qtri3or - Intuitionistic Logic Explorer

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Theorem qtri3or 9861

Description: Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.)

**Assertion**
Ref	Expression
qtri3or	$/\$ $\/$ $\/$

Proof of Theorem qtri3or Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elq 9264	. . . 4
2	1	biimpi 119	. . 3
3	2	adantl 273	. 2 $/\$
4		elq 9264	. . . . . . 7
5	4	biimpi 119	. . . . . 6
6	5	ad3antrrr 479	. . . . 5 $/\$ $/\$ $/\$ $/\$
7		simplrl 505	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simplrr 506	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
9	8	ad2antrr 475	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10	9	nnzd 9024	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
11	7, 10	zmulcld 9031	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
12		simplrl 505	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
13	12	ad2antrr 475	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14		simplrr 506	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15	14	nnzd 9024	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16	13, 15	zmulcld 9031	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
17		ztri3or 8949	. . . . . . . . 9 $/\$ $\/$ $\/$
18	11, 16, 17	syl2anc 406	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
19		simpllr 504	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20	19	breq2d 3887	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21		breq1 3878	. . . . . . . . . . 11
22	21	adantl 273	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23	7	zred 9025	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24	9	nnrpd 9329	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25	13	zred 9025	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	14	nnrpd 9329	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27	23, 24, 25, 26	lt2mul2divd 9393	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	20, 22, 27	3bitr4rd 220	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29		simpr 109	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	29, 19	eqeq12d 2114	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31	7	zcnd 9026	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	13	zcnd 9026	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	14	nncnd 8592	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	14	nnap0d 8624	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
35	33, 34	jca 302	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
36	9	nncnd 8592	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	9	nnap0d 8624	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
38	36, 37	jca 302	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
39		divmuleqap 8338	. . . . . . . . . . 11 $/\$ $/\$ $/\$ # $/\$ $/\$ #
40	31, 32, 35, 38, 39	syl22anc 1185	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	30, 40	bitr2d 188	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	25, 26, 23, 24	lt2mul2divd 9393	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	19, 29	breq12d 3888	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44	42, 43	bitr4d 190	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45	28, 41, 44	3orbi123d 1257	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
46	18, 45	mpbid 146	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
47	46	ex 114	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
48	47	rexlimdvva 2516	. . . . 5 $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
49	6, 48	mpd 13	. . . 4 $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
50	49	ex 114	. . 3 $/\$ $/\$ $/\$ $\/$ $\/$
51	50	rexlimdvva 2516	. 2 $/\$ $\/$ $\/$
52	3, 51	mpd 13	1 $/\$ $\/$ $\/$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $\/$ w3o 929

wceq 1299

wcel 1448

wrex 2376 class class class wbr 3875 (class class class)co 5706

cc 7498

cc0 7500

cmul 7505

clt 7672 # cap 8209

cdiv 8293

cn 8578

cz 8906

cq 9261

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulrcl 7594 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-precex 7605 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-apti 7610 ax-pre-ltadd 7611 ax-pre-mulgt0 7612 ax-pre-mulext 7613

This theorem depends on definitions: df-bi 116 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rmo 2383 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-po 4156 df-iso 4157 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-reap 8203 df-ap 8210 df-div 8294 df-inn 8579 df-n0 8830 df-z 8907 df-q 9262 df-rp 9292

This theorem is referenced by: qletric 9862 qlelttric 9863 qltnle 9864 qdceq 9865 fimaxq 10414

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