Theorem ztri3or 9622

Description: Integer trichotomy. (Contributed by Jim Kingdon, 14-Mar-2020.)

Assertion

Ref	Expression
ztri3or	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N \/ M = N \/ N < M ) )

Proof of Theorem ztri3or

Step	Hyp	Ref	Expression
1		zsubcl 9620	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
2		ztri3or0 9621	. . 3 |- ( ( M - N ) e. ZZ -> ( ( M - N ) < 0 \/ ( M - N ) = 0 \/ 0 < ( M - N ) ) )
3	1, 2	syl 14	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M - N ) < 0 \/ ( M - N ) = 0 \/ 0 < ( M - N ) ) )
4		zre 9583	. . . . . 6 |- ( M e. ZZ -> M e. RR )
5	4	adantr 276	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> M e. RR )
6		zre 9583	. . . . . 6 |- ( N e. ZZ -> N e. RR )
7	6	adantl 277	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> N e. RR )
8	5, 7	posdifd 8808	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> 0 < ( N - M ) ) )
9	7, 5	resubcld 8656	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N - M ) e. RR )
10	9	lt0neg2d 8792	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( 0 < ( N - M ) <-> -u ( N - M ) < 0 ) )
11	7	recnd 8304	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> N e. CC )
12	5	recnd 8304	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> M e. CC )
13	11, 12	negsubdi2d 8602	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> -u ( N - M ) = ( M - N ) )
14	13	breq1d 4121	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -u ( N - M ) < 0 <-> ( M - N ) < 0 ) )
15	8, 10, 14	3bitrd 214	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( M - N ) < 0 ) )
16	12, 11	subeq0ad 8596	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M - N ) = 0 <-> M = N ) )
17	16	bicomd 141	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M = N <-> ( M - N ) = 0 ) )
18	7, 5	posdifd 8808	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N < M <-> 0 < ( M - N ) ) )
19	15, 17, 18	3orbi123d 1348	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M < N \/ M = N \/ N < M ) <-> ( ( M - N ) < 0 \/ ( M - N ) = 0 \/ 0 < ( M - N ) ) ) )
20	3, 19	mpbird 167	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N \/ M = N \/ N < M ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ w3o 1004   = wceq 1398   e. wcel 2205   class class class wbr 4111  (class class class)co 6052   RRcr 8128   0cc0 8129   < clt 8310   - cmin 8446   -ucneg 8447   ZZcz 9579

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-n0 9499  df-z 9580

This theorem is referenced by:  zletric  9623  zlelttric  9624  zltnle  9625  zleloe  9626  zapne  9654  zdceq  9655  zdcle  9656  zdclt  9657  uzm1  9888  qtri3or  10604  iseqf1olemkle  10863  iseqf1olemklt  10864  iswrdiz  11235  cvgratz  12222  divalglemeunn  12611  divalglemeuneg  12613  znege1  12879  lgsdilem  15917