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| Mirrors > Home > ILE Home > Th. List > ltmul1 | Unicode version | ||
| Description: Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
| Ref | Expression |
|---|---|
| ltmul1 |
     ![/\](wedge.gif "/\"){kind=small} 
      |
is distinct from all other
variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ltmul1a 7582 |
. . 3
| |
| 2 | 1 | ex 108 |
. 2
|
| 3 | recexgt0 7571 |
. . . 4
  | |
| 4 | 3 | 3ad2ant3 927 |
. . 3
|
| 5 | simpl1 907 |
. . . . . . . . . 10
| |
| 6 | simpl3l 959 |
. . . . . . . . . 10
| |
| 7 | 5, 6 | remulcld 7056 |
. . . . . . . . 9
|
| 8 | simpl2 908 |
. . . . . . . . . 10
| |
| 9 | 8, 6 | remulcld 7056 |
. . . . . . . . 9
|
| 10 | simprl 483 |
. . . . . . . . . 10
| |
| 11 | simprrl 491 |
. . . . . . . . . 10
| |
| 12 | 10, 11 | jca 290 |
. . . . . . . . 9
|
| 13 | 7, 9, 12 | 3jca 1084 |
. . . . . . . 8
|
| 14 | ltmul1a 7582 |
. . . . . . . 8
| |
| 15 | 13, 14 | sylan 267 |
. . . . . . 7
|
| 16 | 5 | recnd 7054 |
. . . . . . . . 9
 |
| 17 | 16 | adantr 261 |
. . . . . . . 8
|
| 18 | 6 | recnd 7054 |
. . . . . . . . 9
|
| 19 | 18 | adantr 261 |
. . . . . . . 8
|
| 20 | 10 | recnd 7054 |
. . . . . . . . 9
|
| 21 | 20 | adantr 261 |
. . . . . . . 8
|
| 22 | 17, 19, 21 | mulassd 7050 |
. . . . . . 7
|
| 23 | 8 | recnd 7054 |
. . . . . . . . 9
|
| 24 | 23 | adantr 261 |
. . . . . . . 8
|
| 25 | 24, 19, 21 | mulassd 7050 |
. . . . . . 7
|
| 26 | 15, 22, 25 | 3brtr3d 3793 |
. . . . . 6
|
| 27 | simprrr 492 |
. . . . . . . 8
| |
| 28 | 27 | adantr 261 |
. . . . . . 7
|
| 29 | 28 | oveq2d 5528 |
. . . . . 6
|
| 30 | 28 | oveq2d 5528 |
. . . . . 6
|
| 31 | 26, 29, 30 | 3brtr3d 3793 |
. . . . 5
|
| 32 | 17 | mulid1d 7044 |
. . . . 5
|
| 33 | 24 | mulid1d 7044 |
. . . . 5
|
| 34 | 31, 32, 33 | 3brtr3d 3793 |
. . . 4
|
| 35 | 34 | ex 108 |
. . 3
|
| 36 | 4, 35 | rexlimddv 2437 |
. 2
|
| 37 | 2, 36 | impbid 120 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 97
wb 98 w3a 885 wceq 1243 wcel 1393 wrex 2307 class class class wbr 3764
(class class class)co 5512 cc 6887 cr 6888 cc0 6889 c1 6890 cmul 6894 clt 7060 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulrcl 6983 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-mulass 6987 ax-distr 6988 ax-i2m1 6989 ax-1rid 6991 ax-0id 6992 ax-rnegex 6993 ax-precex 6994 ax-cnre 6995 ax-pre-ltadd 7000 ax-pre-mulgt0 7001 |
| This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-pnf 7062 df-mnf 7063 df-ltxr 7065 df-sub 7184 df-neg 7185 |
| This theorem is referenced by: lemul1 7584 reapmul1lem 7585 ltmul2 7822 ltdiv1 7834 ltdiv23 7858 recp1lt1 7865 ltmul1i 7886 ltmul1d 8664 |
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