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Theorem rpnnen1lem5 10362
Description: Lemma for rpnnen1 10363. (Contributed by Mario Carneiro, 12-May-2013.)
Hypotheses
Ref Expression
rpnnen1.1  |-  T  =  { n  e.  ZZ  |  ( n  / 
k )  <  x }
rpnnen1.2  |-  F  =  ( x  e.  RR  |->  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) ) )
Assertion
Ref Expression
rpnnen1lem5  |-  ( x  e.  RR  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  =  x )
Distinct variable groups:    k, F, n, x    T, n
Allowed substitution hints:    T( x, k)

Proof of Theorem rpnnen1lem5
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 rpnnen1.1 . . . 4  |-  T  =  { n  e.  ZZ  |  ( n  / 
k )  <  x }
2 rpnnen1.2 . . . 4  |-  F  =  ( x  e.  RR  |->  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) ) )
31, 2rpnnen1lem3 10360 . . 3  |-  ( x  e.  RR  ->  A. n  e.  ran  ( F `  x ) n  <_  x )
41, 2rpnnen1lem1 10358 . . . . . 6  |-  ( x  e.  RR  ->  ( F `  x )  e.  ( QQ  ^m  NN ) )
5 qexALT 10347 . . . . . . 7  |-  QQ  e.  _V
6 nnexALT 9764 . . . . . . 7  |-  NN  e.  _V
75, 6elmap 6812 . . . . . 6  |-  ( ( F `  x )  e.  ( QQ  ^m  NN )  <->  ( F `  x ) : NN --> QQ )
84, 7sylib 188 . . . . 5  |-  ( x  e.  RR  ->  ( F `  x ) : NN --> QQ )
9 frn 5411 . . . . . 6  |-  ( ( F `  x ) : NN --> QQ  ->  ran  ( F `  x
)  C_  QQ )
10 qssre 10342 . . . . . 6  |-  QQ  C_  RR
119, 10syl6ss 3204 . . . . 5  |-  ( ( F `  x ) : NN --> QQ  ->  ran  ( F `  x
)  C_  RR )
128, 11syl 15 . . . 4  |-  ( x  e.  RR  ->  ran  ( F `  x ) 
C_  RR )
13 1nn 9773 . . . . . . . 8  |-  1  e.  NN
14 ne0i 3474 . . . . . . . 8  |-  ( 1  e.  NN  ->  NN  =/=  (/) )
1513, 14ax-mp 8 . . . . . . 7  |-  NN  =/=  (/)
16 fdm 5409 . . . . . . . 8  |-  ( ( F `  x ) : NN --> QQ  ->  dom  ( F `  x
)  =  NN )
1716neeq1d 2472 . . . . . . 7  |-  ( ( F `  x ) : NN --> QQ  ->  ( dom  ( F `  x )  =/=  (/)  <->  NN  =/=  (/) ) )
1815, 17mpbiri 224 . . . . . 6  |-  ( ( F `  x ) : NN --> QQ  ->  dom  ( F `  x
)  =/=  (/) )
19 dm0rn0 4911 . . . . . . 7  |-  ( dom  ( F `  x
)  =  (/)  <->  ran  ( F `
 x )  =  (/) )
2019necon3bii 2491 . . . . . 6  |-  ( dom  ( F `  x
)  =/=  (/)  <->  ran  ( F `
 x )  =/=  (/) )
2118, 20sylib 188 . . . . 5  |-  ( ( F `  x ) : NN --> QQ  ->  ran  ( F `  x
)  =/=  (/) )
228, 21syl 15 . . . 4  |-  ( x  e.  RR  ->  ran  ( F `  x )  =/=  (/) )
23 breq2 4043 . . . . . . 7  |-  ( y  =  x  ->  (
n  <_  y  <->  n  <_  x ) )
2423ralbidv 2576 . . . . . 6  |-  ( y  =  x  ->  ( A. n  e.  ran  ( F `  x ) n  <_  y  <->  A. n  e.  ran  ( F `  x ) n  <_  x ) )
2524rspcev 2897 . . . . 5  |-  ( ( x  e.  RR  /\  A. n  e.  ran  ( F `  x )
n  <_  x )  ->  E. y  e.  RR  A. n  e.  ran  ( F `  x )
n  <_  y )
263, 25mpdan 649 . . . 4  |-  ( x  e.  RR  ->  E. y  e.  RR  A. n  e. 
