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Theorem opnmbllem 18956
Description: Lemma for opnmbl 18957. (Contributed by Mario Carneiro, 26-Mar-2015.)
Hypothesis
Ref Expression
dyadmbl.1  |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  (
2 ^ y ) ) ,  ( ( x  +  1 )  /  ( 2 ^ y ) ) >.
)
Assertion
Ref Expression
opnmbllem  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  e.  dom  vol )
Distinct variable groups:    x, y, A    x, F, y

Proof of Theorem opnmbllem
Dummy variables  c 
a  b  n  w  z  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . . . . . . 9  |-  ( z  =  w  ->  ( [,] `  z )  =  ( [,] `  w
) )
21sseq1d 3205 . . . . . . . 8  |-  ( z  =  w  ->  (
( [,] `  z
)  C_  A  <->  ( [,] `  w )  C_  A
) )
32elrab 2923 . . . . . . 7  |-  ( w  e.  { z  e. 
ran  F  |  ( [,] `  z )  C_  A }  <->  ( w  e. 
ran  F  /\  ( [,] `  w )  C_  A ) )
4 simprr 733 . . . . . . . 8  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  (
w  e.  ran  F  /\  ( [,] `  w
)  C_  A )
)  ->  ( [,] `  w )  C_  A
)
5 fvex 5539 . . . . . . . . 9  |-  ( [,] `  w )  e.  _V
65elpw 3631 . . . . . . . 8  |-  ( ( [,] `  w )  e.  ~P A  <->  ( [,] `  w )  C_  A
)
74, 6sylibr 203 . . . . . . 7  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  (
w  e.  ran  F  /\  ( [,] `  w
)  C_  A )
)  ->  ( [,] `  w )  e.  ~P A )
83, 7sylan2b 461 . . . . . 6  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  w  e.  { z  e.  ran  F  |  ( [,] `  z
)  C_  A }
)  ->  ( [,] `  w )  e.  ~P A )
98ralrimiva 2626 . . . . 5  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A. w  e.  { z  e.  ran  F  |  ( [,] `  z
)  C_  A } 
( [,] `  w
)  e.  ~P A
)
10 iccf 10742 . . . . . . 7  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
11 ffun 5391 . . . . . . 7  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  Fun  [,] )
1210, 11ax-mp 8 . . . . . 6  |-  Fun  [,]
13 ssrab2 3258 . . . . . . . 8  |-  { z  e.  ran  F  | 
( [,] `  z
)  C_  A }  C_ 
ran  F
14 dyadmbl.1 . . . . . . . . . . 11  |-  F  =  ( x  e.  ZZ ,  y  e.  NN0  |->  <. ( x  /  (
2 ^ y ) ) ,  ( ( x  +  1 )  /  ( 2 ^ y ) ) >.
)
1514dyadf 18946 . . . . . . . . . 10  |-  F :
( ZZ  X.  NN0 )
--> (  <_  i^i  ( RR  X.  RR ) )
16 frn 5395 . . . . . . . . . 10  |-  ( F : ( ZZ  X.  NN0 ) --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  F 
C_  (  <_  i^i  ( RR  X.  RR ) ) )
1715, 16ax-mp 8 . . . . . . . . 9  |-  ran  F  C_  (  <_  i^i  ( RR  X.  RR ) )
18 inss2 3390 . . . . . . . . . 10  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
19 ressxr 8876 . . . . . . . . . . 11  |-  RR  C_  RR*
20 xpss12 4792 . . . . . . . . . . 11  |-  ( ( RR  C_  RR*  /\  RR  C_ 
RR* )  ->  ( RR  X.  RR )  C_  ( RR*  X.  RR* )
)
2119, 19, 20mp2an 653 . . . . . . . . . 10  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
2218, 21sstri 3188 . . . . . . . . 9  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
2317, 22sstri 3188 . . . . . . . 8  |-  ran  F  C_  ( RR*  X.  RR* )
2413, 23sstri 3188 . . . . . . 7  |-  { z  e.  ran  F  | 
( [,] `  z
)  C_  A }  C_  ( RR*  X.  RR* )
