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Theorem lebnumii 21229
Description: Specialize the Lebesgue number lemma lebnum 21227 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 21144 . . 3  |-  II  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) )
2 cnmet 21042 . . . . 5  |-  ( abs 
o.  -  )  e.  ( Met `  CC )
3 unitssre 11667 . . . . . 6  |-  ( 0 [,] 1 )  C_  RR
4 ax-resscn 9549 . . . . . 6  |-  RR  C_  CC
53, 4sstri 3513 . . . . 5  |-  ( 0 [,] 1 )  C_  CC
6 metres2 20629 . . . . 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 21153 . . . 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 21227 . 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 11243 . . . . . . . 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 11256 . . . . . 6  |-  ( ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  /\  r  e.  RR+ )  ->  ( 1  /  r )  e.  RR )
1715rpge0d 11260 . . . . . 6  |-  ( ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  /\  r  e.  RR+ )  ->  0  <_  ( 1  /  r ) )
18 flge0nn0 11922 . . . . . 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 10835 . . . . 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 11714 . . . . . . . . . . . 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 11255 . . . . . . . . . 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 11255 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( |_ `  (
1  /  r ) )  +  1 )  e.  RR+ )
2724, 26rpdivcld 11273 . . . . . . . . 9  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  e.  RR+ )
2827rpred 11256 . . . . . . . 8  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  e.  RR )
2927rpge0d 11260 . . . . . . . 8  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  <_  ( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) )
30 elfzle2 11690 . . . . . . . . . . 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 10551 . . . . . . . . . . . 12  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( |_ `  (
1  /  r ) )  +  1 )  e.  RR )
3332recnd 9622 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( |_ `  (
1  /  r ) )  +  1 )  e.  CC )
3433mulid1d 9613 . . . . . . . . . 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 4473 . . . . . . . . 9  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  <_  ( ( ( |_
`  ( 1  / 
r ) )  +  1 )  x.  1 ) )
3623nnred 10551 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  e.  RR )
37 1re 9595 . . . . . . . . . . 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 10579 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  <  ( ( |_ `  ( 1  /  r
) )  +  1 ) )
40 ledivmul 10418 . . . . . . . . . 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 1232 . . . . . . . . 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 9596 . . . . . . . . 9  |-  0  e.  RR
4443, 37elicc2i 11590 . . . . . . . 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 1180 . . . . . . 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 6291 . . . . . . . . . 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 3531 . . . . . . . . 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 2973 . . . . . . . 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 3210 . . . . . . 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 11256 . . . . . . . . . . . . 13  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  r  e.  RR )
5328, 52resubcld 9987 . . . . . . . . . . . 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 9643 . . . . . . . . . . 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 9623 . . . . . . . . . . . 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 9643 . . . . . . . . . . 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 10837 . . . . . . . . . . . . . . 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 10853 . . . . . . . . . . . . 13  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  -  1 )  e.  RR )
6059, 25nndivred 10584 . . . . . . . . . . . 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 9622 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  e.  CC )
6259recnd 9622 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  -  1 )  e.  CC )
6325nnne0d 10580 . . . . . . . . . . . . . . 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 10359 . . . . . . . . . . . . . 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 9550 . . . . . . . . . . . . . . . 16  |-  1  e.  CC
66 nncan 9848 . . . . . . . . . . . . . . . 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 6299 . . . . . . . . . . . . . 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 2510 . . . . . . . . . . . . 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 11267 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
1  /  r )  e.  RR )
71 flltp1 11905 . . . . . . . . . . . . . . 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 11231 . . . . . . . . . . . . . . . 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 10436 . . . . . . . . . . . . . . 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 1237 . . . . . . . . . . . . . 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 4467 . . . . . . . . . . . 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 10157 . . . . . . . . . . 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 11285 . . . . . . . . . . 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 11593 . . . . . . . . . . 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 1229 . . . . . . . . . 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 9597 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  e.  RR )
