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Mirrors > Home > ILE Home > Th. List > lemul1 | Unicode version |
Description: Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 21-Feb-2005.) |
Ref | Expression |
---|---|
lemul1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltmul1 8354 | . . . 4 | |
2 | 1 | notbid 656 | . . 3 |
3 | 2 | 3com12 1185 | . 2 |
4 | lenlt 7840 | . . 3 | |
5 | 4 | 3adant3 1001 | . 2 |
6 | simp1 981 | . . . 4 | |
7 | simp3l 1009 | . . . 4 | |
8 | 6, 7 | remulcld 7796 | . . 3 |
9 | simp2 982 | . . . 4 | |
10 | 9, 7 | remulcld 7796 | . . 3 |
11 | 8, 10 | lenltd 7880 | . 2 |
12 | 3, 5, 11 | 3bitr4d 219 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3a 962 wcel 1480 class class class wbr 3929 (class class class)co 5774 cr 7619 cc0 7620 cmul 7625 clt 7800 cle 7801 |
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 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 |
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-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-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 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 |
This theorem is referenced by: lemul2 8615 lediv23 8651 lemul1i 8682 div4p1lem1div2 8973 lemul1d 9527 iccdil 9781 expgt1 10331 facubnd 10491 eirraplem 11483 |
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