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Theorem rpnnen1lem5 11296
Description: Lemma for rpnnen1 11297. (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 11294 . . 3  |-  ( x  e.  RR  ->  A. n  e.  ran  ( F `  x ) n  <_  x )
41, 2rpnnen1lem1 11292 . . . . . 6  |-  ( x  e.  RR  ->  ( F `  x )  e.  ( QQ  ^m  NN ) )
5 qex 11278 . . . . . . 7  |-  QQ  e.  _V
6 nnex 10617 . . . . . . 7  |-  NN  e.  _V
75, 6elmap 7506 . . . . . 6  |-  ( ( F `  x )  e.  ( QQ  ^m  NN )  <->  ( F `  x ) : NN --> QQ )
84, 7sylib 200 . . . . 5  |-  ( x  e.  RR  ->  ( F `  x ) : NN --> QQ )
9 frn 5750 . . . . . 6  |-  ( ( F `  x ) : NN --> QQ  ->  ran  ( F `  x
)  C_  QQ )
10 qssre 11276 . . . . . 6  |-  QQ  C_  RR
119, 10syl6ss 3477 . . . . 5  |-  ( ( F `  x ) : NN --> QQ  ->  ran  ( F `  x
)  C_  RR )
128, 11syl 17 . . . 4  |-  ( x  e.  RR  ->  ran  ( F `  x ) 
C_  RR )
13 1nn 10622 . . . . . . . 8  |-  1  e.  NN
1413ne0ii 3769 . . . . . . 7  |-  NN  =/=  (/)
15 fdm 5748 . . . . . . . 8  |-  ( ( F `  x ) : NN --> QQ  ->  dom  ( F `  x
)  =  NN )
1615neeq1d 2702 . . . . . . 7  |-  ( ( F `  x ) : NN --> QQ  ->  ( dom  ( F `  x )  =/=  (/)  <->  NN  =/=  (/) ) )
1714, 16mpbiri 237 . . . . . 6  |-  ( ( F `  x ) : NN --> QQ  ->  dom  ( F `  x
)  =/=  (/) )
18 dm0rn0 5068 . . . . . . 7  |-  ( dom  ( F `  x
)  =  (/)  <->  ran  ( F `
 x )  =  (/) )
1918necon3bii 2693 . . . . . 6  |-  ( dom  ( F `  x
)  =/=  (/)  <->  ran  ( F `
 x )  =/=  (/) )
2017, 19sylib 200 . . . . 5  |-  ( ( F `  x ) : NN --> QQ  ->  ran  ( F `  x
)  =/=  (/) )
218, 20syl 17 . . . 4  |-  ( x  e.  RR  ->  ran  ( F `  x )  =/=  (/) )
22 breq2 4425 . . . . . . 7  |-  ( y  =  x  ->  (
n  <_  y  <->  n  <_  x ) )
2322ralbidv 2865 . . . . . 6  |-  ( y  =  x  ->  ( A. n  e.  ran  ( F `  x ) n  <_  y  <->  A. n  e.  ran  ( F `  x ) n  <_  x ) )
2423rspcev 3183 . . . . 5  |-  ( ( x  e.  RR  /\  A. n  e.  ran  ( F `  x )
n  <_  x )  ->  E. y  e.  RR  A. n  e.  ran  ( F `  x )
n  <_  y )
253, 24mpdan 673 . . . 4  |-  ( x  e.  RR  ->  E. y  e.  RR  A. n  e. 
