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Theorem fztri3or 9211
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 1108 . . 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 1110 . . . 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 943 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  M  e.  ZZ )
76zred 8627 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  M  e.  RR )
8 simpl1 942 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  K  e.  ZZ )
98zred 8627 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  K  e.  RR )
107, 9lenltd 7371 . . . . . . 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 980 . . . . . . . 8  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  N  e.  ZZ )
1615zred 8627 . . . . . . 7  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  N  e.  RR )
1714, 16lenltd 7371 . . . . . 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 9188 . . . . . . 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 886 . . . 4  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  K  e.  ( M ... N ) )
23223mix2d 1115 . . 3  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  ( K  < 
M  \/  K  e.  ( M ... N
)  \/  N  < 
K ) )
24 zdclt 8583 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  -> DECID  N  <  K )
2524ancoms 264 . . . . . 6  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  -> DECID  N  <  K )
26253adant2 958 . . . . 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 777 . . . 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 745 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  ( K  <  M  \/  K  e.  ( M ... N
)  \/  N  < 
K ) )
31 zdclt 8583 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  -> DECID  K  <  M )
32313adant3 959 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  K  <  M )
33 df-dc 777 . . 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 745 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 662  DECID wdc 776    \/ w3o 919    /\ w3a 920    e. wcel 1434   class class class wbr 3806  (class class class)co 5565   RRcr 7119    < clt 7292    <_ cle 7293   ZZcz 8509   ...cfz 9182
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-cnex 7206  ax-resscn 7207  ax-1cn 7208  ax-1re 7209  ax-icn 7210  ax-addcl 7211  ax-addrcl 7212  ax-mulcl 7213  ax-addcom 7215  ax-addass 7217  ax-distr 7219  ax-i2m1 7220  ax-0lt1 7221  ax-0id 7223  ax-rnegex 7224  ax-cnre 7226  ax-pre-ltirr 7227  ax-pre-ltwlin 7228  ax-pre-lttrn 7229  ax-pre-ltadd 7231
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2613  df-sbc 2826  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-br 3807  df-opab 3861  df-id 4077  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-iota 4918  df-fun 4955  df-fv 4961  df-riota 5521  df-ov 5568  df-oprab 5569  df-mpt2 5570  df-pnf 7294  df-mnf 7295  df-xr 7296  df-ltxr 7297  df-le 7298  df-sub 7425  df-neg 7426  df-inn 8184  df-n0 8433  df-z 8510  df-fz 9183
This theorem is referenced by:  fzdcel  9212  hashfiv01gt1  9883
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