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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 8489 |
. . 3
| |
| 2 | 1 | ex 114 |
. 2
|
| 3 | recexgt0 8478 |
. . . 4
  | |
| 4 | 3 | 3ad2ant3 1010 |
. . 3
|
| 5 | simpl1 990 |
. . . . . . . . . 10
| |
| 6 | simpl3l 1042 |
. . . . . . . . . 10
| |
| 7 | 5, 6 | remulcld 7929 |
. . . . . . . . 9
|
| 8 | simpl2 991 |
. . . . . . . . . 10
| |
| 9 | 8, 6 | remulcld 7929 |
. . . . . . . . 9
|
| 10 | simprl 521 |
. . . . . . . . . 10
| |
| 11 | simprrl 529 |
. . . . . . . . . 10
| |
| 12 | 10, 11 | jca 304 |
. . . . . . . . 9
|
| 13 | 7, 9, 12 | 3jca 1167 |
. . . . . . . 8
|
| 14 | ltmul1a 8489 |
. . . . . . . 8
| |
| 15 | 13, 14 | sylan 281 |
. . . . . . 7
|
| 16 | 5 | recnd 7927 |
. . . . . . . . 9
 |
| 17 | 16 | adantr 274 |
. . . . . . . 8
|
| 18 | 6 | recnd 7927 |
. . . . . . . . 9
|
| 19 | 18 | adantr 274 |
. . . . . . . 8
|
| 20 | 10 | recnd 7927 |
. . . . . . . . 9
|
| 21 | 20 | adantr 274 |
. . . . . . . 8
|
| 22 | 17, 19, 21 | mulassd 7922 |
. . . . . . 7
|
| 23 | 8 | recnd 7927 |
. . . . . . . . 9
|
| 24 | 23 | adantr 274 |
. . . . . . . 8
|
| 25 | 24, 19, 21 | mulassd 7922 |
. . . . . . 7
|
| 26 | 15, 22, 25 | 3brtr3d 4013 |
. . . . . 6
|
| 27 | simprrr 530 |
. . . . . . . 8
| |
| 28 | 27 | adantr 274 |
. . . . . . 7
|
| 29 | 28 | oveq2d 5858 |
. . . . . 6
|
| 30 | 28 | oveq2d 5858 |
. . . . . 6
|
| 31 | 26, 29, 30 | 3brtr3d 4013 |
. . . . 5
|
| 32 | 17 | mulid1d 7916 |
. . . . 5
|
| 33 | 24 | mulid1d 7916 |
. . . . 5
|
| 34 | 31, 32, 33 | 3brtr3d 4013 |
. . . 4
|
| 35 | 34 | ex 114 |
. . 3
|
| 36 | 4, 35 | rexlimddv 2588 |
. 2
|
| 37 | 2, 36 | impbid 128 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 w3a 968 wceq 1343 wcel 2136 wrex 2445 class class class wbr 3982
(class class class)co 5842 cc 7751 cr 7752 cc0 7753 c1 7754 cmul 7758 clt 7933 |
| 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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 |
| This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-iota 5153 df-fun 5190 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-pnf 7935 df-mnf 7936 df-ltxr 7938 df-sub 8071 df-neg 8072 |
| This theorem is referenced by: lemul1 8491 reapmul1lem 8492 ltmul2 8751 ltdiv1 8763 ltdiv23 8787 recp1lt1 8794 ltmul1i 8815 ltmul1d 9674 mertenslemi1 11476 flodddiv4t2lthalf 11874 qnumgt0 12130 tangtx 13399 |
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