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Mirrors > Home > ILE Home > Th. List > ltdcnq | Unicode version |
Description: Less-than for positive fractions is decidable. (Contributed by Jim Kingdon, 12-Dec-2019.) |
Ref | Expression |
---|---|
ltdcnq | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nqpi 7298 | . . . 4 | |
2 | nqpi 7298 | . . . 4 | |
3 | 1, 2 | anim12i 336 | . . 3 |
4 | ee4anv 1914 | . . 3 | |
5 | 3, 4 | sylibr 133 | . 2 |
6 | mulclpi 7248 | . . . . . . . . 9 | |
7 | mulclpi 7248 | . . . . . . . . 9 | |
8 | ltdcpi 7243 | . . . . . . . . 9 DECID | |
9 | 6, 7, 8 | syl2an 287 | . . . . . . . 8 DECID |
10 | 9 | an42s 579 | . . . . . . 7 DECID |
11 | ordpipqqs 7294 | . . . . . . . 8 | |
12 | 11 | dcbid 824 | . . . . . . 7 DECID DECID |
13 | 10, 12 | mpbird 166 | . . . . . 6 DECID |
14 | 13 | ad2ant2r 501 | . . . . 5 DECID |
15 | breq12 3970 | . . . . . . 7 | |
16 | 15 | ad2ant2l 500 | . . . . . 6 |
17 | 16 | dcbid 824 | . . . . 5 DECID DECID |
18 | 14, 17 | mpbird 166 | . . . 4 DECID |
19 | 18 | exlimivv 1876 | . . 3 DECID |
20 | 19 | exlimivv 1876 | . 2 DECID |
21 | 5, 20 | syl 14 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 DECID wdc 820 wceq 1335 wex 1472 wcel 2128 cop 3563 class class class wbr 3965 (class class class)co 5824 cec 6478 cnpi 7192 cmi 7194 clti 7195 ceq 7199 cnq 7200 cltq 7205 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-eprel 4249 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-irdg 6317 df-oadd 6367 df-omul 6368 df-er 6480 df-ec 6482 df-qs 6486 df-ni 7224 df-mi 7226 df-lti 7227 df-enq 7267 df-nqqs 7268 df-ltnqqs 7273 |
This theorem is referenced by: distrlem4prl 7504 distrlem4pru 7505 |
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