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Mirrors > Home > ILE Home > Th. List > lerec2 | Unicode version |
Description: Reciprocal swap in a 'less than or equal to' relation. (Contributed by NM, 24-Feb-2005.) |
Ref | Expression |
---|---|
lerec2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . 3 | |
2 | simprl 520 | . . . 4 | |
3 | simprr 521 | . . . . 5 | |
4 | 2, 3 | gt0ap0d 8398 | . . . 4 # |
5 | 2, 4 | rerecclapd 8600 | . . 3 |
6 | recgt0 8615 | . . . 4 | |
7 | 6 | adantl 275 | . . 3 |
8 | lerec 8649 | . . 3 | |
9 | 1, 5, 7, 8 | syl12anc 1214 | . 2 |
10 | 2 | recnd 7801 | . . . 4 |
11 | 10, 4 | recrecapd 8552 | . . 3 |
12 | 11 | breq1d 3939 | . 2 |
13 | 9, 12 | bitrd 187 | 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 7626 cc0 7627 c1 7628 clt 7807 cle 7808 cdiv 8439 |
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 7718 ax-resscn 7719 ax-1cn 7720 ax-1re 7721 ax-icn 7722 ax-addcl 7723 ax-addrcl 7724 ax-mulcl 7725 ax-mulrcl 7726 ax-addcom 7727 ax-mulcom 7728 ax-addass 7729 ax-mulass 7730 ax-distr 7731 ax-i2m1 7732 ax-0lt1 7733 ax-1rid 7734 ax-0id 7735 ax-rnegex 7736 ax-precex 7737 ax-cnre 7738 ax-pre-ltirr 7739 ax-pre-ltwlin 7740 ax-pre-lttrn 7741 ax-pre-apti 7742 ax-pre-ltadd 7743 ax-pre-mulgt0 7744 ax-pre-mulext 7745 |
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 7809 df-mnf 7810 df-xr 7811 df-ltxr 7812 df-le 7813 df-sub 7942 df-neg 7943 df-reap 8344 df-ap 8351 df-div 8440 |
This theorem is referenced by: lerec2d 9512 |
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