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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 7922 | . . . 4 | |
2 | simprl 526 | . . . 4 | |
3 | simpll 524 | . . . 4 | |
4 | simplr 525 | . . . 4 | |
5 | ltmuldiv 8777 | . . . 4 | |
6 | 1, 2, 3, 4, 5 | syl112anc 1237 | . . 3 |
7 | 3 | recnd 7935 | . . . . 5 |
8 | 7 | mulid2d 7925 | . . . 4 |
9 | 8 | breq1d 3997 | . . 3 |
10 | 2 | recnd 7935 | . . . . 5 |
11 | 3, 4 | gt0ap0d 8535 | . . . . 5 # |
12 | 10, 7, 11 | divrecapd 8697 | . . . 4 |
13 | 12 | breq2d 3999 | . . 3 |
14 | 6, 9, 13 | 3bitr3d 217 | . 2 |
15 | 3, 11 | rerecclapd 8738 | . . 3 |
16 | simprr 527 | . . 3 | |
17 | ltdivmul 8779 | . . 3 | |
18 | 1, 15, 2, 16, 17 | syl112anc 1237 | . 2 |
19 | 14, 18 | bitr4d 190 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 2141 class class class wbr 3987 (class class class)co 5850 cr 7760 cc0 7761 c1 7762 cmul 7766 clt 7941 cdiv 8576 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-mulrcl 7860 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-precex 7871 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-apti 7876 ax-pre-ltadd 7877 ax-pre-mulgt0 7878 ax-pre-mulext 7879 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-po 4279 df-iso 4280 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-reap 8481 df-ap 8488 df-div 8577 |
This theorem is referenced by: lerec 8787 ltdiv2 8790 ltrec1 8791 reclt1 8799 recgt1 8800 ltreci 8815 nnrecl 9120 ltrecd 9659 |
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