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Theorem opnmbllem 22176
Description: Lemma for opnmbl 22177. (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 5848 . . . . . . . . 9  |-  ( z  =  w  ->  ( [,] `  z )  =  ( [,] `  w
) )
21sseq1d 3516 . . . . . . . 8  |-  ( z  =  w  ->  (
( [,] `  z
)  C_  A  <->  ( [,] `  w )  C_  A
) )
32elrab 3254 . . . . . . 7  |-  ( w  e.  { z  e. 
ran  F  |  ( [,] `  z )  C_  A }  <->  ( w  e. 
ran  F  /\  ( [,] `  w )  C_  A ) )
4 simprr 755 . . . . . . . 8  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  (
w  e.  ran  F  /\  ( [,] `  w
)  C_  A )
)  ->  ( [,] `  w )  C_  A
)
5 fvex 5858 . . . . . . . . 9  |-  ( [,] `  w )  e.  _V
65elpw 4005 . . . . . . . 8  |-  ( ( [,] `  w )  e.  ~P A  <->  ( [,] `  w )  C_  A
)
74, 6sylibr 212 . . . . . . 7  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  (
w  e.  ran  F  /\  ( [,] `  w
)  C_  A )
)  ->  ( [,] `  w )  e.  ~P A )
83, 7sylan2b 473 . . . . . 6  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  w  e.  { z  e.  ran  F  |  ( [,] `  z
)  C_  A }
)  ->  ( [,] `  w )  e.  ~P A )
98ralrimiva 2868 . . . . 5  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A. w  e.  { z  e.  ran  F  |  ( [,] `  z
)  C_  A } 
( [,] `  w
)  e.  ~P A
)
10 iccf 11626 . . . . . . 7  |-  [,] :
( RR*  X.  RR* ) --> ~P RR*
11 ffun 5715 . . . . . . 7  |-  ( [,]
: ( RR*  X.  RR* )
--> ~P RR*  ->  Fun  [,] )
1210, 11ax-mp 5 . . . . . 6  |-  Fun  [,]
13 ssrab2 3571 . . . . . . . 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 22166 . . . . . . . . . 10  |-  F :
( ZZ  X.  NN0 )
--> (  <_  i^i  ( RR  X.  RR ) )
16 frn 5719 . . . . . . . . . 10  |-  ( F : ( ZZ  X.  NN0 ) --> (  <_  i^i  ( RR  X.  RR ) )  ->  ran  F 
C_  (  <_  i^i  ( RR  X.  RR ) ) )
1715, 16ax-mp 5 . . . . . . . . 9  |-  ran  F  C_  (  <_  i^i  ( RR  X.  RR ) )
18 inss2 3705 . . . . . . . . . 10  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
19 rexpssxrxp 9627 . . . . . . . . . 10  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
2018, 19sstri 3498 . . . . . . . . 9  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
2117, 20sstri 3498 . . . . . . . 8  |-  ran  F  C_  ( RR*  X.  RR* )
2213, 21sstri 3498 . . . . . . 7  |-  { z  e.  ran  F  | 
( [,] `  z
)  C_  A }  C_  ( RR*  X.  RR* )
2310fdmi 5718 . . . . . . 7  |-  dom  [,]  =  ( RR*  X.  RR* )
2422, 23sseqtr4i 3522 . . . . . 6  |-  { z  e.  ran  F  | 
( [,] `  z
)  C_  A }  C_ 
dom  [,]
25 funimass4 5899 . . . . . 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
) )
2612, 24, 25mp2an 670 . . . . 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
)
279, 26sylibr 212 . . . 4  |-  ( A  e.  ( topGen `  ran  (,) )  ->  ( [,] " { z  e.  ran  F  |  ( [,] `  z
)  C_  A }
)  C_  ~P A
)
28 sspwuni 4404 . . . 4  |-  ( ( [,] " { z  e.  ran  F  | 
( [,] `  z
)  C_  A }
)  C_  ~P A  <->  U. ( [,] " {
z  e.  ran  F  |  ( [,] `  z
)  C_  A }
)  C_  A )
2927, 28sylib 196 . . 3  |-  ( A  e.  ( topGen `  ran  (,) )  ->  U. ( [,] " { z  e. 
