qtri3or - Intuitionistic Logic Explorer

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Theorem qtri3or 10499

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 9855	. . . 4
2	1	biimpi 120	. . 3
3	2	adantl 277	. 2 $/\$
4		elq 9855	. . . . . . 7
5	4	biimpi 120	. . . . . 6
6	5	ad3antrrr 492	. . . . 5 $/\$ $/\$ $/\$ $/\$
7		simplrl 537	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simplrr 538	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
9	8	ad2antrr 488	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10	9	nnzd 9600	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
11	7, 10	zmulcld 9607	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
12		simplrl 537	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
13	12	ad2antrr 488	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
14		simplrr 538	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15	14	nnzd 9600	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16	13, 15	zmulcld 9607	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
17		ztri3or 9521	. . . . . . . . 9 $/\$ $\/$ $\/$
18	11, 16, 17	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
19		simpllr 536	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20	19	breq2d 4100	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
21		breq1 4091	. . . . . . . . . . 11
22	21	adantl 277	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23	7	zred 9601	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24	9	nnrpd 9928	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25	13	zred 9601	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	14	nnrpd 9928	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27	23, 24, 25, 26	lt2mul2divd 9999	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28	20, 22, 27	3bitr4rd 221	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29		simpr 110	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	29, 19	eqeq12d 2246	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31	7	zcnd 9602	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	13	zcnd 9602	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	14	nncnd 9156	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	14	nnap0d 9188	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
35	33, 34	jca 306	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
36	9	nncnd 9156	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	9	nnap0d 9188	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
38	36, 37	jca 306	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
39		divmuleqap 8896	. . . . . . . . . . 11 $/\$ $/\$ $/\$ # $/\$ $/\$ #
40	31, 32, 35, 38, 39	syl22anc 1274	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	30, 40	bitr2d 189	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42	25, 26, 23, 24	lt2mul2divd 9999	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	19, 29	breq12d 4101	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44	42, 43	bitr4d 191	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45	28, 41, 44	3orbi123d 1347	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$ $\/$ $\/$
46	18, 45	mpbid 147	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
47	46	ex 115	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
48	47	rexlimdvva 2658	. . . . 5 $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
49	6, 48	mpd 13	. . . 4 $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
50	49	ex 115	. . 3 $/\$ $/\$ $/\$ $\/$ $\/$
51	50	rexlimdvva 2658	. 2 $/\$ $\/$ $\/$
52	3, 51	mpd 13	1 $/\$ $\/$ $\/$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ w3o 1003

wceq 1397

wcel 2202

wrex 2511 class class class wbr 4088 (class class class)co 6017

cc 8029

cc0 8031

cmul 8036

clt 8213 # cap 8760

cdiv 8851

cn 9142

cz 9478

cq 9852

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-mulrcl 8130 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-precex 8141 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-apti 8146 ax-pre-ltadd 8147 ax-pre-mulgt0 8148 ax-pre-mulext 8149

This theorem depends on definitions: df-bi 117 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-id 4390 df-po 4393 df-iso 4394 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-fv 5334 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-sub 8351 df-neg 8352 df-reap 8754 df-ap 8761 df-div 8852 df-inn 9143 df-n0 9402 df-z 9479 df-q 9853 df-rp 9888

This theorem is referenced by: qletric 10500 qlelttric 10501 qltnle 10502 qdceq 10503 qdclt 10504 fimaxq 11090

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