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Mirrors > Home > ILE Home > Th. List > ltdivmul | Unicode version |
Description: 'Less than' relationship between division and multiplication. (Contributed by NM, 18-Nov-2004.) |
Ref | Expression |
---|---|
ltdivmul |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | remulcl 7938 |
. . . . . 6
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2 | 1 | ancoms 268 |
. . . . 5
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3 | 2 | adantrr 479 |
. . . 4
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4 | 3 | 3adant1 1015 |
. . 3
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5 | ltdiv1 8823 |
. . 3
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6 | 4, 5 | syld3an2 1285 |
. 2
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7 | recn 7943 |
. . . . . 6
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8 | 7 | adantr 276 |
. . . . 5
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9 | recn 7943 |
. . . . . 6
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10 | 9 | ad2antrl 490 |
. . . . 5
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11 | gt0ap0 8581 |
. . . . . 6
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12 | 11 | adantl 277 |
. . . . 5
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13 | 8, 10, 12 | divcanap3d 8750 |
. . . 4
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14 | 13 | 3adant1 1015 |
. . 3
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15 | 14 | breq2d 4015 |
. 2
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16 | 6, 15 | bitr2d 189 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-po 4296 df-iso 4297 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537 df-div 8628 |
This theorem is referenced by: ltdivmul2 8833 lt2mul2div 8834 ltrec 8838 avglt2 9156 3halfnz 9348 ltdivmuld 9746 modqid 10346 expnbnd 10640 mertenslemi1 11538 eirraplem 11779 fldivp1 12340 pcfaclem 12341 dveflem 14118 coseq0negpitopi 14188 tangtx 14190 cosordlem 14201 cos02pilt1 14203 2sqlem8 14390 ex-fl 14397 |
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