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Theorem lebnumii 20518
Description: Specialize the Lebesgue number lemma lebnum 20516 to the unit interval. (Contributed by Mario Carneiro, 14-Feb-2015.)
Assertion
Ref Expression
lebnumii  |-  ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  ->  E. n  e.  NN  A. k  e.  ( 1 ... n
) E. u  e.  U  ( ( ( k  -  1 )  /  n ) [,] ( k  /  n
) )  C_  u
)
Distinct variable group:    k, n, u, U

Proof of Theorem lebnumii
Dummy variables  r  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ii 20433 . . 3  |-  II  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) )
2 cnmet 20331 . . . . 5  |-  ( abs 
o.  -  )  e.  ( Met `  CC )
3 unitssre 11424 . . . . . 6  |-  ( 0 [,] 1 )  C_  RR
4 ax-resscn 9331 . . . . . 6  |-  RR  C_  CC
53, 4sstri 3360 . . . . 5  |-  ( 0 [,] 1 )  C_  CC
6 metres2 19918 . . . . 5  |-  ( ( ( abs  o.  -  )  e.  ( Met `  CC )  /\  (
0 [,] 1 ) 
C_  CC )  -> 
( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) )  e.  ( Met `  (
0 [,] 1 ) ) )
72, 5, 6mp2an 672 . . . 4  |-  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  e.  ( Met `  ( 0 [,] 1 ) )
87a1i 11 . . 3  |-  ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  ->  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  e.  ( Met `  ( 0 [,] 1 ) ) )
9 iicmp 20442 . . . 4  |-  II  e.  Comp
109a1i 11 . . 3  |-  ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  ->  II  e.  Comp )
11 simpl 457 . . 3  |-  ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  ->  U  C_  II )
12 simpr 461 . . 3  |-  ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  ->  ( 0 [,] 1 )  = 
U. U )
131, 8, 10, 11, 12lebnum 20516 . 2  |-  ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  ->  E. r  e.  RR+  A. x  e.  ( 0 [,] 1
) E. u  e.  U  ( x (
ball `  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) ) r )  C_  u )
14 rpreccl 11006 . . . . . . . 8  |-  ( r  e.  RR+  ->  ( 1  /  r )  e.  RR+ )
1514adantl 466 . . . . . . 7  |-  ( ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  /\  r  e.  RR+ )  ->  ( 1  /  r )  e.  RR+ )
1615rpred 11019 . . . . . 6  |-  ( ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  /\  r  e.  RR+ )  ->  ( 1  /  r )  e.  RR )
1715rpge0d 11023 . . . . . 6  |-  ( ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  /\  r  e.  RR+ )  ->  0  <_  ( 1  /  r ) )
18 flge0nn0 11658 . . . . . 6  |-  ( ( ( 1  /  r
)  e.  RR  /\  0  <_  ( 1  / 
r ) )  -> 
( |_ `  (
1  /  r ) )  e.  NN0 )
1916, 17, 18syl2anc 661 . . . . 5  |-  ( ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  /\  r  e.  RR+ )  ->  ( |_ `  ( 1  /  r
) )  e.  NN0 )
20 nn0p1nn 10611 . . . . 5  |-  ( ( |_ `  ( 1  /  r ) )  e.  NN0  ->  ( ( |_ `  ( 1  /  r ) )  +  1 )  e.  NN )
2119, 20syl 16 . . . 4  |-  ( ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  /\  r  e.  RR+ )  ->  ( ( |_ `  ( 1  / 
r ) )  +  1 )  e.  NN )
22 elfznn 11470 . . . . . . . . . . . 12  |-  ( k  e.  ( 1 ... ( ( |_ `  ( 1  /  r
) )  +  1 ) )  ->  k  e.  NN )
2322adantl 466 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  e.  NN )
2423nnrpd 11018 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  e.  RR+ )
2521adantr 465 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( |_ `  (
1  /  r ) )  +  1 )  e.  NN )
2625nnrpd 11018 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( |_ `  (
1  /  r ) )  +  1 )  e.  RR+ )
2724, 26rpdivcld 11036 . . . . . . . . 9  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  e.  RR+ )
2827rpred 11019 . . . . . . . 8  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  e.  RR )
2927rpge0d 11023 . . . . . . . 8  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  <_  ( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) )
30 elfzle2 11447 . . . . . . . . . . 11  |-  ( k  e.  ( 1 ... ( ( |_ `  ( 1  /  r
) )  +  1 ) )  ->  k  <_  ( ( |_ `  ( 1  /  r
) )  +  1 ) )
3130adantl 466 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  <_  ( ( |_ `  ( 1  /  r
) )  +  1 ) )
3225nnred 10329 . . . . . . . . . . . 12  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( |_ `  (
1  /  r ) )  +  1 )  e.  RR )
3332recnd 9404 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( |_ `  (
1  /  r ) )  +  1 )  e.  CC )
3433mulid1d 9395 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( ( |_ `  ( 1  /  r
) )  +  1 )  x.  1 )  =  ( ( |_
`  ( 1  / 
r ) )  +  1 ) )
3531, 34breqtrrd 4313 . . . . . . . . 9  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  <_  ( ( ( |_
`  ( 1  / 
r ) )  +  1 )  x.  1 ) )
3623nnred 10329 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  e.  RR )
37 1re 9377 . . . . . . . . . . 11  |-  1  e.  RR
3837a1i 11 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  1  e.  RR )
3925nngt0d 10357 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  <  ( ( |_ `  ( 1  /  r
) )  +  1 ) )
40 ledivmul 10197 . . . . . . . . . 10  |-  ( ( k  e.  RR  /\  1  e.  RR  /\  (
( ( |_ `  ( 1  /  r
) )  +  1 )  e.  RR  /\  0  <  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) )  ->  ( ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) )  <_ 
1  <->  k  <_  (
( ( |_ `  ( 1  /  r