2510fdmi 5394 . . . . . . 7  |-  dom  [,]  =  ( RR*  X.  RR* )
2624, 25sseqtr4i 3211 . . . . . 6  |-  { z  e.  ran  F  | 
( [,] `  z
)  C_  A }  C_ 
dom  [,]
27 funimass4 5573 . . . . . 6  |-  ( ( Fun  [,]  /\  { z  e.  ran  F  | 
( [,] `  z
)  C_  A }  C_ 
dom  [,] )  ->  (
( [,] " {
z  e.  ran  F  |  ( [,] `  z
)  C_  A }
)  C_  ~P A  <->  A. w  e.  { z  e.  ran  F  | 
( [,] `  z
)  C_  A } 
( [,] `  w
)  e.  ~P A
) )
2812, 26, 27mp2an 653 . . . . 5  |-  ( ( [,] " { z  e.  ran  F  | 
( [,] `  z
)  C_  A }
)  C_  ~P A  <->  A. w  e.  { z  e.  ran  F  | 
( [,] `  z
)  C_  A } 
( [,] `  w
)  e.  ~P A
)
299, 28sylibr 203 . . . 4  |-  ( A  e.  ( topGen `  ran  (,) )  ->  ( [,] " { z  e.  ran  F  |  ( [,] `  z
)  C_  A }
)  C_  ~P A
)
30 sspwuni 3987 . . . 4  |-  ( ( [,] " { z  e.  ran  F  | 
( [,] `  z
)  C_  A }
)  C_  ~P A  <->  U. ( [,] " {
z  e.  ran  F  |  ( [,] `  z
)  C_  A }
)  C_  A )
3129, 30sylib 188 . . 3  |-  ( A  e.  ( topGen `  ran  (,) )  ->  U. ( [,] " { z  e. 
ran  F  |  ( [,] `  z )  C_  A } )  C_  A
)
32 eqid 2283 . . . . . . . 8  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
3332rexmet 18297 . . . . . . 7  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )
34 eqid 2283 . . . . . . . . 9  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
3532, 34tgioo 18302 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
3635mopni2 18039 . . . . . . 7  |-  ( ( ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )  e.  ( * Met `  RR )  /\  A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  ->  E. r  e.  RR+  ( w (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) r )  C_  A )
3733, 36mp3an1 1264 . . . . . 6  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  w  e.  A )  ->  E. r  e.  RR+  ( w (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) r )  C_  A )
38 elssuni 3855 . . . . . . . . . . . 12  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  C_  U. ( topGen `
 ran  (,) )
)
39 uniretop 18271 . . . . . . . . . . . 12  |-  RR  =  U. ( topGen `  ran  (,) )
4038, 39syl6sseqr 3225 . . . . . . . . . . 11  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  C_  RR )
4140sselda 3180 . . . . . . . . . 10  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  w  e.  A )  ->  w  e.  RR )
42 rpre 10360 . . . . . . . . . 10  |-  ( r  e.  RR+  ->  r  e.  RR )
4332bl2ioo 18298 . . . . . . . . . 10  |-  ( ( w  e.  RR  /\  r  e.  RR )  ->  ( w ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  =  ( ( w  -  r ) (,) (
w  +  r ) ) )
4441, 42, 43syl2an 463 . . . . . . . . 9  |-  ( ( ( A  e.  (
topGen `  ran  (,) )  /\  w  e.  A
)  /\  r  e.  RR+ )  ->  ( w
( ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) r )  =  ( ( w  -  r ) (,) ( w  +  r ) ) )
4544sseq1d 3205 . . . . . . . 8  |-  ( ( ( A  e.  (
topGen `  ran  (,) )  /\  w  e.  A
)  /\  r  e.  RR+ )  ->  ( (
w ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) r )  C_  A  <->  ( (
w  -  r ) (,) ( w  +  r ) )  C_  A ) )
46 2re 9815 . . . . . . . . . . . 12  |-  2  e.  RR
47 1lt2 9886 . . . . . . . . . . . 12  |-  1  <  2