ran  ( F `  x ) n  <_ 
y )
27 id 19 . . . 4  |-  ( x  e.  RR  ->  x  e.  RR )
28 suprleub 9734 . . . 4  |-  ( ( ( ran  ( F `
 x )  C_  RR  /\  ran  ( F `
 x )  =/=  (/)  /\  E. y  e.  RR  A. n  e. 
ran  ( F `  x ) n  <_ 
y )  /\  x  e.  RR )  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  <_  x  <->  A. n  e.  ran  ( F `  x ) n  <_  x )
)
2912, 22, 26, 27, 28syl31anc 1185 . . 3  |-  ( x  e.  RR  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  <_  x  <->  A. n  e.  ran  ( F `  x ) n  <_  x )
)
303, 29mpbird 223 . 2  |-  ( x  e.  RR  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  <_  x
)
311, 2rpnnen1lem4 10361 . . . . . . . . 9  |-  ( x  e.  RR  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR )
32 resubcl 9127 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  sup ( ran  ( F `
 x ) ,  RR ,  <  )  e.  RR )  ->  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) )  e.  RR )
3331, 32mpdan 649 . . . . . . . 8  |-  ( x  e.  RR  ->  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) )  e.  RR )
3433adantr 451 . . . . . . 7  |-  ( ( x  e.  RR  /\  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )  ->  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) )  e.  RR )
35 posdif 9283 . . . . . . . . . 10  |-  ( ( sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR  /\  x  e.  RR )  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x  <->  0  <  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) ) )
3631, 35mpancom 650 . . . . . . . . 9  |-  ( x  e.  RR  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x  <->  0  <  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) ) )
3736biimpa 470 . . . . . . . 8  |-  ( ( x  e.  RR  /\  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )  ->  0  <  ( x  -  sup ( ran  ( F `  x ) ,  RR ,  <  ) ) )
3837gt0ne0d 9353 . . . . . . 7  |-  ( ( x  e.  RR  /\  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )  ->  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) )  =/=  0
)
3934, 38rereccld 9603 . . . . . 6  |-  ( ( x  e.  RR  /\  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )  ->  (
1  /  ( x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  e.  RR )
40 arch 9978 . . . . . 6  |-  ( ( 1  /  ( x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  e.  RR  ->  E. k  e.  NN  ( 1  / 
( x  -  sup ( ran  ( F `  x ) ,  RR ,  <  ) ) )  <  k )
4139, 40syl 15 . . . . 5  |-  ( ( x  e.  RR  /\  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )  ->  E. k  e.  NN  ( 1  / 
( x  -  sup ( ran  ( F `  x ) ,  RR ,  <  ) ) )  <  k )
4241ex 423 . . . 4  |-  ( x  e.  RR  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x  ->  E. k  e.  NN  ( 1  / 
( x  -  sup ( ran  ( F `  x ) ,  RR ,  <  ) ) )  <  k ) )
431, 2rpnnen1lem2 10359 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  ZZ )
4443zred 10133 . . . . . . . 8  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  RR )
45443adant3 975 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  NN  /\  (
1  /  ( x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  < 
k )  ->  sup ( T ,  RR ,  <  )  e.  RR )
4645ltp1d 9703 . . . . . 6  |-  ( ( x  e.  RR  /\  k  e.  NN  /\  (
1  /  ( x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  < 
k )  ->  sup ( T ,  RR ,  <  )  <  ( sup ( T ,  RR ,  <  )  +  1 ) )
4734, 37jca 518 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )  ->  (
( x  -  sup ( ran  ( F `  x ) ,  RR ,  <  ) )  e.  RR  /\  0  < 