) )  +  1 )  x.  1 ) ) )
4136, 38, 32, 39, 40syl112anc 1222 . . . . . . . . 9  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) )  <_  1  <->  k  <_  ( ( ( |_ `  ( 1  /  r
) )  +  1 )  x.  1 ) ) )
4235, 41mpbird 232 . . . . . . . 8  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  <_  1 )
43 0re 9378 . . . . . . . . 9  |-  0  e.  RR
4443, 37elicc2i 11353 . . . . . . . 8  |-  ( ( k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  e.  ( 0 [,] 1 )  <->  ( (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  e.  RR  /\  0  <_  ( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) )  /\  ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) )  <_ 
1 ) )
4528, 29, 42, 44syl3anbrc 1172 . . . . . . 7  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  e.  ( 0 [,] 1 ) )
46 oveq1 6093 . . . . . . . . . 10  |-  ( x  =  ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) )  ->  (
x ( ball `  (
( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) ) r )  =  ( ( k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) ) ( ball `  (
( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) ) r ) )
4746sseq1d 3378 . . . . . . . . 9  |-  ( x  =  ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) )  ->  (
( x ( ball `  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ) r )  C_  u 
<->  ( ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) ( ball `  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ) r )  C_  u ) )
4847rexbidv 2731 . . . . . . . 8  |-  ( x  =  ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) )  ->  ( E. u  e.  U  ( x ( ball `  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ) r )  C_  u 
<->  E. u  e.  U  ( ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) ( ball `  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ) r )  C_  u ) )
4948rspcv 3064 . . . . . . 7  |-  ( ( k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  e.  ( 0 [,] 1 )  ->  ( A. x  e.  (
0 [,] 1 ) E. u  e.  U  ( x ( ball `  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ) r )  C_  u  ->  E. u  e.  U  ( ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) ( ball `  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ) r )  C_  u ) )
5045, 49syl 16 . . . . . 6  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  ( A. x  e.  (
0 [,] 1 ) E. u  e.  U  ( x ( ball `  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ) r )  C_  u  ->  E. u  e.  U  ( ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) ( ball `  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ) r )  C_  u ) )
51 simplr 754 . . . . . . . . . . . . . 14  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  r  e.  RR+ )
5251rpred 11019 . . . . . . . . . . . . 13  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  r  e.  RR )
5328, 52resubcld 9768 . . . . . . . . . . . 12  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) )  -  r )  e.  RR )
5453rexrd 9425 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) )  -  r )  e.  RR* )
5528, 52readdcld 9405 . . . . . . . . . . . 12  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) )  +  r )  e.  RR )
5655rexrd 9425 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) )  +  r )  e.  RR* )
57 nnm1nn0 10613 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  (
k  -  1 )  e.  NN0 )
5823, 57syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  -  1 )  e.  NN0 )
5958nn0red 10629 . . . . . . . . . . . . 13  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  -  1 )  e.  RR )
6059, 25nndivred 10362 . . . . . . . . . . . 12  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( k  -  1 )  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  e.  RR )
6136recnd 9404 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  e.  CC )
6259recnd 9404 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  -  1 )  e.  CC )
6325nnne0d 10358 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( |_ `  (
1  /  r ) )  +  1 )  =/=  0 )
6461, 62, 33, 63divsubdird 10138 . . . . . . . . . . . . . 14  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( k  -  (
k  -  1 ) )  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  =  ( ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) )  -  ( ( k  - 
1 )  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) ) )
65 ax-1cn 9332 . . . . . . . . . . . . . . . 16  |-  1  e.  CC
66 nncan 9630 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  CC  /\  1  e.  CC )  ->  ( k  -  (
k  -  1 ) )  =  1 )
6761, 65, 66sylancl 662 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  -  ( k  -  1 ) )  =  1 )
6867oveq1d 6101 . . . . . . . . . . . . . 14  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( k  -  (
k  -  1 ) )  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  =  ( 1  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) )
6964, 68eqtr3d 2472 . . . . . . . . . . . . 13  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) )  -  ( ( k  -  1 )  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) )  =  ( 1  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) )
7051rprecred 11030 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