ran  F  |  ( [,] `  z )  C_  A } )  C_  A
)
30 eqid 2454 . . . . . . . 8  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
3130rexmet 21462 . . . . . . 7  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( *Met `  RR )
32 eqid 2454 . . . . . . . . 9  |-  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) )
3330, 32tgioo 21467 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  =  (
MetOpen `  ( ( abs 
o.  -  )  |`  ( RR  X.  RR ) ) )
3433mopni2 21162 . . . . . . 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 )
3531, 34mp3an1 1309 . . . . . 6  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  w  e.  A )  ->  E. r  e.  RR+  ( w (
ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) r )  C_  A )
36 elssuni 4264 . . . . . . . . . . . 12  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  C_  U. ( topGen `
 ran  (,) )
)
37 uniretop 21435 . . . . . . . . . . . 12  |-  RR  =  U. ( topGen `  ran  (,) )
3836, 37syl6sseqr 3536 . . . . . . . . . . 11  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  C_  RR )
3938sselda 3489 . . . . . . . . . 10  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  w  e.  A )  ->  w  e.  RR )
40 rpre 11227 . . . . . . . . . 10  |-  ( r  e.  RR+  ->  r  e.  RR )
4130bl2ioo 21463 . . . . . . . . . 10  |-  ( ( w  e.  RR  /\  r  e.  RR )  ->  ( w ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  =  ( ( w  -  r ) (,) (
w  +  r ) ) )
4239, 40, 41syl2an 475 . . . . . . . . 9  |-  ( ( ( A  e.  (
topGen `  ran  (,) )  /\  w  e.  A
)  /\  r  e.  RR+ )  ->  ( w
( ball `  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) r )  =  ( ( w  -  r ) (,) ( w  +  r ) ) )
4342sseq1d 3516 . . . . . . . 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 ) )
44 2re 10601 . . . . . . . . . . . 12  |-  2  e.  RR
45 1lt2 10698 . . . . . . . . . . . 12  |-  1  <  2
46 expnlbnd 12278 . . . . . . . . . . . 12  |-  ( ( r  e.  RR+  /\  2  e.  RR  /\  1  <  2 )  ->  E. n  e.  NN  ( 1  / 
( 2 ^ n
) )  <  r
)
4744, 45, 46mp3an23 1314 . . . . . . . . . . 11  |-  ( r  e.  RR+  ->  E. n  e.  NN  ( 1  / 
( 2 ^ n
) )  <  r
)
4847ad2antrl 725 . . . . . . . . . 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 )
4939ad2antrr 723 . . . . . . . . . . . . 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.  RR )
50 2nn 10689 . . . . . . . . . . . . . . . . . . 19  |-  2  e.  NN
51 nnnn0 10798 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  NN  ->  n  e.  NN0 )
5251ad2antrl 725 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( 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 )
53 nnexpcl 12161 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 2  e.  NN  /\  n  e.  NN0 )  -> 
( 2 ^ n
)  e.  NN )
5450, 52, 53sylancr 661 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( 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 )
5554nnred 10546 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( 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 )
5649, 55remulcld 9613 . . . . . . . . . . . . . . . 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
) )  e.  RR )
57 fllelt 11915 . . . . . . . . . . . . . . . 16  |-  ( ( w  x.  ( 2 ^ n ) )  e.  RR  ->  (
( |_ `  (
w  x.  ( 2 ^ n ) ) )  <_  ( w  x.  ( 2 ^ n
) )  /\  (
w  x.  ( 2 ^ n ) )  <  ( ( |_
`  ( w  x.  ( 2 ^ n
) ) )  +  1 ) ) )
5856, 57syl 16 . . . . . . . . . . . . . . 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
) ) )  <_ 
( w  x.  (
2 ^ n ) )  /\  ( w  x.  ( 2 ^ n ) )  < 
( ( |_ `  ( w  x.  (
2 ^ n ) ) )  +  1 ) ) )
5958simpld 457 . . . . . . . . . . . . . 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 ) ) )  <_  (