ran  ( F `  x ) n  <_ 
y )
26 id 23 . . . 4  |-  ( x  e.  RR  ->  x  e.  RR )
27 suprleub 10575 . . . 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 )
)
2812, 21, 25, 26, 27syl31anc 1268 . . 3  |-  ( x  e.  RR  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  <_  x  <->  A. n  e.  ran  ( F `  x ) n  <_  x )
)
293, 28mpbird 236 . 2  |-  ( x  e.  RR  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  <_  x
)
301, 2rpnnen1lem4 11295 . . . . . . . . 9  |-  ( x  e.  RR  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR )
31 resubcl 9940 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  sup ( ran  ( F `
 x ) ,  RR ,  <  )  e.  RR )  ->  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) )  e.  RR )
3230, 31mpdan 673 . . . . . . . 8  |-  ( x  e.  RR  ->  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) )  e.  RR )
3332adantr 467 . . . . . . 7  |-  ( ( x  e.  RR  /\  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )  ->  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) )  e.  RR )
34 posdif 10109 . . . . . . . . . 10  |-  ( ( sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR  /\  x  e.  RR )  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x  <->  0  <  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) ) )
3530, 34mpancom 674 . . . . . . . . 9  |-  ( x  e.  RR  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x  <->  0  <  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) ) )
3635biimpa 487 . . . . . . . 8  |-  ( ( x  e.  RR  /\  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )  ->  0  <  ( x  -  sup ( ran  ( F `  x ) ,  RR ,  <  ) ) )
3736gt0ne0d 10180 . . . . . . 7  |-  ( ( x  e.  RR  /\  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )  ->  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) )  =/=  0
)
3833, 37rereccld 10436 . . . . . 6  |-  ( ( x  e.  RR  /\  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )  ->  (
1  /  ( x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  e.  RR )
39 arch 10868 . . . . . 6  |-  ( ( 1  /  ( x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  e.  RR  ->  E. k  e.  NN  ( 1  / 
( x  -  sup ( ran  ( F `  x ) ,  RR ,  <  ) ) )  <  k )
4038, 39syl 17 . . . . 5  |-  ( ( x  e.  RR  /\  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )  ->  E. k  e.  NN  ( 1  / 
( x  -  sup ( ran  ( F `  x ) ,  RR ,  <  ) ) )  <  k )
4140ex 436 . . . 4  |-  ( x  e.  RR  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x  ->  E. k  e.  NN  ( 1  / 
( x  -  sup ( ran  ( F `  x ) ,  RR ,  <  ) ) )  <  k ) )
421, 2rpnnen1lem2 11293 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  ZZ )
4342zred 11042 . . . . . . . 8  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  RR )
44433adant3 1026 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  NN  /\  (
1  /  ( x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  < 
k )  ->  sup ( T ,  RR ,  <  )  e.  RR )
4544ltp1d 10539 . . . . . 6  |-  ( ( x  e.  RR  /\  k  e.  NN  /\  (
1  /  ( x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  < 
k )  ->  sup ( T ,  RR ,  <  )  <  ( sup ( T ,  RR ,  <  )  +  1 ) )
4633, 36jca 535 . . . . . . . . . . . . 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 ,  <  ) ) ) )
47 nnre 10618 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  k  e.  RR )
48 nngt0 10640 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  0  <  k )
4947, 48jca 535 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  (
k  e.  RR  /\  0  <  k ) )
50 ltrec1 10495 . . . . . . . . . . . . 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 ,  <  )
) ) )
5146, 49, 50syl2an 480 . . . . . . . . . . . 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 ,  <  ) ) ) )
5230ad2antrr 731 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR  /\ 
sup ( ran  ( F `  x ) ,  RR ,  <  )  <  x )  /\  k  e.  NN )  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR )
53 nnrecre 10648 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR )
5453adantl 468 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR  /\ 
sup ( ran  ( F `  x ) ,  RR ,  <  )  <  x )  /\  k  e.  NN )  ->  (
1  /  k )  e.  RR )
55 simpll 759 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR  /\ 
sup ( ran  ( F `  x ) ,  RR ,  <  )  <  x )  /\  k  e.  NN )  ->  x  e.  RR )
5652, 54, 55ltaddsub2d 10216 . . . . . . . . . . . . 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 ,  <  ) ) ) )
5712adantr 467 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ran  ( F `  x )  C_  RR )
58 ffn 5744 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F `  x ) : NN --> QQ  ->  ( F `  x )  Fn  NN )
598, 58syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  RR  ->  ( F `  x )  Fn  NN )
60 fnfvelrn 6032 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( F `  x
)  Fn  NN  /\  k  e.  NN )  ->  ( ( F `  x ) `  k
)  e.  ran  ( F `  x )
)
6159, 60sylan 474 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( F `  x ) `  k
)  e.  ran  ( F `  x )
)
6257, 61sseldd 3466 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( F `  x ) `  k
)  e.  RR )
6330adantr 467 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR )
6453adantl 468 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( 1  /  k
)  e.  RR )
6512, 21, 253jca 1186 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  RR  ->  ( ran  ( F `  x
)  C_  RR  /\  ran  ( F `  x )  =/=  (/)  /\  E. y  e.  RR  A. n  e. 