( x  -  sup ( ran  ( F `  x ) ,  RR ,  <  ) ) ) )
48 nnre 9769 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  k  e.  RR )
49 nngt0 9791 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  0  <  k )
5048, 49jca 518 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  (
k  e.  RR  /\  0  <  k ) )
51 ltrec1 9659 . . . . . . . . . . . . 13  |-  ( ( ( ( x  -  sup ( ran  ( F `
 x ) ,  RR ,  <  )
)  e.  RR  /\  0  <  ( x  -  sup ( ran  ( F `
 x ) ,  RR ,  <  )
) )  /\  (
k  e.  RR  /\  0  <  k ) )  ->  ( ( 1  /  ( x  -  sup ( ran  ( F `
 x ) ,  RR ,  <  )
) )  <  k  <->  ( 1  /  k )  <  ( x  -  sup ( ran  ( F `
 x ) ,  RR ,  <  )
) ) )
5247, 50, 51syl2an 463 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\ 
sup ( ran  ( F `  x ) ,  RR ,  <  )  <  x )  /\  k  e.  NN )  ->  (
( 1  /  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  < 
k  <->  ( 1  / 
k )  <  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) ) )
5331ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR  /\ 
sup ( ran  ( F `  x ) ,  RR ,  <  )  <  x )  /\  k  e.  NN )  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR )
54 nnrecre 9798 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR )
5554adantl 452 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR  /\ 
sup ( ran  ( F `  x ) ,  RR ,  <  )  <  x )  /\  k  e.  NN )  ->  (
1  /  k )  e.  RR )
56 simpll 730 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR  /\ 
sup ( ran  ( F `  x ) ,  RR ,  <  )  <  x )  /\  k  e.  NN )  ->  x  e.  RR )
57 ltaddsub2 9265 . . . . . . . . . . . . . 14  |-  ( ( sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR  /\  ( 1  /  k )  e.  RR  /\  x  e.  RR )  ->  (
( sup ( ran  ( F `  x
) ,  RR ,  <  )  +  ( 1  /  k ) )  <  x  <->  ( 1  /  k )  < 
( x  -  sup ( ran  ( F `  x ) ,  RR ,  <  ) ) ) )
5853, 55, 56, 57syl3anc 1182 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  RR  /\ 
sup ( ran  ( F `  x ) ,  RR ,  <  )  <  x )  /\  k  e.  NN )  ->  (
( sup ( ran  ( F `  x
) ,  RR ,  <  )  +  ( 1  /  k ) )  <  x  <->  ( 1  /  k )  < 
( x  -  sup ( ran  ( F `  x ) ,  RR ,  <  ) ) ) )
5912adantr 451 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ran  ( F `  x )  C_  RR )
60 ffn 5405 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F `  x ) : NN --> QQ  ->  ( F `  x )  Fn  NN )
618, 60syl 15 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  RR  ->  ( F `  x )  Fn  NN )
62 fnfvelrn 5678 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( F `  x
)  Fn  NN  /\  k  e.  NN )  ->  ( ( F `  x ) `  k
)  e.  ran  ( F `  x )
)
6361, 62sylan 457 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( F `  x ) `  k
)  e.  ran  ( F `  x )
)
6459, 63sseldd 3194 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( F `  x ) `  k
)  e.  RR )
6531adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR )
6654adantl 452 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( 1  /  k
)  e.  RR )
6712, 22, 263jca 1132 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  RR  ->  ( ran  ( F `  x
)  C_  RR  /\  ran  ( F `  x )  =/=  (/)  /\  E. y  e.  RR  A. n  e. 
ran  ( F `  x ) n  <_ 
y ) )
6867adantr 451 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ran  ( F `
 x )  C_  RR  /\  ran  ( F `
 x )  =/=  (/)  /\  E. y  e.  RR  A. n  e. 
ran  ( F `  x ) n  <_ 
y ) )
69 suprub 9731 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ran  ( F `
 x )  C_  RR  /\  ran  ( F `
 x )  =/=  (/)  /\  E. y  e.  RR  A. n  e. 
