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Theorem rpnnen1lem5 10983
Description: Lemma for rpnnen1 10984. (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 10981 . . 3  |-  ( x  e.  RR  ->  A. n  e.  ran  ( F `  x ) n  <_  x )
41, 2rpnnen1lem1 10979 . . . . . 6  |-  ( x  e.  RR  ->  ( F `  x )  e.  ( QQ  ^m  NN ) )
5 qexALT 10968 . . . . . . 7  |-  QQ  e.  _V
6 nnexALT 10324 . . . . . . 7  |-  NN  e.  _V
75, 6elmap 7241 . . . . . 6  |-  ( ( F `  x )  e.  ( QQ  ^m  NN )  <->  ( F `  x ) : NN --> QQ )
84, 7sylib 196 . . . . 5  |-  ( x  e.  RR  ->  ( F `  x ) : NN --> QQ )
9 frn 5565 . . . . . 6  |-  ( ( F `  x ) : NN --> QQ  ->  ran  ( F `  x
)  C_  QQ )
10 qssre 10963 . . . . . 6  |-  QQ  C_  RR
119, 10syl6ss 3368 . . . . 5  |-  ( ( F `  x ) : NN --> QQ  ->  ran  ( F `  x
)  C_  RR )
128, 11syl 16 . . . 4  |-  ( x  e.  RR  ->  ran  ( F `  x ) 
C_  RR )
13 1nn 10333 . . . . . . . 8  |-  1  e.  NN
14 ne0i 3643 . . . . . . . 8  |-  ( 1  e.  NN  ->  NN  =/=  (/) )
1513, 14ax-mp 5 . . . . . . 7  |-  NN  =/=  (/)
16 fdm 5563 . . . . . . . 8  |-  ( ( F `  x ) : NN --> QQ  ->  dom  ( F `  x
)  =  NN )
1716neeq1d 2621 . . . . . . 7  |-  ( ( F `  x ) : NN --> QQ  ->  ( dom  ( F `  x )  =/=  (/)  <->  NN  =/=  (/) ) )
1815, 17mpbiri 233 . . . . . 6  |-  ( ( F `  x ) : NN --> QQ  ->  dom  ( F `  x
)  =/=  (/) )
19 dm0rn0 5056 . . . . . . 7  |-  ( dom  ( F `  x
)  =  (/)  <->  ran  ( F `
 x )  =  (/) )
2019necon3bii 2640 . . . . . 6  |-  ( dom  ( F `  x
)  =/=  (/)  <->  ran  ( F `
 x )  =/=  (/) )
2118, 20sylib 196 . . . . 5  |-  ( ( F `  x ) : NN --> QQ  ->  ran  ( F `  x
)  =/=  (/) )
228, 21syl 16 . . . 4  |-  ( x  e.  RR  ->  ran  ( F `  x )  =/=  (/) )
23 breq2 4296 . . . . . . 7  |-  ( y  =  x  ->  (
n  <_  y  <->  n  <_  x ) )
2423ralbidv 2735 . . . . . 6  |-  ( y  =  x  ->  ( A. n  e.  ran  ( F `  x ) n  <_  y  <->  A. n  e.  ran  ( F `  x ) n  <_  x ) )
2524rspcev 3073 . . . . 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 668 . . . 4  |-  ( x  e.  RR  ->  E. y  e.  RR  A. n  e. 