ran  ( F `  x ) n  <_ 
y ) )
6665adantr 467 . . . . . . . . . . . . . . . . 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 ) )
67 suprub 10572 . . . . . . . . . . . . . . . . 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 ,  <  ) )
6866, 61, 67syl2anc 666 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( F `  x ) `  k
)  <_  sup ( ran  ( F `  x
) ,  RR ,  <  ) )
6962, 63, 64, 68leadd1dd 10229 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( F `
 x ) `  k )  +  ( 1  /  k ) )  <_  ( sup ( ran  ( F `  x ) ,  RR ,  <  )  +  ( 1  /  k ) ) )
7062, 64readdcld 9672 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( F `
 x ) `  k )  +  ( 1  /  k ) )  e.  RR )
71 readdcl 9624 . . . . . . . . . . . . . . . . 17  |-  ( ( sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR  /\  ( 1  /  k )  e.  RR )  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  +  ( 1  / 
k ) )  e.  RR )
7230, 53, 71syl2an 480 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( sup ( ran  ( F `  x
) ,  RR ,  <  )  +  ( 1  /  k ) )  e.  RR )
73 simpl 459 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  x  e.  RR )
74 lelttr 9726 . . . . . . . . . . . . . . . . 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 ) )
7574expd 438 . . . . . . . . . . . . . . . 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 ) ) )
7670, 72, 73, 75syl3anc 1265 . . . . . . . . . . . . . . 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 ) ) )
7769, 76mpd 15 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( sup ( ran  ( F `  x
) ,  RR ,  <  )  +  ( 1  /  k ) )  <  x  ->  (
( ( F `  x ) `  k
)  +  ( 1  /  k ) )  <  x ) )
7877adantlr 720 . . . . . . . . . . . . 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 ) )
7956, 78sylbird 239 . . . . . . . . . . . 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 ) )
8051, 79sylbid 219 . . . . . . . . . . 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 ) )
8142peano2zd 11045 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( sup ( T ,  RR ,  <  )  +  1 )  e.  ZZ )
82 oveq1 6310 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  ( sup ( T ,  RR ,  <  )  +  1 )  ->  ( n  / 
k )  =  ( ( sup ( T ,  RR ,  <  )  +  1 )  / 
k ) )
8382breq1d 4431 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  ( sup ( T ,  RR ,  <  )  +  1 )  ->  ( ( n  /  k )  < 
x  <->  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  <  x ) )
8483, 1elrab2 3232 . . . . . . . . . . . . . . . . 17  |-  ( ( sup ( T ,  RR ,  <  )  +  1 )  e.  T  <->  ( ( sup ( T ,  RR ,  <  )  +  1 )  e.  ZZ  /\  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k
)  <  x )
)
8584biimpri 210 . . . . . . . . . . . . . . . 16  |-  ( ( ( sup ( T ,  RR ,  <  )  +  1 )  e.  ZZ  /\  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k
)  <  x )  ->  ( sup ( T ,  RR ,  <  )  +  1 )  e.  T )
8681, 85sylan 474 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  <  x )  -> 
( sup ( T ,  RR ,  <  )  +  1 )  e.  T )
87 ssrab2 3547 . . . . . . . . . . . . . . . . . . . 20  |-  { n  e.  ZZ  |  ( n  /  k )  < 
x }  C_  ZZ
881, 87eqsstri 3495 . . . . . . . . . . . . . . . . . . 19  |-  T  C_  ZZ
89 zssre 10946 . . . . . . . . . . . . . . . . . . 19  |-  ZZ  C_  RR
9088, 89sstri 3474 . . . . . . . . . . . . . . . . . 18  |-  T  C_  RR
9190a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  C_  RR )
92 remulcl 9626 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( k  e.  RR  /\  x  e.  RR )  ->  ( k  x.  x
)  e.  RR )
9392ancoms 455 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  RR  /\  k  e.  RR )  ->  ( k  x.  x
)  e.  RR )
9447, 93sylan2 477 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( k  x.  x
)  e.  RR )
95 btwnz 11039 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( k  x.  x )  e.  RR  ->  ( E. n  e.  ZZ  n  <  ( k  x.  x )  /\  E. n  e.  ZZ  (
k  x.  x )  <  n ) )
9695simpld 461 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( k  x.  x )  e.  RR  ->  E. n  e.  ZZ  n  <  (
k  x.  x ) )
9794, 96syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. n  e.  ZZ  n  <  ( k  x.  x ) )
98 zre 10943 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( n  e.  ZZ  ->  n  e.  RR )
9998adantl 468 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  n  e.  RR )
100 simpll 759 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  x  e.  RR )
10149ad2antlr 732 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( k  e.  RR  /\  0  < 