48 expnlbnd 11231 . . . . . . . . . . . 12  |-  ( ( r  e.  RR+  /\  2  e.  RR  /\  1  <  2 )  ->  E. n  e.  NN  ( 1  / 
( 2 ^ n
) )  <  r
)
4946, 47, 48mp3an23 1269 . . . . . . . . . . 11  |-  ( r  e.  RR+  ->  E. n  e.  NN  ( 1  / 
( 2 ^ n
) )  <  r
)
5049ad2antrl 708 . . . . . . . . . 10  |-  ( ( ( A  e.  (
topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  ->  E. n  e.  NN  ( 1  /  (
2 ^ n ) )  <  r )
5141ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  w  e.  RR )
52 2nn 9877 . . . . . . . . . . . . . . . . . . . . 21  |-  2  e.  NN
53 nnnn0 9972 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( n  e.  NN  ->  n  e.  NN0 )
5453ad2antrl 708 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  n  e.  NN0 )
55 nnexpcl 11116 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 2  e.  NN  /\  n  e.  NN0 )  -> 
( 2 ^ n
)  e.  NN )
5652, 54, 55sylancr 644 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( 2 ^ n )  e.  NN )
5756nnred 9761 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( 2 ^ n )  e.  RR )
5851, 57remulcld 8863 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( w  x.  ( 2 ^ n
) )  e.  RR )
59 fllelt 10929 . . . . . . . . . . . . . . . . . 18  |-  ( ( w  x.  ( 2 ^ n ) )  e.  RR  ->  (
( |_ `  (
w  x.  ( 2 ^ n ) ) )  <_  ( w  x.  ( 2 ^ n
) )  /\  (
w  x.  ( 2 ^ n ) )  <  ( ( |_
`  ( w  x.  ( 2 ^ n
) ) )  +  1 ) ) )
6058, 59syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( ( |_ `  ( w  x.  ( 2 ^ n
) ) )  <_ 
( w  x.  (
2 ^ n ) )  /\  ( w  x.  ( 2 ^ n ) )  < 
( ( |_ `  ( w  x.  (
2 ^ n ) ) )  +  1 ) ) )
6160simpld 445 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( |_ `  ( w  x.  (
2 ^ n ) ) )  <_  (
w  x.  ( 2 ^ n ) ) )
62 reflcl 10928 . . . . . . . . . . . . . . . . . 18  |-  ( ( w  x.  ( 2 ^ n ) )  e.  RR  ->  ( |_ `  ( w  x.  ( 2 ^ n
) ) )  e.  RR )
6358, 62syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( |_ `  ( w  x.  (
2 ^ n ) ) )  e.  RR )
6456nngt0d 9789 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  0  <  ( 2 ^ n ) )
65 ledivmul2 9633 . . . . . . . . . . . . . . . . 17  |-  ( ( ( |_ `  (
w  x.  ( 2 ^ n ) ) )  e.  RR  /\  w  e.  RR  /\  (
( 2 ^ n
)  e.  RR  /\  0  <  ( 2 ^ n ) ) )  ->  ( ( ( |_ `  ( w  x.  ( 2 ^ n ) ) )  /  ( 2 ^ n ) )  <_  w 
<->  ( |_ `  (
w  x.  ( 2 ^ n ) ) )  <_  ( w  x.  ( 2 ^ n
) ) ) )
6663, 51, 57, 64, 65syl112anc 1186 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( (
( |_ `  (
w  x.  ( 2 ^ n ) ) )  /  ( 2 ^ n ) )  <_  w  <->  ( |_ `  ( w  x.  (
2 ^ n ) ) )  <_  (
w  x.  ( 2 ^ n ) ) ) )
6761, 66mpbird 223 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( ( |_ `  ( w  x.  ( 2 ^ n
) ) )  / 
( 2 ^ n
) )  <_  w
)
68 peano2re 8985 . . . . . . . . . . . . . . . . . 18  |-  ( ( |_ `  ( w  x.  ( 2 ^ n ) ) )  e.  RR  ->  (
( |_ `  (
w  x.  ( 2 ^ n ) ) )  +  1 )  e.  RR )
6963, 68syl 15 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( ( |_ `  ( w  x.  ( 2 ^ n
) ) )  +  1 )  e.  RR )
7069, 56nndivred 9794 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( (
( |_ `  (
w  x.  ( 2 ^ n ) ) )  +  1 )  /  ( 2 ^ n ) )  e.  RR )
7160simprd 449 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( w  x.  ( 2 ^ n
