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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 7292 | . 2 | |
2 | enqer 7290 | . 2 | |
3 | df-nqqs 7280 | . 2 | |
4 | df-ltnqqs 7285 | . 2 | |
5 | enqeceq 7291 | . . . . 5 | |
6 | enqeceq 7291 | . . . . . 6 | |
7 | eqcom 2166 | . . . . . 6 | |
8 | 6, 7 | bitrdi 195 | . . . . 5 |
9 | 5, 8 | bi2anan9 596 | . . . 4 |
10 | oveq12 5845 | . . . . 5 | |
11 | simplll 523 | . . . . . . 7 | |
12 | simprlr 528 | . . . . . . 7 | |
13 | simplrr 526 | . . . . . . 7 | |
14 | mulcompig 7263 | . . . . . . . 8 | |
15 | 14 | adantl 275 | . . . . . . 7 |
16 | mulasspig 7264 | . . . . . . . 8 | |
17 | 16 | adantl 275 | . . . . . . 7 |
18 | simprrl 529 | . . . . . . 7 | |
19 | mulclpi 7260 | . . . . . . . 8 | |
20 | 19 | adantl 275 | . . . . . . 7 |
21 | 11, 12, 13, 15, 17, 18, 20 | caov4d 6017 | . . . . . 6 |
22 | simpllr 524 | . . . . . . 7 | |
23 | simprll 527 | . . . . . . 7 | |
24 | simplrl 525 | . . . . . . 7 | |
25 | simprrr 530 | . . . . . . 7 | |
26 | 22, 23, 24, 15, 17, 25, 20 | caov4d 6017 | . . . . . 6 |
27 | 21, 26 | eqeq12d 2179 | . . . . 5 |
28 | 10, 27 | syl5ibr 155 | . . . 4 |
29 | 9, 28 | sylbid 149 | . . 3 |
30 | ltmpig 7271 | . . . . 5 | |
31 | 30 | adantl 275 | . . . 4 |
32 | 20, 11, 12 | caovcld 5986 | . . . 4 |
33 | 20, 13, 18 | caovcld 5986 | . . . 4 |
34 | 20, 22, 23 | caovcld 5986 | . . . 4 |
35 | 20, 24, 25 | caovcld 5986 | . . . 4 |
36 | 31, 32, 33, 34, 15, 35 | caovord3d 6003 | . . 3 |
37 | 29, 36 | syld 45 | . 2 |
38 | 1, 2, 3, 4, 37 | brecop 6582 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wcel 2135 cop 3573 class class class wbr 3976 (class class class)co 5836 cec 6490 cnpi 7204 cmi 7206 clti 7207 ceq 7211 cnq 7212 cltq 7217 |
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-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 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-ral 2447 df-rex 2448 df-reu 2449 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-nul 3405 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-tr 4075 df-eprel 4261 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 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-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-mi 7238 df-lti 7239 df-enq 7279 df-nqqs 7280 df-ltnqqs 7285 |
This theorem is referenced by: nqtri3or 7328 ltdcnq 7329 ltsonq 7330 ltanqg 7332 ltmnqg 7333 1lt2nq 7338 ltexnqq 7340 archnqq 7349 prarloclemarch2 7351 ltnnnq 7355 prarloclemlt 7425 |
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