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Theorem lebnumii 20669
Description: Specialize the Lebesgue number lemma lebnum 20667 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 20584 . . 3  |-  II  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) )
2 cnmet 20482 . . . . 5  |-  ( abs 
o.  -  )  e.  ( Met `  CC )
3 unitssre 11548 . . . . . 6  |-  ( 0 [,] 1 )  C_  RR
4 ax-resscn 9449 . . . . . 6  |-  RR  C_  CC
53, 4sstri 3472 . . . . 5  |-  ( 0 [,] 1 )  C_  CC
6 metres2 20069 . . . . 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 20593 . . . 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 20667 . 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 11124 . . . . . . . 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 11137 . . . . . 6  |-  ( ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  /\  r  e.  RR+ )  ->  ( 1  /  r )  e.  RR )
1715rpge0d 11141 . . . . . 6  |-  ( ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  /\  r  e.  RR+ )  ->  0  <_  ( 1  /  r ) )
18 flge0nn0 11782 . . . . . 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 10729 . . . . 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 11594 . . . . . . . . . . . 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 11136 . . . . . . . . . 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 11136 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( |_ `  (
1  /  r ) )  +  1 )  e.  RR+ )
2724, 26rpdivcld 11154 . . . . . . . . 9  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  e.  RR+ )
2827rpred 11137 . . . . . . . 8  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  e.  RR )
2927rpge0d 11141 . . . . . . . 8  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  <_  ( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) )
30 elfzle2 11571 . . . . . . . . . . 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 10447 . . . . . . . . . . . 12  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( |_ `  (
1  /  r ) )  +  1 )  e.  RR )
3332recnd 9522 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( |_ `  (
1  /  r ) )  +  1 )  e.  CC )
3433mulid1d 9513 . . . . . . . . . 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 4425 . . . . . . . . 9  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  <_  ( ( ( |_
`  ( 1  / 
r ) )  +  1 )  x.  1 ) )
3623nnred 10447 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  e.  RR )
37 1re 9495 . . . . . . . . . . 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 10475 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  <  ( ( |_ `  ( 1  /  r
) )  +  1 ) )
40 ledivmul 10315 . . . . . . . . . 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 1223 . . . . . . . . 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 9496 . . . . . . . . 9  |-  0  e.  RR
4443, 37elicc2i 11471 . . . . . . . 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 6206 . . . . . . . . . 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 3490 . . . . . . . . 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 2864 . . . . . . . 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 3173 . . . . . . 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 11137 . . . . . . . . . . . . 13  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  r  e.  RR )
5328, 52resubcld 9886 . . . . . . . . . . . 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 9543 . . . . . . . . . . 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 9523 . . . . . . . . . . . 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 9543 . . . . . . . . . . 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 10731 . . . . . . . . . . . . . . 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 10747 . . . . . . . . . . . . 13  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  -  1 )  e.  RR )
6059, 25nndivred 10480 . . . . . . . . . . . 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 9522 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  e.  CC )
6259recnd 9522 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  -  1 )  e.  CC )
6325nnne0d 10476 . . . . . . . . . . . . . . 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 10256 . . . . . . . . . . . . . 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 9450 . . . . . . . . . . . . . . . 16  |-  1  e.  CC
66 nncan 9748 . . . . . . . . . . . . . . . 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 6214 . . . . . . . . . . . . . 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 2497 . . . . . . . . . . . . 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 11148 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
1  /  r )  e.  RR )
71 flltp1 11766 . . . . . . . . . . . . . . 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 11112 . . . . . . . . . . . . . . . 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 10333 . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . 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 4419 . . . . . . . . . . . 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 10054 . . . . . . . . . . 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 11166 . . . . . . . . . . 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 11474 . . . . . . . . . . 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 1220 . . . . . . . . . 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 9497 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  e.  RR )
