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Theorem lebnumii 18480
Description: Specialize the Lebesgue number lemma lebnum 18478 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 18397 . . 3  |-  II  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) ) )
2 cnmet 18297 . . . . 5  |-  ( abs 
o.  -  )  e.  ( Met `  CC )
3 0re 8854 . . . . . . 7  |-  0  e.  RR
4 1re 8853 . . . . . . 7  |-  1  e.  RR
5 iccssre 10747 . . . . . . 7  |-  ( ( 0  e.  RR  /\  1  e.  RR )  ->  ( 0 [,] 1
)  C_  RR )
63, 4, 5mp2an 653 . . . . . 6  |-  ( 0 [,] 1 )  C_  RR
7 ax-resscn 8810 . . . . . 6  |-  RR  C_  CC
86, 7sstri 3201 . . . . 5  |-  ( 0 [,] 1 )  C_  CC
9 metres2 17943 . . . . 5  |-  ( ( ( abs  o.  -  )  e.  ( Met `  CC )  /\  (
0 [,] 1 ) 
C_  CC )  -> 
( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) ) )  e.  ( Met `  (
0 [,] 1 ) ) )
102, 8, 9mp2an 653 . . . 4  |-  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  e.  ( Met `  ( 0 [,] 1 ) )
1110a1i 10 . . 3  |-  ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  ->  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  e.  ( Met `  ( 0 [,] 1 ) ) )
12 iicmp 18406 . . . 4  |-  II  e.  Comp
1312a1i 10 . . 3  |-  ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  ->  II  e.  Comp )
14 simpl 443 . . 3  |-  ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  ->  U  C_  II )
15 simpr 447 . . 3  |-  ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  ->  ( 0 [,] 1 )  = 
U. U )
161, 11, 13, 14, 15lebnum 18478 . 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 )
17 rpreccl 10393 . . . . . . . 8  |-  ( r  e.  RR+  ->  ( 1  /  r )  e.  RR+ )
1817adantl 452 . . . . . . 7  |-  ( ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  /\  r  e.  RR+ )  ->  ( 1  /  r )  e.  RR+ )
1918rpred 10406 . . . . . 6  |-  ( ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  /\  r  e.  RR+ )  ->  ( 1  /  r )  e.  RR )
2018rpge0d 10410 . . . . . 6  |-  ( ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  /\  r  e.  RR+ )  ->  0  <_  ( 1  /  r ) )
21 flge0nn0 10964 . . . . . 6  |-  ( ( ( 1  /  r
)  e.  RR  /\  0  <_  ( 1  / 
r ) )  -> 
( |_ `  (
1  /  r ) )  e.  NN0 )
2219, 20, 21syl2anc 642 . . . . 5  |-  ( ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  /\  r  e.  RR+ )  ->  ( |_ `  ( 1  /  r
) )  e.  NN0 )
23 nn0p1nn 10019 . . . . 5  |-  ( ( |_ `  ( 1  /  r ) )  e.  NN0  ->  ( ( |_ `  ( 1  /  r ) )  +  1 )  e.  NN )
2422, 23syl 15 . . . 4  |-  ( ( ( U  C_  II  /\  ( 0 [,] 1
)  =  U. U
)  /\  r  e.  RR+ )  ->  ( ( |_ `  ( 1  / 
r ) )  +  1 )  e.  NN )
25 elfznn 10835 . . . . . . . . . . . 12  |-  ( k  e.  ( 1 ... ( ( |_ `  ( 1  /  r
) )  +  1 ) )  ->  k  e.  NN )
2625adantl 452 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  e.  NN )
2726nnrpd 10405 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  e.  RR+ )
2824adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( |_ `  (
1  /  r ) )  +  1 )  e.  NN )
2928nnrpd 10405 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( |_ `  (
1  /  r ) )  +  1 )  e.  RR+ )
3027, 29rpdivcld 10423 . . . . . . . . 9  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  e.  RR+ )
3130rpred 10406 . . . . . . . 8  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  e.  RR )
3230rpge0d 10410 . . . . . . . 8  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  <_  ( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) )
33 elfzle2 10816 . . . . . . . . . . 11  |-  ( k  e.  ( 1 ... ( ( |_ `  ( 1  /  r
) )  +  1 ) )  ->  k  <_  ( ( |_ `  ( 1  /  r
) )  +  1 ) )
3433adantl 452 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  <_  ( ( |_ `  ( 1  /  r
) )  +  1 ) )
3528nnred 9777 . . . . . . . . . . . 12  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( |_ `  (
1  /  r ) )  +  1 )  e.  RR )
3635recnd 8877 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( |_ `  (
1  /  r ) )  +  1 )  e.  CC )
3736mulid1d 8868 . . . . . . . . . 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 ) )
3834, 37breqtrrd 4065 . . . . . . . . 9  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  <_  ( ( ( |_
`  ( 1  / 
r ) )  +  1 )  x.  1 ) )
3926nnred 9777 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  e.  RR )
404a1i 10 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  1  e.  RR )
4128nngt0d 9805 . . . . . . . . . 10  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  <  ( ( |_ `  ( 1  /  r
) )  +  1 ) )
42 ledivmul 9645 . . . . . . . . . 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 ) ) )
4339, 40, 35, 41, 42syl112anc 1186 . . . . . . . . 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 ) ) )
4438, 43mpbird 223 . . . . . . . 8  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  <_  1 )
453, 4elicc2i 10732 . . . . . . . 8  |-  ( ( k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  e.  ( 0 [,] 1 )  <->  ( (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  e.  RR  /\  0  <_  ( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) )  /\  ( k  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) )  <_ 
1 ) )
4631, 32, 44, 45syl3anbrc 1136 . . . . . . 7  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  e.  ( 0 [,] 1 ) )
47 oveq1 5881 . . . . . . . . . 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 ) )
4847sseq1d 3218 . . . . . . . . 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 ) )
