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Theorem dya2icoseg 28226
Description: For any point and any closed-below, open-above interval of  RR centered on that point, there is a closed-below open-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 6309 . . . . 5  |-  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) )  e. 
_V
31, 2fnmpt2i 6854 . . . 4  |-  I  Fn  ( ZZ  X.  ZZ )
43a1i 11 . . 3  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  I  Fn  ( ZZ  X.  ZZ ) )
5 simpl 457 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  X  e.  RR )
6 2rp 11236 . . . . . . 7  |-  2  e.  RR+
7 dya2icoseg.1 . . . . . . . 8  |-  N  =  ( |_ `  (
1  -  ( 2logb D ) ) )
8 1red 9614 . . . . . . . . . 10  |-  ( D  e.  RR+  ->  1  e.  RR )
9 2z 10903 . . . . . . . . . . . 12  |-  2  e.  ZZ
10 uzid 11106 . . . . . . . . . . . 12  |-  ( 2  e.  ZZ  ->  2  e.  ( ZZ>= `  2 )
)
119, 10ax-mp 5 . . . . . . . . . . 11  |-  2  e.  ( ZZ>= `  2 )
12 rnlogbcl 27995 . . . . . . . . . . 11  |-  ( ( 2  e.  ( ZZ>= ` 
2 )  /\  D  e.  RR+ )  ->  (
2logb D )  e.  RR )
1311, 12mpan 670 . . . . . . . . . 10  |-  ( D  e.  RR+  ->  ( 2logb D )  e.  RR )
148, 13resubcld 9994 . . . . . . . . 9  |-  ( D  e.  RR+  ->  ( 1  -  ( 2logb D
) )  e.  RR )
1514flcld 11917 . . . . . . . 8  |-  ( D  e.  RR+  ->  ( |_
`  ( 1  -  ( 2logb D ) ) )  e.  ZZ )
167, 15syl5eqel 2535 . . . . . . 7  |-  ( D  e.  RR+  ->  N  e.  ZZ )
17 rpexpcl 12167 . . . . . . . 8  |-  ( ( 2  e.  RR+  /\  N  e.  ZZ )  ->  (
2 ^ N )  e.  RR+ )
1817rpred 11267 . . . . . . 7  |-  ( ( 2  e.  RR+  /\  N  e.  ZZ )  ->  (
2 ^ N )  e.  RR )
196, 16, 18sylancr 663 . . . . . 6  |-  ( D  e.  RR+  ->  ( 2 ^ N )  e.  RR )
2019adantl 466 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 2 ^ N
)  e.  RR )
215, 20remulcld 9627 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  x.  (
2 ^ N ) )  e.  RR )
2221flcld 11917 . . 3  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( X  x.  ( 2 ^ N ) ) )  e.  ZZ )
2316adantl 466 . . 3  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  N  e.  ZZ )
24 fnovrn 6435 . . 3  |-  ( ( I  Fn  ( ZZ 
X.  ZZ )  /\  ( |_ `  ( X  x.  ( 2 ^ N ) ) )  e.  ZZ  /\  N  e.  ZZ )  ->  (
( |_ `  ( X  x.  ( 2 ^ N ) ) ) I N )  e.  ran  I )
254, 22, 23, 24syl3anc 1229 . 2  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N )  e.  ran  I
)
2622zred 10976 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( X  x.  ( 2 ^ N ) ) )  e.  RR )
276, 23, 17sylancr 663 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 2 ^ N
)  e.  RR+ )
28 fllelt 11916 . . . . . . . 8  |-  ( ( X  x.  ( 2 ^ N ) )  e.  RR  ->  (
( |_ `  ( X  x.  ( 2 ^ N ) ) )  <_  ( X  x.  ( 2 ^ N
) )  /\  ( X  x.  ( 2 ^ N ) )  <  ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 ) ) )
2921, 28syl 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 ) ) )
3029simpld 459 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( X  x.  ( 2 ^ N ) ) )  <_  ( X  x.  ( 2 ^ N
) ) )
3126, 21, 27, 30lediv1dd 11321 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  /  (
2 ^ N ) )  <_  ( ( X  x.  ( 2 ^ N ) )  /  ( 2 ^ N ) ) )
325recnd 9625 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  X  e.  CC )
3320recnd 9625 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 2 ^ N
)  e.  CC )
34 2cnd 10615 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
2  e.  CC )
35 2ne0 10635 . . . . . . . 8  |-  2  =/=  0
3635a1i 11 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
2  =/=  0 )
3734, 36, 23expne0d 12298 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 2 ^ N
)  =/=  0 )
3832, 33, 37divcan4d 10333 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  /  (
2 ^ N ) )  =  X )
3931, 38breqtrd 4461 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  /  (
2 ^ N ) )  <_  X )
40 1red 9614 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
1  e.  RR )
4126, 40readdcld 9626 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  +  1 )  e.  RR )
4229simprd 463 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  x.  (
2 ^ N ) )  <  ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 ) )
4321, 41, 27, 42ltdiv1dd 11320 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  /  (
2 ^ N ) )  <  ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  /  ( 2 ^ N ) ) )
4438, 43eqbrtrrd 4459 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  X  <  ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  / 
