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Mirrors > Home > ILE Home > Th. List > ltrnqg | Unicode version |
Description: Ordering property of reciprocal for positive fractions. For a simplified version of the forward implication, see ltrnqi 6519. (Contributed by Jim Kingdon, 29-Dec-2019.) |
Ref | Expression |
---|---|
ltrnqg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recclnq 6490 | . . . 4 | |
2 | recclnq 6490 | . . . 4 | |
3 | mulclnq 6474 | . . . 4 | |
4 | 1, 2, 3 | syl2an 273 | . . 3 |
5 | ltmnqg 6499 | . . 3 | |
6 | 4, 5 | mpd3an3 1233 | . 2 |
7 | simpl 102 | . . . . . 6 | |
8 | mulcomnqg 6481 | . . . . . 6 | |
9 | 4, 7, 8 | syl2anc 391 | . . . . 5 |
10 | 1 | adantr 261 | . . . . . 6 |
11 | 2 | adantl 262 | . . . . . 6 |
12 | mulassnqg 6482 | . . . . . 6 | |
13 | 7, 10, 11, 12 | syl3anc 1135 | . . . . 5 |
14 | mulclnq 6474 | . . . . . . 7 | |
15 | 7, 10, 14 | syl2anc 391 | . . . . . 6 |
16 | mulcomnqg 6481 | . . . . . 6 | |
17 | 15, 11, 16 | syl2anc 391 | . . . . 5 |
18 | 9, 13, 17 | 3eqtr2d 2078 | . . . 4 |
19 | recidnq 6491 | . . . . . 6 | |
20 | 19 | oveq2d 5528 | . . . . 5 |
21 | mulidnq 6487 | . . . . . 6 | |
22 | 2, 21 | syl 14 | . . . . 5 |
23 | 20, 22 | sylan9eq 2092 | . . . 4 |
24 | 18, 23 | eqtrd 2072 | . . 3 |
25 | simpr 103 | . . . . 5 | |
26 | mulassnqg 6482 | . . . . 5 | |
27 | 10, 11, 25, 26 | syl3anc 1135 | . . . 4 |
28 | mulcomnqg 6481 | . . . . . 6 | |
29 | 11, 25, 28 | syl2anc 391 | . . . . 5 |
30 | 29 | oveq2d 5528 | . . . 4 |
31 | recidnq 6491 | . . . . . 6 | |
32 | 31 | oveq2d 5528 | . . . . 5 |
33 | mulidnq 6487 | . . . . . 6 | |
34 | 1, 33 | syl 14 | . . . . 5 |
35 | 32, 34 | sylan9eqr 2094 | . . . 4 |
36 | 27, 30, 35 | 3eqtrd 2076 | . . 3 |
37 | 24, 36 | breq12d 3777 | . 2 |
38 | 6, 37 | bitrd 177 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 class class class wbr 3764 cfv 4902 (class class class)co 5512 cnq 6378 c1q 6379 cmq 6381 crq 6382 cltq 6383 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-mi 6404 df-lti 6405 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 |
This theorem is referenced by: ltrnqi 6519 recexprlemloc 6729 archrecnq 6761 |
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