Theorem ztri3or 9698

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 9696	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M - N ) e. ZZ )
2		ztri3or0 9697	. . 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 9659	. . . . . 6 |- ( M e. ZZ -> M e. RR )
5	4	adantr 276	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> M e. RR )
6		zre 9659	. . . . . 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 8867	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> 0 < ( N - M ) ) )
9	7, 5	resubcld 8715	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N - M ) e. RR )
10	9	lt0neg2d 8851	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( 0 < ( N - M ) <-> -u ( N - M ) < 0 ) )
11	7	recnd 8360	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> N e. CC )
12	5	recnd 8360	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> M e. CC )
13	11, 12	negsubdi2d 8660	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> -u ( N - M ) = ( M - N ) )
14	13	breq1d 4141	. . . 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 8654	. . . 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 8867	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N < M <-> 0 < ( M - N ) ) )
19	15, 17, 18	3orbi123d 1352	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   \/ w3o 1008   = wceq 1402   e. wcel 2209   class class class wbr 4131  (class class class)co 6088   RRcr 8184   0cc0 8185   < clt 8366   - cmin 8504   -ucneg 8505   ZZcz 9655

This proof depends on 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 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-iota 5340  df-fun 5382  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-n0 9575  df-z 9656

This theorem is used by:  zletric  9699  zlelttric  9700  zltnle  9701  zleloe  9702  zapne  9730  zdceq  9731  zdcle  9732  zdclt  9733  uzm1  9969  qtri3or  10692  iseqf1olemkle  10955  iseqf1olemklt  10956  iswrdiz  11333  cvgratz  12324  divalglemeunn  12713  divalglemeuneg  12715  znege1  12983  lgsdilem  16351