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Theorem dya2icoseg 28606
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  -  ( 2 logb  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 6260 . . . . 5  |-  ( ( x  /  ( 2 ^ n ) ) [,) ( ( x  +  1 )  / 
( 2 ^ n
) ) )  e. 
_V
31, 2fnmpt2i 6805 . . . 4  |-  I  Fn  ( ZZ  X.  ZZ )
43a1i 11 . . 3  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  I  Fn  ( ZZ  X.  ZZ ) )
5 simpl 455 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  X  e.  RR )
6 2rp 11186 . . . . . . 7  |-  2  e.  RR+
7 dya2icoseg.1 . . . . . . . 8  |-  N  =  ( |_ `  (
1  -  ( 2 logb  D ) ) )
8 1red 9559 . . . . . . . . . 10  |-  ( D  e.  RR+  ->  1  e.  RR )
9 2z 10855 . . . . . . . . . . . 12  |-  2  e.  ZZ
10 uzid 11057 . . . . . . . . . . . 12  |-  ( 2  e.  ZZ  ->  2  e.  ( ZZ>= `  2 )
)
119, 10ax-mp 5 . . . . . . . . . . 11  |-  2  e.  ( ZZ>= `  2 )
12 relogbzcl 23331 . . . . . . . . . . 11  |-  ( ( 2  e.  ( ZZ>= ` 
2 )  /\  D  e.  RR+ )  ->  (
2 logb  D )  e.  RR )
1311, 12mpan 668 . . . . . . . . . 10  |-  ( D  e.  RR+  ->  ( 2 logb  D )  e.  RR )
148, 13resubcld 9946 . . . . . . . . 9  |-  ( D  e.  RR+  ->  ( 1  -  ( 2 logb  D ) )  e.  RR )
1514flcld 11883 . . . . . . . 8  |-  ( D  e.  RR+  ->  ( |_
`  ( 1  -  ( 2 logb  D ) ) )  e.  ZZ )
167, 15syl5eqel 2492 . . . . . . 7  |-  ( D  e.  RR+  ->  N  e.  ZZ )
17 rpexpcl 12137 . . . . . . . 8  |-  ( ( 2  e.  RR+  /\  N  e.  ZZ )  ->  (
2 ^ N )  e.  RR+ )
1817rpred 11220 . . . . . . 7  |-  ( ( 2  e.  RR+  /\  N  e.  ZZ )  ->  (
2 ^ N )  e.  RR )
196, 16, 18sylancr 661 . . . . . 6  |-  ( D  e.  RR+  ->  ( 2 ^ N )  e.  RR )
2019adantl 464 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 2 ^ N
)  e.  RR )
215, 20remulcld 9572 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  x.  (
2 ^ N ) )  e.  RR )
2221flcld 11883 . . 3  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( X  x.  ( 2 ^ N ) ) )  e.  ZZ )
2316adantl 464 . . 3  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  N  e.  ZZ )
24 fnovrn 6385 . . 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 1228 . 2  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N )  e.  ran  I
)
2622zred 10926 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( X  x.  ( 2 ^ N ) ) )  e.  RR )
276, 23, 17sylancr 661 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 2 ^ N
)  e.  RR+ )
28 fllelt 11882 . . . . . . . 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 17 . . . . . . 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 457 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( X  x.  ( 2 ^ N ) ) )  <_  ( X  x.  ( 2 ^ N
) ) )
3126, 21, 27, 30lediv1dd 11274 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  /  (
2 ^ N ) )  <_  ( ( X  x.  ( 2 ^ N ) )  /  ( 2 ^ N ) ) )
325recnd 9570 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  X  e.  CC )
3320recnd 9570 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 2 ^ N
)  e.  CC )
34 2cnd 10567 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
2  e.  CC )
35 2ne0 10587 . . . . . . . 8  |-  2  =/=  0
3635a1i 11 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
2  =/=  0 )
3734, 36, 23expne0d 12268 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 2 ^ N
)  =/=  0 )
3832, 33, 37divcan4d 10285 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  /  (
2 ^ N ) )  =  X )
3931, 38breqtrd 4416 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  /  (
2 ^ N ) )  <_  X )
40 1red 9559 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
1  e.  RR )
4126, 40readdcld 9571 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  +  1 )  e.  RR )
4229simprd 461 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  x.  (
2 ^ N ) )  <  ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 ) )
4321, 41, 27, 42ltdiv1dd 11273 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  /  (
2 ^ N ) )  <  ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  /  ( 2 ^ N ) ) )
4438, 43eqbrtrrd 4414 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  X  <  ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  / 
( 2 ^ N
) ) )
4526, 20, 37redivcld 10331 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  /  (
2 ^ N ) )  e.  RR )
4641, 20, 37redivcld 10331 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  e.  RR )
4746rexrd 9591 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  e.  RR* )