w  x.  ( 2 ^ n ) ) )
60 reflcl 11914 . . . . . . . . . . . . . . . 16  |-  ( ( w  x.  ( 2 ^ n ) )  e.  RR  ->  ( |_ `  ( w  x.  ( 2 ^ n
) ) )  e.  RR )
6156, 60syl 16 . . . . . . . . . . . . . . 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 ) ) )  e.  RR )
6254nngt0d 10575 . . . . . . . . . . . . . . 15  |-  ( ( ( ( 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 ) )
63 ledivmul2 10417 . . . . . . . . . . . . . . 15  |-  ( ( ( |_ `  (
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
) ) ) )
6461, 49, 55, 62, 63syl112anc 1230 . . . . . . . . . . . . . 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 ) ) )  /  ( 2 ^ n ) )  <_  w  <->  ( |_ `  ( w  x.  (
2 ^ n ) ) )  <_  (
w  x.  ( 2 ^ n ) ) ) )
6559, 64mpbird 232 . . . . . . . . . . . . 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
) ) )  / 
( 2 ^ n
) )  <_  w
)
66 peano2re 9742 . . . . . . . . . . . . . . . 16  |-  ( ( |_ `  ( w  x.  ( 2 ^ n ) ) )  e.  RR  ->  (
( |_ `  (
w  x.  ( 2 ^ n ) ) )  +  1 )  e.  RR )
6761, 66syl 16 . . . . . . . . . . . . . . 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
) ) )  +  1 )  e.  RR )
6867, 54nndivred 10580 . . . . . . . . . . . . . 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 ) ) )  +  1 )  /  ( 2 ^ n ) )  e.  RR )
6958simprd 461 . . . . . . . . . . . . . . 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
) )  <  (
( |_ `  (
w  x.  ( 2 ^ n ) ) )  +  1 ) )
70 ltmuldiv 10411 . . . . . . . . . . . . . . . 16  |-  ( ( 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 ) ) ) )
7149, 67, 55, 62, 70syl112anc 1230 . . . . . . . . . . . . . . 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 ) )  <  ( ( |_
`  ( w  x.  ( 2 ^ n
) ) )  +  1 )  <->  w  <  ( ( ( |_ `  ( w  x.  (
2 ^ n ) ) )  +  1 )  /  ( 2 ^ n ) ) ) )
7269, 71mpbid 210 . . . . . . . . . . . . . 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  <  ( ( ( |_ `  ( w  x.  (
2 ^ n ) ) )  +  1 )  /  ( 2 ^ n ) ) )
7349, 68, 72ltled 9722 . . . . . . . . . . . . 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  <_  ( ( ( |_ `  ( w  x.  (
2 ^ n ) ) )  +  1 )  /  ( 2 ^ n ) ) )
7461, 54nndivred 10580 . . . . . . . . . . . . . 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
) ) )  / 
( 2 ^ n
) )  e.  RR )
75 elicc2 11592 . . . . . . . . . . . . . 14  |-  ( ( ( ( |_ `  ( 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 ) ) ) ) )
7674, 68, 75syl2anc 659 . . . . . . . . . . . . 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
) ) )  / 
( 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 ) ) ) ) )
7749, 65, 73, 76mpbir3and 1177 . . . . . . . . . . . 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.  ( ( ( |_
`  ( w  x.  ( 2 ^ n
) ) )  / 
( 2 ^ n
) ) [,] (
( ( |_ `  ( w  x.  (
2 ^ n ) ) )  +  1 )  /  ( 2 ^ n ) ) ) )
7856flcld 11916 . . . . . . . . . . . . . . 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 ) ) )  e.  ZZ )
7914dyadval 22167 . . . . . . . . . . . . . . 15  |-  ( ( ( |_ `  (
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
) ) >. )
8078, 52, 79syl2anc 659 . . . . . . . . . . . . . 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
) ) >. )
8180fveq2d 5852 . . . . . . . . . . . . 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 ) )  =  ( [,] `  <. (
( |_ `  (
w  x.  ( 2 ^ n ) ) )  /  ( 2 ^ n ) ) ,  ( ( ( |_ `  ( w  x.  ( 2 ^ n ) ) )  +  1 )  / 
( 2 ^ n
) ) >. )
)
82 df-ov 6273 . . . . . . . . . . . . 13  |-  ( ( ( |_ `  (
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
) ) >. )
8381, 82syl6eqr 2513 . . . . . . . . . . . 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  x.  (
