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Theorem fztri3or 9770
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 1133 . . 3  |-  ( K  <  M  ->  ( K  <  M  \/  K  e.  ( M ... N
)  \/  N  < 
K ) )
21adantl 273 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  K  <  M )  ->  ( K  < 
M  \/  K  e.  ( M ... N
)  \/  N  < 
K ) )
3 3mix3 1135 . . . 4  |-  ( N  <  K  ->  ( K  <  M  \/  K  e.  ( M ... N
)  \/  N  < 
K ) )
43adantl 273 . . 3  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  N  <  K )  -> 
( K  <  M  \/  K  e.  ( M ... N )  \/  N  <  K ) )
5 simpr 109 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  -.  K  <  M )
6 simpl2 968 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  M  e.  ZZ )
76zred 9127 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  M  e.  RR )
8 simpl1 967 . . . . . . . . 9  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  K  e.  ZZ )
98zred 9127 . . . . . . . 8  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  K  e.  RR )
107, 9lenltd 7844 . . . . . . 7  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  ( M  <_  K  <->  -.  K  <  M ) )
115, 10mpbird 166 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  M  <_  K )
1211adantr 272 . . . . 5  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  M  <_  K
)
13 simpr 109 . . . . . 6  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  -.  N  <  K )
149adantr 272 . . . . . . 7  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  K  e.  RR )
15 simpll3 1005 . . . . . . . 8  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  N  e.  ZZ )
1615zred 9127 . . . . . . 7  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  N  e.  RR )
1714, 16lenltd 7844 . . . . . 6  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  ( K  <_  N 
<->  -.  N  <  K
) )
1813, 17mpbird 166 . . . . 5  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  K  <_  N
)
19 elfz 9747 . . . . . . 7  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N )  <->  ( M  <_  K  /\  K  <_  N ) ) )
2019adantr 272 . . . . . 6  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  ( K  e.  ( M ... N
)  <->  ( M  <_  K  /\  K  <_  N
) ) )
2120adantr 272 . . . . 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 911 . . . 4  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  K  e.  ( M ... N ) )
23223mix2d 1140 . . 3  |-  ( ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M )  /\  -.  N  <  K )  ->  ( K  < 
M  \/  K  e.  ( M ... N
)  \/  N  < 
K ) )
24 zdclt 9082 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  K  e.  ZZ )  -> DECID  N  <  K )
2524ancoms 266 . . . . . 6  |-  ( ( K  e.  ZZ  /\  N  e.  ZZ )  -> DECID  N  <  K )
26253adant2 983 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  N  <  K )
2726adantr 272 . . . 4  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  -> DECID  N  <  K )
28 df-dc 803 . . . 4  |-  (DECID  N  < 
K  <->  ( N  < 
K  \/  -.  N  <  K ) )
2927, 28sylib 121 . . 3  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  ( N  <  K  \/  -.  N  <  K ) )
304, 23, 29mpjaodan 770 . 2  |-  ( ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  /\  -.  K  <  M
)  ->  ( K  <  M  \/  K  e.  ( M ... N
)  \/  N  < 
K ) )
31 zdclt 9082 . . . 4  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ )  -> DECID  K  <  M )
32313adant3 984 . . 3  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  -> DECID  K  <  M )
33 df-dc 803 . . 3  |-  (DECID  K  < 
M  <->  ( K  < 
M  \/  -.  K  <  M ) )
3432, 33sylib 121 . 2  |-  ( ( K  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  <  M  \/  -.  K  <  M ) )
352, 30, 34mpjaodan 770 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 103    <-> wb 104    \/ wo 680  DECID wdc 802    \/ w3o 944    /\ w3a 945    e. wcel 1463   class class class wbr 3897  (class class class)co 5740   RRcr 7583    < clt 7764    <_ cle 7765   ZZcz 9008   ...cfz 9741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-ltadd 7700
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-inn 8681  df-n0 8932  df-z 9009  df-fz 9742
This theorem is referenced by:  fzdcel  9771  hashfiv01gt1  10479
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