Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ledivdiv | Unicode version |
Description: Invert ratios of positive numbers and swap their ordering. (Contributed by NM, 9-Jan-2006.) |
Ref | Expression |
---|---|
ledivdiv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplll 507 | . . . 4 | |
2 | simplrl 509 | . . . 4 | |
3 | simplrr 510 | . . . . 5 | |
4 | 2, 3 | gt0ap0d 8359 | . . . 4 # |
5 | 1, 2, 4 | redivclapd 8562 | . . 3 |
6 | divgt0 8598 | . . . 4 | |
7 | 6 | adantr 274 | . . 3 |
8 | simprll 511 | . . . 4 | |
9 | simprrl 513 | . . . 4 | |
10 | simprrr 514 | . . . . 5 | |
11 | 9, 10 | gt0ap0d 8359 | . . . 4 # |
12 | 8, 9, 11 | redivclapd 8562 | . . 3 |
13 | divgt0 8598 | . . . 4 | |
14 | 13 | adantl 275 | . . 3 |
15 | lerec 8610 | . . 3 | |
16 | 5, 7, 12, 14, 15 | syl22anc 1202 | . 2 |
17 | 8 | recnd 7762 | . . . 4 |
18 | 9 | recnd 7762 | . . . 4 |
19 | simprlr 512 | . . . . 5 | |
20 | 8, 19 | gt0ap0d 8359 | . . . 4 # |
21 | 17, 18, 20, 11 | recdivapd 8535 | . . 3 |
22 | 1 | recnd 7762 | . . . 4 |
23 | 2 | recnd 7762 | . . . 4 |
24 | simpllr 508 | . . . . 5 | |
25 | 1, 24 | gt0ap0d 8359 | . . . 4 # |
26 | 22, 23, 25, 4 | recdivapd 8535 | . . 3 |
27 | 21, 26 | breq12d 3912 | . 2 |
28 | 16, 27 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 1465 class class class wbr 3899 (class class class)co 5742 cr 7587 cc0 7588 c1 7589 clt 7768 cle 7769 cdiv 8400 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 |
This theorem is referenced by: ledivdivd 9477 |
Copyright terms: Public domain | W3C validator |