ran  ( F `  x ) n  <_ 
y )  /\  (
( F `  x
) `  k )  e.  ran  ( F `  x ) )  -> 
( ( F `  x ) `  k
)  <_  sup ( ran  ( F `  x
) ,  RR ,  <  ) )
7068, 63, 69syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( F `  x ) `  k
)  <_  sup ( ran  ( F `  x
) ,  RR ,  <  ) )
7164, 65, 66, 70leadd1dd 9402 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( F `
 x ) `  k )  +  ( 1  /  k ) )  <_  ( sup ( ran  ( F `  x ) ,  RR ,  <  )  +  ( 1  /  k ) ) )
7264, 66readdcld 8878 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( F `
 x ) `  k )  +  ( 1  /  k ) )  e.  RR )
73 readdcl 8836 . . . . . . . . . . . . . . . . 17  |-  ( ( sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR  /\  ( 1  /  k )  e.  RR )  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  +  ( 1  / 
k ) )  e.  RR )
7431, 54, 73syl2an 463 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( sup ( ran  ( F `  x
) ,  RR ,  <  )  +  ( 1  /  k ) )  e.  RR )
75 simpl 443 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  x  e.  RR )
76 lelttr 8928 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( F `
 x ) `  k )  +  ( 1  /  k ) )  e.  RR  /\  ( sup ( ran  ( F `  x ) ,  RR ,  <  )  +  ( 1  / 
k ) )  e.  RR  /\  x  e.  RR )  ->  (
( ( ( ( F `  x ) `
 k )  +  ( 1  /  k
) )  <_  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  +  ( 1  / 
k ) )  /\  ( sup ( ran  ( F `  x ) ,  RR ,  <  )  +  ( 1  / 
k ) )  < 
x )  ->  (
( ( F `  x ) `  k
)  +  ( 1  /  k ) )  <  x ) )
7776exp3a 425 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( F `
 x ) `  k )  +  ( 1  /  k ) )  e.  RR  /\  ( sup ( ran  ( F `  x ) ,  RR ,  <  )  +  ( 1  / 
k ) )  e.  RR  /\  x  e.  RR )  ->  (
( ( ( F `
 x ) `  k )  +  ( 1  /  k ) )  <_  ( sup ( ran  ( F `  x ) ,  RR ,  <  )  +  ( 1  /  k ) )  ->  ( ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  +  ( 1  / 
k ) )  < 
x  ->  ( (
( F `  x
) `  k )  +  ( 1  / 
k ) )  < 
x ) ) )
7872, 74, 75, 77syl3anc 1182 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( ( F `  x ) `
 k )  +  ( 1  /  k
) )  <_  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  +  ( 1  / 
k ) )  -> 
( ( sup ( ran  ( F `  x
) ,  RR ,  <  )  +  ( 1  /  k ) )  <  x  ->  (
( ( F `  x ) `  k
)  +  ( 1  /  k ) )  <  x ) ) )
7971, 78mpd 14 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( sup ( ran  ( F `  x
) ,  RR ,  <  )  +  ( 1  /  k ) )  <  x  ->  (
( ( F `  x ) `  k
)  +  ( 1  /  k ) )  <  x ) )
8079adantlr 695 . . . . . . . . . . . . 13  |-  ( ( ( x  e.  RR  /\ 
sup ( ran  ( F `  x ) ,  RR ,  <  )  <  x )  /\  k  e.  NN )  ->  (
( sup ( ran  ( F `  x
) ,  RR ,  <  )  +  ( 1  /  k ) )  <  x  ->  (
( ( F `  x ) `  k
)  +  ( 1  /  k ) )  <  x ) )
8158, 80sylbird 226 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR  /\ 
sup ( ran  ( F `  x ) ,  RR ,  <  )  <  x )  /\  k  e.  NN )  ->  (
( 1  /  k
)  <  ( x  -  sup ( ran  ( F `  x ) ,  RR ,  <  )
)  ->  ( (
( F `  x
) `  k )  +  ( 1  / 
k ) )  < 
x ) )
8252, 81sylbid 206 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR  /\ 
sup ( ran  ( F `  x ) ,  RR ,  <  )  <  x )  /\  k  e.  NN )  ->  (
( 1  /  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  < 
k  ->  ( (
( F `  x
) `  k )  +  ( 1  / 
k ) )  < 
x ) )
8343peano2zd 10136 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( sup ( T ,  RR ,  <  )  +  1 )  e.  ZZ )
84 oveq1 5881 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  ( sup ( T ,  RR ,  <  )  +  1 )  ->  ( n  / 
k )  =  ( ( sup ( T ,  RR ,  <  )  +  1 )  / 
k ) )
8584breq1d 4049 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  ( sup ( T ,  RR ,  <  )  +  1 )  ->  ( ( n  /  k )  < 
x  <->  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  <  x ) )
8685, 1elrab2 2938 . . . . . . . . . . . . . . . . 17  |-  ( ( sup ( T ,  RR ,  <  )  +  1 )  e.  T  <->  ( ( sup ( T ,  RR ,  <  )  +  1 )  e.  ZZ  /\  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k
)  <  x )
)
8786biimpri 197 . . . . . . . . . . . . . . . 16  |-  ( ( ( sup ( T ,  RR ,  <  )  +  1 )  e.  ZZ  /\  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k
