ) ) )  e.  ZZ  /\  N  e.  ZZ )  ->  (
( |_ `  ( X  x.  ( 2 ^ N ) ) ) I N )  e.  ran  I )
264, 23, 24, 25syl3anc 1184 . 2  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N )  e.  ran  I
)
2723zred 10331 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( X  x.  ( 2 ^ N ) ) )  e.  RR )
286, 24, 18sylancr 645 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 2 ^ N
)  e.  RR+ )
29 fllelt 11161 . . . . . . . 8  |-  ( ( X  x.  ( 2 ^ N ) )  e.  RR  ->  (
( |_ `  ( X  x.  ( 2 ^ N ) ) )  <_  ( X  x.  ( 2 ^ N
) )  /\  ( X  x.  ( 2 ^ N ) )  <  ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 ) ) )
3022, 29syl 16 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  <_  ( X  x.  ( 2 ^ N ) )  /\  ( X  x.  ( 2 ^ N
) )  <  (
( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 ) ) )
3130simpld 446 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( X  x.  ( 2 ^ N ) ) )  <_  ( X  x.  ( 2 ^ N
) ) )
3227, 22, 28, 31lediv1dd 10658 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  /  (
2 ^ N ) )  <_  ( ( X  x.  ( 2 ^ N ) )  /  ( 2 ^ N ) ) )
335recnd 9070 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  X  e.  CC )
3421recnd 9070 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 2 ^ N
)  e.  CC )
35 2cn 10026 . . . . . . . 8  |-  2  e.  CC
3635a1i 11 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
2  e.  CC )
37 2ne0 10039 . . . . . . . 8  |-  2  =/=  0
3837a1i 11 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
2  =/=  0 )
3936, 38, 24expne0d 11484 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 2 ^ N
)  =/=  0 )
4033, 34, 39divcan4d 9752 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  /  (
2 ^ N ) )  =  X )
4132, 40breqtrd 4196 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  /  (
2 ^ N ) )  <_  X )
428a1i 11 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
1  e.  RR )
4327, 42readdcld 9071 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  +  1 )  e.  RR )
4430simprd 450 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  x.  (
2 ^ N ) )  <  ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 ) )
4522, 43, 28, 44ltdiv1dd 10657 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  /  (
2 ^ N ) )  <  ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  /  ( 2 ^ N ) ) )
4640, 45eqbrtrrd 4194 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  X  <  ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  / 
( 2 ^ N
) ) )
4727, 21, 39redivcld 9798 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  /  (
2 ^ N ) )  e.  RR )
4843, 21, 39redivcld 9798 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  e.  RR )
4948rexrd 9090 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  e.  RR* )
50 elico2 10930 . . . . 5  |-  ( ( ( ( |_ `  ( 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
) ) ) ) )
5147, 49, 50syl2anc 643 . . . 4  |-  ( ( 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
) ) ) ) )
525, 41, 46, 51mpbir3and 1137 . . 3  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  X  e.  ( (
( |_ `  ( X  x.  ( 2 ^ N ) ) )  /  ( 2 ^ N ) ) [,) ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  / 
( 2 ^ N
) ) ) )
53 sxbrsiga.0 . . . . 5  |-  J  =  ( topGen `  ran  (,) )
5453, 1dya2iocival 24576 . . . 4  |-  ( ( 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
) ) ) )
5524, 23, 54syl2anc 643 . . 3  |-  ( ( 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
) ) ) )
5652, 55eleqtrrd 2481 . 2  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  X  e.  ( ( |_ `  ( X  x.  ( 2 ^ N
) ) ) I N ) )
57 simpr 448 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  D  e.  RR+ )
5857rpred 10604 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  D  e.  RR )
595, 58resubcld 9421 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  D
)  e.  RR )
6059rexrd 9090 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  D
)  e.  RR* )
615, 58readdcld 9071 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  +  D
)  e.  RR )
6261rexrd 9090 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  +  D
)  e.  RR* )
6321, 39rereccld 9797 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 1  /  (
2 ^ N ) )  e.  RR )
645, 63resubcld 9421 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  (
1  /  ( 2 ^ N ) ) )  e.  RR )
657oveq2i 6051 . . . . . . . 8  |-  ( 2 ^ N )  =  ( 2 ^ ( |_ `  ( 1  -  ( 2logb D ) ) ) )
6665oveq2i 6051 . . . . . . 7  |-  ( 1  /  ( 2 ^ N ) )  =  ( 1  /  (
2 ^ ( |_
`  ( 1  -  ( 2logb D ) ) ) ) )
67 dya2ub 24573 . . . . . . . 8  |-  ( D  e.  RR+  ->  ( 1  /  ( 2 ^ ( |_ `  (
1  -  ( 2logb D ) ) ) ) )  <  D
)
6867adantl 453 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 1  /  (
2 ^ ( |_
`  ( 1  -  ( 2logb D ) ) ) ) )  <  D )
6966, 68syl5eqbr 4205 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 1  /  (
2 ^ N ) )  <  D )
7063, 58, 5, 69ltsub2dd 9595 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  D
)  <  ( X  -  ( 1  / 
( 2 ^ N
) ) ) )
7133, 34mulcld 9064 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  x.  (
2 ^ N ) )  e.  CC )
72 ax-1cn 9004 . . . . . . . . 9  |-  1  e.  CC
7372a1i 11 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
1  e.  CC )
