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Theorem fztri3or 9974
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 1156 . . 3 |- ( K < M -> ( K < M \/ K e. ( M ... N ) \/ N < K ) )
21adantl 275 . 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ K < M ) -> ( K < M \/ K e. ( M ... N ) \/ N < K ) )
3 3mix3 1158 . . . 4 |- ( N < K -> ( K < M \/ K e. ( M ... N ) \/ N < K ) )
43adantl 275 . . 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 991 . . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> M e. ZZ )
76zred 9313 . . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> M e. RR )
8 simpl1 990 . . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> K e. ZZ )
98zred 9313 . . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> K e. RR )
107, 9lenltd 8016 . . . . . . 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 274 . . . . 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 274 . . . . . . 7 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ -. N < K ) -> K e. RR )
15 simpll3 1028 . . . . . . . 8 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ -. N < K ) -> N e. ZZ )
1615zred 9313 . . . . . . 7 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ -. N < K ) -> N e. RR )
1714, 16lenltd 8016 . . . . . 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 9950 . . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ( M ... N ) <-> ( M <_ K /\ K <_ N ) ) )
2019adantr 274 . . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> ( K e. ( M ... N ) <-> ( M <_ K /\ K <_ N ) ) )
2120adantr 274 . . . . 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 934 . . . 4 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ -. N < K ) -> K e. ( M ... N ) )
23223mix2d 1163 . . 3 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ -. N < K ) -> ( K < M \/ K e. ( M ... N ) \/ N < K ) )
24 zdclt 9268 . . . . . . 7 |- ( ( N e. ZZ /\ K e. ZZ ) -> DECID N < K )
2524ancoms 266 . . . . . 6 |- ( ( K e. ZZ /\ N e. ZZ ) -> DECID N < K )
26253adant2 1006 . . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID N < K )
2726adantr 274 . . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> DECID N < K )
28 df-dc 825 . . . 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 788 . 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> ( K < M \/ K e. ( M ... N ) \/ N < K ) )
31 zdclt 9268 . . . 4 |- ( ( K e. ZZ /\ M e. ZZ ) -> DECID K < M )
32313adant3 1007 . . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID K < M )
33 df-dc 825 . . 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 788 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 698  DECID wdc 824   \/ w3o 967   /\ w3a 968   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   RRcr 7752   < clt 7933   <_ cle 7934   ZZcz 9191   ...cfz 9944
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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-fz 9945
This theorem is referenced by:  fzdcel  9975  hashfiv01gt1  10695
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