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Theorem rpnnen1lem2 10359
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
rpnnen1lem2  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  ZZ )
Distinct variable groups:    k, F, n, x    T, n
Allowed substitution hints:    T( x, k)

Proof of Theorem rpnnen1lem2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 rpnnen1.1 . . 3  |-  T  =  { n  e.  ZZ  |  ( n  / 
k )  <  x }
2 ssrab2 3271 . . 3  |-  { n  e.  ZZ  |  ( n  /  k )  < 
x }  C_  ZZ
31, 2eqsstri 3221 . 2  |-  T  C_  ZZ
43a1i 10 . . 3  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  C_  ZZ )
5 nnre 9769 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  RR )
6 remulcl 8838 . . . . . . . . 9  |-  ( ( k  e.  RR  /\  x  e.  RR )  ->  ( k  x.  x
)  e.  RR )
76ancoms 439 . . . . . . . 8  |-  ( ( x  e.  RR  /\  k  e.  RR )  ->  ( k  x.  x
)  e.  RR )
85, 7sylan2 460 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( k  x.  x
)  e.  RR )
9 btwnz 10130 . . . . . . . 8  |-  ( ( k  x.  x )  e.  RR  ->  ( E. n  e.  ZZ  n  <  ( k  x.  x )  /\  E. n  e.  ZZ  (
k  x.  x )  <  n ) )
109simpld 445 . . . . . . 7  |-  ( ( k  x.  x )  e.  RR  ->  E. n  e.  ZZ  n  <  (
k  x.  x ) )
118, 10syl 15 . . . . . 6  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. n  e.  ZZ  n  <  ( k  x.  x ) )
12 zre 10044 . . . . . . . . 9  |-  ( n  e.  ZZ  ->  n  e.  RR )
1312adantl 452 . . . . . . . 8  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  n  e.  RR )
14 simpll 730 . . . . . . . 8  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  x  e.  RR )
15 nngt0 9791 . . . . . . . . . 10  |-  ( k  e.  NN  ->  0  <  k )
165, 15jca 518 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
k  e.  RR  /\  0  <  k ) )
1716ad2antlr 707 . . . . . . . 8  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( k  e.  RR  /\  0  < 
k ) )
18 ltdivmul 9644 . . . . . . . 8  |-  ( ( n  e.  RR  /\  x  e.  RR  /\  (
k  e.  RR  /\  0  <  k ) )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
1913, 14, 17, 18syl3anc 1182 . . . . . . 7  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
2019rexbidva 2573 . . . . . 6  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( E. n  e.  ZZ  ( n  / 
k )  <  x  <->  E. n  e.  ZZ  n  <  ( k  x.  x
) ) )
2111, 20mpbird 223 . . . . 5  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. n  e.  ZZ  ( n  /  k
)  <  x )
22 rabn0 3487 . . . . 5  |-  ( { n  e.  ZZ  | 
( n  /  k
)  <  x }  =/=  (/)  <->  E. n  e.  ZZ  ( n  /  k
)  <  x )
2321, 22sylibr 203 . . . 4  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  { n  e.  ZZ  |  ( n  / 
k )  <  x }  =/=  (/) )
241neeq1i 2469 . . . 4  |-  ( T  =/=  (/)  <->  { n  e.  ZZ  |  ( n  / 
k )  <  x }  =/=  (/) )
2523, 24sylibr 203 . . 3  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  =/=  (/) )
261rabeq2i 2798 . . . . . 6  |-  ( n  e.  T  <->  ( n  e.  ZZ  /\  ( n  /  k )  < 
x ) )
275ad2antlr 707 . . . . . . . . . 10  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  k  e.  RR )
2827, 14, 6syl2anc 642 . . . . . . . . 9  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( k  x.  x )  e.  RR )
29 ltle 8926 . . . . . . . . 9  |-  ( ( n  e.  RR  /\  ( k  x.  x
)  e.  RR )  ->  ( n  < 
( k  x.  x
)  ->  n  <_  ( k  x.  x ) ) )
3013, 28, 29syl2anc 642 . . . . . . . 8  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( n  < 
( k  x.  x
)  ->  n  <_  ( k  x.  x ) ) )
3119, 30sylbid 206 . . . . . . 7  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( ( n  /  k )  < 
x  ->  n  <_  ( k  x.  x ) ) )
3231impr 602 . . . . . 6  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  ( n  e.  ZZ  /\  ( n  /  k )  < 
x ) )  ->  n  <_  ( k  x.  x ) )
3326, 32sylan2b 461 . . . . 5  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  T
)  ->  n  <_  ( k  x.  x ) )
3433ralrimiva 2639 . . . 4  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  A. n  e.  T  n  <_  ( k  x.  x ) )
35 breq2 4043 . . . . . 6  |-  ( y  =  ( k  x.  x )  ->  (
n  <_  y  <->  n  <_  ( k  x.  x ) ) )
3635ralbidv 2576 . . . . 5  |-  ( y  =  ( k  x.  x )  ->  ( A. n  e.  T  n  <_  y  <->  A. n  e.  T  n  <_  ( k  x.  x ) ) )
3736rspcev 2897 . . . 4  |-  ( ( ( k  x.  x
)  e.  RR  /\  A. n  e.  T  n  <_  ( k  x.  x ) )  ->  E. y  e.  RR  A. n  e.  T  n  <_  y )
388, 34, 37syl2anc 642 . . 3  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. y  e.  RR  A. n  e.  T  n  <_  y )
39 suprzcl 10107 . . 3  |-  ( ( T  C_  ZZ  /\  T  =/=  (/)  /\  E. y  e.  RR  A. n  e.  T  n  <_  y
)  ->  sup ( T ,  RR ,  <  )  e.  T )
404, 25, 38, 39syl3anc 1182 . 2  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  T )
413, 40sseldi 3191 1  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  ZZ )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560    C_ wss 3165   (/)c0 3468   class class class wbr 4039    e. cmpt 4093  (class class class)co 5874   supcsup 7209   RRcr 8752   0cc0 8753    x. cmul 8758    < clt 8883    <_ cle 8884    / cdiv 9439   NNcn 9762   ZZcz 10040
This theorem is referenced by:  rpnnen1lem3  10360  rpnnen1lem5  10362
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-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  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-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  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
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