2 ^ n ) ) ) F n ) )  =  ( ( ( |_ `  ( w  x.  (
2 ^ n ) ) )  /  (
2 ^ n ) ) [,] ( ( ( |_ `  (
w  x.  ( 2 ^ n ) ) )  +  1 )  /  ( 2 ^ n ) ) ) )
8477, 83eleqtrrd 2545 . . . . . . . . . . 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.  ( [,] `  ( ( |_ `  ( w  x.  ( 2 ^ n ) ) ) F n ) ) )
85 ffn 5713 . . . . . . . . . . . . . . . 16  |-  ( F : ( ZZ  X.  NN0 ) --> (  <_  i^i  ( RR  X.  RR ) )  ->  F  Fn  ( ZZ  X.  NN0 ) )
8615, 85ax-mp 5 . . . . . . . . . . . . . . 15  |-  F  Fn  ( ZZ  X.  NN0 )
87 fnovrn 6423 . . . . . . . . . . . . . . 15  |-  ( ( F  Fn  ( ZZ 
X.  NN0 )  /\  ( |_ `  ( w  x.  ( 2 ^ n
) ) )  e.  ZZ  /\  n  e. 
NN0 )  ->  (
( |_ `  (
w  x.  ( 2 ^ n ) ) ) F n )  e.  ran  F )
8886, 87mp3an1 1309 . . . . . . . . . . . . . 14  |-  ( ( ( |_ `  (
w  x.  ( 2 ^ n ) ) )  e.  ZZ  /\  n  e.  NN0 )  -> 
( ( |_ `  ( w  x.  (
2 ^ n ) ) ) F n )  e.  ran  F
)
8978, 52, 88syl2anc 659 . . . . . . . . . . . . 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.  ran  F )
90 simplrl 759 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( 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+ )
9190rpred 11259 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( 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 )
9249, 91resubcld 9983 . . . . . . . . . . . . . . . . 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  -  r )  e.  RR )
9392rexrd 9632 . . . . . . . . . . . . . . . 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 )  e. 
RR* )
9449, 91readdcld 9612 . . . . . . . . . . . . . . . . 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  +  r )  e.  RR )
9594rexrd 9632 . . . . . . . . . . . . . . . 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 )  e. 
RR* )
9674, 91readdcld 9612 . . . . . . . . . . . . . . . . . 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 ) ) )  /  ( 2 ^ n ) )  +  r )  e.  RR )
9761recnd 9611 . . . . . . . . . . . . . . . . . . . 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 ) ) )  e.  CC )
98 1cnd 9601 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( 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 )
9955recnd 9611 . . . . . . . . . . . . . . . . . . . 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.  CC )
10054nnne0d 10576 . . . . . . . . . . . . . . . . . . . 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 )  =/=  0 )
10197, 98, 99, 100divdird 10354 . . . . . . . . . . . . . . . . . . 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  x.  ( 2 ^ n
) ) )  / 
( 2 ^ n
) )  +  ( 1  /  ( 2 ^ n ) ) ) )
10254nnrecred 10577 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( 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 )
103 simprr 755 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( 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 )
104102, 91, 74, 103ltadd2dd 9730 . . . . . . . . . . . . . . . . . . 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 ) ) )  /  ( 2 ^ n ) )  +  ( 1  / 
( 2 ^ n
) ) )  < 
( ( ( |_
`  ( w  x.  ( 2 ^ n
) ) )  / 
( 2 ^ n
) )  +  r ) )
105101, 104eqbrtrd 4459 . . . . . . . . . . . . . . . . . 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  x.  ( 2 ^ n
) ) )  / 
( 2 ^ n
) )  +  r ) )
10649, 68, 96, 72, 105lttrd 9732 . . . . . . . . . . . . . . . . 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  <  ( ( ( |_ `  ( w  x.  (
2 ^ n ) ) )  /  (
2 ^ n ) )  +  r ) )
10749, 91, 74ltsubaddd 10144 . . . . . . . . . . . . . . . . 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  -  r )  <  ( ( |_
`  ( w  x.  ( 2 ^ n
) ) )  / 
( 2 ^ n
) )  <->  w  <  ( ( ( |_ `  ( w  x.  (
2 ^ n ) ) )  /  (
2 ^ n ) )  +  r ) ) )