( 2 ^ N
) ) )
4526, 20, 37redivcld 10379 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  /  (
2 ^ N ) )  e.  RR )
4641, 20, 37redivcld 10379 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  e.  RR )
4746rexrd 9646 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  e.  RR* )
48 elico2 11599 . . . . 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
) ) ) ) )
4945, 47, 48syl2anc 661 . . . 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
) ) ) ) )
505, 39, 44, 49mpbir3and 1180 . . 3  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  X  e.  ( (
( |_ `  ( X  x.  ( 2 ^ N ) ) )  /  ( 2 ^ N ) ) [,) ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  / 
( 2 ^ N
) ) ) )
51 sxbrsiga.0 . . . . 5  |-  J  =  ( topGen `  ran  (,) )
5251, 1dya2iocival 28222 . . . 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
) ) ) )
5323, 22, 52syl2anc 661 . . 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
) ) ) )
5450, 53eleqtrrd 2534 . 2  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  X  e.  ( ( |_ `  ( X  x.  ( 2 ^ N
) ) ) I N ) )
55 simpr 461 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  D  e.  RR+ )
5655rpred 11267 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  D  e.  RR )
575, 56resubcld 9994 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  D
)  e.  RR )
5857rexrd 9646 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  D
)  e.  RR* )
595, 56readdcld 9626 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  +  D
)  e.  RR )
6059rexrd 9646 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  +  D
)  e.  RR* )
6120, 37rereccld 10378 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 1  /  (
2 ^ N ) )  e.  RR )
625, 61resubcld 9994 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  (
1  /  ( 2 ^ N ) ) )  e.  RR )
637oveq2i 6292 . . . . . . . 8  |-  ( 2 ^ N )  =  ( 2 ^ ( |_ `  ( 1  -  ( 2logb D ) ) ) )
6463oveq2i 6292 . . . . . . 7  |-  ( 1  /  ( 2 ^ N ) )  =  ( 1  /  (
2 ^ ( |_
`  ( 1  -  ( 2logb D ) ) ) ) )
65 dya2ub 28219 . . . . . . . 8  |-  ( D  e.  RR+  ->  ( 1  /  ( 2 ^ ( |_ `  (
1  -  ( 2logb D ) ) ) ) )  <  D
)
6665adantl 466 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 1  /  (
2 ^ ( |_
`  ( 1  -  ( 2logb D ) ) ) ) )  <  D )
6764, 66syl5eqbr 4470 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 1  /  (
2 ^ N ) )  <  D )
6861, 56, 5, 67ltsub2dd 10172 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  D
)  <  ( X  -  ( 1  / 
( 2 ^ N
) ) ) )
6932, 33mulcld 9619 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  x.  (
2 ^ N ) )  e.  CC )
70 1cnd 9615 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
1  e.  CC )
7169, 70, 33, 37divsubdird 10366 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  - 
1 )  /  (
2 ^ N ) )  =  ( ( ( X  x.  (
2 ^ N ) )  /  ( 2 ^ N ) )  -  ( 1  / 
( 2 ^ N
) ) ) )
7238oveq1d 6296 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  / 
( 2 ^ N
) )  -  (
1  /  ( 2 ^ N ) ) )  =  ( X  -  ( 1  / 
( 2 ^ N
) ) ) )
7371, 72eqtrd 2484 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  - 
1 )  /  (
2 ^ N ) )  =  ( X  -  ( 1  / 
( 2 ^ N
) ) ) )
7421, 40resubcld 9994 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  -  1 )  e.  RR )
7521, 41, 40, 42ltsub1dd 10171 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  -  1 )  <  ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  -  1 ) )
7626recnd 9625 . . . . . . . . . 10  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( X  x.  ( 2 ^ N ) ) )  e.  CC )
7776, 70pncand 9937 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  -  1 )  =  ( |_
`  ( X  x.  ( 2 ^ N
) ) ) )
7875, 77breqtrd 4461 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  -  1 )  <  ( |_
`  ( X  x.  ( 2 ^ N
) ) ) )
7974, 26, 78ltled 9736 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  -  1 )  <_  ( |_ `  ( X  x.  (
2 ^ N ) ) ) )
8074, 26, 27, 79lediv1dd 11321 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  - 
1 )  /  (
2 ^ N ) )  <_  ( ( |_ `  ( X  x.  ( 2 ^ N
) ) )  / 
( 2 ^ N
) ) )
8173, 80eqbrtrrd 4459 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  (
1  /  ( 2 ^ N ) ) )  <_  ( ( |_ `  ( X  x.  ( 2 ^ N
) ) )  / 
( 2 ^ N
) ) )
8257, 62, 45, 68, 81ltletrd 9745 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  D
)  <  ( ( |_ `  ( X  x.  ( 2 ^ N
) ) )  / 
( 2 ^ N
