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Mirrors > Home > ILE Home > Th. List > ltrec | Unicode version |
Description: The reciprocal of both sides of 'less than'. (Contributed by NM, 26-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
ltrec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1red 7784 | . . . 4 | |
2 | simprl 520 | . . . 4 | |
3 | simpll 518 | . . . 4 | |
4 | simplr 519 | . . . 4 | |
5 | ltmuldiv 8635 | . . . 4 | |
6 | 1, 2, 3, 4, 5 | syl112anc 1220 | . . 3 |
7 | 3 | recnd 7797 | . . . . 5 |
8 | 7 | mulid2d 7787 | . . . 4 |
9 | 8 | breq1d 3939 | . . 3 |
10 | 2 | recnd 7797 | . . . . 5 |
11 | 3, 4 | gt0ap0d 8394 | . . . . 5 # |
12 | 10, 7, 11 | divrecapd 8556 | . . . 4 |
13 | 12 | breq2d 3941 | . . 3 |
14 | 6, 9, 13 | 3bitr3d 217 | . 2 |
15 | 3, 11 | rerecclapd 8596 | . . 3 |
16 | simprr 521 | . . 3 | |
17 | ltdivmul 8637 | . . 3 | |
18 | 1, 15, 2, 16, 17 | syl112anc 1220 | . 2 |
19 | 14, 18 | bitr4d 190 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 1480 class class class wbr 3929 (class class class)co 5774 cr 7622 cc0 7623 c1 7624 cmul 7628 clt 7803 cdiv 8435 |
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-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-mulrcl 7722 ax-addcom 7723 ax-mulcom 7724 ax-addass 7725 ax-mulass 7726 ax-distr 7727 ax-i2m1 7728 ax-0lt1 7729 ax-1rid 7730 ax-0id 7731 ax-rnegex 7732 ax-precex 7733 ax-cnre 7734 ax-pre-ltirr 7735 ax-pre-ltwlin 7736 ax-pre-lttrn 7737 ax-pre-apti 7738 ax-pre-ltadd 7739 ax-pre-mulgt0 7740 ax-pre-mulext 7741 |
This theorem depends on definitions: df-bi 116 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-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-le 7809 df-sub 7938 df-neg 7939 df-reap 8340 df-ap 8347 df-div 8436 |
This theorem is referenced by: lerec 8645 ltdiv2 8648 ltrec1 8649 reclt1 8657 recgt1 8658 ltreci 8673 nnrecl 8978 ltrecd 9505 |
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