)  <  x )  ->  ( sup ( T ,  RR ,  <  )  +  1 )  e.  T )
8883, 87sylan 457 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  <  x )  -> 
( sup ( T ,  RR ,  <  )  +  1 )  e.  T )
89 ssrab2 3271 . . . . . . . . . . . . . . . . . . . 20  |-  { n  e.  ZZ  |  ( n  /  k )  < 
x }  C_  ZZ
901, 89eqsstri 3221 . . . . . . . . . . . . . . . . . . 19  |-  T  C_  ZZ
91 zssre 10047 . . . . . . . . . . . . . . . . . . 19  |-  ZZ  C_  RR
9290, 91sstri 3201 . . . . . . . . . . . . . . . . . 18  |-  T  C_  RR
9392a1i 10 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  C_  RR )
94 remulcl 8838 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( k  e.  RR  /\  x  e.  RR )  ->  ( k  x.  x
)  e.  RR )
9594ancoms 439 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  RR  /\  k  e.  RR )  ->  ( k  x.  x
)  e.  RR )
9648, 95sylan2 460 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( k  x.  x
)  e.  RR )
97 btwnz 10130 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( k  x.  x )  e.  RR  ->  ( E. n  e.  ZZ  n  <  ( k  x.  x )  /\  E. n  e.  ZZ  (
k  x.  x )  <  n ) )
9897simpld 445 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( k  x.  x )  e.  RR  ->  E. n  e.  ZZ  n  <  (
k  x.  x ) )
9996, 98syl 15 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. n  e.  ZZ  n  <  ( k  x.  x ) )
100 zre 10044 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( n  e.  ZZ  ->  n  e.  RR )
101100adantl 452 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  n  e.  RR )
102 simpll 730 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  x  e.  RR )
10350ad2antlr 707 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( k  e.  RR  /\  0  < 
k ) )
104 ltdivmul 9644 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( n  e.  RR  /\  x  e.  RR  /\  (
k  e.  RR  /\  0  <  k ) )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
105101, 102, 103, 104syl3anc 1182 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
106105rexbidva 2573 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( E. n  e.  ZZ  ( n  / 
k )  <  x  <->  E. n  e.  ZZ  n  <  ( k  x.  x
) ) )
10799, 106mpbird 223 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. n  e.  ZZ  ( n  /  k
)  <  x )
108 rabn0 3487 . . . . . . . . . . . . . . . . . . 19  |-  ( { n  e.  ZZ  | 
( n  /  k
)  <  x }  =/=  (/)  <->  E. n  e.  ZZ  ( n  /  k
)  <  x )
109107, 108sylibr 203 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  { n  e.  ZZ  |  ( n  / 
k )  <  x }  =/=  (/) )
1101neeq1i 2469 . . . . . . . . . . . . . . . . . 18  |-  ( T  =/=  (/)  <->  { n  e.  ZZ  |  ( n  / 
k )  <  x }  =/=  (/) )
111109, 110sylibr 203 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  =/=  (/) )
1121rabeq2i 2798 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  T  <->  ( n  e.  ZZ  /\  ( n  /  k )  < 
x ) )
11348ad2antlr 707 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  k  e.  RR )
114113, 102, 94syl2anc 642 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( k  x.  x )  e.  RR )
115 ltle 8926 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( n  e.  RR  /\  ( k  x.  x
)  e.  RR )  ->  ( n  < 
( k  x.  x
)  ->  n  <_  ( k  x.  x ) ) )
116101, 114, 115syl2anc 642 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( n  < 
( k  x.  x
)  ->  n  <_  ( k  x.  x ) ) )
117105, 116sylbid 206 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( ( n  /  k )  < 
x  ->  n  <_  ( k  x.  x ) ) )
118117impr 602 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  ( n  e.  ZZ  /\  ( n  /  k )  < 
x ) )  ->  n  <_  ( k  x.  x ) )
119112, 118sylan2b 461 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  T
)  ->  n  <_  ( k  x.  x ) )
120119ralrimiva 2639 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  A. n  e.  T  n  <_  ( k  x.  x ) )
121 breq2 4043 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  ( k  x.  x )  ->  (
n  <_  y  <->  n  <_  ( k  x.  x ) ) )
122121ralbidv 2576 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  ( k  x.  x )  ->  ( A. n  e.  T  n  <_  y  <->  A. n  e.  T  n  <_  ( k  x.  x ) ) )