k ) )
102 ltdivmul 10482 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( n  e.  RR  /\  x  e.  RR  /\  (
k  e.  RR  /\  0  <  k ) )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
10399, 100, 101, 102syl3anc 1265 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
104103rexbidva 2937 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( E. n  e.  ZZ  ( n  / 
k )  <  x  <->  E. n  e.  ZZ  n  <  ( k  x.  x
) ) )
10597, 104mpbird 236 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. n  e.  ZZ  ( n  /  k
)  <  x )
106 rabn0 3783 . . . . . . . . . . . . . . . . . . 19  |-  ( { n  e.  ZZ  | 
( n  /  k
)  <  x }  =/=  (/)  <->  E. n  e.  ZZ  ( n  /  k
)  <  x )
107105, 106sylibr 216 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  { n  e.  ZZ  |  ( n  / 
k )  <  x }  =/=  (/) )
1081neeq1i 2710 . . . . . . . . . . . . . . . . . 18  |-  ( T  =/=  (/)  <->  { n  e.  ZZ  |  ( n  / 
k )  <  x }  =/=  (/) )
109107, 108sylibr 216 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  =/=  (/) )
1101rabeq2i 3079 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  T  <->  ( n  e.  ZZ  /\  ( n  /  k )  < 
x ) )
11147ad2antlr 732 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  k  e.  RR )
112111, 100, 92syl2anc 666 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( k  x.  x )  e.  RR )
113 ltle 9724 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( n  e.  RR  /\  ( k  x.  x
)  e.  RR )  ->  ( n  < 
( k  x.  x
)  ->  n  <_  ( k  x.  x ) ) )
11499, 112, 113syl2anc 666 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( n  < 
( k  x.  x
)  ->  n  <_  ( k  x.  x ) ) )
115103, 114sylbid 219 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( ( n  /  k )  < 
x  ->  n  <_  ( k  x.  x ) ) )
116115impr 624 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  ( n  e.  ZZ  /\  ( n  /  k )  < 
x ) )  ->  n  <_  ( k  x.  x ) )
117110, 116sylan2b 478 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  T
)  ->  n  <_  ( k  x.  x ) )
118117ralrimiva 2840 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  A. n  e.  T  n  <_  ( k  x.  x ) )
119 breq2 4425 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  ( k  x.  x )  ->  (
n  <_  y  <->  n  <_  ( k  x.  x ) ) )
120119ralbidv 2865 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  ( k  x.  x )  ->  ( A. n  e.  T  n  <_  y  <->  A. n  e.  T  n  <_  ( k  x.  x ) ) )
121120rspcev 3183 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( k  x.  x
)  e.  RR  /\  A. n  e.  T  n  <_  ( k  x.  x ) )  ->  E. y  e.  RR  A. n  e.  T  n  <_  y )
12294, 118, 121syl2anc 666 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. y  e.  RR  A. n  e.  T  n  <_  y )
12391, 109, 1223jca 1186 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( T  C_  RR  /\  T  =/=  (/)  /\  E. y  e.  RR  A. n  e.  T  n  <_  y ) )
124 suprub 10572 . . . . . . . . . . . . . . . 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 ,  <  ) )
125123, 124sylan 474 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  ( sup ( T ,  RR ,  <  )  +  1 )  e.  T )  -> 
( sup ( T ,  RR ,  <  )  +  1 )  <_  sup ( T ,  RR ,  <  ) )
12686, 125syldan 473 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  <  x )  -> 
( sup ( T ,  RR ,  <  )  +  1 )  <_  sup ( T ,  RR ,  <  ) )
127126ex 436 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  <  x  ->  ( sup ( T ,  RR ,  <  )  +  1 )  <_  sup ( T ,  RR ,  <  ) ) )
12842zcnd 11043 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  CC )
129 1cnd 9661 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  1  e.  CC )
130 nncn 10619 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN  ->  k  e.  CC )
131 nnne0 10644 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN  ->  k  =/=  0 )
132130, 131jca 535 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  (
k  e.  CC  /\  k  =/=  0 ) )
133132adantl 468 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( k  e.  CC  /\  k  =/=  0 ) )
134 divdir 10295 . . . . . . . . . . . . . . . 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 ) ) )
135128, 129, 133, 134syl3anc 1265 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  =  ( ( sup ( T ,  RR ,  <  )  /  k )  +  ( 1  / 
k ) ) )
1366mptex 6149 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k
) )  e.  _V
1372fvmpt2 5971 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  RR  /\  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )  e. 