ran  ( F `  x ) n  <_ 
y )
27 id 22 . . . 4  |-  ( x  e.  RR  ->  x  e.  RR )
28 suprleub 10294 . . . 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 1221 . . 3  |-  ( x  e.  RR  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  <_  x  <->  A. n  e.  ran  ( F `  x ) n  <_  x )
)
303, 29mpbird 232 . 2  |-  ( x  e.  RR  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  <_  x
)
311, 2rpnnen1lem4 10982 . . . . . . . . 9  |-  ( x  e.  RR  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR )
32 resubcl 9673 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  sup ( ran  ( F `
 x ) ,  RR ,  <  )  e.  RR )  ->  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) )  e.  RR )
3331, 32mpdan 668 . . . . . . . 8  |-  ( x  e.  RR  ->  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) )  e.  RR )
3433adantr 465 . . . . . . 7  |-  ( ( x  e.  RR  /\  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )  ->  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) )  e.  RR )
35 posdif 9832 . . . . . . . . . 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 669 . . . . . . . . 9  |-  ( x  e.  RR  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x  <->  0  <  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) ) )
3736biimpa 484 . . . . . . . 8  |-  ( ( x  e.  RR  /\  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )  ->  0  <  ( x  -  sup ( ran  ( F `  x ) ,  RR ,  <  ) ) )
3837gt0ne0d 9904 . . . . . . 7  |-  ( ( x  e.  RR  /\  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )  ->  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) )  =/=  0
)
3934, 38rereccld 10158 . . . . . 6  |-  ( ( x  e.  RR  /\  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )  ->  (
1  /  ( x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  e.  RR )
40 arch 10576 . . . . . 6  |-  ( ( 1  /  ( x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  e.  RR  ->  E. k  e.  NN  ( 1  / 
( x  -  sup ( ran  ( F `  x ) ,  RR ,  <  ) ) )  <  k )
4139, 40syl 16 . . . . 5  |-  ( ( x  e.  RR  /\  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )  ->  E. k  e.  NN  ( 1  / 
( x  -  sup ( ran  ( F `  x ) ,  RR ,  <  ) ) )  <  k )
4241ex 434 . . . 4  |-  ( x  e.  RR  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x  ->  E. k  e.  NN  ( 1  / 
( x  -  sup ( ran  ( F `  x ) ,  RR ,  <  ) ) )  <  k ) )
431, 2rpnnen1lem2 10980 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  ZZ )
4443zred 10747 . . . . . . . 8  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  RR )
45443adant3 1008 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  NN  /\  (
1  /  ( x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  < 
k )  ->  sup ( T ,  RR ,  <  )  e.  RR )
4645ltp1d 10263 . . . . . 6  |-  ( ( x  e.  RR  /\  k  e.  NN  /\  (
1  /  ( x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  < 
k )  ->  sup ( T ,  RR ,  <  )  <  ( sup ( T ,  RR ,  <  )  +  1 ) )
4734, 37jca 532 . . . . . . . . . . . . 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 10329 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  k  e.  RR )
49 nngt0 10351 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  0  <  k )
5048, 49jca 532 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  (
k  e.  RR  /\  0  <  k ) )
51 ltrec1 10219 . . . . . . . . . . . . 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 477 . . . . . . . . . . . 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 725 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR  /\ 
sup ( ran  ( F `  x ) ,  RR ,  <  )  <  x )  /\  k  e.  NN )  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR )
54 nnrecre 10358 . . . . . . . . . . . . . . 15  |-  ( k  e.  NN  ->  (
1  /  k )  e.  RR )
5554adantl 466 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR  /\ 
sup ( ran  ( F `  x ) ,  RR ,  <  )  <  x )  /\  k  e.  NN )  ->  (
1  /  k )  e.  RR )
56 simpll 753 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR  /\ 
sup ( ran  ( F `  x ) ,  RR ,  <  )  <  x )  /\  k  e.  NN )  ->  x  e.  RR )
5753, 55, 56ltaddsub2d 9940 . . . . . . . . . . . . 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 ,  <  ) ) ) )
5812adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ran  ( F `  x )  C_  RR )
59 ffn 5559 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F `  x ) : NN --> QQ  ->  ( F `  x )  Fn  NN )
608, 59syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  RR  ->  ( F `  x )  Fn  NN )
61 fnfvelrn 5840 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( F `  x
)  Fn  NN  /\  k  e.  NN )  ->  ( ( F `  x ) `  k
)  e.  ran  ( F `  x )
)
6260, 61sylan 471 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( F `  x ) `  k
)  e.  ran  ( F `  x )
)
6358, 62sseldd 3357 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( F `  x ) `  k
)  e.  RR )
6431adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR )
6554adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( 1  /  k
)  e.  RR )
6612, 22, 263jca 1168 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  RR  ->  ( ran  ( F `  x
)  C_  RR  /\  ran  ( F `  x )  =/=  (/)  /\  E. y  e.  RR  A. n  e. 