) )  <  (
( |_ `  (
w  x.  ( 2 ^ n ) ) )  +  1 ) )
72 ltmuldiv 9626 . . . . . . . . . . . . . . . . . 18  |-  ( ( w  e.  RR  /\  ( ( |_ `  ( w  x.  (
2 ^ n ) ) )  +  1 )  e.  RR  /\  ( ( 2 ^ n )  e.  RR  /\  0  <  ( 2 ^ n ) ) )  ->  ( (
w  x.  ( 2 ^ n ) )  <  ( ( |_
`  ( w  x.  ( 2 ^ n
) ) )  +  1 )  <->  w  <  ( ( ( |_ `  ( w  x.  (
2 ^ n ) ) )  +  1 )  /  ( 2 ^ n ) ) ) )
7351, 69, 57, 64, 72syl112anc 1186 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( (
w  x.  ( 2 ^ n ) )  <  ( ( |_
`  ( w  x.  ( 2 ^ n
) ) )  +  1 )  <->  w  <  ( ( ( |_ `  ( w  x.  (
2 ^ n ) ) )  +  1 )  /  ( 2 ^ n ) ) ) )
7471, 73mpbid 201 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  w  <  ( ( ( |_ `  ( w  x.  (
2 ^ n ) ) )  +  1 )  /  ( 2 ^ n ) ) )
7551, 70, 74ltled 8967 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  w  <_  ( ( ( |_ `  ( w  x.  (
2 ^ n ) ) )  +  1 )  /  ( 2 ^ n ) ) )
7663, 56nndivred 9794 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( ( |_ `  ( w  x.  ( 2 ^ n
) ) )  / 
( 2 ^ n
) )  e.  RR )
77 elicc2 10715 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( |_ `  ( w  x.  (
2 ^ n ) ) )  /  (
2 ^ n ) )  e.  RR  /\  ( ( ( |_
`  ( w  x.  ( 2 ^ n
) ) )  +  1 )  /  (
2 ^ n ) )  e.  RR )  ->  ( w  e.  ( ( ( |_
`  ( w  x.  ( 2 ^ n
) ) )  / 
( 2 ^ n
) ) [,] (
( ( |_ `  ( w  x.  (
2 ^ n ) ) )  +  1 )  /  ( 2 ^ n ) ) )  <->  ( w  e.  RR  /\  ( ( |_ `  ( w  x.  ( 2 ^ n ) ) )  /  ( 2 ^ n ) )  <_  w  /\  w  <_  (
( ( |_ `  ( w  x.  (
2 ^ n ) ) )  +  1 )  /  ( 2 ^ n ) ) ) ) )
7876, 70, 77syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( w  e.  ( ( ( |_
`  ( w  x.  ( 2 ^ n
) ) )  / 
( 2 ^ n
) ) [,] (
( ( |_ `  ( w  x.  (
2 ^ n ) ) )  +  1 )  /  ( 2 ^ n ) ) )  <->  ( w  e.  RR  /\  ( ( |_ `  ( w  x.  ( 2 ^ n ) ) )  /  ( 2 ^ n ) )  <_  w  /\  w  <_  (
( ( |_ `  ( w  x.  (
2 ^ n ) ) )  +  1 )  /  ( 2 ^ n ) ) ) ) )
7951, 67, 75, 78mpbir3and 1135 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  w  e.  ( ( ( |_
`  ( w  x.  ( 2 ^ n
) ) )  / 
( 2 ^ n
) ) [,] (
( ( |_ `  ( w  x.  (
2 ^ n ) ) )  +  1 )  /  ( 2 ^ n ) ) ) )
8058flcld 10930 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( |_ `  ( w  x.  (
2 ^ n ) ) )  e.  ZZ )
8114dyadval 18947 . . . . . . . . . . . . . . . . 17  |-  ( ( ( |_ `  (
w  x.  ( 2 ^ n ) ) )  e.  ZZ  /\  n  e.  NN0 )  -> 
( ( |_ `  ( w  x.  (
2 ^ n ) ) ) F n )  =  <. (
( |_ `  (
w  x.  ( 2 ^ n ) ) )  /  ( 2 ^ n ) ) ,  ( ( ( |_ `  ( w  x.  ( 2 ^ n ) ) )  +  1 )  / 
( 2 ^ n
) ) >. )
8280, 54, 81syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( ( |_ `  ( w  x.  ( 2 ^ n
) ) ) F n )  =  <. ( ( |_ `  (
w  x.  ( 2 ^ n ) ) )  /  ( 2 ^ n ) ) ,  ( ( ( |_ `  ( w  x.  ( 2 ^ n ) ) )  +  1 )  / 
( 2 ^ n
) ) >. )
8382fveq2d 5529 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( [,] `  ( ( |_ `  ( w  x.  (
2 ^ n ) ) ) F n ) )  =  ( [,] `  <. (
( |_ `  (
w  x.  ( 2 ^ n ) ) )  /  ( 2 ^ n ) ) ,  ( ( ( |_ `  ( w  x.  ( 2 ^ n ) ) )  +  1 )  / 
( 2 ^ n
) ) >. )
)
84 df-ov 5861 . . . . . . . . . . . . . . 15  |-  ( ( ( |_ `  (
w  x.  ( 2 ^ n ) ) )  /  ( 2 ^ n ) ) [,] ( ( ( |_ `  ( w  x.  ( 2 ^ n ) ) )  +  1 )  / 