1  /  r )  e.  RR )
71 flltp1 11642 . . . . . . . . . . . . . . 15  |-  ( ( 1  /  r )  e.  RR  ->  (
1  /  r )  <  ( ( |_
`  ( 1  / 
r ) )  +  1 ) )
7270, 71syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
1  /  r )  <  ( ( |_
`  ( 1  / 
r ) )  +  1 ) )
73 rpgt0 10994 . . . . . . . . . . . . . . . 16  |-  ( r  e.  RR+  ->  0  < 
r )
7473ad2antlr 726 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  <  r )
75 ltdiv23 10215 . . . . . . . . . . . . . . 15  |-  ( ( 1  e.  RR  /\  ( r  e.  RR  /\  0  <  r )  /\  ( ( ( |_ `  ( 1  /  r ) )  +  1 )  e.  RR  /\  0  < 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) )  -> 
( ( 1  / 
r )  <  (
( |_ `  (
1  /  r ) )  +  1 )  <-> 
( 1  /  (
( |_ `  (
1  /  r ) )  +  1 ) )  <  r ) )
7638, 52, 74, 32, 39, 75syl122anc 1227 . . . . . . . . . . . . . 14  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( 1  /  r
)  <  ( ( |_ `  ( 1  / 
r ) )  +  1 )  <->  ( 1  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) )  < 
r ) )
7772, 76mpbid 210 . . . . . . . . . . . . 13  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
1  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  <  r )
7869, 77eqbrtrd 4307 . . . . . . . . . . . 12  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) )  -  ( ( k  -  1 )  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) )  <  r )
7928, 60, 52, 78ltsub23d 9936 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) )  -  r )  <  ( ( k  -  1 )  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) )
8028, 51ltaddrpd 11048 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  <  ( ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) )  +  r ) )
81 iccssioo 11356 . . . . . . . . . . 11  |-  ( ( ( ( ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) )  -  r )  e.  RR*  /\  ( ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) )  +  r )  e.  RR* )  /\  ( ( ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) )  -  r )  <  (
( k  -  1 )  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  /\  ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) )  <  (
( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) )  +  r ) ) )  ->  (
( ( k  - 
1 )  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) [,] ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) ) 
C_  ( ( ( k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  -  r ) (,) ( ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) )  +  r ) ) )
8254, 56, 79, 80, 81syl22anc 1219 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( ( k  - 
1 )  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) [,] ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) ) 
C_  ( ( ( k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  -  r ) (,) ( ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) )  +  r ) ) )
83 0red 9379 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  e.  RR )
8458nn0ge0d 10631 . . . . . . . . . . . 12  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  <_  ( k  -  1 ) )
85 divge0 10190 . . . . . . . . . . . 12  |-  ( ( ( ( k  - 
1 )  e.  RR  /\  0  <_  ( k  -  1 ) )  /\  ( ( ( |_ `  ( 1  /  r ) )  +  1 )  e.  RR  /\  0  < 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) )  -> 
0  <_  ( (
k  -  1 )  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) )
8659, 84, 32, 39, 85syl22anc 1219 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  <_  ( ( k  - 
1 )  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) )
87 iccss 11355 . . . . . . . . . . 11  |-  ( ( ( 0  e.  RR  /\  1  e.  RR )  /\  ( 0  <_ 
( ( k  - 
1 )  /  (
( |_ `  (
1  /  r ) )  +  1 ) )  /\  ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) )  <_ 
1 ) )  -> 
( ( ( k  -  1 )  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) [,] (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) ) )  C_  ( 0 [,] 1 ) )
8883, 38, 86, 42, 87syl22anc 1219 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( ( k  - 
1 )  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) [,] ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) ) 
C_  ( 0 [,] 1 ) )
8982, 88ssind 3569 . . . . . . . . 9  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( ( k  - 
1 )  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) [,] ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) ) 
C_  ( ( ( ( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) )  -  r ) (,) ( ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) )  +  r ) )  i^i  ( 0 [,] 1
) ) )
90 eqid 2438 . . . . . . . . . . . . 13  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
9190rexmet 20348 . . . . . . . . . . . 12  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( *Met `  RR )
9291a1i 11 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( *Met `  RR ) )
93 dfss1 3550 . . . . . . . . . . . . 13  |-  ( ( 0 [,] 1 ) 
C_  RR  <->  ( RR  i^i  ( 0 [,] 1
) )  =  ( 0 [,] 1 ) )
943, 93mpbi 208 . . . . . . . . . . . 12  |-  ( RR 
i^i  ( 0 [,] 1 ) )  =  ( 0 [,] 1
)
9545, 94syl6eleqr 2529 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  e.  ( RR  i^i  ( 0 [,] 1
) ) )
96 rpxr 10990 . . . . . . . . . . . 12  |-  ( r  e.  RR+  ->  r  e. 