ran  ( F `  x ) n  <_ 
y ) )
6766adantr 465 . . . . . . . . . . . . . . . . 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 ) )
68 suprub 10291 . . . . . . . . . . . . . . . . 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 ,  <  ) )
6967, 62, 68syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( F `  x ) `  k
)  <_  sup ( ran  ( F `  x
) ,  RR ,  <  ) )
7063, 64, 65, 69leadd1dd 9953 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( F `
 x ) `  k )  +  ( 1  /  k ) )  <_  ( sup ( ran  ( F `  x ) ,  RR ,  <  )  +  ( 1  /  k ) ) )
7163, 65readdcld 9413 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( F `
 x ) `  k )  +  ( 1  /  k ) )  e.  RR )
72 readdcl 9365 . . . . . . . . . . . . . . . . 17  |-  ( ( sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR  /\  ( 1  /  k )  e.  RR )  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  +  ( 1  / 
k ) )  e.  RR )
7331, 54, 72syl2an 477 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( sup ( ran  ( F `  x
) ,  RR ,  <  )  +  ( 1  /  k ) )  e.  RR )
74 simpl 457 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  x  e.  RR )
75 lelttr 9465 . . . . . . . . . . . . . . . . 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 ) )
7675expd 436 . . . . . . . . . . . . . . . 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 ) ) )
7771, 73, 74, 76syl3anc 1218 . . . . . . . . . . . . . . 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 ) ) )
7870, 77mpd 15 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( sup ( ran  ( F `  x
) ,  RR ,  <  )  +  ( 1  /  k ) )  <  x  ->  (
( ( F `  x ) `  k
)  +  ( 1  /  k ) )  <  x ) )
7978adantlr 714 . . . . . . . . . . . . 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 ) )
8057, 79sylbird 235 . . . . . . . . . . . 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 ) )
8152, 80sylbid 215 . . . . . . . . . . 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 ) )
8243peano2zd 10750 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( sup ( T ,  RR ,  <  )  +  1 )  e.  ZZ )
83 oveq1 6098 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  ( sup ( T ,  RR ,  <  )  +  1 )  ->  ( n  / 
k )  =  ( ( sup ( T ,  RR ,  <  )  +  1 )  / 
k ) )
8483breq1d 4302 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  ( sup ( T ,  RR ,  <  )  +  1 )  ->  ( ( n  /  k )  < 
x  <->  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  <  x ) )
8584, 1elrab2 3119 . . . . . . . . . . . . . . . . 17  |-  ( ( sup ( T ,  RR ,  <  )  +  1 )  e.  T  <->  ( ( sup ( T ,  RR ,  <  )  +  1 )  e.  ZZ  /\  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k
)  <  x )
)
8685biimpri 206 . . . . . . . . . . . . . . . 16  |-  ( ( ( sup ( T ,  RR ,  <  )  +  1 )  e.  ZZ  /\  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k
)  <  x )  ->  ( sup ( T ,  RR ,  <  )  +  1 )  e.  T )
8782, 86sylan 471 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  <  x )  -> 
( sup ( T ,  RR ,  <  )  +  1 )  e.  T )
88 ssrab2 3437 . . . . . . . . . . . . . . . . . . . 20  |-  { n  e.  ZZ  |  ( n  /  k )  < 
x }  C_  ZZ
891, 88eqsstri 3386 . . . . . . . . . . . . . . . . . . 19  |-  T  C_  ZZ
90 zssre 10653 . . . . . . . . . . . . . . . . . . 19  |-  ZZ  C_  RR
9189, 90sstri 3365 . . . . . . . . . . . . . . . . . 18  |-  T  C_  RR
9291a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  C_  RR )
93 remulcl 9367 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( k  e.  RR  /\  x  e.  RR )  ->  ( k  x.  x
)  e.  RR )
9493ancoms 453 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( x  e.  RR  /\  k  e.  RR )  ->  ( k  x.  x
)  e.  RR )
9548, 94sylan2 474 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( k  x.  x
)  e.  RR )
96 btwnz 10744 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( k  x.  x )  e.  RR  ->  ( E. n  e.  ZZ  n  <  ( k  x.  x )  /\  E. n  e.  ZZ  (
k  x.  x )  <  n ) )
9796simpld 459 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( k  x.  x )  e.  RR  ->  E. n  e.  ZZ  n  <  (
k  x.  x ) )
9895, 97syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. n  e.  ZZ  n  <  ( k  x.  x ) )
99 zre 10650 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( n  e.  ZZ  ->  n  e.  RR )
10099adantl 466 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  n  e.  RR )
101 simpll 753 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  x  e.  RR )
10250ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( k  e.  RR  /\  0  < 
k ) )
