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Theorem fztri3or 9453
Description: Trichotomy in terms of a finite interval of integers. (Contributed by Jim Kingdon, 1-Jun-2020.)
Assertion
Ref Expression
fztri3or  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  <  M  \/  K  e.  ( M ... N
)  \/  N  < 
K ) )

Proof of Theorem fztri3or
StepHypRef Expression
1 3mix1 1112 . . 3  |-  ( K  <  M  ->  ( K  <  M  \/  K  e.  ( M ... N
)  \/  N  < 
K ) )
21adantl 271 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  K  <  M )  ->  ( K  < 
M  \/  K  e.  ( M ... N
)  \/  N  < 
K ) )
3 3mix3 1114 . . . 4  |-  ( N  <  K  ->  ( K  <  M  \/  K  e.  ( M ... N
)  \/  N  < 
K ) )
43adantl 271 . . 3  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  N  <  K )  -> 
( K  <  M  \/  K  e.  ( M ... N )  \/  N  <  K ) )
5 simpr 108 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  -.  K  <  M )
6 simpl2 947 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  M  e.  ZZ )
76zred 8868 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  M  e.  RR )
8 simpl1 946 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  K  e.  ZZ )
98zred 8868 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  K  e.  RR )
107, 9lenltd 7601 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  ( M  <_  K  <->  -.  K  <  M ) )
115, 10mpbird 165 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  M  <_  K )
1211adantr 270 . . . . 5  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  M  <_  K
)
13 simpr 108 . . . . . 6  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  -.  N  <  K )
149adantr 270 . . . . . . 7  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  K  e.  RR )
15 simpll3 984 . . . . . . . 8  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  N  e.  ZZ )
1615zred 8868 . . . . . . 7  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  N  e.  RR )
1714, 16lenltd 7601 . . . . . 6  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  ( K  <_  N 
<->  -.  N  <  K
) )
1813, 17mpbird 165 . . . . 5  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  K  <_  N
)
19 elfz 9430 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N )  <->  ( M  <_  K  /\  K  <_  N ) ) )
2019adantr 270 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  ( K  e.  ( M ... N
)  <->  ( M  <_  K  /\  K  <_  N
) ) )
2120adantr 270 . . . . 5  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  ( K  e.  ( M ... N
)  <->  ( M  <_  K  /\  K  <_  N
) ) )
2212, 18, 21mpbir2and 890 . . . 4  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  K  e.  ( M ... N ) )
23223mix2d 1119 . . 3  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  ( K  < 
M  \/  K  e.  ( M ... N
)  \/  N  < 
K ) )
24 zdclt 8824 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  -> DECID  N  <  K )
2524ancoms 264 . . . . . 6  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  -> DECID  N  <  K )
26253adant2 962 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  N  <  K )
2726adantr 270 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  -> DECID  N  <  K )
28 df-dc 781 . . . 4  |-  (DECID  N  < 
K  <->  ( N  < 
K  \/  -.  N  <  K ) )
2927, 28sylib 120 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  ( N  <  K  \/  -.  N  <  K ) )
304, 23, 29mpjaodan 747 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  ( K  <  M  \/  K  e.  ( M ... N
)  \/  N  < 
K ) )
31 zdclt 8824 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  -> DECID  K  <  M )
32313adant3 963 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  K  <  M )
33 df-dc 781 . . 3  |-  (DECID  K  < 
M  <->  ( K  < 
M  \/  -.  K  <  M ) )
3432, 33sylib 120 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  <  M  \/  -.  K  <  M ) )
352, 30, 34mpjaodan 747 1  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  <  M  \/  K  e.  ( M ... N
)  \/  N  < 
K ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 664  DECID wdc 780    \/ w3o 923    /\ w3a 924    e. wcel 1438   class class class wbr 3845  (class class class)co 5652   RRcr 7349    < clt 7522    <_ cle 7523   ZZcz 8750   ...cfz 9424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-addass 7447  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-ltadd 7461
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-inn 8423  df-n0 8674  df-z 8751  df-fz 9425
This theorem is referenced by:  fzdcel  9454  hashfiv01gt1  10190
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