qtri3or - Intuitionistic Logic Explorer

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Theorem qtri3or 10178

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 9560	. . . 4
2	1	biimpi 119	. . 3
3	2	adantl 275	. 2 $/\$
4		elq 9560	. . . . . . 7
5	4	biimpi 119	. . . . . 6
6	5	ad3antrrr 484	. . . . 5 $/\$ $/\$ $/\$ $/\$
7		simplrl 525	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simplrr 526	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
9	8	ad2antrr 480	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10	9	nnzd 9312	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
11	7, 10	zmulcld 9319	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
12		simplrl 525	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
13	12	ad2antrr 480	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14		simplrr 526	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15	14	nnzd 9312	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16	13, 15	zmulcld 9319	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
17		ztri3or 9234	. . . . . . . . 9 $/\$ $\/$ $\/$
18	11, 16, 17	syl2anc 409	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
19		simpllr 524	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20	19	breq2d 3994	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21		breq1 3985	. . . . . . . . . . 11
22	21	adantl 275	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23	7	zred 9313	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24	9	nnrpd 9630	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25	13	zred 9313	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	14	nnrpd 9630	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27	23, 24, 25, 26	lt2mul2divd 9701	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	20, 22, 27	3bitr4rd 220	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29		simpr 109	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	29, 19	eqeq12d 2180	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31	7	zcnd 9314	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	13	zcnd 9314	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	14	nncnd 8871	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	14	nnap0d 8903	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
35	33, 34	jca 304	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
36	9	nncnd 8871	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	9	nnap0d 8903	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
38	36, 37	jca 304	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
39		divmuleqap 8613	. . . . . . . . . . 11 $/\$ $/\$ $/\$ # $/\$ $/\$ #
40	31, 32, 35, 38, 39	syl22anc 1229	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	30, 40	bitr2d 188	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	25, 26, 23, 24	lt2mul2divd 9701	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	19, 29	breq12d 3995	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44	42, 43	bitr4d 190	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45	28, 41, 44	3orbi123d 1301	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
46	18, 45	mpbid 146	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
47	46	ex 114	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
48	47	rexlimdvva 2591	. . . . 5 $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
49	6, 48	mpd 13	. . . 4 $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
50	49	ex 114	. . 3 $/\$ $/\$ $/\$ $\/$ $\/$
51	50	rexlimdvva 2591	. 2 $/\$ $\/$ $\/$
52	3, 51	mpd 13	1 $/\$ $\/$ $\/$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $\/$ w3o 967

wceq 1343

wcel 2136

wrex 2445 class class class wbr 3982 (class class class)co 5842

cc 7751

cc0 7753

cmul 7758

clt 7933 # cap 8479

cdiv 8568

cn 8857

cz 9191

cq 9557

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871

This theorem depends on definitions: df-bi 116 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-id 4271 df-po 4274 df-iso 4275 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-n0 9115 df-z 9192 df-q 9558 df-rp 9590

This theorem is referenced by: qletric 10179 qlelttric 10180 qltnle 10181 qdceq 10182 fimaxq 10740

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