( 2 ^ n
) ) )  =  ( [,] `  <. ( ( |_ `  (
w  x.  ( 2 ^ n ) ) )  /  ( 2 ^ n ) ) ,  ( ( ( |_ `  ( w  x.  ( 2 ^ n ) ) )  +  1 )  / 
( 2 ^ n
) ) >. )
8583, 84syl6eqr 2333 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( [,] `  ( ( |_ `  ( w  x.  (
2 ^ n ) ) ) F n ) )  =  ( ( ( |_ `  ( w  x.  (
2 ^ n ) ) )  /  (
2 ^ n ) ) [,] ( ( ( |_ `  (
w  x.  ( 2 ^ n ) ) )  +  1 )  /  ( 2 ^ n ) ) ) )
8679, 85eleqtrrd 2360 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  w  e.  ( [,] `  ( ( |_ `  ( w  x.  ( 2 ^ n ) ) ) F n ) ) )
87 ffn 5389 . . . . . . . . . . . . . . . . . 18  |-  ( F : ( ZZ  X.  NN0 ) --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  Fn  ( ZZ  X.  NN0 ) )
8815, 87ax-mp 8 . . . . . . . . . . . . . . . . 17  |-  F  Fn  ( ZZ  X.  NN0 )
89 fnovrn 5995 . . . . . . . . . . . . . . . . 17  |-  ( ( F  Fn  ( ZZ 
X.  NN0 )  /\  ( |_ `  ( w  x.  ( 2 ^ n
) ) )  e.  ZZ  /\  n  e. 
NN0 )  ->  (
( |_ `  (
w  x.  ( 2 ^ n ) ) ) F n )  e.  ran  F )
9088, 89mp3an1 1264 . . . . . . . . . . . . . . . 16  |-  ( ( ( |_ `  (
w  x.  ( 2 ^ n ) ) )  e.  ZZ  /\  n  e.  NN0 )  -> 
( ( |_ `  ( w  x.  (
2 ^ n ) ) ) F n )  e.  ran  F
)
9180, 54, 90syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( ( |_ `  ( w  x.  ( 2 ^ n
) ) ) F n )  e.  ran  F )
92 simplrl 736 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  r  e.  RR+ )
9392rpred 10390 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  r  e.  RR )
9451, 93resubcld 9211 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( w  -  r )  e.  RR )
9594rexrd 8881 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( w  -  r )  e. 
RR* )
9651, 93readdcld 8862 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( w  +  r )  e.  RR )
9796rexrd 8881 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( w  +  r )  e. 
RR* )
9876, 93readdcld 8862 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( (
( |_ `  (
w  x.  ( 2 ^ n ) ) )  /  ( 2 ^ n ) )  +  r )  e.  RR )
9963recnd 8861 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( |_ `  ( w  x.  (
2 ^ n ) ) )  e.  CC )
100 ax-1cn 8795 . . . . . . . . . . . . . . . . . . . . . . 23  |-  1  e.  CC
101100a1i 10 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  1  e.  CC )
10257recnd 8861 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( 2 ^ n )  e.  CC )
10356nnne0d 9790 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( 2 ^ n )  =/=  0 )
10499, 101, 102, 103divdird 9574 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( (
( |_ `  (
w  x.  ( 2 ^ n ) ) )  +  1 )  /  ( 2 ^ n ) )  =  ( ( ( |_
`  ( w  x.  ( 2 ^ n
) ) )  / 
( 2 ^ n
) )  +  ( 1  /  ( 2 ^ n ) ) ) )
10556nnrecred 9791 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( 1  /  ( 2 ^ n ) )  e.  RR )
106 simprr 733 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( 1  /  ( 2 ^ n ) )  < 
r )
107105, 93, 76, 106ltadd2dd 8975 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( (
( |_ `  (
w  x.  ( 2 ^ n ) ) )  /  ( 2 ^ n ) )  +  ( 1  / 
( 2 ^ n
) ) )  < 
( ( ( |_
`  ( w  x.  ( 2 ^ n
) ) )  / 
( 2 ^ n
) )  +  r ) )