108106, 107mpbird 232 . . . . . . . . . . . . . . . 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  x.  (
2 ^ n ) ) )  /  (
2 ^ n ) ) )
10949, 102readdcld 9612 . . . . . . . . . . . . . . . . 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  +  ( 1  / 
( 2 ^ n
) ) )  e.  RR )
11074, 49, 102, 65leadd1dd 10162 . . . . . . . . . . . . . . . . . 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 ) ) )  /  ( 2 ^ n ) )  +  ( 1  / 
( 2 ^ n
) ) )  <_ 
( w  +  ( 1  /  ( 2 ^ n ) ) ) )
111101, 110eqbrtrd 4459 . . . . . . . . . . . . . . . . 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 )  /  ( 2 ^ n ) )  <_ 
( w  +  ( 1  /  ( 2 ^ n ) ) ) )
112102, 91, 49, 103ltadd2dd 9730 . . . . . . . . . . . . . . . . 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  +  ( 1  / 
( 2 ^ n
) ) )  < 
( w  +  r ) )
11368, 109, 94, 111, 112lelttrd 9729 . . . . . . . . . . . . . . . 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 ) )  < 
( w  +  r ) )
114 iccssioo 11596 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( 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 ) ) )
11593, 95, 108, 113, 114syl22anc 1227 . . . . . . . . . . . . . . 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  x.  ( 2 ^ n ) ) )  +  1 )  / 
( 2 ^ n
) ) )  C_  ( ( w  -  r ) (,) (
w  +  r ) ) )
11683, 115eqsstrd 3523 . . . . . . . . . . . . . 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 ) )  C_  (
( w  -  r
) (,) ( w  +  r ) ) )
117 simplrr 760 . . . . . . . . . . . . . 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  -  r ) (,) ( w  +  r ) )  C_  A )
118116, 117sstrd 3499 . . . . . . . . . . . . 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 ) )  C_  A
)
119 fveq2 5848 . . . . . . . . . . . . . . 15  |-  ( z  =  ( ( |_
`  ( w  x.  ( 2 ^ n
) ) ) F n )  ->  ( [,] `  z )  =  ( [,] `  (
( |_ `  (
w  x.  ( 2 ^ n ) ) ) F n ) ) )
120119sseq1d 3516 . . . . . . . . . . . . . 14  |-  ( z  =  ( ( |_
`  ( w  x.  ( 2 ^ n
) ) ) F n )  ->  (
( [,] `  z
)  C_  A  <->  ( [,] `  ( ( |_ `  ( w  x.  (
2 ^ n ) ) ) F n ) )  C_  A
) )
121120elrab 3254 . . . . . . . . . . . . 13  |-  ( ( ( |_ `  (
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 ) )
12289, 118, 121sylanbrc 662 . . . . . . . . . . . 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  x.  ( 2 ^ n
) ) ) F n )  e.  {
z  e.  ran  F  |  ( [,] `  z
)  C_  A }
)
123 funfvima2 6123 . . . . . . . . . . . . 13  |-  ( ( 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 }
) ) )
12412, 24, 123mp2an 670 . . . . . . . . . . . 12  |-  ( ( ( |_ `  (
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 }
) )
125122, 124syl 16 . . . . . . . . . . 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  x.  (
2 ^ n ) ) ) F n ) )  e.  ( [,] " { z  e.  ran  F  | 
( [,] `  z
)  C_  A }
) )
126 elunii 4240 . . . . . . . . . . 11  |-  ( ( 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 }
) )
12784, 125, 126syl2anc 659 . . . . . . . . . 10  |-  ( ( ( ( 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 }
) )
12848, 127rexlimddv 2950 . . . . . . . . 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 } ) )
129128expr 613 . . . . . . . 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 }
) ) )
13043, 129sylbid 215 . . . . . . 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 }
) ) )
131130rexlimdva 2946 . . . . . 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 }
) ) )
13235, 131mpd 15 . . . . 5  |-  ( ( A  e.  ( topGen ` 
ran  (,) )  /\  w  e.  A )  ->  w  e.  U. ( [,] " {
z  e.  ran  F  |  ( [,] `  z
)  C_  A }
) )
133132ex 432 . . . 4  |-  ( A  e.  ( topGen `  ran  (,) )  ->  ( w  e.  A  ->  w  e. 