8458nn0ge0d 10855 . . . . . . . . . . . 12  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  <_  ( k  -  1 ) )
85 divge0 10411 . . . . . . . . . . . 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 1229 . . . . . . . . . . 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 11592 . . . . . . . . . . 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 1229 . . . . . . . . . 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 3722 . . . . . . . . 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 2467 . . . . . . . . . . . . 13  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
9190rexmet 21059 . . . . . . . . . . . 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 3703 . . . . . . . . . . . . 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 2566 . . . . . . . . . . 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 11227 . . . . . . . . . . . 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 5108 . . . . . . . . . . . . . . 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 5302 . . . . . . . . . . . . . 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 2480 . . . . . . . . . . . 12  |-  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  =  ( ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )
103102blres 20697 . . . . . . . . . . 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 1228 . . . . . . . . . 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 21060 . . . . . . . . . . . 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 3699 . . . . . . . . . 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 2508 . . . . . . . . 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 3541 . . . . . . . 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 3511 . . . . . . . 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 2937 . . . . . 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 2882 . . . 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 6292 . . . . . 6  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
1 ... n )  =  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )
116 oveq2 6292 . . . . . . . . 9  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
( k  -  1 )  /  n )  =  ( ( k  -  1 )  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) )
117 oveq2 6292 . . . . . . . . 9  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
k  /  n )  =  ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) )
118116, 117oveq12d 6302 . . . . . . . 8  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
( ( k  - 
1 )  /  n
) [,] ( k  /  n ) )  =  ( ( ( k  -  1 )  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) [,] ( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) ) )
119118sseq1d 3531 . . . . . . 7  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
( ( ( k  -  1 )  /  n ) [,] (
k  /  n ) )  C_  u  <->  ( (
( k  -  1 )  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) ) [,] ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) )  C_  u ) )
120119rexbidv 2973 . . . . . 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 3072 . . . . 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 3214 . . . 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 2955 . 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 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    i^i cin 3475    C_ wss 3476   U.cuni 4245   class class class wbr 4447    X. cxp 4997    |` cres 5001    o. ccom 5003   ` cfv 5588  (class class class)co 6284   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497   RR*cxr 9627    < clt 9628    <_ cle 9629    - cmin 9805    / cdiv 10206   NNcn 10536   NN0cn0 10795   RR+crp 11220   (,)cioo 11529   [,]cicc 11532   ...cfz 11672   |_cfl 11895   abscabs 13030   *Metcxmt 18202   Metcme 18203   ballcbl 18204   Compccmp 19680   IIcii 21142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-ec 7313  df-map 7422  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-fi 7871  df-sup 7901  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-7 10599  df-8 10600  df-9 10601  df-10 10602  df-n0 10796  df-z 10865  df-dec 10977  df-uz 11083  df-q 11183  df-rp 11221  df-xneg 11318  df-xadd 11319  df-xmul 11320  df-ioo 11533  df-ico 11535  df-icc 11536  df-fz 11673  df-fzo 11793  df-fl 11897  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-sum 13472  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-starv 14570  df-sca 14571  df-vsca 14572  df-ip 14573  df-tset 14574  df-ple 14575  df-ds 14577  df-unif 14578  df-hom 14579  df-cco 14580  df-rest 14678  df-topn 14679  df-0g 14697  df-gsum 14698  df-topgen 14699  df-pt 14700  df-prds 14703  df-xrs 14757  df-qtop 14762  df-imas 14763  df-xps 14765  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-submnd 15787  df-mulg 15870  df-cntz 16160  df-cmn 16606  df-psmet 18210  df-xmet 18211  df-met 18212  df-bl 18213  df-mopn 18214  df-cnfld 18220  df-top 19194  df-bases 19196  df-topon 19197  df-topsp 19198  df-cld 19314  df-ntr 19315  df-cls 19316  df-cn 19522  df-cnp 19523  df-cmp 19681  df-tx 19826  df-hmeo 20019  df-xms 20586  df-ms 20587  df-tms 20588  df-ii 21144
This theorem is referenced by:  cvmliftlem15  28411
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