108104, 107eqbrtrd 4043 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( (
( |_ `  (
w  x.  ( 2 ^ n ) ) )  +  1 )  /  ( 2 ^ n ) )  < 
( ( ( |_
`  ( w  x.  ( 2 ^ n
) ) )  / 
( 2 ^ n
) )  +  r ) )
10951, 70, 98, 74, 108lttrd 8977 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  w  <  ( ( ( |_ `  ( w  x.  (
2 ^ n ) ) )  /  (
2 ^ n ) )  +  r ) )
11051, 93, 76ltsubaddd 9368 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( (
w  -  r )  <  ( ( |_
`  ( w  x.  ( 2 ^ n
) ) )  / 
( 2 ^ n
) )  <->  w  <  ( ( ( |_ `  ( w  x.  (
2 ^ n ) ) )  /  (
2 ^ n ) )  +  r ) ) )
111109, 110mpbird 223 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( w  -  r )  < 
( ( |_ `  ( w  x.  (
2 ^ n ) ) )  /  (
2 ^ n ) ) )
11251, 105readdcld 8862 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( w  +  ( 1  / 
( 2 ^ n
) ) )  e.  RR )
11376, 51, 105, 67leadd1dd 9386 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( (
( |_ `  (
w  x.  ( 2 ^ n ) ) )  /  ( 2 ^ n ) )  +  ( 1  / 
( 2 ^ n
) ) )  <_ 
( w  +  ( 1  /  ( 2 ^ n ) ) ) )
114104, 113eqbrtrd 4043 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( (
( |_ `  (
w  x.  ( 2 ^ n ) ) )  +  1 )  /  ( 2 ^ n ) )  <_ 
( w  +  ( 1  /  ( 2 ^ n ) ) ) )
115105, 93, 51, 106ltadd2dd 8975 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( w  +  ( 1  / 
( 2 ^ n
) ) )  < 
( w  +  r ) )
11670, 112, 96, 114, 115lelttrd 8974 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( (
( |_ `  (
w  x.  ( 2 ^ n ) ) )  +  1 )  /  ( 2 ^ n ) )  < 
( w  +  r ) )
117 iccssioo 10719 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( w  -  r )  e.  RR*  /\  ( w  +  r )  e.  RR* )  /\  ( ( w  -  r )  <  (
( |_ `  (
w  x.  ( 2 ^ n ) ) )  /  ( 2 ^ n ) )  /\  ( ( ( |_ `  ( w  x.  ( 2 ^ n ) ) )  +  1 )  / 
( 2 ^ n
) )  <  (
w  +  r ) ) )  ->  (
( ( |_ `  ( w  x.  (
2 ^ n ) ) )  /  (
2 ^ n ) ) [,] ( ( ( |_ `  (
w  x.  ( 2 ^ n ) ) )  +  1 )  /  ( 2 ^ n ) ) ) 
C_  ( ( w  -  r ) (,) ( w  +  r ) ) )
11895, 97, 111, 116, 117syl22anc 1183 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( (
( |_ `  (
w  x.  ( 2 ^ n ) ) )  /  ( 2 ^ n ) ) [,] ( ( ( |_ `  ( w  x.  ( 2 ^ n ) ) )  +  1 )  / 
( 2 ^ n
) ) )  C_  ( ( w  -  r ) (,) (
w  +  r ) ) )
11985, 118eqsstrd 3212 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( [,] `  ( ( |_ `  ( w  x.  (
2 ^ n ) ) ) F n ) )  C_  (
( w  -  r
) (,) ( w  +  r ) ) )
120 simplrr 737 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( (
w  -  r ) (,) ( w  +  r ) )  C_  A )
121119, 120sstrd 3189 . . . . . . . . . . . . . . 15  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( [,] `  ( ( |_ `  ( w  x.  (
2 ^ n ) ) ) F n ) )  C_  A
)
122 fveq2 5525 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( ( |_
`  ( w  x.  ( 2 ^ n
) ) ) F n )  ->  ( [,] `  z )  =  ( [,] `  (
( |_ `  (
w  x.  ( 2 ^ n ) ) ) F n ) ) )
123122sseq1d 3205 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( ( |_
`  ( w  x.  ( 2 ^ n
) ) ) F n )  ->  (
( [,] `  z
)  C_  A  <->  ( [,] `  ( ( |_ `  ( w  x.  (
2 ^ n ) ) ) F n ) )  C_  A
) )
124123elrab 2923 . . . . . . . . . . . . . . 15  |-  ( ( ( |_ `  (
w  x.  ( 2 ^ n ) ) ) F n )  e.  { z  e. 