103 ltdivmul 10204 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( n  e.  RR  /\  x  e.  RR  /\  (
k  e.  RR  /\  0  <  k ) )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
104100, 101, 102, 103syl3anc 1218 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
105104rexbidva 2732 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( E. n  e.  ZZ  ( n  / 
k )  <  x  <->  E. n  e.  ZZ  n  <  ( k  x.  x
) ) )
10698, 105mpbird 232 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. n  e.  ZZ  ( n  /  k
)  <  x )
107 rabn0 3657 . . . . . . . . . . . . . . . . . . 19  |-  ( { n  e.  ZZ  | 
( n  /  k
)  <  x }  =/=  (/)  <->  E. n  e.  ZZ  ( n  /  k
)  <  x )
108106, 107sylibr 212 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  { n  e.  ZZ  |  ( n  / 
k )  <  x }  =/=  (/) )
1091neeq1i 2618 . . . . . . . . . . . . . . . . . 18  |-  ( T  =/=  (/)  <->  { n  e.  ZZ  |  ( n  / 
k )  <  x }  =/=  (/) )
110108, 109sylibr 212 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  =/=  (/) )
1111rabeq2i 2969 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  T  <->  ( n  e.  ZZ  /\  ( n  /  k )  < 
x ) )
11248ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  k  e.  RR )
113112, 101, 93syl2anc 661 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( k  x.  x )  e.  RR )
114 ltle 9463 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( n  e.  RR  /\  ( k  x.  x
)  e.  RR )  ->  ( n  < 
( k  x.  x
)  ->  n  <_  ( k  x.  x ) ) )
115100, 113, 114syl2anc 661 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( n  < 
( k  x.  x
)  ->  n  <_  ( k  x.  x ) ) )
116104, 115sylbid 215 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( ( n  /  k )  < 
x  ->  n  <_  ( k  x.  x ) ) )
117116impr 619 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  ( n  e.  ZZ  /\  ( n  /  k )  < 
x ) )  ->  n  <_  ( k  x.  x ) )
118111, 117sylan2b 475 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  T
)  ->  n  <_  ( k  x.  x ) )
119118ralrimiva 2799 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  A. n  e.  T  n  <_  ( k  x.  x ) )
120 breq2 4296 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  ( k  x.  x )  ->  (
n  <_  y  <->  n  <_  ( k  x.  x ) ) )
121120ralbidv 2735 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  ( k  x.  x )  ->  ( A. n  e.  T  n  <_  y  <->  A. n  e.  T  n  <_  ( k  x.  x ) ) )
122121rspcev 3073 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( k  x.  x
)  e.  RR  /\  A. n  e.  T  n  <_  ( k  x.  x ) )  ->  E. y  e.  RR  A. n  e.  T  n  <_  y )
12395, 119, 122syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. y  e.  RR  A. n  e.  T  n  <_  y )
12492, 110, 1233jca 1168 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( T  C_  RR  /\  T  =/=  (/)  /\  E. y  e.  RR  A. n  e.  T  n  <_  y ) )
125 suprub 10291 . . . . . . . . . . . . . . . 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 ,  <  ) )
126124, 125sylan 471 . . . . . . . . . . . . . . 15  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  ( sup ( T ,  RR ,  <  )  +  1 )  e.  T )  -> 
( sup ( T ,  RR ,  <  )  +  1 )  <_  sup ( T ,  RR ,  <  ) )
12787, 126syldan 470 . . . . . . . . . . . . . 14  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  <  x )  -> 
( sup ( T ,  RR ,  <  )  +  1 )  <_  sup ( T ,  RR ,  <  ) )
128127ex 434 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  <  x  ->  ( sup ( T ,  RR ,  <  )  +  1 )  <_  sup ( T ,  RR ,  <  ) ) )
12943zcnd 10748 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  CC )
130 1cnd 9402 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  1  e.  CC )
131 nncn 10330 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN  ->  k  e.  CC )
132 nnne0 10354 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN  ->  k  =/=  0 )
133131, 132jca 532 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  (
k  e.  CC  /\  k  =/=  0 ) )
134133adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( k  e.  CC  /\  k  =/=  0 ) )
135 divdir 10017 . . . . . . . . . . . . . . . 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 ) ) )
136129, 130, 134, 135syl3anc 1218 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  =  ( ( sup ( T ,  RR ,  <  )  /  k )  +  ( 1  / 
k ) ) )
1376mptex 5948 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k
) )  e.  _V
1382fvmpt2 5781 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  RR  /\  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )  e. 