_V )  ->  ( F `  x )  =  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k ) ) )
138136, 137mpan2 676 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  RR  ->  ( F `  x )  =  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k ) ) )
139138fveq1d 5881 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  RR  ->  (
( F `  x
) `  k )  =  ( ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k
) ) `  k
) )
140 ovex 6331 . . . . . . . . . . . . . . . . . 18  |-  ( sup ( T ,  RR ,  <  )  /  k
)  e.  _V
141 eqid 2423 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k
) )  =  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )
142141fvmpt2 5971 . . . . . . . . . . . . . . . . . 18  |-  ( ( k  e.  NN  /\  ( sup ( T ,  RR ,  <  )  / 
k )  e.  _V )  ->  ( ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k
) ) `  k
)  =  ( sup ( T ,  RR ,  <  )  /  k
) )
143140, 142mpan2 676 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  (
( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) ) `  k )  =  ( sup ( T ,  RR ,  <  )  / 
k ) )
144139, 143sylan9eq 2484 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( F `  x ) `  k
)  =  ( sup ( T ,  RR ,  <  )  /  k
) )
145144oveq1d 6318 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( F `
 x ) `  k )  +  ( 1  /  k ) )  =  ( ( sup ( T ,  RR ,  <  )  / 
k )  +  ( 1  /  k ) ) )
146135, 145eqtr4d 2467 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  =  ( ( ( F `
 x ) `  k )  +  ( 1  /  k ) ) )
147146breq1d 4431 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  <  x  <->  ( (
( F `  x
) `  k )  +  ( 1  / 
k ) )  < 
x ) )
14881zred 11042 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( sup ( T ,  RR ,  <  )  +  1 )  e.  RR )
149148, 43lenltd 9783 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( sup ( T ,  RR ,  <  )  +  1 )  <_  sup ( T ,  RR ,  <  )  <->  -.  sup ( T ,  RR ,  <  )  <  ( sup ( T ,  RR ,  <  )  +  1 ) ) )
150127, 147, 1493imtr3d 271 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( ( F `  x ) `
 k )  +  ( 1  /  k
) )  <  x  ->  -.  sup ( T ,  RR ,  <  )  <  ( sup ( T ,  RR ,  <  )  +  1 ) ) )
151150adantlr 720 . . . . . . . . . . 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 ) ) )
15280, 151syld 46 . . . . . . . . . 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 ) ) )
153152exp31 608 . . . . . . . . 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 ) ) ) ) )
154153com4l 88 . . . . . . . 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 ) ) ) ) )
155154com14 92 . . . . . . 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 ) ) ) ) )
1561553imp 1200 . . . . . 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 ) ) )
15745, 156mt2d 121 . . . . 5  |-  ( ( x  e.  RR  /\  k  e.  NN  /\  (
1  /  ( x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  < 
k )  ->  -.  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )
158157rexlimdv3a 2920 . . . 4  |-  ( x  e.  RR  ->  ( E. k  e.  NN  ( 1  /  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  < 
k  ->  -.  sup ( ran  ( F `  x
) ,  RR ,  <  )  <  x ) )
15941, 158syld 46 . . 3  |-  ( x  e.  RR  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x  ->  -.  sup ( ran  ( F `  x
) ,  RR ,  <  )  <  x ) )
160159pm2.01d 173 . 2  |-  ( x  e.  RR  ->  -.  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )
161 eqlelt 9723 . . 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 ) ) )
16230, 161mpancom 674 . 2  |-  ( x  e.  RR  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  =  x  <->  ( sup ( ran  ( F `  x
) ,  RR ,  <  )  <_  x  /\  -.  sup ( ran  ( F `  x ) ,  RR ,  <  )  <  x ) ) )
16329, 160, 162mpbir2and 931 1  |-  ( x  e.  RR  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  =  x )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    =/= wne 2619   A.wral 2776   E.wrex 2777   {crab 2780   _Vcvv 3082    C_ wss 3437   (/)c0 3762   class class class wbr 4421    |-> cmpt 4480   dom cdm 4851   ran crn 4852    Fn wfn 5594   -->wf 5595   ` cfv 5599  (class class class)co 6303    ^m cmap 7478   supcsup 7958   CCcc 9539   RRcr 9540   0cc0 9541   1c1 9542    + caddc 9544    x. cmul 9546    < clt 9677    <_ cle 9678    - cmin 9862    / cdiv 10271   NNcn 10611   ZZcz 10939   QQcq 11266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-er 7369  df-map 7480  df-en 7576  df-dom 7577  df-sdom 7578  df-sup 7960  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-n0 10872  df-z 10940  df-q 11267
This theorem is referenced by:  rpnnen1  11297
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