123122rspcev 2897 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( k  x.  x
)  e.  RR  /\  A. n  e.  T  n  <_  ( k  x.  x ) )  ->  E. y  e.  RR  A. n  e.  T  n  <_  y )
12496, 120, 123syl2anc 642 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. y  e.  RR  A. n  e.  T  n  <_  y )
12593, 111, 1243jca 1132 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( T  C_  RR  /\  T  =/=  (/)  /\  E. y  e.  RR  A. n  e.  T  n  <_  y ) )
126 suprub 9731 . . . . . . . . . . . . . . . 16  |-  ( ( ( T  C_  RR  /\  T  =/=  (/)  /\  E. y  e.  RR  A. n  e.  T  n  <_  y )  /\  ( sup ( T ,  RR ,  <  )  +  1 )  e.  T )  ->  ( sup ( T ,  RR ,  <  )  +  1 )  <_  sup ( T ,  RR ,  <  ) )
127125, 126sylan 457 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  ( sup ( T ,  RR ,  <  )  +  1 )  e.  T )  -> 
( sup ( T ,  RR ,  <  )  +  1 )  <_  sup ( T ,  RR ,  <  ) )
12888, 127syldan 456 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  <  x )  -> 
( sup ( T ,  RR ,  <  )  +  1 )  <_  sup ( T ,  RR ,  <  ) )
129128ex 423 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  <  x  ->  ( sup ( T ,  RR ,  <  )  +  1 )  <_  sup ( T ,  RR ,  <  ) ) )
13043zcnd 10134 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  CC )
131 ax-1cn 8811 . . . . . . . . . . . . . . . . 17  |-  1  e.  CC
132131a1i 10 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  1  e.  CC )
133 nncn 9770 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN  ->  k  e.  CC )
134 nnne0 9794 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN  ->  k  =/=  0 )
135133, 134jca 518 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  (
k  e.  CC  /\  k  =/=  0 ) )
136135adantl 452 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( k  e.  CC  /\  k  =/=  0 ) )
137 divdir 9463 . . . . . . . . . . . . . . . 16  |-  ( ( sup ( T ,  RR ,  <  )  e.  CC  /\  1  e.  CC  /\  ( k  e.  CC  /\  k  =/=  0 ) )  -> 
( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  =  ( ( sup ( T ,  RR ,  <  )  /  k )  +  ( 1  / 
k ) ) )
138130, 132, 136, 137syl3anc 1182 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  =  ( ( sup ( T ,  RR ,  <  )  /  k )  +  ( 1  / 
k ) ) )
1396mptex 5762 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k
) )  e.  _V
1402fvmpt2 5624 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  RR  /\  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )  e. 
_V )  ->  ( F `  x )  =  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k ) ) )
141139, 140mpan2 652 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  RR  ->  ( F `  x )  =  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k ) ) )
142141fveq1d 5543 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  RR  ->  (
( F `  x
) `  k )  =  ( ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k
) ) `  k
) )
143 ovex 5899 . . . . . . . . . . . . . . . . . 18  |-  ( sup ( T ,  RR ,  <  )  /  k
)  e.  _V
144 eqid 2296 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k
) )  =  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )
145144fvmpt2 5624 . . . . . . . . . . . . . . . . . 18  |-  ( ( k  e.  NN  /\  ( sup ( T ,  RR ,  <  )  / 
k )  e.  _V )  ->  ( ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k
) ) `  k
)  =  ( sup ( T ,  RR ,  <  )  /  k
) )
146143, 145mpan2 652 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  (
( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) ) `  k )  =  ( sup ( T ,  RR ,  <  )  / 
k ) )
147142, 146sylan9eq 2348 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( F `  x ) `  k
)  =  ( sup ( T ,  RR ,  <  )  /  k
) )
148147oveq1d 5889 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( F `
 x ) `  k )  +  ( 1  /  k ) )  =  ( ( sup ( T ,  RR ,  <  )  / 
k )  +  ( 1  /  k ) ) )
149138, 148eqtr4d 2331 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  =  ( ( ( F `
 x ) `  k )  +  ( 1  /  k ) ) )
150149breq1d 4049 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  <  x  <->  ( (
( F `  x
) `  k )  +  ( 1  / 
k ) )  < 
x ) )
15183zred 10133 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( sup ( T ,  RR ,  <  )  +  1 )  e.  RR )
