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Theorem opnmbllem 18972
Description: Lemma for opnmbl 18973. (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 5541 . . . . . . . . 9  |-  ( z  =  w  ->  ( [,] `  z )  =  ( [,] `  w
) )
21sseq1d 3218 . . . . . . . 8  |-  ( z  =  w  ->  (
( [,] `  z
)  C_  A  <->  ( [,] `  w )  C_  A
) )
32elrab 2936 . . . . . . 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 5555 . . . . . . . . 9  |-  ( [,] `  w )  e.  _V
65elpw 3644 . . . . . . . 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 2639 . . . . 5  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A. w  e.  { z  e.  ran  F  |  ( [,] `  z
)  C_  A } 
( [,] `  w
)  e.  ~P A
)
10 iccf 10758 . . . . . . 7  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
11 ffun 5407 . . . . . . 7  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  Fun  [,] )
1210, 11ax-mp 8 . . . . . 6  |-  Fun  [,]
13 ssrab2 3271 . . . . . . . 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 18962 . . . . . . . . . 10  |-  F :
( ZZ  X.  NN0 )
--> (  <_  i^i  ( RR  X.  RR ) )
16 frn 5411 . . . . . . . . . 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 3403 . . . . . . . . . 10  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
19 ressxr 8892 . . . . . . . . . . 11  |-  RR  C_  RR*
20 xpss12 4808 . . . . . . . . . . 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 3201 . . . . . . . . 9  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
2317, 22sstri 3201 . . . . . . . 8  |-  ran  F  C_  ( RR*  X.  RR* )
2413, 23sstri 3201 . . . . . . 7  |-  { z  e.  ran  F  | 
( [,] `  z
)  C_  A }  C_  ( RR*  X.  RR* )
2510fdmi 5410 . . . . . . 7  |-  dom  [,]  =  ( RR*  X.  RR* )
2624, 25sseqtr4i 3224 . . . . . 6  |-  { z  e.  ran  F  | 
( [,] `  z
)  C_  A }  C_ 
dom  [,]
27 funimass4 5589 . . . . . 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 4003 . . . 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 2296 . . . . . . . 8  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
3332rexmet 18313 . . . . . . 7  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )
34 eqid 2296 . . . . . . . . 9  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
3532, 34tgioo 18318 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
3635mopni2 18055 . . . . . . 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 3871 . . . . . . . . . . . 12  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  C_  U. ( topGen `
 ran  (,) )
)
39 uniretop 18287 . . . . . . . . . . . 12  |-  RR  =  U. ( topGen `  ran  (,) )
4038, 39syl6sseqr 3238 . . . . . . . . . . 11  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  C_  RR )
4140sselda 3193 . . . . . . . . . 10  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  w  e.  A )  ->  w  e.  RR )
42 rpre 10376 . . . . . . . . . 10  |-  ( r  e.  RR+  ->  r  e.  RR )
4332bl2ioo 18314 . . . . . . . . . 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 3218 . . . . . . . 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 9831 . . . . . . . . . . . 12  |-  2  e.  RR
47 1lt2 9902 . . . . . . . . . . . 12  |-  1  <  2
48 expnlbnd 11247 . . . . . . . . . . . 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 9893 . . . . . . . . . . . . . . . . . . . . 21  |-  2  e.  NN
53 nnnn0 9988 . . . . . . . . . . . . . . . . . . . . . 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 11132 . . . . . . . . . . . . . . . . . . . . 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 9777 . . . . . . . . . . . . . . . . . . 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 8879 . . . . . . . . . . . . . . . . . 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 10945 . . . . . . . . . . . . . . . . . 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 10944 . . . . . . . . . . . . . . . . . 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 9805 . . . . . . . . . . . . . . . . 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 9649 . . . . . . . . . . . . . . . . 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 9001 . . . . . . . . . . . . . . . . . 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 9810 . . . . . . . . . . . . . . . 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 9642 . . . . . . . . . . . . . . . . . 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 8983 . . . . . . . . . . . . . . 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 9810 . . . . . . . . . . . . . . . 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 10731 . . . . . . . . . . . . . . . 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 10946 . . . . . . . . . . . . . . . . 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 18963 . . . . . . . . . . . . . . . . 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 5545 . . . . . . . . . . . . . . 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 5877 . . . . . . . . . . . . . . 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 2346 . . . . . . . . . . . . . 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 2373 . . . . . . . . . . . . 