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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 7691 |
. . 3
| |
| 2 | 1 | ex 113 |
. 2
|
| 3 | recexgt0 7680 |
. . . 4
  | |
| 4 | 3 | 3ad2ant3 961 |
. . 3
|
| 5 | simpl1 941 |
. . . . . . . . . 10
| |
| 6 | simpl3l 993 |
. . . . . . . . . 10
| |
| 7 | 5, 6 | remulcld 7149 |
. . . . . . . . 9
|
| 8 | simpl2 942 |
. . . . . . . . . 10
| |
| 9 | 8, 6 | remulcld 7149 |
. . . . . . . . 9
|
| 10 | simprl 497 |
. . . . . . . . . 10
| |
| 11 | simprrl 505 |
. . . . . . . . . 10
| |
| 12 | 10, 11 | jca 300 |
. . . . . . . . 9
|
| 13 | 7, 9, 12 | 3jca 1118 |
. . . . . . . 8
|
| 14 | ltmul1a 7691 |
. . . . . . . 8
| |
| 15 | 13, 14 | sylan 277 |
. . . . . . 7
|
| 16 | 5 | recnd 7147 |
. . . . . . . . 9
 |
| 17 | 16 | adantr 270 |
. . . . . . . 8
|
| 18 | 6 | recnd 7147 |
. . . . . . . . 9
|
| 19 | 18 | adantr 270 |
. . . . . . . 8
|
| 20 | 10 | recnd 7147 |
. . . . . . . . 9
|
| 21 | 20 | adantr 270 |
. . . . . . . 8
|
| 22 | 17, 19, 21 | mulassd 7142 |
. . . . . . 7
|
| 23 | 8 | recnd 7147 |
. . . . . . . . 9
|
| 24 | 23 | adantr 270 |
. . . . . . . 8
|
| 25 | 24, 19, 21 | mulassd 7142 |
. . . . . . 7
|
| 26 | 15, 22, 25 | 3brtr3d 3814 |
. . . . . 6
|
| 27 | simprrr 506 |
. . . . . . . 8
| |
| 28 | 27 | adantr 270 |
. . . . . . 7
|
| 29 | 28 | oveq2d 5548 |
. . . . . 6
|
| 30 | 28 | oveq2d 5548 |
. . . . . 6
|
| 31 | 26, 29, 30 | 3brtr3d 3814 |
. . . . 5
|
| 32 | 17 | mulid1d 7136 |
. . . . 5
|
| 33 | 24 | mulid1d 7136 |
. . . . 5
|
| 34 | 31, 32, 33 | 3brtr3d 3814 |
. . . 4
|
| 35 | 34 | ex 113 |
. . 3
|
| 36 | 4, 35 | rexlimddv 2481 |
. 2
|
| 37 | 2, 36 | impbid 127 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 102
wb 103 w3a 919 wceq 1284 wcel 1433 wrex 2349 class class class wbr 3785
(class class class)co 5532 cc 6979 cr 6980 cc0 6981 c1 6982 cmul 6986 clt 7153 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 |
| This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-ltxr 7158 df-sub 7281 df-neg 7282 |
| This theorem is referenced by: lemul1 7693 reapmul1lem 7694 ltmul2 7934 ltdiv1 7946 ltdiv23 7970 recp1lt1 7977 ltmul1i 7998 ltmul1d 8815 flodddiv4t2lthalf 10337 |
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