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Mirrors > Home > ILE Home > Th. List > ltdiv2 | Unicode version |
Description: Division of a positive number by both sides of 'less than'. (Contributed by NM, 27-Apr-2005.) |
Ref | Expression |
---|---|
ltdiv2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2l 1012 | . . . 4 | |
2 | gt0ap0 8515 | . . . . 5 # | |
3 | 2 | 3ad2ant2 1008 | . . . 4 # |
4 | 1, 3 | rerecclapd 8721 | . . 3 |
5 | gt0ap0 8515 | . . . . 5 # | |
6 | rerecclap 8617 | . . . . 5 # | |
7 | 5, 6 | syldan 280 | . . . 4 |
8 | 7 | 3ad2ant1 1007 | . . 3 |
9 | simp3 988 | . . 3 | |
10 | ltmul2 8742 | . . 3 | |
11 | 4, 8, 9, 10 | syl3anc 1227 | . 2 |
12 | ltrec 8769 | . . 3 | |
13 | 12 | 3adant3 1006 | . 2 |
14 | simp3l 1014 | . . . . 5 | |
15 | 14 | recnd 7918 | . . . 4 |
16 | 1 | recnd 7918 | . . . 4 |
17 | 15, 16, 3 | divrecapd 8680 | . . 3 |
18 | simp1l 1010 | . . . . 5 | |
19 | 18 | recnd 7918 | . . . 4 |
20 | 5 | 3ad2ant1 1007 | . . . 4 # |
21 | 15, 19, 20 | divrecapd 8680 | . . 3 |
22 | 17, 21 | breq12d 3989 | . 2 |
23 | 11, 13, 22 | 3bitr4d 219 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wcel 2135 class class class wbr 3976 (class class class)co 5836 cr 7743 cc0 7744 c1 7745 cmul 7749 clt 7924 # cap 8470 cdiv 8559 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 |
This theorem is referenced by: ltdiv2d 9647 sin01gt0 11688 coseq0negpitopi 13304 sincos6thpi 13310 cos02pilt1 13319 |
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