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Mirrors > Home > ILE Home > Th. List > ledivmul | Unicode version |
Description: 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005.) |
Ref | Expression |
---|---|
ledivmul |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3l 1010 | . . . 4 | |
2 | simp2 983 | . . . 4 | |
3 | 1, 2 | remulcld 7909 | . . 3 |
4 | lediv1 8741 | . . 3 | |
5 | 3, 4 | syld3an2 1267 | . 2 |
6 | 2 | recnd 7907 | . . . 4 |
7 | 1 | recnd 7907 | . . . 4 |
8 | simp3r 1011 | . . . . 5 | |
9 | 1, 8 | gt0ap0d 8505 | . . . 4 # |
10 | 6, 7, 9 | divcanap3d 8669 | . . 3 |
11 | 10 | breq2d 3978 | . 2 |
12 | 5, 11 | bitr2d 188 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 963 wcel 2128 class class class wbr 3966 (class class class)co 5825 cr 7732 cc0 7733 cmul 7738 clt 7913 cle 7914 cdiv 8546 |
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 4083 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-mulrcl 7832 ax-addcom 7833 ax-mulcom 7834 ax-addass 7835 ax-mulass 7836 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-1rid 7840 ax-0id 7841 ax-rnegex 7842 ax-precex 7843 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-apti 7848 ax-pre-ltadd 7849 ax-pre-mulgt0 7850 ax-pre-mulext 7851 |
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 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-id 4254 df-po 4257 df-iso 4258 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-iota 5136 df-fun 5173 df-fv 5179 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-reap 8451 df-ap 8458 df-div 8547 |
This theorem is referenced by: ledivmul2 8752 ledivmuld 9658 divelunit 9907 faclbnd2 10620 eftlub 11591 eflegeo 11602 oddge22np1 11776 sinhalfpilem 13154 sincosq1lem 13188 |
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