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Theorem lebnumii 21890
Description: Specialize the Lebesgue number lemma lebnum 21888 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 21805 . . 3  |-  II  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) )
2 cnmet 21703 . . . . 5  |-  ( abs 
o.  -  )  e.  ( Met `  CC )
3 unitssre 11777 . . . . . 6  |-  ( 0 [,] 1 )  C_  RR
4 ax-resscn 9595 . . . . . 6  |-  RR  C_  CC
53, 4sstri 3479 . . . . 5  |-  ( 0 [,] 1 )  C_  CC
6 metres2 21309 . . . . 5  |-  ( ( ( abs  o.  -  )  e.  ( Met `  CC )  /\  (
0 [,] 1 ) 
C_  CC )  -> 
( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) )  e.  ( Met `  (
0 [,] 1 ) ) )
72, 5, 6mp2an 676 . . . 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 21814 . . . 4  |-  II  e.  Comp
109a1i 11 . . 3  |-  ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  ->  II  e.  Comp )
11 simpl 458 . . 3  |-  ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  ->  U  C_  II )
12 simpr 462 . . 3  |-  ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  ->  ( 0 [,] 1 )  = 
U. U )
131, 8, 10, 11, 12lebnum 21888 . 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 11326 . . . . . . . 8  |-  ( r  e.  RR+  ->  ( 1  /  r )  e.  RR+ )
1514adantl 467 . . . . . . 7  |-  ( ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  /\  r  e.  RR+ )  ->  ( 1  /  r )  e.  RR+ )
1615rpred 11341 . . . . . 6  |-  ( ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  /\  r  e.  RR+ )  ->  ( 1  /  r )  e.  RR )
1715rpge0d 11345 . . . . . 6  |-  ( ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  /\  r  e.  RR+ )  ->  0  <_  ( 1  /  r ) )
18 flge0nn0 12051 . . . . . 6  |-  ( ( ( 1  /  r
)  e.  RR  /\  0  <_  ( 1  / 
r ) )  -> 
( |_ `  (
1  /  r ) )  e.  NN0 )
1916, 17, 18syl2anc 665 . . . . 5  |-  ( ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  /\  r  e.  RR+ )  ->  ( |_ `  ( 1  /  r
) )  e.  NN0 )
20 nn0p1nn 10909 . . . . 5  |-  ( ( |_ `  ( 1  /  r ) )  e.  NN0  ->  ( ( |_ `  ( 1  /  r ) )  +  1 )  e.  NN )
2119, 20syl 17 . . . 4  |-  ( ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  /\  r  e.  RR+ )  ->  ( ( |_ `  ( 1  / 
r ) )  +  1 )  e.  NN )
22 elfznn 11826 . . . . . . . . . . . 12  |-  ( k  e.  ( 1 ... ( ( |_ `  ( 1  /  r
) )  +  1 ) )  ->  k  e.  NN )
2322adantl 467 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  e.  NN )
2423nnrpd 11339 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  e.  RR+ )
2521adantr 466 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( |_ `  (
1  /  r ) )  +  1 )  e.  NN )
2625nnrpd 11339 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( |_ `  (
1  /  r ) )  +  1 )  e.  RR+ )
2724, 26rpdivcld 11358 . . . . . . . . 9  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  e.  RR+ )
2827rpred 11341 . . . . . . . 8  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  e.  RR )
2927rpge0d 11345 . . . . . . . 8  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  <_  ( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) )
30 elfzle2 11801 . . . . . . . . . . 11  |-  ( k  e.  ( 1 ... ( ( |_ `  ( 1  /  r
) )  +  1 ) )  ->  k  <_  ( ( |_ `  ( 1  /  r
) )  +  1 ) )
3130adantl 467 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  <_  ( ( |_ `  ( 1  /  r
) )  +  1 ) )
3225nnred 10624 . . . . . . . . . . . 12  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( |_ `  (
1  /  r ) )  +  1 )  e.  RR )
3332recnd 9668 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( |_ `  (
1  /  r ) )  +  1 )  e.  CC )
3433mulid1d 9659 . . . . . . . . . 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 4452 . . . . . . . . 9  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  <_  ( ( ( |_
`  ( 1  / 
r ) )  +  1 )  x.  1 ) )
3623nnred 10624 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  e.  RR )
37 1re 9641 . . . . . . . . . . 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 10653 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  <  ( ( |_ `  ( 1  /  r
) )  +  1 ) )
40 ledivmul 10480 . . . . . . . . . 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 1268 . . . . . . . . 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 235 . . . . . . . 8  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  <_  1 )
43 0re 9642 . . . . . . . . 9  |-  0  e.  RR
4443, 37elicc2i 11700 . . . . . . . 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 1189 . . . . . . 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 6312 . . . . . . . . . 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 3497 . . . . . . . . 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 2946 . . . . . . . 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 3184 . . . . . . 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 17 . . . . . 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 760 . . . . . . . . . . . . . 14  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  r  e.  RR+ )
5251rpred 11341 . . . . . . . . . . . . 13  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  r  e.  RR )
5328, 52resubcld 10046 . . . . . . . . . . . 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 9689 . . . . . . . . . . 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 9669 . . . . . . . . . . . 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 9689 . . . . . . . . . . 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 10911 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  (
k  -  1 )  e.  NN0 )
5823, 57syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  -  1 )  e.  NN0 )
5958nn0red 10926 . . . . . . . . . . . . 13  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  -  1 )  e.  RR )
6059, 25nndivred 10658 . . . . . . . . . . . 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 9668 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  e.  CC )
6259recnd 9668 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  -  1 )  e.  CC )
6325nnne0d 10654 . . . . . . . . . . . . . . 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 10421 . . . . . . . . . . . . . 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 9596 . . . . . . . . . . . . . . . 16  |-  1  e.  CC
66 nncan 9902 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  CC  /\  1  e.  CC )  ->  ( k  -  (
k  -  1 ) )  =  1 )
6761, 65, 66sylancl 666 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  -  ( k  -  1 ) )  =  1 )