152151, 44lenltd 8981 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( sup ( T ,  RR ,  <  )  +  1 )  <_  sup ( T ,  RR ,  <  )  <->  -.  sup ( T ,  RR ,  <  )  <  ( sup ( T ,  RR ,  <  )  +  1 ) ) )
153129, 150, 1523imtr3d 258 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( ( F `  x ) `
 k )  +  ( 1  /  k
) )  <  x  ->  -.  sup ( T ,  RR ,  <  )  <  ( sup ( T ,  RR ,  <  )  +  1 ) ) )
154153adantlr 695 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR  /\ 
sup ( ran  ( F `  x ) ,  RR ,  <  )  <  x )  /\  k  e.  NN )  ->  (
( ( ( F `
 x ) `  k )  +  ( 1  /  k ) )  <  x  ->  -.  sup ( T ,  RR ,  <  )  < 
( sup ( T ,  RR ,  <  )  +  1 ) ) )
15582, 154syld 40 . . . . . . . . . 10  |-  ( ( ( x  e.  RR  /\ 
sup ( ran  ( F `  x ) ,  RR ,  <  )  <  x )  /\  k  e.  NN )  ->  (
( 1  /  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  < 
k  ->  -.  sup ( T ,  RR ,  <  )  <  ( sup ( T ,  RR ,  <  )  +  1 ) ) )
156155exp31 587 . . . . . . . . 9  |-  ( x  e.  RR  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x  ->  ( k  e.  NN  ->  ( (
1  /  ( x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  < 
k  ->  -.  sup ( T ,  RR ,  <  )  <  ( sup ( T ,  RR ,  <  )  +  1 ) ) ) ) )
157156com4l 78 . . . . . . . 8  |-  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x  ->  ( k  e.  NN  ->  ( (
1  /  ( x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  < 
k  ->  ( x  e.  RR  ->  -.  sup ( T ,  RR ,  <  )  <  ( sup ( T ,  RR ,  <  )  +  1 ) ) ) ) )
158157com14 82 . . . . . . 7  |-  ( x  e.  RR  ->  (
k  e.  NN  ->  ( ( 1  /  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  < 
k  ->  ( sup ( ran  ( F `  x ) ,  RR ,  <  )  <  x  ->  -.  sup ( T ,  RR ,  <  )  <  ( sup ( T ,  RR ,  <  )  +  1 ) ) ) ) )
1591583imp 1145 . . . . . 6  |-  ( ( x  e.  RR  /\  k  e.  NN  /\  (
1  /  ( x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  < 
k )  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x  ->  -.  sup ( T ,  RR ,  <  )  <  ( sup ( T ,  RR ,  <  )  +  1 ) ) )
16046, 159mt2d 109 . . . . 5  |-  ( ( x  e.  RR  /\  k  e.  NN  /\  (
1  /  ( x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  < 
k )  ->  -.  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )
161160rexlimdv3a 2682 . . . 4  |-  ( x  e.  RR  ->  ( E. k  e.  NN  ( 1  /  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  < 
k  ->  -.  sup ( ran  ( F `  x
) ,  RR ,  <  )  <  x ) )
16242, 161syld 40 . . 3  |-  ( x  e.  RR  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x  ->  -.  sup ( ran  ( F `  x
) ,  RR ,  <  )  <  x ) )
163162pm2.01d 161 . 2  |-  ( x  e.  RR  ->  -.  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )
164 eqlelt 8925 . . 3  |-  ( ( sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR  /\  x  e.  RR )  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  =  x  <->  ( sup ( ran  ( F `  x
) ,  RR ,  <  )  <_  x  /\  -.  sup ( ran  ( F `  x ) ,  RR ,  <  )  <  x ) ) )
16531, 164mpancom 650 . 2  |-  ( x  e.  RR  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  =  x  <->  ( sup ( ran  ( F `  x
) ,  RR ,  <  )  <_  x  /\  -.  sup ( ran  ( F `  x ) ,  RR ,  <  )  <  x ) ) )
16630, 163, 165mpbir2and 888 1  |-  ( x  e.  RR  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  =  x )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    C_ wss 3165   (/)c0 3468   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   supcsup 7209   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    - cmin 9053    / cdiv 9439   NNcn 9762   ZZcz 10040   QQcq 10332
This theorem is referenced by:  rpnnen1  10363
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-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
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-iun 3923  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-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-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  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-n0 9982  df-z 10041  df-q 10333
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