Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ordpipqqs | Unicode version |
Description: Ordering of positive fractions in terms of positive integers. (Contributed by Jim Kingdon, 14-Sep-2019.) |
Ref | Expression |
---|---|
ordpipqqs |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enqex 7168 | . 2 | |
2 | enqer 7166 | . 2 | |
3 | df-nqqs 7156 | . 2 | |
4 | df-ltnqqs 7161 | . 2 | |
5 | enqeceq 7167 | . . . . 5 | |
6 | enqeceq 7167 | . . . . . 6 | |
7 | eqcom 2141 | . . . . . 6 | |
8 | 6, 7 | syl6bb 195 | . . . . 5 |
9 | 5, 8 | bi2anan9 595 | . . . 4 |
10 | oveq12 5783 | . . . . 5 | |
11 | simplll 522 | . . . . . . 7 | |
12 | simprlr 527 | . . . . . . 7 | |
13 | simplrr 525 | . . . . . . 7 | |
14 | mulcompig 7139 | . . . . . . . 8 | |
15 | 14 | adantl 275 | . . . . . . 7 |
16 | mulasspig 7140 | . . . . . . . 8 | |
17 | 16 | adantl 275 | . . . . . . 7 |
18 | simprrl 528 | . . . . . . 7 | |
19 | mulclpi 7136 | . . . . . . . 8 | |
20 | 19 | adantl 275 | . . . . . . 7 |
21 | 11, 12, 13, 15, 17, 18, 20 | caov4d 5955 | . . . . . 6 |
22 | simpllr 523 | . . . . . . 7 | |
23 | simprll 526 | . . . . . . 7 | |
24 | simplrl 524 | . . . . . . 7 | |
25 | simprrr 529 | . . . . . . 7 | |
26 | 22, 23, 24, 15, 17, 25, 20 | caov4d 5955 | . . . . . 6 |
27 | 21, 26 | eqeq12d 2154 | . . . . 5 |
28 | 10, 27 | syl5ibr 155 | . . . 4 |
29 | 9, 28 | sylbid 149 | . . 3 |
30 | ltmpig 7147 | . . . . 5 | |
31 | 30 | adantl 275 | . . . 4 |
32 | 20, 11, 12 | caovcld 5924 | . . . 4 |
33 | 20, 13, 18 | caovcld 5924 | . . . 4 |
34 | 20, 22, 23 | caovcld 5924 | . . . 4 |
35 | 20, 24, 25 | caovcld 5924 | . . . 4 |
36 | 31, 32, 33, 34, 15, 35 | caovord3d 5941 | . . 3 |
37 | 29, 36 | syld 45 | . 2 |
38 | 1, 2, 3, 4, 37 | brecop 6519 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 cop 3530 class class class wbr 3929 (class class class)co 5774 cec 6427 cnpi 7080 cmi 7082 clti 7083 ceq 7087 cnq 7088 cltq 7093 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-mi 7114 df-lti 7115 df-enq 7155 df-nqqs 7156 df-ltnqqs 7161 |
This theorem is referenced by: nqtri3or 7204 ltdcnq 7205 ltsonq 7206 ltanqg 7208 ltmnqg 7209 1lt2nq 7214 ltexnqq 7216 archnqq 7225 prarloclemarch2 7227 ltnnnq 7231 prarloclemlt 7301 |
Copyright terms: Public domain | W3C validator |