RR* )
9796ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  r  e.  RR* )
98 xpss12 4940 . . . . . . . . . . . . . . 15  |-  ( ( ( 0 [,] 1
)  C_  RR  /\  (
0 [,] 1 ) 
C_  RR )  -> 
( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  C_  ( RR  X.  RR ) )
993, 3, 98mp2an 672 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  C_  ( RR  X.  RR )
100 resabs1 5134 . . . . . . . . . . . . . 14  |-  ( ( ( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) 
C_  ( RR  X.  RR )  ->  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )  =  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) )
10199, 100ax-mp 5 . . . . . . . . . . . . 13  |-  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )  =  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )
102101eqcomi 2442 . . . . . . . . . . . 12  |-  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  =  ( ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )
103102blres 19986 . . . . . . . . . . 11  |-  ( ( ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )  e.  ( *Met `  RR )  /\  (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  e.  ( RR  i^i  ( 0 [,] 1
) )  /\  r  e.  RR* )  ->  (
( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) ( ball `  (
( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) ) r )  =  ( ( ( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) r )  i^i  ( 0 [,] 1 ) ) )
10492, 95, 97, 103syl3anc 1218 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) ( ball `  (
( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) ) r )  =  ( ( ( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) r )  i^i  ( 0 [,] 1 ) ) )
10590bl2ioo 20349 . . . . . . . . . . . 12  |-  ( ( ( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) )  e.  RR  /\  r  e.  RR )  ->  ( ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  =  ( ( ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) )  -  r ) (,) (
( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) )  +  r ) ) )
10628, 52, 105syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) ( ball `  (
( abs  o.  -  )  |`  ( RR  X.  RR ) ) ) r )  =  ( ( ( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) )  -  r ) (,) ( ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) )  +  r ) ) )
107106ineq1d 3546 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) ( ball `  ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) ) ) r )  i^i  ( 0 [,] 1
) )  =  ( ( ( ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) )  -  r ) (,) (
( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) )  +  r ) )  i^i  ( 0 [,] 1 ) ) )
108104, 107eqtrd 2470 . . . . . . . . 9  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) ( ball `  (
( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) ) r )  =  ( ( ( ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) )  -  r
) (,) ( ( k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  +  r ) )  i^i  ( 0 [,] 1 ) ) )
10989, 108sseqtr4d 3388 . . . . . . . 8  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( ( k  - 
1 )  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) [,] ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) ) 
C_  ( ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) (
ball `  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) ) r ) )
110 sstr2 3358 . . . . . . . 8  |-  ( ( ( ( k  - 
1 )  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) [,] ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) ) 
C_  ( ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) (
ball `  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) ) r )  ->  ( (
( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) ( ball `  (
( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) ) r )  C_  u  ->  ( ( ( k  - 
1 )  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) [,] ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) ) 
C_  u ) )
111109, 110syl 16 . . . . . . 7  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) ( ball `  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ) r )  C_  u  ->  ( ( ( k  -  1 )  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) [,] ( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) )  C_  u
) )
112111reximdv 2822 . . . . . 6  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  ( E. u  e.  U  ( ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) ( ball `  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ) r )  C_  u  ->  E. u  e.  U  ( ( ( k  -  1 )  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) [,] (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) ) )  C_  u )
)
11350, 112syld 44 . . . . 5  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  ( A. x  e.  (