48 elico2 11557 . . . . 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 659 . . . 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 1178 . . 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 28602 . . . 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 659 . . 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 2491 . 2  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  X  e.  ( ( |_ `  ( X  x.  ( 2 ^ N
) ) ) I N ) )
55 simpr 459 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  D  e.  RR+ )
5655rpred 11220 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  ->  D  e.  RR )
575, 56resubcld 9946 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  D
)  e.  RR )
5857rexrd 9591 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  D
)  e.  RR* )
595, 56readdcld 9571 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  +  D
)  e.  RR )
6059rexrd 9591 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  +  D
)  e.  RR* )
6120, 37rereccld 10330 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 1  /  (
2 ^ N ) )  e.  RR )
625, 61resubcld 9946 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  (
1  /  ( 2 ^ N ) ) )  e.  RR )
637oveq2i 6243 . . . . . . . 8  |-  ( 2 ^ N )  =  ( 2 ^ ( |_ `  ( 1  -  ( 2 logb  D ) ) ) )
6463oveq2i 6243 . . . . . . 7  |-  ( 1  /  ( 2 ^ N ) )  =  ( 1  /  (
2 ^ ( |_
`  ( 1  -  ( 2 logb  D ) ) ) ) )
65 dya2ub 28599 . . . . . . . 8  |-  ( D  e.  RR+  ->  ( 1  /  ( 2 ^ ( |_ `  (
1  -  ( 2 logb  D ) ) ) ) )  <  D )
6665adantl 464 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 1  /  (
2 ^ ( |_
`  ( 1  -  ( 2 logb  D ) ) ) ) )  < 
D )
6764, 66syl5eqbr 4425 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( 1  /  (
2 ^ N ) )  <  D )
6861, 56, 5, 67ltsub2dd 10123 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  D
)  <  ( X  -  ( 1  / 
( 2 ^ N
) ) ) )
6932, 33mulcld 9564 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  x.  (
2 ^ N ) )  e.  CC )
70 1cnd 9560 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
1  e.  CC )
7169, 70, 33, 37divsubdird 10318 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  - 
1 )  /  (
2 ^ N ) )  =  ( ( ( X  x.  (
2 ^ N ) )  /  ( 2 ^ N ) )  -  ( 1  / 
( 2 ^ N
) ) ) )
7238oveq1d 6247 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  / 
( 2 ^ N
) )  -  (
1  /  ( 2 ^ N ) ) )  =  ( X  -  ( 1  / 
( 2 ^ N
) ) ) )
7371, 72eqtrd 2441 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  - 
1 )  /  (
2 ^ N ) )  =  ( X  -  ( 1  / 
( 2 ^ N
) ) ) )
7421, 40resubcld 9946 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  -  1 )  e.  RR )
7521, 41, 40, 42ltsub1dd 10122 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  -  1 )  <  ( ( ( |_ `  ( X  x.  ( 2 ^ N ) ) )  +  1 )  -  1 ) )
7626recnd 9570 . . . . . . . . . 10  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( |_ `  ( X  x.  ( 2 ^ N ) ) )  e.  CC )
7776, 70pncand 9886 . . . . . . . . 9  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  -  1 )  =  ( |_
`  ( X  x.  ( 2 ^ N
) ) ) )
7875, 77breqtrd 4416 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  -  1 )  <  ( |_
`  ( X  x.  ( 2 ^ N
) ) ) )
7974, 26, 78ltled 9683 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  -  1 )  <_  ( |_ `  ( X  x.  (
2 ^ N ) ) ) )
8074, 26, 27, 79lediv1dd 11274 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  - 
1 )  /  (
2 ^ N ) )  <_  ( ( |_ `  ( X  x.  ( 2 ^ N
) ) )  / 
( 2 ^ N
) ) )
8173, 80eqbrtrrd 4414 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  (
1  /  ( 2 ^ N ) ) )  <_  ( ( |_ `  ( X  x.  ( 2 ^ N
) ) )  / 
( 2 ^ N
) ) )
8257, 62, 45, 68, 81ltletrd 9694 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  -  D
)  <  ( ( |_ `  ( X  x.  ( 2 ^ N
) ) )  / 
( 2 ^ N
) ) )
835, 61readdcld 9571 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  +  ( 1  /  ( 2 ^ N ) ) )  e.  RR )
8421, 40readdcld 9571 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( X  x.  ( 2 ^ N
) )  +  1 )  e.  RR )
8526, 21, 40, 30leadd1dd 10124 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) )  +  1 )  <_  ( ( X  x.  ( 2 ^ N ) )  +  1 ) )
8641, 84, 27, 85lediv1dd 11274 . . . . . . 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 10317 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  +  1 )  /  (
2 ^ N ) )  =  ( ( ( X  x.  (
2 ^ N ) )  /  ( 2 ^ N ) )  +  ( 1  / 
( 2 ^ N
) ) ) )
8838oveq1d 6247 . . . . . . . 8  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  / 
( 2 ^ N
) )  +  ( 1  /  ( 2 ^ N ) ) )  =  ( X  +  ( 1  / 
( 2 ^ N
) ) ) )