7471, 73, 34, 39divsubdird 9785 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  - 
1 )  /  (
2 ^ N ) )  =  ( ( ( X  x.  (
2 ^ N ) )  /  ( 2 ^ N ) )  -  ( 1  / 
( 2 ^ N
) ) ) )
7540oveq1d 6055 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  / 
( 2 ^ N
) )  -  (
1  /  ( 2 ^ N ) ) )  =  ( X  -  ( 1  / 
( 2 ^ N
) ) ) )
7674, 75eqtrd 2436 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  - 
1 )  /  (
2 ^ N ) )  =  ( X  -  ( 1  / 
( 2 ^ N
) ) ) )
7722, 42resubcld 9421 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  -  1 )  e.  RR )
7822, 43, 42, 44ltsub1dd 9594 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  -  1 )  <  ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  -  1 ) )
7927recnd 9070 . . . . . . . . . 10  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( X  x.  ( 2 ^ N ) ) )  e.  CC )
8079, 73pncand 9368 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  -  1 )  =  ( |_
`  ( X  x.  ( 2 ^ N
) ) ) )
8178, 80breqtrd 4196 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  -  1 )  <  ( |_
`  ( X  x.  ( 2 ^ N
) ) ) )
8277, 27, 81ltled 9177 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  -  1 )  <_  ( |_ `  ( X  x.  (
2 ^ N ) ) ) )
8377, 27, 28, 82lediv1dd 10658 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  - 
1 )  /  (
2 ^ N ) )  <_  ( ( |_ `  ( X  x.  ( 2 ^ N
) ) )  / 
( 2 ^ N
) ) )
8476, 83eqbrtrrd 4194 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  (
1  /  ( 2 ^ N ) ) )  <_  ( ( |_ `  ( X  x.  ( 2 ^ N
) ) )  / 
( 2 ^ N
) ) )
8559, 64, 47, 70, 84ltletrd 9186 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  D
)  <  ( ( |_ `  ( X  x.  ( 2 ^ N
) ) )  / 
( 2 ^ N
) ) )
865, 63readdcld 9071 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  +  ( 1  /  ( 2 ^ N ) ) )  e.  RR )
8722, 42readdcld 9071 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  +  1 )  e.  RR )
8827, 22, 42, 31leadd1dd 9596 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  +  1 )  <_  ( ( X  x.  ( 2 ^ N ) )  +  1 ) )
8943, 87, 28, 88lediv1dd 10658 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  <_  ( (
( X  x.  (
2 ^ N ) )  +  1 )  /  ( 2 ^ N ) ) )
9071, 73, 34, 39divdird 9784 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  +  1 )  /  (
2 ^ N ) )  =  ( ( ( X  x.  (
2 ^ N ) )  /  ( 2 ^ N ) )  +  ( 1  / 
( 2 ^ N
) ) ) )
9140oveq1d 6055 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  / 
( 2 ^ N
) )  +  ( 1  /  ( 2 ^ N ) ) )  =  ( X  +  ( 1  / 
( 2 ^ N
) ) ) )
9290, 91eqtrd 2436 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  +  1 )  /  (
2 ^ N ) )  =  ( X  +  ( 1  / 
( 2 ^ N
) ) ) )
9389, 92breqtrd 4196 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  <_  ( X  +  ( 1  / 
( 2 ^ N
) ) ) )
9463, 58, 5, 69ltadd2dd 9185 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  +  ( 1  /  ( 2 ^ N ) ) )  <  ( X  +  D ) )
9548, 86, 61, 93, 94lelttrd 9184 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  <  ( X  +  D ) )
9648, 61, 95ltled 9177 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  <_  ( X  +  D ) )
97 icossioo 24086 . . . 4  |-  ( ( ( ( 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
) ) )
9860, 62, 85, 96, 97syl22anc 1185 . . 3  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  / 
( 2 ^ N
) ) [,) (
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  +  1 )  /  ( 2 ^ N ) ) )  C_  ( ( X  -  D ) (,) ( X  +  D
) ) )
9955, 98eqsstrd 3342 . 2  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N )  C_  ( ( X  -  D ) (,) ( X  +  D
) ) )
100 eleq2 2465 . . . 4  |-  ( b  =  ( ( |_
`  ( X  x.  ( 2 ^ N
) ) ) I N )  ->  ( X  e.  b  <->  X  e.  ( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N ) ) )
101 sseq1 3329 . . . 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 ) ) ) )
102100, 101anbi12d 692 . . 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 )
) ) ) )
103102rspcev 3012 . 2  |-  ( ( ( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N )  e.  ran  I  /\  ( X  e.  ( ( |_ `  ( X  x.  ( 2 ^ N ) ) ) I N )  /\  ( ( |_
`  ( X  x.  ( 2 ^ N
) ) ) I N )  C_  (
( X  -  D
) (,) ( X  +  D ) ) ) )  ->  E. b  e.  ran  I ( X  e.  b  /\  b  C_  ( ( X  -  D ) (,) ( X  +  D )
) ) )
10426, 56, 99, 103syl12anc 1182 1  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  E. b  e.  ran  I ( X  e.  b  /\  b  C_  ( ( X  -  D ) (,) ( X  +  D )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667    C_ wss 3280   class class class wbr 4172    X. cxp 4835   ran crn 4838    Fn wfn 5408   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951   RR*cxr 9075    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   2c2 10005   ZZcz 10238   ZZ>=cuz 10444   RR+crp 10568   (,)cioo 10872   [,)cico 10874   |_cfl 11156   ^cexp 11337   topGenctg 13620  logbclogb 24341
This theorem is referenced by:  dya2icoseg2  24581
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-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ioc 10877  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625  df-sin 12627  df-cos 12628  df-pi 12630  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707  df-log 20407  df-cxp 20408  df-logb 24342
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