4948rexbidv 2577 . . . . . . . 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 ) )
5049rspcv 2893 . . . . . . 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 ) )
5146, 50syl 15 . . . . . 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 ) )
52 simplr 731 . . . . . . . . . . . . . 14  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  r  e.  RR+ )
5352rpred 10406 . . . . . . . . . . . . 13  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  r  e.  RR )
5431, 53resubcld 9227 . . . . . . . . . . . 12  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) )  -  r )  e.  RR )
5554rexrd 8897 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) )  -  r )  e.  RR* )
5631, 53readdcld 8878 . . . . . . . . . . . 12  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) )  +  r )  e.  RR )
5756rexrd 8897 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) )  +  r )  e.  RR* )
58 nnm1nn0 10021 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  (
k  -  1 )  e.  NN0 )
5926, 58syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  -  1 )  e.  NN0 )
6059nn0red 10035 . . . . . . . . . . . . 13  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  -  1 )  e.  RR )
6160, 28nndivred 9810 . . . . . . . . . . . 12  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( k  -  1 )  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  e.  RR )
6239recnd 8877 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  k  e.  CC )
6360recnd 8877 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  -  1 )  e.  CC )
6428nnne0d 9806 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
( |_ `  (
1  /  r ) )  +  1 )  =/=  0 )
6562, 63, 36, 64divsubdird 9591 . . . . . . . . . . . . . 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 ) ) ) )
66 ax-1cn 8811 . . . . . . . . . . . . . . . 16  |-  1  e.  CC
67 nncan 9092 . . . . . . . . . . . . . . . 16  |-  ( ( k  e.  CC  /\  1  e.  CC )  ->  ( k  -  (
k  -  1 ) )  =  1 )
6862, 66, 67sylancl 643 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
k  -  ( k  -  1 ) )  =  1 )
6968oveq1d 5889 . . . . . . . . . . . . . 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 ) ) )
7065, 69eqtr3d 2330 . . . . . . . . . . . . 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 ) ) )
7152rprecred 10417 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
1  /  r )  e.  RR )
72 flltp1 10948 . . . . . . . . . . . . . . 15  |-  ( ( 1  /  r )  e.  RR  ->  (
1  /  r )  <  ( ( |_
`  ( 1  / 
r ) )  +  1 ) )
7371, 72syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
1  /  r )  <  ( ( |_
`  ( 1  / 
r ) )  +  1 ) )
74 rpgt0 10381 . . . . . . . . . . . . . . . 16  |-  ( r  e.  RR+  ->  0  < 
r )
7574ad2antlr 707 . . . . . . . . . . . . . . 15  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  <  r )
76 ltdiv23 9663 . . . . . . . . . . . . . . 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 ) )
7740, 53, 75, 35, 41, 76syl122anc 1191 . . . . . . . . . . . . . 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 ) )
7873, 77mpbid 201 . . . . . . . . . . . . 13  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  (
1  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) )  <  r )
7970, 78eqbrtrd 4059 . . . . . . . . . . . 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 )
8031, 61, 53, 79ltsub23d 9393 . . . . . . . . . . 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 ) ) )
8131, 52ltaddrpd 10435 . . . . . . . . . . 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 ) )
82 iccssioo 10735 . . . . . . . . . . 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 ) ) )
8355, 57, 80, 81, 82syl22anc 1183 . . . . . . . . . 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 ) ) )
843a1i 10 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  e.  RR )
8559nn0ge0d 10037 . . . . . . . . . . . 12  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  <_  ( k  -  1 ) )
86 divge0 9641 . . . . . . . . . . . 12  |-  ( ( ( ( k  - 
1 )  e.  RR  /\  0  <_  ( k  -  1 ) )  /\  ( ( ( |_ `  ( 1  /  r ) )  +  1 )  e.  RR  /\  0  < 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) )  -> 
0  <_  ( (
k  -  1 )  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) )
8760, 85, 35, 41, 86syl22anc 1183 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  0  <_  ( ( k  - 
1 )  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) )
88 iccss 10734 . . . . . . . . . . 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 ) )
8984, 40, 87, 44, 88syl22anc 1183 . . . . . . . . . 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 ) )
9083, 89ssind 3406 . . . . . . . . 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
) ) )
91 eqid 2296 . . . . . . . . . . . . 13  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )
9291rexmet 18313 . . . . . . . . . . . 12  |-  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  e.  ( * Met `  RR )
9392a1i 10 . . . . . . . . . . 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 ) )
94 dfss1 3386 . . . . . . . . . . . . 13  |-  ( ( 0 [,] 1 ) 
C_  RR  <->  ( RR  i^i  ( 0 [,] 1
) )  =  ( 0 [,] 1 ) )
956, 94mpbi 199 . . . . . . . . . . . 12  |-  ( RR 
i^i  ( 0 [,] 1 ) )  =  ( 0 [,] 1
)
9646, 95syl6eleqr 2387 . . . . . . . . . . 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
) ) )
97 rpxr 10377 . . . . . . . . . . . 12  |-  ( r  e.  RR+  ->  r  e. 
