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Theorem rpnnen1lem2 11210
Description: Lemma for rpnnen1 11214. (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 3571 . . 3  |-  { n  e.  ZZ  |  ( n  /  k )  < 
x }  C_  ZZ
31, 2eqsstri 3519 . 2  |-  T  C_  ZZ
43a1i 11 . . 3  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  C_  ZZ )
5 nnre 10538 . . . . . . . 8  |-  ( k  e.  NN  ->  k  e.  RR )
6 remulcl 9566 . . . . . . . . 9  |-  ( ( k  e.  RR  /\  x  e.  RR )  ->  ( k  x.  x
)  e.  RR )
76ancoms 451 . . . . . . . 8  |-  ( ( x  e.  RR  /\  k  e.  RR )  ->  ( k  x.  x
)  e.  RR )
85, 7sylan2 472 . . . . . . 7  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  ( k  x.  x
)  e.  RR )
9 btwnz 10962 . . . . . . . 8  |-  ( ( k  x.  x )  e.  RR  ->  ( E. n  e.  ZZ  n  <  ( k  x.  x )  /\  E. n  e.  ZZ  (
k  x.  x )  <  n ) )
109simpld 457 . . . . . . 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 10864 . . . . . . . . 9  |-  ( n  e.  ZZ  ->  n  e.  RR )
1312adantl 464 . . . . . . . 8  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  n  e.  RR )
14 simpll 751 . . . . . . . 8  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  x  e.  RR )
15 nngt0 10560 . . . . . . . . . 10  |-  ( k  e.  NN  ->  0  <  k )
165, 15jca 530 . . . . . . . . 9  |-  ( k  e.  NN  ->  (
k  e.  RR  /\  0  <  k ) )
1716ad2antlr 724 . . . . . . . 8  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( k  e.  RR  /\  0  < 
k ) )
18 ltdivmul 10413 . . . . . . . 8  |-  ( ( n  e.  RR  /\  x  e.  RR  /\  (
k  e.  RR  /\  0  <  k ) )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
1913, 14, 17, 18syl3anc 1226 . . . . . . 7  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( ( n  /  k )  < 
x  <->  n  <  ( k  x.  x ) ) )
2019rexbidva 2962 . . . . . 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 3804 . . . . 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 2739 . . . 4  |-  ( T  =/=  (/)  <->  { n  e.  ZZ  |  ( n  / 
k )  <  x }  =/=  (/) )
2523, 24sylibr 212 . . 3  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  T  =/=  (/) )
261rabeq2i 3103 . . . . . 6  |-  ( n  e.  T  <->  ( n  e.  ZZ  /\  ( n  /  k )  < 
x ) )
275ad2antlr 724 . . . . . . . . . 10  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  k  e.  RR )
2827, 14, 6syl2anc 659 . . . . . . . . 9  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  ZZ )  ->  ( k  x.  x )  e.  RR )
29 ltle 9662 . . . . . . . . 9  |-  ( ( n  e.  RR  /\  ( k  x.  x
)  e.  RR )  ->  ( n  < 
( k  x.  x
)  ->  n  <_  ( k  x.  x ) ) )
3013, 28, 29syl2anc 659 . . . . . . . 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 617 . . . . . 6  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  ( n  e.  ZZ  /\  ( n  /  k )  < 
x ) )  ->  n  <_  ( k  x.  x ) )
3326, 32sylan2b 473 . . . . 5  |-  ( ( ( x  e.  RR  /\  k  e.  NN )  /\  n  e.  T
)  ->  n  <_  ( k  x.  x ) )
3433ralrimiva 2868 . . . 4  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  A. n  e.  T  n  <_  ( k  x.  x ) )
35 breq2 4443 . . . . . 6  |-  ( y  =  ( k  x.  x )  ->  (
n  <_  y  <->  n  <_  ( k  x.  x ) ) )
3635ralbidv 2893 . . . . 5  |-  ( y  =  ( k  x.  x )  ->  ( A. n  e.  T  n  <_  y  <->  A. n  e.  T  n  <_  ( k  x.  x ) ) )
3736rspcev 3207 . . . 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 659 . . 3  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  E. y  e.  RR  A. n  e.  T  n  <_  y )
39 suprzcl 10938 . . 3  |-  ( ( T  C_  ZZ  /\  T  =/=  (/)  /\  E. y  e.  RR  A. n  e.  T  n  <_  y
)  ->  sup ( T ,  RR ,  <  )  e.  T )
404, 25, 38, 39syl3anc 1226 . 2  |-  ( ( x  e.  RR  /\  k  e.  NN )  ->  sup ( T ,  RR ,  <  )  e.  T )
413, 40sseldi 3487 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 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   {crab 2808    C_ wss 3461   (/)c0 3783   class class class wbr 4439    |-> cmpt 4497  (class class class)co 6270   supcsup 7892   RRcr 9480   0cc0 9481    x. cmul 9486    < clt 9617    <_ cle 9618    / cdiv 10202   NNcn 10531   ZZcz 10860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-n0 10792  df-z 10861
This theorem is referenced by:  rpnnen1lem3  11211  rpnnen1lem5  11213
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