_V )  ->  ( F `  x )  =  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k ) ) )
139137, 138mpan2 671 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  RR  ->  ( F `  x )  =  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k ) ) )
140139fveq1d 5693 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  RR  ->  (
( F `  x
) `  k )  =  ( ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k
) ) `  k
) )
141 ovex 6116 . . . . . . . . . . . . . . . . . 18  |-  ( sup ( T ,  RR ,  <  )  /  k
)  e.  _V
142 eqid 2443 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k
) )  =  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )
143142fvmpt2 5781 . . . . . . . . . . . . . . . . . 18  |-  ( ( k  e.  NN  /\  ( sup ( T ,  RR ,  <  )  / 
k )  e.  _V )  ->  ( ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k
) ) `  k
)  =  ( sup ( T ,  RR ,  <  )  /  k
) )
144141, 143mpan2 671 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  NN  ->  (
( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) ) `  k )  =  ( sup ( T ,  RR ,  <  )  / 
k ) )
145140, 144sylan9eq 2495 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( F `  x ) `  k
)  =  ( sup ( T ,  RR ,  <  )  /  k
) )
146145oveq1d 6106 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( F `
 x ) `  k )  +  ( 1  /  k ) )  =  ( ( sup ( T ,  RR ,  <  )  / 
k )  +  ( 1  /  k ) ) )
147136, 146eqtr4d 2478 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  =  ( ( ( F `
 x ) `  k )  +  ( 1  /  k ) ) )
148147breq1d 4302 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  <  x  <->  ( (
( F `  x
) `  k )  +  ( 1  / 
k ) )  < 
x ) )
14982zred 10747 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( sup ( T ,  RR ,  <  )  +  1 )  e.  RR )
150149, 44lenltd 9520 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( sup ( T ,  RR ,  <  )  +  1 )  <_  sup ( T ,  RR ,  <  )  <->  -.  sup ( T ,  RR ,  <  )  <  ( sup ( T ,  RR ,  <  )  +  1 ) ) )
151128, 148, 1503imtr3d 267 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( ( F `  x ) `
 k )  +  ( 1  /  k
) )  <  x  ->  -.  sup ( T ,  RR ,  <  )  <  ( sup ( T ,  RR ,  <  )  +  1 ) ) )
152151adantlr 714 . . . . . . . . . . 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 ) ) )
15381, 152syld 44 . . . . . . . . . 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 ) ) )
154153exp31 604 . . . . . . . . 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 ) ) ) ) )
155154com4l 84 . . . . . . . 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 ) ) ) ) )
156155com14 88 . . . . . . 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 ) ) ) ) )
1571563imp 1181 . . . . . 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 ) ) )
15846, 157mt2d 117 . . . . 5  |-  ( ( x  e.  RR  /\  k  e.  NN  /\  (
1  /  ( x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  < 
k )  ->  -.  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )
159158rexlimdv3a 2843 . . . 4  |-  ( x  e.  RR  ->  ( E. k  e.  NN  ( 1  /  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  < 
k  ->  -.  sup ( ran  ( F `  x
) ,  RR ,  <  )  <  x ) )
16042, 159syld 44 . . 3  |-  ( x  e.  RR  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x  ->  -.  sup ( ran  ( F `  x
) ,  RR ,  <  )  <  x ) )
161160pm2.01d 169 . 2  |-  ( x  e.  RR  ->  -.  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )
162 eqlelt 9462 . . 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 ) ) )
16331, 162mpancom 669 . 2  |-  ( x  e.  RR  ->  ( sup ( ran  ( F `
 x ) ,  RR ,  <  )  =  x  <->  ( sup ( ran  ( F `  x
) ,  RR ,  <  )  <_  x  /\  -.  sup ( ran  ( F `  x ) ,  RR ,  <  )  <  x ) ) )
16430, 161, 163mpbir2and 913 1  |-  ( x  e.  RR  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  =  x )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   E.wrex 2716   {crab 2719   _Vcvv 2972    C_ wss 3328   (/)c0 3637   class class class wbr 4292    e. cmpt 4350   dom cdm 4840   ran crn 4841    Fn wfn 5413   -->wf 5414   ` cfv 5418  (class class class)co 6091    ^m cmap 7214   supcsup 7690   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285    x. cmul 9287    < clt 9418    <_ cle 9419    - cmin 9595    / cdiv 9993   NNcn 10322   ZZcz 10646   QQcq 10953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-n0 10580  df-z 10647  df-q 10954
This theorem is referenced by:  rpnnen1  10984
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