RR* )
9897ad2antlr 707 . . . . . . . . . . 11  |-  ( ( ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  /\  r  e.  RR+ )  /\  k  e.  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )  ->  r  e.  RR* )
99 xpss12 4808 . . . . . . . . . . . . . . 15  |-  ( ( ( 0 [,] 1
)  C_  RR  /\  (
0 [,] 1 ) 
C_  RR )  -> 
( ( 0 [,] 1 )  X.  (
0 [,] 1 ) )  C_  ( RR  X.  RR ) )
1006, 6, 99mp2an 653 . . . . . . . . . . . . . 14  |-  ( ( 0 [,] 1 )  X.  ( 0 [,] 1 ) )  C_  ( RR  X.  RR )
101 resabs1 5000 . . . . . . . . . . . . . 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 ) ) ) )
102100, 101ax-mp 8 . . . . . . . . . . . . 13  |-  ( ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )  |`  (
( 0 [,] 1
)  X.  ( 0 [,] 1 ) ) )  =  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )
103102eqcomi 2300 . . . . . . . . . . . 12  |-  ( ( abs  o.  -  )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )  =  ( ( ( abs  o.  -  )  |`  ( RR 
X.  RR ) )  |`  ( ( 0 [,] 1 )  X.  (
0 [,] 1 ) ) )
104103blres 17993 . . . . . . . . . . 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 ) ) )
10593, 96, 98, 104syl3anc 1182 . . . . . . . . . 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 ) ) )
10691bl2ioo 18314 . . . . . . . . . . . 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 ) ) )
10731, 53, 106syl2anc 642 . . . . . . . . . . 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 ) ) )
108107ineq1d 3382 . . . . . . . . . 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 ) ) )
109105, 108eqtrd 2328 . . . . . . . . 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 ) ) )
11090, 109sseqtr4d 3228 . . . . . . . 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 ) )
111 sstr2 3199 . . . . . . . 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 ) )
112110, 111syl 15 . . . . . . 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
) )
113112reximdv 2667 . . . . . 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 )
)
11451, 113syld 40 . . . . 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 )
)
115114ralrimdva 2646 . . . 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 ) )
116 oveq2 5882 . . . . . 6  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
1 ... n )  =  ( 1 ... (
( |_ `  (
1  /  r ) )  +  1 ) ) )
117 oveq2 5882 . . . . . . . . 9  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
( k  -  1 )  /  n )  =  ( ( k  -  1 )  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) )
118 oveq2 5882 . . . . . . . . 9  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
k  /  n )  =  ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) )
119117, 118oveq12d 5892 . . . . . . . 8  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
( ( k  - 
1 )  /  n
) [,] ( k  /  n ) )  =  ( ( ( k  -  1 )  /  ( ( |_
`  ( 1  / 
r ) )  +  1 ) ) [,] ( k  /  (
( |_ `  (
1  /  r ) )  +  1 ) ) ) )
120119sseq1d 3218 . . . . . . 7  |-  ( n  =  ( ( |_
`  ( 1  / 
r ) )  +  1 )  ->  (
( ( ( k  -  1 )  /  n ) [,] (
k  /  n ) )  C_  u  <->  ( (
( k  -  1 )  /  ( ( |_ `  ( 1  /  r ) )  +  1 ) ) [,] ( k  / 
( ( |_ `  ( 1  /  r
) )  +  1 ) ) )  C_  u ) )
121120rexbidv 2577 . . . . . 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 ) )
122116, 121raleqbidv 2761 . . . . 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
) )
123122rspcev 2897 . . . 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 )
12424, 115, 123ee12an 1353 . . 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 ) )
125124rexlimdva 2680 . 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 ) )
12616, 125mpd 14 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    i^i cin 3164    C_ wss 3165   U.cuni 3843   class class class wbr 4039    X. cxp 4703    |` cres 4707    o. ccom 4709   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   NNcn 9762   NN0cn0 9981   RR+crp 10370   (,)cioo 10672   [,]cicc 10675   ...cfz 10798   |_cfl 10940   abscabs 11735   * Metcxmt 16385   Metcme 16386   ballcbl 16387   Compccmp 17129   IIcii 18395
This theorem is referenced by:  cvmliftlem15  23844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-ec 6678  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-cn 16973  df-cnp 16974  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-xms 17901  df-ms 17902  df-tms 17903  df-ii 18397
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