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Theorem fztri3or 10422
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 1197 . . 3 |- ( K < M -> ( K < M \/ K e. ( M ... N ) \/ N < K ) )
21adantl 277 . 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ K < M ) -> ( K < M \/ K e. ( M ... N ) \/ N < K ) )
3 3mix3 1199 . . . 4 |- ( N < K -> ( K < M \/ K e. ( M ... N ) \/ N < K ) )
43adantl 277 . . 3 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ N < K ) -> ( K < M \/ K e. ( M ... N ) \/ N < K ) )
5 simpr 110 . . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> -. K < M )
6 simpl2 1032 . . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> M e. ZZ )
76zred 9747 . . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> M e. RR )
8 simpl1 1031 . . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> K e. ZZ )
98zred 9747 . . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> K e. RR )
107, 9lenltd 8434 . . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> ( M <_ K <-> -. K < M ) )
115, 10mpbird 167 . . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> M <_ K )
1211adantr 276 . . . . 5 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ -. N < K ) -> M <_ K )
13 simpr 110 . . . . . 6 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ -. N < K ) -> -. N < K )
149adantr 276 . . . . . . 7 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ -. N < K ) -> K e. RR )
15 simpll3 1069 . . . . . . . 8 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ -. N < K ) -> N e. ZZ )
1615zred 9747 . . . . . . 7 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ -. N < K ) -> N e. RR )
1714, 16lenltd 8434 . . . . . 6 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ -. N < K ) -> ( K <_ N <-> -. N < K ) )
1813, 17mpbird 167 . . . . 5 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ -. N < K ) -> K <_ N )
19 elfz 10396 . . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ( M ... N ) <-> ( M <_ K /\ K <_ N ) ) )
2019adantr 276 . . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> ( K e. ( M ... N ) <-> ( M <_ K /\ K <_ N ) ) )
2120adantr 276 . . . . 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 957 . . . 4 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ -. N < K ) -> K e. ( M ... N ) )
23223mix2d 1204 . . 3 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ -. N < K ) -> ( K < M \/ K e. ( M ... N ) \/ N < K ) )
24 zdclt 9701 . . . . . . 7 |- ( ( N e. ZZ /\ K e. ZZ ) -> DECID N < K )
2524ancoms 268 . . . . . 6 |- ( ( K e. ZZ /\ N e. ZZ ) -> DECID N < K )
26253adant2 1047 . . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID N < K )
2726adantr 276 . . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> DECID N < K )
28 df-dc 847 . . . 4 |- (DECID N < K <-> ( N < K \/ -. N < K ) )
2927, 28sylib 122 . . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> ( N < K \/ -. N < K ) )
304, 23, 29mpjaodan 810 . 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> ( K < M \/ K e. ( M ... N ) \/ N < K ) )
31 zdclt 9701 . . . 4 |- ( ( K e. ZZ /\ M e. ZZ ) -> DECID K < M )
32313adant3 1048 . . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID K < M )
33 df-dc 847 . . 3 |- (DECID K < M <-> ( K < M \/ -. K < M ) )
3432, 33sylib 122 . 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K < M \/ -. K < M ) )
352, 30, 34mpjaodan 810 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 104   <-> wb 105   \/ wo 720  DECID wdc 846   \/ w3o 1008   /\ w3a 1009   e. wcel 2209   class class class wbr 4125  (class class class)co 6075   RRcr 8168   < clt 8350   <_ cle 8351   ZZcz 9623   ...cfz 10390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285
This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-fz 10391
This theorem is referenced by:  fzdcel  10423  hashfiv01gt1  11199
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