) ) )
835, 61readdcld 9626 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  +  ( 1  /  ( 2 ^ N ) ) )  e.  RR )
8421, 40readdcld 9626 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  +  1 )  e.  RR )
8526, 21, 40, 30leadd1dd 10173 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  +  1 )  <_  ( ( X  x.  ( 2 ^ N ) )  +  1 ) )
8641, 84, 27, 85lediv1dd 11321 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  <_  ( (
( X  x.  (
2 ^ N ) )  +  1 )  /  ( 2 ^ N ) ) )
8769, 70, 33, 37divdird 10365 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  +  1 )  /  (
2 ^ N ) )  =  ( ( ( X  x.  (
2 ^ N ) )  /  ( 2 ^ N ) )  +  ( 1  / 
( 2 ^ N
) ) ) )
8838oveq1d 6296 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  / 
( 2 ^ N
) )  +  ( 1  /  ( 2 ^ N ) ) )  =  ( X  +  ( 1  / 
( 2 ^ N
) ) ) )
8987, 88eqtrd 2484 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  +  1 )  /  (
2 ^ N ) )  =  ( X  +  ( 1  / 
( 2 ^ N
) ) ) )
9086, 89breqtrd 4461 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  <_  ( X  +  ( 1  / 
( 2 ^ N
) ) ) )
9161, 56, 5, 67ltadd2dd 9744 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  +  ( 1  /  ( 2 ^ N ) ) )  <  ( X  +  D ) )
9246, 83, 59, 90, 91lelttrd 9743 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  <  ( X  +  D ) )
9346, 59, 92ltled 9736 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  <_  ( X  +  D ) )
94 icossioo 11626 . . . 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
) ) )
9558, 60, 82, 93, 94syl22anc 1230 . . 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
) ) )
9653, 95eqsstrd 3523 . 2  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N )  C_  ( ( X  -  D ) (,) ( X  +  D
) ) )
97 eleq2 2516 . . . 4  |-  ( b  =  ( ( |_
`  ( X  x.  ( 2 ^ N
) ) ) I N )  ->  ( X  e.  b  <->  X  e.  ( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N ) ) )
98 sseq1 3510 . . . 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 ) ) ) )
9997, 98anbi12d 710 . . 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 )
) ) ) )
10099rspcev 3196 . 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 )
) ) )
10125, 54, 96, 100syl12anc 1227 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   E.wrex 2794    C_ wss 3461   class class class wbr 4437    X. cxp 4987   ran crn 4990    Fn wfn 5573   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   RRcr 9494   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500   RR*cxr 9630    < clt 9631    <_ cle 9632    - cmin 9810    / cdiv 10213   2c2 10592   ZZcz 10871   ZZ>=cuz 11092   RR+crp 11231   (,)cioo 11540   [,)cico 11542   |_cfl 11909   ^cexp 12148   topGenctg 14817  logbclogb 27984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-fi 7873  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-uz 11093  df-q 11194  df-rp 11232  df-xneg 11329  df-xadd 11330  df-xmul 11331  df-ioo 11544  df-ioc 11545  df-ico 11546  df-icc 11547  df-fz 11684  df-fzo 11807  df-fl 11911  df-mod 11979  df-seq 12090  df-exp 12149  df-fac 12336  df-bc 12363  df-hash 12388  df-shft 12882  df-cj 12914  df-re 12915  df-im 12916  df-sqrt 13050  df-abs 13051  df-limsup 13276  df-clim 13293  df-rlim 13294  df-sum 13491  df-ef 13785  df-sin 13787  df-cos 13788  df-pi 13790  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-mulr 14693  df-starv 14694  df-sca 14695  df-vsca 14696  df-ip 14697  df-tset 14698  df-ple 14699  df-ds 14701  df-unif 14702  df-hom 14703  df-cco 14704  df-rest 14802  df-topn 14803  df-0g 14821  df-gsum 14822  df-topgen 14823  df-pt 14824  df-prds 14827  df-xrs 14881  df-qtop 14886  df-imas 14887  df-xps 14889  df-mre 14965  df-mrc 14966  df-acs 14968  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-submnd 15946  df-mulg 16039  df-cntz 16334  df-cmn 16779  df-psmet 18390  df-xmet 18391  df-met 18392  df-bl 18393  df-mopn 18394  df-fbas 18395  df-fg 18396  df-cnfld 18400  df-top 19377  df-bases 19379  df-topon 19380  df-topsp 19381  df-cld 19498  df-ntr 19499  df-cls 19500  df-nei 19577  df-lp 19615  df-perf 19616  df-cn 19706  df-cnp 19707  df-haus 19794  df-tx 20041  df-hmeo 20234  df-fil 20325  df-fm 20417  df-flim 20418  df-flf 20419  df-xms 20801  df-ms 20802  df-tms 20803  df-cncf 21360  df-limc 22248  df-dv 22249  df-log 22922  df-cxp 22923  df-logb 27985
This theorem is referenced by:  dya2icoseg2  28227
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