ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  apbtwnz Unicode version

Theorem apbtwnz 10288
Description: There is a unique greatest integer less than or equal to a real number which is apart from all integers. (Contributed by Jim Kingdon, 11-May-2022.)
Assertion
Ref Expression
apbtwnz  |-  ( ( A  e.  RR  /\  A. n  e.  ZZ  A #  n )  ->  E! x  e.  ZZ  (
x  <_  A  /\  A  <  ( x  + 
1 ) ) )
Distinct variable group:    A, n, x

Proof of Theorem apbtwnz
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 simpl 109 . . 3  |-  ( ( A  e.  RR  /\  A. n  e.  ZZ  A #  n )  ->  A  e.  RR )
2 simpr 110 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  /\  A  <  m
)  ->  A  <  m )
32olcd 735 . . . 4  |-  ( ( ( ( A  e.  RR  /\  A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  /\  A  <  m
)  ->  ( m  <_  A  \/  A  < 
m ) )
4 simpr 110 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\ 
A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  ->  m  e.  ZZ )
54zred 9389 . . . . . . 7  |-  ( ( ( A  e.  RR  /\ 
A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  ->  m  e.  RR )
65adantr 276 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  /\  m  <  A
)  ->  m  e.  RR )
71adantr 276 . . . . . . 7  |-  ( ( ( A  e.  RR  /\ 
A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  ->  A  e.  RR )
87adantr 276 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  /\  m  <  A
)  ->  A  e.  RR )
9 simpr 110 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  /\  m  <  A
)  ->  m  <  A )
106, 8, 9ltled 8090 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  /\  m  <  A
)  ->  m  <_  A )
1110orcd 734 . . . 4  |-  ( ( ( ( A  e.  RR  /\  A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  /\  m  <  A
)  ->  ( m  <_  A  \/  A  < 
m ) )
12 breq2 4019 . . . . . 6  |-  ( n  =  m  ->  ( A #  n  <->  A #  m )
)
13 simplr 528 . . . . . 6  |-  ( ( ( A  e.  RR  /\ 
A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  ->  A. n  e.  ZZ  A #  n )
1412, 13, 4rspcdva 2858 . . . . 5  |-  ( ( ( A  e.  RR  /\ 
A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  ->  A #  m )
15 reaplt 8559 . . . . . 6  |-  ( ( A  e.  RR  /\  m  e.  RR )  ->  ( A #  m  <->  ( A  <  m  \/  m  < 
A ) ) )
167, 5, 15syl2anc 411 . . . . 5  |-  ( ( ( A  e.  RR  /\ 
A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  ->  ( A #  m  <->  ( A  < 
m  \/  m  < 
A ) ) )
1714, 16mpbid 147 . . . 4  |-  ( ( ( A  e.  RR  /\ 
A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  ->  ( A  <  m  \/  m  <  A ) )
183, 11, 17mpjaodan 799 . . 3  |-  ( ( ( A  e.  RR  /\ 
A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  ->  (
m  <_  A  \/  A  <  m ) )
191, 18exbtwnzlemex 10264 . 2  |-  ( ( A  e.  RR  /\  A. n  e.  ZZ  A #  n )  ->  E. x  e.  ZZ  ( x  <_  A  /\  A  <  (
x  +  1 ) ) )
2019, 1exbtwnz 10265 1  |-  ( ( A  e.  RR  /\  A. n  e.  ZZ  A #  n )  ->  E! x  e.  ZZ  (
x  <_  A  /\  A  <  ( x  + 
1 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    e. wcel 2158   A.wral 2465   E!wreu 2467   class class class wbr 4015  (class class class)co 5888   RRcr 7824   1c1 7826    + caddc 7828    < clt 8006    <_ cle 8007   # cap 8552   ZZcz 9267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-arch 7944
This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-inn 8934  df-n0 9191  df-z 9268
This theorem is referenced by:  flapcl  10289
  Copyright terms: Public domain W3C validator