qtri3or - Intuitionistic Logic Explorer

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Theorem qtri3or 10455

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 9813	. . . 4
2	1	biimpi 120	. . 3
3	2	adantl 277	. 2 $/\$
4		elq 9813	. . . . . . 7
5	4	biimpi 120	. . . . . 6
6	5	ad3antrrr 492	. . . . 5 $/\$ $/\$ $/\$ $/\$
7		simplrl 535	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simplrr 536	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
9	8	ad2antrr 488	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10	9	nnzd 9564	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
11	7, 10	zmulcld 9571	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
12		simplrl 535	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
13	12	ad2antrr 488	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14		simplrr 536	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15	14	nnzd 9564	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16	13, 15	zmulcld 9571	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
17		ztri3or 9485	. . . . . . . . 9 $/\$ $\/$ $\/$
18	11, 16, 17	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
19		simpllr 534	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20	19	breq2d 4094	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21		breq1 4085	. . . . . . . . . . 11
22	21	adantl 277	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23	7	zred 9565	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24	9	nnrpd 9886	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25	13	zred 9565	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	14	nnrpd 9886	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27	23, 24, 25, 26	lt2mul2divd 9957	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	20, 22, 27	3bitr4rd 221	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29		simpr 110	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	29, 19	eqeq12d 2244	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31	7	zcnd 9566	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	13	zcnd 9566	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	14	nncnd 9120	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	14	nnap0d 9152	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
35	33, 34	jca 306	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
36	9	nncnd 9120	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	9	nnap0d 9152	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
38	36, 37	jca 306	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
39		divmuleqap 8860	. . . . . . . . . . 11 $/\$ $/\$ $/\$ # $/\$ $/\$ #
40	31, 32, 35, 38, 39	syl22anc 1272	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	30, 40	bitr2d 189	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	25, 26, 23, 24	lt2mul2divd 9957	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	19, 29	breq12d 4095	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44	42, 43	bitr4d 191	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45	28, 41, 44	3orbi123d 1345	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
46	18, 45	mpbid 147	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
47	46	ex 115	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
48	47	rexlimdvva 2656	. . . . 5 $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
49	6, 48	mpd 13	. . . 4 $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
50	49	ex 115	. . 3 $/\$ $/\$ $/\$ $\/$ $\/$
51	50	rexlimdvva 2656	. 2 $/\$ $\/$ $\/$
52	3, 51	mpd 13	1 $/\$ $\/$ $\/$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ w3o 1001

wceq 1395

wcel 2200

wrex 2509 class class class wbr 4082 (class class class)co 6000

cc 7993

cc0 7995

cmul 8000

clt 8177 # cap 8724

cdiv 8815

cn 9106

cz 9442

cq 9810

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4201 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-cnex 8086 ax-resscn 8087 ax-1cn 8088 ax-1re 8089 ax-icn 8090 ax-addcl 8091 ax-addrcl 8092 ax-mulcl 8093 ax-mulrcl 8094 ax-addcom 8095 ax-mulcom 8096 ax-addass 8097 ax-mulass 8098 ax-distr 8099 ax-i2m1 8100 ax-0lt1 8101 ax-1rid 8102 ax-0id 8103 ax-rnegex 8104 ax-precex 8105 ax-cnre 8106 ax-pre-ltirr 8107 ax-pre-ltwlin 8108 ax-pre-lttrn 8109 ax-pre-apti 8110 ax-pre-ltadd 8111 ax-pre-mulgt0 8112 ax-pre-mulext 8113

This theorem depends on definitions: df-bi 117 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-id 4383 df-po 4386 df-iso 4387 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-fv 5325 df-riota 5953 df-ov 6003 df-oprab 6004 df-mpo 6005 df-1st 6284 df-2nd 6285 df-pnf 8179 df-mnf 8180 df-xr 8181 df-ltxr 8182 df-le 8183 df-sub 8315 df-neg 8316 df-reap 8718 df-ap 8725 df-div 8816 df-inn 9107 df-n0 9366 df-z 9443 df-q 9811 df-rp 9846

This theorem is referenced by: qletric 10456 qlelttric 10457 qltnle 10458 qdceq 10459 qdclt 10460 fimaxq 11044

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