MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rpnnen1lem2 Structured version   Unicode version

Theorem rpnnen1lem2 11094
Description: Lemma for rpnnen1 11098. (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 3548 . . 3  |-  { n  e.  ZZ  |  ( n  /  k )  < 
x }  C_  ZZ
31, 2eqsstri 3497 . 2  |-  T  C_  ZZ
43a1i 11 . . 3  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  C_  ZZ )
5 nnre 10443 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  RR )
6 remulcl 9481 . . . . . . . . 9  |-  ( ( k  e.  RR  /\  x  e.  RR )  ->  ( k  x.  x
)  e.  RR )
76ancoms 453 . . . . . . . 8  |-  ( ( x  e.  RR  /\  k  e.  RR )  ->  ( k  x.  x
)  e.  RR )
85, 7sylan2 474 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( k  x.  x
)  e.  RR )
9 btwnz 10858 . . . . . . . 8  |-  ( ( k  x.  x )  e.  RR  ->  ( E. n  e.  ZZ  n  <  ( k  x.  x )  /\  E. n  e.  ZZ  (
k  x.  x )  <  n ) )
109simpld 459 . . . . . . 7  |-  ( ( k  x.  x )  e.  RR  ->  E. n  e.  ZZ  n  <  (
k  x.  x ) )
118, 10syl 16 . . . . . 6  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. n  e.  ZZ  n  <  ( k  x.  x ) )
12 zre 10764 . . . . . . . . 9  |-  ( n  e.  ZZ  ->  n  e.  RR )
1312adantl 466 . . . . . . . 8  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  n  e.  RR )
14 simpll 753 . . . . . . . 8  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  x  e.  RR )
15 nngt0 10465 . . . . . . . . . 10  |-  ( k  e.  NN  ->  0  <  k )
165, 15jca 532 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
k  e.  RR  /\  0  <  k ) )
1716ad2antlr 726 . . . . . . . 8  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( k  e.  RR  /\  0  < 
k ) )
18 ltdivmul 10318 . . . . . . . 8  |-  ( ( n  e.  RR  /\  x  e.  RR  /\  (
k  e.  RR  /\  0  <  k ) )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
1913, 14, 17, 18syl3anc 1219 . . . . . . 7  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
2019rexbidva 2865 . . . . . 6  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( E. n  e.  ZZ  ( n  / 
k )  <  x  <->  E. n  e.  ZZ  n  <  ( k  x.  x
) ) )
2111, 20mpbird 232 . . . . 5  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. n  e.  ZZ  ( n  /  k
)  <  x )
22 rabn0 3768 . . . . 5  |-  ( { n  e.  ZZ  | 
( n  /  k
)  <  x }  =/=  (/)  <->  E. n  e.  ZZ  ( n  /  k
)  <  x )
2321, 22sylibr 212 . . . 4  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  { n  e.  ZZ  |  ( n  / 
k )  <  x }  =/=  (/) )
241neeq1i 2737 . . . 4  |-  ( T  =/=  (/)  <->  { n  e.  ZZ  |  ( n  / 
k )  <  x }  =/=  (/) )
2523, 24sylibr 212 . . 3  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  =/=  (/) )
261rabeq2i 3075 . . . . . 6  |-  ( n  e.  T  <->  ( n  e.  ZZ  /\  ( n  /  k )  < 
x ) )
275ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  k  e.  RR )
2827, 14, 6syl2anc 661 . . . . . . . . 9  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( k  x.  x )  e.  RR )
29 ltle 9577 . . . . . . . . 9  |-  ( ( n  e.  RR  /\  ( k  x.  x
)  e.  RR )  ->  ( n  < 
( k  x.  x
)  ->  n  <_  ( k  x.  x ) ) )
3013, 28, 29syl2anc 661 . . . . . . . 8  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( n  < 
( k  x.  x
)  ->  n  <_  ( k  x.  x ) ) )
3119, 30sylbid 215 . . . . . . 7  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( ( n  /  k )  < 
x  ->  n  <_  ( k  x.  x ) ) )
3231impr 619 . . . . . 6  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  ( n  e.  ZZ  /\  ( n  /  k )  < 
x ) )  ->  n  <_  ( k  x.  x ) )
3326, 32sylan2b 475 . . . . 5  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  T
)  ->  n  <_  ( k  x.  x ) )
3433ralrimiva 2830 . . . 4  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  A. n  e.  T  n  <_  ( k  x.  x ) )
35 breq2 4407 . . . . . 6  |-  ( y  =  ( k  x.  x )  ->  (
n  <_  y  <->  n  <_  ( k  x.  x ) ) )
3635ralbidv 2846 . . . . 5  |-  ( y  =  ( k  x.  x )  ->  ( A. n  e.  T  n  <_  y  <->  A. n  e.  T  n  <_  ( k  x.  x ) ) )
3736rspcev 3179 . . . 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 661 . . 3  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. y  e.  RR  A. n  e.  T  n  <_  y )
39 suprzcl 10835 . . 3  |-  ( ( T  C_  ZZ  /\  T  =/=  (/)  /\  E. y  e.  RR  A. n  e.  T  n  <_  y
)  ->  sup ( T ,  RR ,  <  )  e.  T )
404, 25, 38, 39syl3anc 1219 . 2  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  T )
413, 40sseldi 3465 1  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  ZZ )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800   {crab 2803    C_ wss 3439   (/)c0 3748   class class class wbr 4403    |-> cmpt 4461  (class class class)co 6203   supcsup 7804   RRcr 9395   0cc0 9396    x. cmul 9401    < clt 9532    <_ cle 9533    / cdiv 10107   NNcn 10436   ZZcz 10760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-n0 10694  df-z 10761
This theorem is referenced by:  rpnnen1lem3  11095  rpnnen1lem5  11097
  Copyright terms: Public domain W3C validator