6867oveq1d 6320 . . . . . . . . . . . . . 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 11352 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
1  /  r )  e.  RR )
71 flltp1 12033 . . . . . . . . . . . . . . 15  |-  ( ( 1  /  r )  e.  RR  ->  (
1  /  r )  <  ( ( |_
`  ( 1  / 
r ) )  +  1 ) )
7270, 71syl 17 . . . . . . . . . . . . . 14  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
1  /  r )  <  ( ( |_
`  ( 1  / 
r ) )  +  1 ) )
73 rpgt0 11313 . . . . . . . . . . . . . . . 16  |-  ( r  e.  RR+  ->  0  < 
r )
7473ad2antlr 731 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  <  r )
75 ltdiv23 10497 . . . . . . . . . . . . . . 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 1273 . . . . . . . . . . . . . 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 213 . . . . . . . . . . . . 13  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
1  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  <  r )
7869, 77eqbrtrd 4446 . . . . . . . . . . . 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 10217 . . . . . . . . . . 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 11371 . . . . . . . . . . 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 11703 . . . . . . . . . . 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 1265 . . . . . . . . . 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 9643 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  e.  RR )
8458nn0ge0d 10928 . . . . . . . . . . . 12  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  <_  ( k  -  1 ) )
85 divge0 10473 . . . . . . . . . . . 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 1265 . . . . . . . . . . 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 11702 . . . . . . . . . . 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 1265 . . . . . . . . . 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 3692 . . . . . . . . 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 2429 . . . . . . . . . . . . 13  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
9190rexmet 21720 . . . . . . . . . . . 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 3673 . . . . . . . . . . . . 13  |-  ( ( 0 [,] 1 ) 
C_  RR  <->  ( RR  i^i  ( 0 [,] 1
) )  =  ( 0 [,] 1 ) )
943, 93mpbi 211 . . . . . . . . . . . 12  |-  ( RR 
i^i  ( 0 [,] 1 ) )  =  ( 0 [,] 1
)
9545, 94syl6eleqr 2528 . . . . . . . . . . 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 11309 . . . . . . . . . . . 12  |-  ( r  e.  RR+  ->  r  e. 
RR* )
9796ad2antlr 731 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  r  e.  RR* )
98 xpss12 4960 . . . . . . . . . . . . . . 15  |-  ( ( ( 0 [,] 1
)  C_  RR  /\  (
0 [,] 1 ) 
C_  RR )  -> 
( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  C_  ( RR  X.  RR ) )
993, 3, 98mp2an 676 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  C_  ( RR  X.  RR )
100 resabs1 5153 . . . . . . . . . . . . . 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 21377 . . . . . . . . . . 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 1264 . . . . . . . . . 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 21721 . . . . . . . . . . . 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 665 . . . . . . . . . . 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 3669 . . . . . . . . . 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 3507 . . . . . . . 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 3477 . . . . . . . 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 17 . . . . . . 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 2906 . . . . . 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 45 . . . . 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 2850 . . . 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 6313 . . . . . 6  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
1 ... n )  =  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )
116 oveq2 6313 . . . . . . . . 9  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
( k  -  1 )  /  n )  =  ( ( k  -  1 )  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) )
117 oveq2 6313 . . . . . . . . 9  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
k  /  n )  =  ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) )
118116, 117oveq12d 6323 . . . . . . . 8  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
( ( k  - 
1 )  /  n
) [,] ( k  /  n ) )  =  ( ( ( k  -  1 )  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) [,] ( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) ) )
119118sseq1d 3497 . . . . . . 7  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
( ( ( k  -  1 )  /  n ) [,] (
k  /  n ) )  C_  u  <->  ( (
( k  -  1 )  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) ) [,] ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) )  C_  u ) )
120119rexbidv 2946 . . . . . 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 3046 . . . . 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 3188 . . . 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 547 . . 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 2924 . 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 187    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782   E.wrex 2783    i^i cin 3441    C_ wss 3442   U.cuni 4222   class class class wbr 4426    X. cxp 4852    |` cres 4856    o. ccom 4858   ` cfv 5601  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539    + caddc 9541    x. cmul 9543   RR*cxr 9673    < clt 9674    <_ cle 9675    - cmin 9859    / cdiv 10268   NNcn 10609   NN0cn0 10869   RR+crp 11302   (,)cioo 11635   [,]cicc 11638   ...cfz 11782   |_cfl 12023   abscabs 13276   *Metcxmt 18890   Metcme 18891   ballcbl 18892   Compccmp 20332   IIcii 21803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-ec 7373  df-map 7482  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-sum 13731  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-rest 15280  df-topn 15281  df-0g 15299  df-gsum 15300  df-topgen 15301  df-pt 15302  df-prds 15305  df-xrs 15359  df-qtop 15364  df-imas 15365  df-xps 15367  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-mulg 16627  df-cntz 16922  df-cmn 17367  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-ntr 19966  df-cls 19967  df-cn 20174  df-cnp 20175  df-cmp 20333  df-tx 20508  df-hmeo 20701  df-xms 21266  df-ms 21267  df-tms 21268  df-ii 21805
This theorem is referenced by:  cvmliftlem15  29809
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