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Mirrors > Home > ILE Home > Th. List > ltrec1 | Unicode version |
Description: Reciprocal swap in a 'less than' relation. (Contributed by NM, 24-Feb-2005.) |
Ref | Expression |
---|---|
ltrec1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 519 | . . . 4 | |
2 | simplr 520 | . . . . 5 | |
3 | 1, 2 | gt0ap0d 8509 | . . . 4 # |
4 | 1, 3 | rerecclapd 8712 | . . 3 |
5 | recgt0 8727 | . . . 4 | |
6 | 5 | adantr 274 | . . 3 |
7 | simpr 109 | . . 3 | |
8 | ltrec 8760 | . . 3 | |
9 | 4, 6, 7, 8 | syl21anc 1219 | . 2 |
10 | 1 | recnd 7909 | . . . 4 |
11 | 10, 3 | recrecapd 8663 | . . 3 |
12 | 11 | breq2d 3979 | . 2 |
13 | 9, 12 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 2128 class class class wbr 3967 (class class class)co 5827 cr 7734 cc0 7735 c1 7736 clt 7915 cdiv 8550 |
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-sep 4085 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 ax-cnex 7826 ax-resscn 7827 ax-1cn 7828 ax-1re 7829 ax-icn 7830 ax-addcl 7831 ax-addrcl 7832 ax-mulcl 7833 ax-mulrcl 7834 ax-addcom 7835 ax-mulcom 7836 ax-addass 7837 ax-mulass 7838 ax-distr 7839 ax-i2m1 7840 ax-0lt1 7841 ax-1rid 7842 ax-0id 7843 ax-rnegex 7844 ax-precex 7845 ax-cnre 7846 ax-pre-ltirr 7847 ax-pre-ltwlin 7848 ax-pre-lttrn 7849 ax-pre-apti 7850 ax-pre-ltadd 7851 ax-pre-mulgt0 7852 ax-pre-mulext 7853 |
This theorem depends on definitions: df-bi 116 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-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4029 df-id 4256 df-po 4259 df-iso 4260 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-iota 5138 df-fun 5175 df-fv 5181 df-riota 5783 df-ov 5830 df-oprab 5831 df-mpo 5832 df-pnf 7917 df-mnf 7918 df-xr 7919 df-ltxr 7920 df-le 7921 df-sub 8053 df-neg 8054 df-reap 8455 df-ap 8462 df-div 8551 |
This theorem is referenced by: recreclt 8777 ltrec1d 9631 expnlbnd 10552 |
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