Theorem fztri3or 9995

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

Step	Hyp	Ref	Expression
1		3mix1 1161	. . 3 |- ( K < M -> ( K < M \/ K e. ( M ... N ) \/ N < K ) )
2	1	adantl 275	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ K < M ) -> ( K < M \/ K e. ( M ... N ) \/ N < K ) )
3		3mix3 1163	. . . 4 |- ( N < K -> ( K < M \/ K e. ( M ... N ) \/ N < K ) )
4	3	adantl 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 996	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> M e. ZZ )
7	6	zred 9334	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> M e. RR )
8		simpl1 995	. . . . . . . . 9 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> K e. ZZ )
9	8	zred 9334	. . . . . . . 8 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> K e. RR )
10	7, 9	lenltd 8037	. . . . . . 7 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> ( M <_ K <-> -. K < M ) )
11	5, 10	mpbird 166	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> M <_ K )
12	11	adantr 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 )
14	9	adantr 274	. . . . . . 7 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ -. N < K ) -> K e. RR )
15		simpll3 1033	. . . . . . . 8 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ -. N < K ) -> N e. ZZ )
16	15	zred 9334	. . . . . . 7 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ -. N < K ) -> N e. RR )
17	14, 16	lenltd 8037	. . . . . 6 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ -. N < K ) -> ( K <_ N <-> -. N < K ) )
18	13, 17	mpbird 166	. . . . 5 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ -. N < K ) -> K <_ N )
19		elfz 9971	. . . . . . 7 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K e. ( M ... N ) <-> ( M <_ K /\ K <_ N ) ) )
20	19	adantr 274	. . . . . 6 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> ( K e. ( M ... N ) <-> ( M <_ K /\ K <_ N ) ) )
21	20	adantr 274	. . . . 5 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ -. N < K ) -> ( K e. ( M ... N ) <-> ( M <_ K /\ K <_ N ) ) )
22	12, 18, 21	mpbir2and 939	. . . 4 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ -. N < K ) -> K e. ( M ... N ) )
23	22	3mix2d 1168	. . 3 |- ( ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) /\ -. N < K ) -> ( K < M \/ K e. ( M ... N ) \/ N < K ) )
24		zdclt 9289	. . . . . . 7 |- ( ( N e. ZZ /\ K e. ZZ ) -> DECID N < K )
25	24	ancoms 266	. . . . . 6 |- ( ( K e. ZZ /\ N e. ZZ ) -> DECID N < K )
26	25	3adant2 1011	. . . . 5 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID N < K )
27	26	adantr 274	. . . 4 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> DECID N < K )
28		df-dc 830	. . . 4 |- (DECID N < K <-> ( N < K \/ -. N < K ) )
29	27, 28	sylib 121	. . 3 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> ( N < K \/ -. N < K ) )
30	4, 23, 29	mpjaodan 793	. 2 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. K < M ) -> ( K < M \/ K e. ( M ... N ) \/ N < K ) )
31		zdclt 9289	. . . 4 |- ( ( K e. ZZ /\ M e. ZZ ) -> DECID K < M )
32	31	3adant3 1012	. . 3 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID K < M )
33		df-dc 830	. . 3 |- (DECID K < M <-> ( K < M \/ -. K < M ) )
34	32, 33	sylib 121	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K < M \/ -. K < M ) )
35	2, 30, 34	mpjaodan 793	1 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( K < M \/ K e. ( M ... N ) \/ N < K ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703  DECID wdc 829   \/ w3o 972   /\ w3a 973   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   RRcr 7773   < clt 7954   <_ cle 7955   ZZcz 9212   ...cfz 9965

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-fz 9966

This theorem is referenced by:  fzdcel  9996  hashfiv01gt1  10716