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Theorem rpnnen1lem5 11212
Description: Lemma for rpnnen1 11213. (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 11210 . . 3  |-  ( x  e.  RR  ->  A. n  e.  ran  ( F `  x ) n  <_  x )
41, 2rpnnen1lem1 11208 . . . . . 6  |-  ( x  e.  RR  ->  ( F `  x )  e.  ( QQ  ^m  NN ) )
5 qexALT 11197 . . . . . . 7  |-  QQ  e.  _V
6 nnexALT 10538 . . . . . . 7  |-  NN  e.  _V
75, 6elmap 7447 . . . . . 6  |-  ( ( F `  x )  e.  ( QQ  ^m  NN )  <->  ( F `  x ) : NN --> QQ )
84, 7sylib 196 . . . . 5  |-  ( x  e.  RR  ->  ( F `  x ) : NN --> QQ )
9 frn 5737 . . . . . 6  |-  ( ( F `  x ) : NN --> QQ  ->  ran  ( F `  x
)  C_  QQ )
10 qssre 11192 . . . . . 6  |-  QQ  C_  RR
119, 10syl6ss 3516 . . . . 5  |-  ( ( F `  x ) : NN --> QQ  ->  ran  ( F `  x
)  C_  RR )
128, 11syl 16 . . . 4  |-  ( x  e.  RR  ->  ran  ( F `  x ) 
C_  RR )
13 1nn 10547 . . . . . . . 8  |-  1  e.  NN
14 ne0i 3791 . . . . . . . 8  |-  ( 1  e.  NN  ->  NN  =/=  (/) )
1513, 14ax-mp 5 . . . . . . 7  |-  NN  =/=  (/)
16 fdm 5735 . . . . . . . 8  |-  ( ( F `  x ) : NN --> QQ  ->  dom  ( F `  x
)  =  NN )
1716neeq1d 2744 . . . . . . 7  |-  ( ( F `  x ) : NN --> QQ  ->  ( dom  ( F `  x )  =/=  (/)  <->  NN  =/=  (/) ) )
1815, 17mpbiri 233 . . . . . 6  |-  ( ( F `  x ) : NN --> QQ  ->  dom  ( F `  x
)  =/=  (/) )
19 dm0rn0 5219 . . . . . . 7  |-  ( dom  ( F `  x
)  =  (/)  <->  ran  ( F `
 x )  =  (/) )
2019necon3bii 2735 . . . . . 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 4451 . . . . . . 7  |-  ( y  =  x  ->  (
n  <_  y  <->  n  <_  x ) )
2423ralbidv 2903 . . . . . 6  |-  ( y  =  x  ->  ( A. n  e.  ran  ( F `  x ) n  <_  y  <->  A. n  e.  ran  ( F `  x ) n  <_  x ) )
2524rspcev 3214 . . . . 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 10507 . . . 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 1231 . . 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 11211 . . . . . . . . 9  |-  ( x  e.  RR  ->  sup ( ran  ( F `  x ) ,  RR ,  <  )  e.  RR )
32 resubcl 9883 . . . . . . . . 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 10045 . . . . . . . . . 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 10117 . . . . . . 7  |-  ( ( x  e.  RR  /\  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )  ->  (
x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) )  =/=  0
)
3934, 38rereccld 10371 . . . . . 6  |-  ( ( x  e.  RR  /\  sup ( ran  ( F `
 x ) ,  RR ,  <  )  <  x )  ->  (
1  /  ( x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  e.  RR )
40 arch 10792 . . . . . 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 11209 . . . . . . . . 9  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  ZZ )
4443zred 10966 . . . . . . . 8  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  RR )
45443adant3 1016 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  NN  /\  (
1  /  ( x  -  sup ( ran  ( F `  x
) ,  RR ,  <  ) ) )  < 
k )  ->  sup ( T ,  RR ,  <  )  e.  RR )
4645ltp1d 10476 . . . . . 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 10543 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  k  e.  RR )
49 nngt0 10565 . . . . . . . . . . . . . 14  |-  ( k  e.  NN  ->  0  <  k )
5048, 49jca 532 . . . . . . . . . . . . 13  |-  ( k  e.  NN  ->  (
k  e.  RR  /\  0  <  k ) )
51 ltrec1 10432 . . . . . . . . . . . . 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 10572 . . . . . . . . . . . . . . 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 10153 . . . . . . . . . . . . 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 5731 . . . . . . . . . . . . . . . . . . 19  |-  ( ( F `  x ) : NN --> QQ  ->  ( F `  x )  Fn  NN )
608, 59syl 16 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  RR  ->  ( F `  x )  Fn  NN )
61 fnfvelrn 6018 . . . . . . . . . . . . . . . . . 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 3505 . . . . . . . . . . . . . . . 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 1176 . . . . . . . . . . . . . . . . . 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 10504 . . . . . . . . . . . . . . . . 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 10166 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( F `
 x ) `  k )  +  ( 1  /  k ) )  <_  ( sup ( ran  ( F `  x ) ,  RR ,  <  )  +  ( 1  /  k ) ) )
7163, 65readdcld 9623 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( F `
 x ) `  k )  +  ( 1  /  k ) )  e.  RR )
72 readdcl 9575 . . . . . . . . . . . . . . . . 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 9675 . . . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . . 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 10969 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( sup ( T ,  RR ,  <  )  +  1 )  e.  ZZ )
83 oveq1 6291 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  ( sup ( T ,  RR ,  <  )  +  1 )  ->  ( n  / 
k )  =  ( ( sup ( T ,  RR ,  <  )  +  1 )  / 
k ) )
8483breq1d 4457 . . . . . . . . . . . . . . . . . 18  |-  ( n  =  ( sup ( T ,  RR ,  <  )  +  1 )  ->  ( ( n  /  k )  < 
x  <->  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  <  x ) )
8584, 1elrab2 3263 . . . . . . . . . . . . . . . . 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 3585 . . . . . . . . . . . . . . . . . . . 20  |-  { n  e.  ZZ  |  ( n  /  k )  < 