U. ( [,] " {
z  e.  ran  F  |  ( [,] `  z
)  C_  A }
) ) )
134133ssrdv 3495 . . 3  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  C_  U. ( [,] " { z  e. 
ran  F  |  ( [,] `  z )  C_  A } ) )
13529, 134eqssd 3506 . 2  |-  ( A  e.  ( topGen `  ran  (,) )  ->  U. ( [,] " { z  e. 
ran  F  |  ( [,] `  z )  C_  A } )  =  A )
136 fveq2 5848 . . . . . . 7  |-  ( c  =  a  ->  ( [,] `  c )  =  ( [,] `  a
) )
137136sseq1d 3516 . . . . . 6  |-  ( c  =  a  ->  (
( [,] `  c
)  C_  ( [,] `  b )  <->  ( [,] `  a )  C_  ( [,] `  b ) ) )
138 equequ1 1803 . . . . . 6  |-  ( c  =  a  ->  (
c  =  b  <->  a  =  b ) )
139137, 138imbi12d 318 . . . . 5  |-  ( c  =  a  ->  (
( ( [,] `  c
)  C_  ( [,] `  b )  ->  c  =  b )  <->  ( ( [,] `  a )  C_  ( [,] `  b )  ->  a  =  b ) ) )
140139ralbidv 2893 . . . 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 ) ) )
141140cbvrabv 3105 . . 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 ) }
14213a1i 11 . . 3  |-  ( A  e.  ( topGen `  ran  (,) )  ->  { z  e.  ran  F  |  ( [,] `  z ) 
C_  A }  C_  ran  F )
14314, 141, 142dyadmbl 22175 . 2  |-  ( A  e.  ( topGen `  ran  (,) )  ->  U. ( [,] " { z  e. 
ran  F  |  ( [,] `  z )  C_  A } )  e.  dom  vol )
144135, 143eqeltrrd 2543 1  |-  ( A  e.  ( topGen `  ran  (,) )  ->  A  e.  dom  vol )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   {crab 2808    i^i cin 3460    C_ wss 3461   ~Pcpw 3999   <.cop 4022   U.cuni 4235   class class class wbr 4439    X. cxp 4986   dom cdm 4988   ran crn 4989    |` cres 4990   "cima 4991    o. ccom 4992   Fun wfun 5564    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486   RR*cxr 9616    < clt 9617    <_ cle 9618    - cmin 9796    / cdiv 10202   NNcn 10531   2c2 10581   NN0cn0 10791   ZZcz 10860   RR+crp 11221   (,)cioo 11532   [,]cicc 11535   |_cfl 11908   ^cexp 12148   abscabs 13149   topGenctg 14927   *Metcxmt 18598   ballcbl 18600   MetOpencmopn 18603   volcvol 22041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-disj 4411  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-omul 7127  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-acn 8314  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-rlim 13394  df-sum 13591  df-rest 14912  df-topgen 14933  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-top 19566  df-bases 19568  df-topon 19569  df-cmp 20054  df-ovol 22042  df-vol 22043
This theorem is referenced by:  opnmbl  22177
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