ran  F  |  ( [,] `  z )  C_  A }  <->  ( ( ( |_ `  ( w  x.  ( 2 ^ n ) ) ) F n )  e. 
ran  F  /\  ( [,] `  ( ( |_
`  ( w  x.  ( 2 ^ n
) ) ) F n ) )  C_  A ) )
12591, 121, 124sylanbrc 645 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( ( |_ `  ( w  x.  ( 2 ^ n
) ) ) F n )  e.  {
z  e.  ran  F  |  ( [,] `  z
)  C_  A }
)
126 funfvima2 5754 . . . . . . . . . . . . . . 15  |-  ( ( Fun  [,]  /\  { z  e.  ran  F  | 
( [,] `  z
)  C_  A }  C_ 
dom  [,] )  ->  (
( ( |_ `  ( w  x.  (
2 ^ n ) ) ) F n )  e.  { z  e.  ran  F  | 
( [,] `  z
)  C_  A }  ->  ( [,] `  (
( |_ `  (
w  x.  ( 2 ^ n ) ) ) F n ) )  e.  ( [,] " { z  e.  ran  F  |  ( [,] `  z
)  C_  A }
) ) )
12712, 26, 126mp2an 653 . . . . . . . . . . . . . 14  |-  ( ( ( |_ `  (
w  x.  ( 2 ^ n ) ) ) F n )  e.  { z  e. 
ran  F  |  ( [,] `  z )  C_  A }  ->  ( [,] `  ( ( |_ `  ( w  x.  (
2 ^ n ) ) ) F n ) )  e.  ( [,] " { z  e.  ran  F  | 
( [,] `  z
)  C_  A }
) )
128125, 127syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  ( [,] `  ( ( |_ `  ( w  x.  (
2 ^ n ) ) ) F n ) )  e.  ( [,] " { z  e.  ran  F  | 
( [,] `  z
)  C_  A }
) )
129 elunii 3832 . . . . . . . . . . . . 13  |-  ( ( w  e.  ( [,] `  ( ( |_ `  ( w  x.  (
2 ^ n ) ) ) F n ) )  /\  ( [,] `  ( ( |_
`  ( w  x.  ( 2 ^ n
) ) ) F n ) )  e.  ( [,] " {
z  e.  ran  F  |  ( [,] `  z
)  C_  A }
) )  ->  w  e.  U. ( [,] " {
z  e.  ran  F  |  ( [,] `  z
)  C_  A }
) )
13086, 128, 129syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  ( n  e.  NN  /\  ( 1  /  (
2 ^ n ) )  <  r ) )  ->  w  e.  U. ( [,] " {
z  e.  ran  F  |  ( [,] `  z
)  C_  A }
) )
131130expr 598 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  ( topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  /\  n  e.  NN )  ->  ( ( 1  / 
( 2 ^ n
) )  <  r  ->  w  e.  U. ( [,] " { z  e. 
ran  F  |  ( [,] `  z )  C_  A } ) ) )
132131rexlimdva 2667 . . . . . . . . . 10  |-  ( ( ( A  e.  (
topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  -> 
( E. n  e.  NN  ( 1  / 
( 2 ^ n
) )  <  r  ->  w  e.  U. ( [,] " { z  e. 
ran  F  |  ( [,] `  z )  C_  A } ) ) )
13350, 132mpd 14 . . . . . . . . 9  |-  ( ( ( A  e.  (
topGen `  ran  (,) )  /\  w  e.  A
)  /\  ( r  e.  RR+  /\  ( ( w  -  r ) (,) ( w  +  r ) )  C_  A ) )  ->  w  e.  U. ( [,] " { z  e. 