0 [,] 1 ) E. u  e.  U  ( x ( ball `  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) ) ) r )  C_  u  ->  E. u  e.  U  ( ( ( k  -  1 )  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) [,] (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) ) )  C_  u )
)
114113ralrimdva 2801 . . . 4  |-  ( ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  /\  r  e.  RR+ )  ->  ( A. x  e.  ( 0 [,] 1 ) E. u  e.  U  ( x ( ball `  (
( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) ) r )  C_  u  ->  A. k  e.  ( 1 ... ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) E. u  e.  U  ( ( ( k  - 
1 )  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) [,] ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) ) 
C_  u ) )
115 oveq2 6094 . . . . . 6  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
1 ... n )  =  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )
116 oveq2 6094 . . . . . . . . 9  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
( k  -  1 )  /  n )  =  ( ( k  -  1 )  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) )
117 oveq2 6094 . . . . . . . . 9  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
k  /  n )  =  ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) )
118116, 117oveq12d 6104 . . . . . . . 8  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
( ( k  - 
1 )  /  n
) [,] ( k  /  n ) )  =  ( ( ( k  -  1 )  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) [,] ( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) ) )
119118sseq1d 3378 . . . . . . 7  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
( ( ( k  -  1 )  /  n ) [,] (
k  /  n ) )  C_  u  <->  ( (
( k  -  1 )  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) ) [,] ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) )  C_  u ) )
120119rexbidv 2731 . . . . . 6  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  ( E. u  e.  U  ( ( ( k  -  1 )  /  n ) [,] (
k  /  n ) )  C_  u  <->  E. u  e.  U  ( (
( k  -  1 )  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) ) [,] ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) )  C_  u ) )
121115, 120raleqbidv 2926 . . . . 5  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  ( A. k  e.  (
1 ... n ) E. u  e.  U  ( ( ( k  - 
1 )  /  n
) [,] ( k  /  n ) ) 
C_  u  <->  A. k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) E. u  e.  U  ( ( ( k  -  1 )  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) [,] ( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) )  C_  u
) )
122121rspcev 3068 . . . 4  |-  ( ( ( ( |_ `  ( 1  /  r
) )  +  1 )  e.  NN  /\  A. k  e.  ( 1 ... ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) E. u  e.  U  ( ( ( k  - 
1 )  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) [,] ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) ) 
C_  u )  ->  E. n  e.  NN  A. k  e.  ( 1 ... n ) E. u  e.  U  ( ( ( k  - 
1 )  /  n
) [,] ( k  /  n ) ) 
C_  u )
12321, 114, 122syl6an 545 . . 3  |-  ( ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  /\  r  e.  RR+ )  ->  ( A. x  e.  ( 0 [,] 1 ) E. u  e.  U  ( x ( ball `  (
( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) ) r )  C_  u  ->  E. n  e.  NN  A. k  e.  ( 1 ... n ) E. u  e.  U  ( ( ( k  - 
1 )  /  n
) [,] ( k  /  n ) ) 
C_  u ) )
124123rexlimdva 2836 . 2  |-  ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  ->  ( E. r  e.  RR+  A. x  e.  ( 0 [,] 1
) E. u  e.  U  ( x (
ball `  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) ) r )  C_  u  ->  E. n  e.  NN  A. k  e.  ( 1 ... n ) E. u  e.  U  ( ( ( k  - 
1 )  /  n
) [,] ( k  /  n ) ) 
C_  u ) )
12513, 124mpd 15 1  |-  ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  ->  E. n  e.  NN  A. k  e.  ( 1 ... n
) E. u  e.  U  ( ( ( k  -  1 )  /  n ) [,] ( k  /  n
) )  C_  u
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2710   E.wrex 2711    i^i cin 3322    C_ wss 3323   U.cuni 4086   class class class wbr 4287    X. cxp 4833    |` cres 4837    o. ccom 4839   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279   RR*cxr 9409    < clt 9410    <_ cle 9411    - cmin 9587    / cdiv 9985   NNcn 10314   NN0cn0 10571   RR+crp 10983   (,)cioo 11292   [,]cicc 11295   ...cfz 11429   |_cfl 11632   abscabs 12715   *Metcxmt 17781   Metcme 17782   ballcbl 17783   Compccmp 18969   IIcii 20431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-ec 7095  df-map 7208  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-sum 13156  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-cnfld 17799  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-cld 18603  df-ntr 18604  df-cls 18605  df-cn 18811  df-cnp 18812  df-cmp 18970  df-tx 19115  df-hmeo 19308  df-xms 19875  df-ms 19876  df-tms 19877  df-ii 20433
This theorem is referenced by:  cvmliftlem15  27156
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