8458nn0ge0d 10749 . . . . . . . . . . . 12  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  <_  ( k  -  1 ) )
85 divge0 10308 . . . . . . . . . . . 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 1220 . . . . . . . . . . 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 11473 . . . . . . . . . . 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 1220 . . . . . . . . . 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 3681 . . . . . . . . 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 2454 . . . . . . . . . . . . 13  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
9190rexmet 20499 . . . . . . . . . . . 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 3662 . . . . . . . . . . . . 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 2553 . . . . . . . . . . 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 11108 . . . . . . . . . . . 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 5052 . . . . . . . . . . . . . . 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 5246 . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . 12  |-  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  =  ( ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )
103102blres 20137 . . . . . . . . . . 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 1219 . . . . . . . . . 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 20500 . . . . . . . . . . . 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 3658 . . . . . . . . . 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 2495 . . . . . . . . 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 3500 . . . . . . . 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 3470 . . . . . . . 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 2931 . . . . . 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 2910 . . . 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 6207 . . . . . 6  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
1 ... n )  =  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )
116 oveq2 6207 . . . . . . . . 9  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
( k  -  1 )  /  n )  =  ( ( k  -  1 )  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) )
117 oveq2 6207 . . . . . . . . 9  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
k  /  n )  =  ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) )
118116, 117oveq12d 6217 . . . . . . . 8  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
( ( k  - 
1 )  /  n
) [,] ( k  /  n ) )  =  ( ( ( k  -  1 )  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) [,] ( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) ) )
119118sseq1d 3490 . . . . . . 7  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
( ( ( k  -  1 )  /  n ) [,] (
k  /  n ) )  C_  u  <->  ( (
( k  -  1 )  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) ) [,] ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) )  C_  u ) )
120119rexbidv 2864 . . . . . 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 3035 . . . . 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 3177 . . . 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 2945 . 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 1370    e. wcel 1758   A.wral 2798   E.wrex 2799    i^i cin 3434    C_ wss 3435   U.cuni 4198   class class class wbr 4399    X. cxp 4945    |` cres 4949    o. ccom 4951   ` cfv 5525  (class class class)co 6199   CCcc 9390   RRcr 9391   0cc0 9392   1c1 9393    + caddc 9395    x. cmul 9397   RR*cxr 9527    < clt 9528    <_ cle 9529    - cmin 9705    / cdiv 10103   NNcn 10432   NN0cn0 10689   RR+crp 11101   (,)cioo 11410   [,]cicc 11413   ...cfz 11553   |_cfl 11756   abscabs 12840   *Metcxmt 17925   Metcme 17926   ballcbl 17927   Compccmp 19120   IIcii 20582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470  ax-addf 9471  ax-mulf 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-iin 4281  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-of 6429  df-om 6586  df-1st 6686  df-2nd 6687  df-supp 6800  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-ec 7212  df-map 7325  df-ixp 7373  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fsupp 7731  df-fi 7771  df-sup 7801  df-oi 7834  df-card 8219  df-cda 8447  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-ioo 11414  df-ico 11416  df-icc 11417  df-fz 11554  df-fzo 11665  df-fl 11758  df-seq 11923  df-exp 11982  df-hash 12220  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-clim 13083  df-sum 13281  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-starv 14371  df-sca 14372  df-vsca 14373  df-ip 14374  df-tset 14375  df-ple 14376  df-ds 14378  df-unif 14379  df-hom 14380  df-cco 14381  df-rest 14479  df-topn 14480  df-0g 14498  df-gsum 14499  df-topgen 14500  df-pt 14501  df-prds 14504  df-xrs 14558  df-qtop 14563  df-imas 14564  df-xps 14566  df-mre 14642  df-mrc 14643  df-acs 14645  df-mnd 15533  df-submnd 15583  df-mulg 15666  df-cntz 15953  df-cmn 16399  df-psmet 17933  df-xmet 17934  df-met 17935  df-bl 17936  df-mopn 17937  df-cnfld 17943  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-cld 18754  df-ntr 18755  df-cls 18756  df-cn 18962  df-cnp 18963  df-cmp 19121  df-tx 19266  df-hmeo 19459  df-xms 20026  df-ms 20027  df-tms 20028  df-ii 20584
This theorem is referenced by:  cvmliftlem15  27330
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