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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 8484 | . . . 4 | |
2 | 1 | notbid 657 | . . 3 |
3 | 2 | 3com12 1196 | . 2 |
4 | lenlt 7968 | . . 3 | |
5 | 4 | 3adant3 1006 | . 2 |
6 | simp1 986 | . . . 4 | |
7 | simp3l 1014 | . . . 4 | |
8 | 6, 7 | remulcld 7923 | . . 3 |
9 | simp2 987 | . . . 4 | |
10 | 9, 7 | remulcld 7923 | . . 3 |
11 | 8, 10 | lenltd 8010 | . 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 967 wcel 2135 class class class wbr 3979 (class class class)co 5839 cr 7746 cc0 7747 cmul 7752 clt 7927 cle 7928 |
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 4097 ax-pow 4150 ax-pr 4184 ax-un 4408 ax-setind 4511 ax-cnex 7838 ax-resscn 7839 ax-1cn 7840 ax-1re 7841 ax-icn 7842 ax-addcl 7843 ax-addrcl 7844 ax-mulcl 7845 ax-mulrcl 7846 ax-addcom 7847 ax-mulcom 7848 ax-addass 7849 ax-mulass 7850 ax-distr 7851 ax-i2m1 7852 ax-1rid 7854 ax-0id 7855 ax-rnegex 7856 ax-precex 7857 ax-cnre 7858 ax-pre-ltadd 7863 ax-pre-mulgt0 7864 |
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-rab 2451 df-v 2726 df-sbc 2950 df-dif 3116 df-un 3118 df-in 3120 df-ss 3127 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-uni 3787 df-br 3980 df-opab 4041 df-id 4268 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-dm 4611 df-iota 5150 df-fun 5187 df-fv 5193 df-riota 5795 df-ov 5842 df-oprab 5843 df-mpo 5844 df-pnf 7929 df-mnf 7930 df-xr 7931 df-ltxr 7932 df-le 7933 df-sub 8065 df-neg 8066 |
This theorem is referenced by: lemul2 8746 lediv23 8782 lemul1i 8813 div4p1lem1div2 9104 lemul1d 9670 iccdil 9928 expgt1 10487 facubnd 10652 eirraplem 11711 |
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