8987, 88eqtrd 2441 . . . . . . 7  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( X  x.  ( 2 ^ N ) )  +  1 )  /  (
2 ^ N ) )  =  ( X  +  ( 1  / 
( 2 ^ N
) ) ) )
9086, 89breqtrd 4416 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  <_  ( X  +  ( 1  / 
( 2 ^ N
) ) ) )
9161, 56, 5, 67ltadd2dd 9693 . . . . . 6  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( X  +  ( 1  /  ( 2 ^ N ) ) )  <  ( X  +  D ) )
9246, 83, 59, 90, 91lelttrd 9692 . . . . 5  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  <  ( X  +  D ) )
9346, 59, 92ltled 9683 . . . 4  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( ( |_
`  ( X  x.  ( 2 ^ N
) ) )  +  1 )  /  (
2 ^ N ) )  <_  ( X  +  D ) )
94 icossioo 11584 . . . 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 1229 . . 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 3473 . 2  |-  ( ( X  e.  RR  /\  D  e.  RR+ )  -> 
( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N )  C_  ( ( X  -  D ) (,) ( X  +  D
) ) )
97 eleq2 2473 . . . 4  |-  ( b  =  ( ( |_
`  ( X  x.  ( 2 ^ N
) ) ) I N )  ->  ( X  e.  b  <->  X  e.  ( ( |_ `  ( X  x.  (
2 ^ N ) ) ) I N ) ) )
98 sseq1 3460 . . . 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 709 . . 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 3157 . 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 1226 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 367    /\ w3a 972    = wceq 1403    e. wcel 1840    =/= wne 2596   E.wrex 2752    C_ wss 3411   class class class wbr 4392    X. cxp 4938   ran crn 4941    Fn wfn 5518   ` cfv 5523  (class class class)co 6232    |-> cmpt2 6234   RRcr 9439   0cc0 9440   1c1 9441    + caddc 9443    x. cmul 9445   RR*cxr 9575    < clt 9576    <_ cle 9577    - cmin 9759    / cdiv 10165   2c2 10544   ZZcz 10823   ZZ>=cuz 11043   RR+crp 11181   (,)cioo 11498   [,)cico 11500   |_cfl 11875   ^cexp 12118   topGenctg 14942   logb clogb 23321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-inf2 8009  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-pre-sup 9518  ax-addf 9519  ax-mulf 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-fal 1409  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-iin 4271  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-isom 5532  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-of 6475  df-om 6637  df-1st 6736  df-2nd 6737  df-supp 6855  df-recs 6997  df-rdg 7031  df-1o 7085  df-2o 7086  df-oadd 7089  df-er 7266  df-map 7377  df-pm 7378  df-ixp 7426  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-fsupp 7782  df-fi 7823  df-sup 7853  df-oi 7887  df-card 8270  df-cda 8498  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-div 10166  df-nn 10495  df-2 10553  df-3 10554  df-4 10555  df-5 10556  df-6 10557  df-7 10558  df-8 10559  df-9 10560  df-10 10561  df-n0 10755  df-z 10824  df-dec 10938  df-uz 11044  df-q 11144  df-rp 11182  df-xneg 11287  df-xadd 11288  df-xmul 11289  df-ioo 11502  df-ioc 11503  df-ico 11504  df-icc 11505  df-fz 11642  df-fzo 11766  df-fl 11877  df-mod 11946  df-seq 12060  df-exp 12119  df-fac 12306  df-bc 12333  df-hash 12358  df-shft 12954  df-cj 12986  df-re 12987  df-im 12988  df-sqrt 13122  df-abs 13123  df-limsup 13348  df-clim 13365  df-rlim 13366  df-sum 13563  df-ef 13902  df-sin 13904  df-cos 13905  df-pi 13907  df-struct 14733  df-ndx 14734  df-slot 14735  df-base 14736  df-sets 14737  df-ress 14738  df-plusg 14812  df-mulr 14813  df-starv 14814  df-sca 14815  df-vsca 14816  df-ip 14817  df-tset 14818  df-ple 14819  df-ds 14821  df-unif 14822  df-hom 14823  df-cco 14824  df-rest 14927  df-topn 14928  df-0g 14946  df-gsum 14947  df-topgen 14948  df-pt 14949  df-prds 14952  df-xrs 15006  df-qtop 15011  df-imas 15012  df-xps 15014  df-mre 15090  df-mrc 15091  df-acs 15093  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-submnd 16181  df-mulg 16274  df-cntz 16569  df-cmn 17014  df-psmet 18621  df-xmet 18622  df-met 18623  df-bl 18624  df-mopn 18625  df-fbas 18626  df-fg 18627  df-cnfld 18631  df-top 19581  df-bases 19583  df-topon 19584  df-topsp 19585  df-cld 19702  df-ntr 19703  df-cls 19704  df-nei 19782  df-lp 19820  df-perf 19821  df-cn 19911  df-cnp 19912  df-haus 19999  df-tx 20245  df-hmeo 20438  df-fil 20529  df-fm 20621  df-flim 20622  df-flf 20623  df-xms 21005  df-ms 21006  df-tms 21007  df-cncf 21564  df-limc 22452  df-dv 22453  df-log 23126  df-cxp 23127  df-logb 23322
This theorem is referenced by:  dya2icoseg2  28607
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