x }  C_  ZZ
891, 88eqsstri 3534 . . . . . . . . . . . . . . . . . . 19  |-  T  C_  ZZ
90 zssre 10871 . . . . . . . . . . . . . . . . . . 19  |-  ZZ  C_  RR
9189, 90sstri 3513 . . . . . . . . . . . . . . . . . 18  |-  T  C_  RR
9291a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  C_  RR )
93 remulcl 9577 . . . . . . . . . . . . . . . . . . . . . . 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 10963 . . . . . . . . . . . . . . . . . . . . . 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 10868 . . . . . . . . . . . . . . . . . . . . . . 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 10417 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( n  e.  RR  /\  x  e.  RR  /\  (
k  e.  RR  /\  0  <  k ) )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
104100, 101, 102, 103syl3anc 1228 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
105104rexbidva 2970 . . . . . . . . . . . . . . . . . . . 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 3805 . . . . . . . . . . . . . . . . . . 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 2752 . . . . . . . . . . . . . . . . . 18  |-  ( T  =/=  (/)  <->  { n  e.  ZZ  |  ( n  / 
k )  <  x }  =/=  (/) )
110108, 109sylibr 212 . . . . . . . . . . . . . . . . 17  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  =/=  (/) )
1111rabeq2i 3110 . . . . . . . . . . . . . . . . . . . 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 9673 . . . . . . . . . . . . . . . . . . . . . . 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 2878 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  A. n  e.  T  n  <_  ( k  x.  x ) )
120 breq2 4451 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  ( k  x.  x )  ->  (
n  <_  y  <->  n  <_  ( k  x.  x ) ) )
121120ralbidv 2903 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  ( k  x.  x )  ->  ( A. n  e.  T  n  <_  y  <->  A. n  e.  T  n  <_  ( k  x.  x ) ) )
122121rspcev 3214 . . . . . . . . . . . . . . . . . 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 1176 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( T  C_  RR  /\  T  =/=  (/)  /\  E. y  e.  RR  A. n  e.  T  n  <_  y ) )
125 suprub 10504 . . . . . . . . . . . . . . . 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 10967 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  CC )
130 1cnd 9612 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  1  e.  CC )
131 nncn 10544 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  NN  ->  k  e.  CC )
132 nnne0 10568 . . . . . . . . . . . . . . . . . 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 10230 . . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  =  ( ( sup ( T ,  RR ,  <  )  /  k )  +  ( 1  / 
k ) ) )
1376mptex 6131 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k
) )  e.  _V
1382fvmpt2 5957 . . . . . . . . . . . . . . . . . . 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 5868 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  RR  ->  (
( F `  x
) `  k )  =  ( ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k
) ) `  k
) )
141 ovex 6309 . . . . . . . . . . . . . . . . . 18  |-  ( sup ( T ,  RR ,  <  )  /  k
)  e.  _V
142 eqid 2467 . . . . . . . . . . . . . . . . . . 19  |-  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  /  k
) )  =  ( k  e.  NN  |->  ( sup ( T ,  RR ,  <  )  / 
k ) )
143142fvmpt2 5957 . . . . . . . . . . . . . . . . . 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 2528 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( F `  x ) `  k
)  =  ( sup ( T ,  RR ,  <  )  /  k
) )
146145oveq1d 6299 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( F `
 x ) `  k )  +  ( 1  /  k ) )  =  ( ( sup ( T ,  RR ,  <  )  / 
k )  +  ( 1  /  k ) ) )
147136, 146eqtr4d 2511 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  =  ( ( ( F `
 x ) `  k )  +  ( 1  /  k ) ) )
148147breq1d 4457 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( ( ( sup ( T ,  RR ,  <  )  +  1 )  /  k )  <  x  <->  ( (
( F `  x
) `  k )  +  ( 1  / 
k ) )  < 
x ) )
14982zred 10966 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( sup ( T ,  RR ,  <  )  +  1 )  e.  RR )
150149, 44lenltd 9730 . . . . . . . . . . . . 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 1190 . . . . . 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 2957 . . . 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 9672 . . 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 920 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113    C_ wss 3476   (/)c0 3785   class class class wbr 4447    |-> cmpt 4505   dom cdm 4999   ran crn 5000    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284    ^m cmap 7420   supcsup 7900   CCcc 9490   RRcr 9491   0cc0 9492   1c1 9493    + caddc 9495    x. cmul 9497    < clt 9628    <_ cle 9629    - cmin 9805    / cdiv 10206   NNcn 10536   ZZcz 10864   QQcq 11182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-sup 7901  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-n0 10796  df-z 10865  df-q 11183
This theorem is referenced by:  rpnnen1  11213
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