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Theorem apbtwnz 9830
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 108 . . 3  |-  ( ( A  e.  RR  /\  A. n  e.  ZZ  A #  n )  ->  A  e.  RR )
2 simpr 109 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  /\  A  <  m
)  ->  A  <  m )
32olcd 691 . . . 4  |-  ( ( ( ( A  e.  RR  /\  A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  /\  A  <  m
)  ->  ( m  <_  A  \/  A  < 
m ) )
4 simpr 109 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\ 
A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  ->  m  e.  ZZ )
54zred 8967 . . . . . . 7  |-  ( ( ( A  e.  RR  /\ 
A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  ->  m  e.  RR )
65adantr 271 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  /\  m  <  A
)  ->  m  e.  RR )
71adantr 271 . . . . . . 7  |-  ( ( ( A  e.  RR  /\ 
A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  ->  A  e.  RR )
87adantr 271 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  /\  m  <  A
)  ->  A  e.  RR )
9 simpr 109 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  /\  m  <  A
)  ->  m  <  A )
106, 8, 9ltled 7699 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  /\  m  <  A
)  ->  m  <_  A )
1110orcd 690 . . . 4  |-  ( ( ( ( A  e.  RR  /\  A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  /\  m  <  A
)  ->  ( m  <_  A  \/  A  < 
m ) )
12 breq2 3871 . . . . . 6  |-  ( n  =  m  ->  ( A #  n  <->  A #  m )
)
13 simplr 498 . . . . . 6  |-  ( ( ( A  e.  RR  /\ 
A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  ->  A. n  e.  ZZ  A #  n )
1412, 13, 4rspcdva 2741 . . . . 5  |-  ( ( ( A  e.  RR  /\ 
A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  ->  A #  m )
15 reaplt 8162 . . . . . 6  |-  ( ( A  e.  RR  /\  m  e.  RR )  ->  ( A #  m  <->  ( A  <  m  \/  m  < 
A ) ) )
167, 5, 15syl2anc 404 . . . . 5  |-  ( ( ( A  e.  RR  /\ 
A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  ->  ( A #  m  <->  ( A  < 
m  \/  m  < 
A ) ) )
1714, 16mpbid 146 . . . 4  |-  ( ( ( A  e.  RR  /\ 
A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  ->  ( A  <  m  \/  m  <  A ) )
183, 11, 17mpjaodan 750 . . 3  |-  ( ( ( A  e.  RR  /\ 
A. n  e.  ZZ  A #  n )  /\  m  e.  ZZ )  ->  (
m  <_  A  \/  A  <  m ) )
191, 18exbtwnzlemex 9810 . 2  |-  ( ( A  e.  RR  /\  A. n  e.  ZZ  A #  n )  ->  E. x  e.  ZZ  ( x  <_  A  /\  A  <  (
x  +  1 ) ) )
2019, 1exbtwnz 9811 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 103    <-> wb 104    \/ wo 667    e. wcel 1445   A.wral 2370   E!wreu 2372   class class class wbr 3867  (class class class)co 5690   RRcr 7446   1c1 7448    + caddc 7450    < clt 7619    <_ cle 7620   # cap 8155   ZZcz 8848
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-arch 7561
This theorem depends on definitions:  df-bi 116  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-br 3868  df-opab 3922  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-inn 8521  df-n0 8772  df-z 8849
This theorem is referenced by:  flapcl  9831
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