ran  F  |  ( [,] `  z )  C_  A } ) )
134133expr 598 . . . . . . . 8  |-  ( ( ( A  e.  (
topGen `  ran  (,) )  /\  w  e.  A
)  /\  r  e.  RR+ )  ->  ( (
( w  -  r
) (,) ( w  +  r ) ) 
C_  A  ->  w  e.  U. ( [,] " {
z  e.  ran  F  |  ( [,] `  z
)  C_  A }
) ) )
13545, 134sylbid 206 . . . . . . 7  |-  ( ( ( A  e.  (
topGen `  ran  (,) )  /\  w  e.  A
)  /\  r  e.  RR+ )  ->  ( (
w ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) r )  C_  A  ->  w  e.  U. ( [,] " { z  e.  ran  F  |  ( [,] `  z
)  C_  A }
) ) )
136135rexlimdva 2667 . . . . . 6  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  w  e.  A )  ->  ( E. r  e.  RR+  (
w ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) r )  C_  A  ->  w  e.  U. ( [,] " { z  e.  ran  F  |  ( [,] `  z
)  C_  A }
) ) )
13737, 136mpd 14 . . . . 5  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  w  e.  A )  ->  w  e.  U. ( [,] " {
z  e.  ran  F  |  ( [,] `  z
)  C_  A }
) )
138137ex 423 . . . 4  |-  ( A  e.  ( topGen `  ran  (,) )  ->  ( w  e.  A  ->  w  e. 
U. ( [,] " {
z  e.  ran  F  |  ( [,] `  z
)  C_  A }
) ) )
139138ssrdv 3185 . . 3  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  C_  U. ( [,] " { z  e. 
ran  F  |  ( [,] `  z )  C_  A } ) )
14031, 139eqssd 3196 . 2  |-  ( A  e.  ( topGen `  ran  (,) )  ->  U. ( [,] " { z  e. 
ran  F  |  ( [,] `  z )  C_  A } )  =  A )
141 fveq2 5525 . . . . . . 7  |-  ( c  =  a  ->  ( [,] `  c )  =  ( [,] `  a
) )
142141sseq1d 3205 . . . . . 6  |-  ( c  =  a  ->  (
( [,] `  c
)  C_  ( [,] `  b )  <->  ( [,] `  a )  C_  ( [,] `  b ) ) )
143 eqeq1 2289 . . . . . 6  |-  ( c  =  a  ->  (
c  =  b  <->  a  =  b ) )
144142, 143imbi12d 311 . . . . 5  |-  ( c  =  a  ->  (
( ( [,] `  c
)  C_  ( [,] `  b )  ->  c  =  b )  <->  ( ( [,] `  a )  C_  ( [,] `  b )  ->  a  =  b ) ) )
145144ralbidv 2563 . . . 4  |-  ( c  =  a  ->  ( A. b  e.  { z  e.  ran  F  | 
( [,] `  z
)  C_  A } 
( ( [,] `  c
)  C_  ( [,] `  b )  ->  c  =  b )  <->  A. b  e.  { z  e.  ran  F  |  ( [,] `  z
)  C_  A } 
( ( [,] `  a
)  C_  ( [,] `  b )  ->  a  =  b ) ) )
146145cbvrabv 2787 . . 3  |-  { c  e.  { z  e. 
ran  F  |  ( [,] `  z )  C_  A }  |  A. b  e.  { z  e.  ran  F  |  ( [,] `  z ) 
C_  A }  (
( [,] `  c
)  C_  ( [,] `  b )  ->  c  =  b ) }  =  { a  e. 
{ z  e.  ran  F  |  ( [,] `  z
)  C_  A }  |  A. b  e.  {
z  e.  ran  F  |  ( [,] `  z
)  C_  A } 
( ( [,] `  a
)  C_  ( [,] `  b )  ->  a  =  b ) }
14713a1i 10 . . 3  |-  ( A  e.  ( topGen `  ran  (,) )  ->  { z  e.  ran  F  |  ( [,] `  z ) 
C_  A }  C_  ran  F )
14814, 146, 147dyadmbl 18955 . 2  |-  ( A  e.  ( topGen `  ran  (,) )  ->  U. ( [,] " { z  e. 
ran  F  |  ( [,] `  z )  C_  A } )  e.  dom  vol )
149140, 148eqeltrrd 2358 1  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  e.  dom  vol )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   <.cop 3643   U.cuni 3827   class class class wbr 4023    X. cxp 4687   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692    o. ccom 4693   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   NNcn 9746   2c2 9795   NN0cn0 9965   ZZcz 10024   RR+crp 10354   (,)cioo 10656   [,]cicc 10659   |_cfl 10924   ^cexp 11104   abscabs 11719   topGenctg 13342   * Metcxmt 16369   ballcbl 16371   MetOpencmopn 16372   volcvol 18823
This theorem is referenced by:  opnmbl  18957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-disj 3994  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-rest 13327  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-cmp 17114  df-ovol 18824  df-vol 18825
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