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Mirrors > Home > ILE Home > Th. List > qtri3or | Unicode version |
Description: Rational trichotomy. (Contributed by Jim Kingdon, 6-Oct-2021.) |
Ref | Expression |
---|---|
qtri3or |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elq 9551 | . . . 4 | |
2 | 1 | biimpi 119 | . . 3 |
3 | 2 | adantl 275 | . 2 |
4 | elq 9551 | . . . . . . 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 9303 | . . . . . . . . . 10 |
11 | 7, 10 | zmulcld 9310 | . . . . . . . . 9 |
12 | simplrl 525 | . . . . . . . . . . 11 | |
13 | 12 | ad2antrr 480 | . . . . . . . . . 10 |
14 | simplrr 526 | . . . . . . . . . . 11 | |
15 | 14 | nnzd 9303 | . . . . . . . . . 10 |
16 | 13, 15 | zmulcld 9310 | . . . . . . . . 9 |
17 | ztri3or 9225 | . . . . . . . . 9 | |
18 | 11, 16, 17 | syl2anc 409 | . . . . . . . 8 |
19 | simpllr 524 | . . . . . . . . . . 11 | |
20 | 19 | breq2d 3988 | . . . . . . . . . 10 |
21 | breq1 3979 | . . . . . . . . . . 11 | |
22 | 21 | adantl 275 | . . . . . . . . . 10 |
23 | 7 | zred 9304 | . . . . . . . . . . 11 |
24 | 9 | nnrpd 9621 | . . . . . . . . . . 11 |
25 | 13 | zred 9304 | . . . . . . . . . . 11 |
26 | 14 | nnrpd 9621 | . . . . . . . . . . 11 |
27 | 23, 24, 25, 26 | lt2mul2divd 9692 | . . . . . . . . . 10 |
28 | 20, 22, 27 | 3bitr4rd 220 | . . . . . . . . 9 |
29 | simpr 109 | . . . . . . . . . . 11 | |
30 | 29, 19 | eqeq12d 2179 | . . . . . . . . . 10 |
31 | 7 | zcnd 9305 | . . . . . . . . . . 11 |
32 | 13 | zcnd 9305 | . . . . . . . . . . 11 |
33 | 14 | nncnd 8862 | . . . . . . . . . . . 12 |
34 | 14 | nnap0d 8894 | . . . . . . . . . . . 12 # |
35 | 33, 34 | jca 304 | . . . . . . . . . . 11 # |
36 | 9 | nncnd 8862 | . . . . . . . . . . . 12 |
37 | 9 | nnap0d 8894 | . . . . . . . . . . . 12 # |
38 | 36, 37 | jca 304 | . . . . . . . . . . 11 # |
39 | divmuleqap 8604 | . . . . . . . . . . 11 # # | |
40 | 31, 32, 35, 38, 39 | syl22anc 1228 | . . . . . . . . . 10 |
41 | 30, 40 | bitr2d 188 | . . . . . . . . 9 |
42 | 25, 26, 23, 24 | lt2mul2divd 9692 | . . . . . . . . . 10 |
43 | 19, 29 | breq12d 3989 | . . . . . . . . . 10 |
44 | 42, 43 | bitr4d 190 | . . . . . . . . 9 |
45 | 28, 41, 44 | 3orbi123d 1300 | . . . . . . . 8 |
46 | 18, 45 | mpbid 146 | . . . . . . 7 |
47 | 46 | ex 114 | . . . . . 6 |
48 | 47 | rexlimdvva 2589 | . . . . 5 |
49 | 6, 48 | mpd 13 | . . . 4 |
50 | 49 | ex 114 | . . 3 |
51 | 50 | rexlimdvva 2589 | . 2 |
52 | 3, 51 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3o 966 wceq 1342 wcel 2135 wrex 2443 class class class wbr 3976 (class class class)co 5836 cc 7742 cc0 7744 cmul 7749 clt 7924 # cap 8470 cdiv 8559 cn 8848 cz 9182 cq 9548 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-n0 9106 df-z 9183 df-q 9549 df-rp 9581 |
This theorem is referenced by: qletric 10169 qlelttric 10170 qltnle 10171 qdceq 10172 fimaxq 10729 |
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