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 5405 . . . . . . . . . . . . . . . . . 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 6011 . . . . . . . . . . . . . . . . 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 10406 . . . . . . . . . . . . . . . . . . . 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 9227 . . . . . . . . . . . . . . . . . . 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 8897 . . . . . . . . . . . . . . . . . 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 8878 . . . . . . . . . . . . . . . . . . 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 8897 . . . . . . . . . . . . . . . . . 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 8878 . . . . . . . . . . . . . . . . . . . 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 8877 . . . . . . . . . . . . . . . . . . . . . 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 8811 . . . . . . . . . . . . . . . . . . . . . . 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 8877 . . . . . . . . . . . . . . . . . . . . . 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 9806 . . . . . . . . . . . . . . . . . . . . . 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 9590 . . . . . . . . . . . . . . . . . . . . 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 9807 . . . . . . . . . . . . . . . . . . . . . 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 8991 . . . . . . . . . . . . . . . . . . . . 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 4059 . . . . . . . . . . . . . . . . . . . 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 8993 . . . . . . . . . . . . . . . . . . 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 9384 . . . . . . . . . . . . . . . . . . 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 8878 . . . . . . . . . . . . . . . . . . 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 9402 . . . . . . . . . . . . . . . . . . . 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 4059 . . . . . . . . . . . . . . . . . . 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 8991 . . . . . . . . . . . . . . . . . . 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 8990 . . . . . . . . . . . . . . . . . 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 10735 . . . . . . . . . . . . . . . . . 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 3225 . . . . . . . . . . . . . . . 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 3202 . . . . . . . . . . . . . . 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 5541 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( ( |_
`  ( w  x.  ( 2 ^ n
) ) ) F n )  ->  ( [,] `  z )  =  ( [,] `  (
( |_ `  (
w  x.  ( 2 ^ n ) ) ) F n ) ) )
123122sseq1d 3218 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( ( |_
`  ( w  x.  ( 2 ^ n
) ) ) F n )  ->  (
( [,] `  z
)  C_  A  <->  ( [,] `  ( ( |_ `  ( w  x.  (
2 ^ n ) ) ) F n ) )  C_  A
) )
124123elrab 2936 . . . . . . . . . . . . . . 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 5770 . . . . . . . . . . . . . . 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 3848 . . . . . . . . . . . . 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 2680 . . . . . . . . . 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 2680 . . . . . 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 3198 . . 3  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  C_  U. ( [,] " { z  e. 
ran  F  |  ( [,] `  z )  C_  A } ) )
14031, 139eqssd 3209 . 2  |-  ( A  e.  ( topGen `  ran  (,) )  ->  U. ( [,] " { z  e. 
ran  F  |  ( [,] `  z )  C_  A } )  =  A )
141 fveq2 5541 . . . . . . 7  |-  ( c  =  a  ->  ( [,] `  c )  =  ( [,] `  a
) )
142141sseq1d 3218 . . . . . 6  |-  ( c  =  a  ->  (
( [,] `  c
)  C_  ( [,] `  b )  <->  ( [,] `  a )  C_  ( [,] `  b ) ) )
143 eqeq1 2302 . . . . . 6  |-  ( c  =  a  ->  (
c  =  b  <->  a  =  b ) )
144142, 143imbi12d 311 . . . . 5  |-  ( c  =  a  ->  (
( ( [,] `  c
)  C_  ( [,] `  b )  ->  c  =  b )  <->  ( ( [,] `  a )  C_  ( [,] `  b )  ->  a  =  b ) ) )
145144ralbidv 2576 . . . 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 2800 . . 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 18971 . 2  |-  ( A  e.  ( topGen `  ran  (,) )  ->  U. ( [,] " { z  e. 
ran  F  |  ( [,] `  z )  C_  A } )  e.  dom  vol )
149140, 148eqeltrrd 2371 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   <.cop 3656   U.cuni 3843   class class class wbr 4039    X. cxp 4703   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708    o. ccom 4709   Fun wfun 5265    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   RR+crp 10370   (,)cioo 10672   [,]cicc 10675   |_cfl 10940   ^cexp 11120   abscabs 11735   topGenctg 13358   * Metcxmt 16385   ballcbl 16387   MetOpencmopn 16388   volcvol 18839
This theorem is referenced by:  opnmbl  18973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-disj 4010  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-rest 13343  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-cmp 17130  df-ovol 18840  df-vol 18841
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