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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 8271 |
. . 3
| |
| 2 | 1 | ex 114 |
. 2
|
| 3 | recexgt0 8260 |
. . . 4
  | |
| 4 | 3 | 3ad2ant3 987 |
. . 3
|
| 5 | simpl1 967 |
. . . . . . . . . 10
| |
| 6 | simpl3l 1019 |
. . . . . . . . . 10
| |
| 7 | 5, 6 | remulcld 7720 |
. . . . . . . . 9
|
| 8 | simpl2 968 |
. . . . . . . . . 10
| |
| 9 | 8, 6 | remulcld 7720 |
. . . . . . . . 9
|
| 10 | simprl 503 |
. . . . . . . . . 10
| |
| 11 | simprrl 511 |
. . . . . . . . . 10
| |
| 12 | 10, 11 | jca 302 |
. . . . . . . . 9
|
| 13 | 7, 9, 12 | 3jca 1144 |
. . . . . . . 8
|
| 14 | ltmul1a 8271 |
. . . . . . . 8
| |
| 15 | 13, 14 | sylan 279 |
. . . . . . 7
|
| 16 | 5 | recnd 7718 |
. . . . . . . . 9
 |
| 17 | 16 | adantr 272 |
. . . . . . . 8
|
| 18 | 6 | recnd 7718 |
. . . . . . . . 9
|
| 19 | 18 | adantr 272 |
. . . . . . . 8
|
| 20 | 10 | recnd 7718 |
. . . . . . . . 9
|
| 21 | 20 | adantr 272 |
. . . . . . . 8
|
| 22 | 17, 19, 21 | mulassd 7713 |
. . . . . . 7
|
| 23 | 8 | recnd 7718 |
. . . . . . . . 9
|
| 24 | 23 | adantr 272 |
. . . . . . . 8
|
| 25 | 24, 19, 21 | mulassd 7713 |
. . . . . . 7
|
| 26 | 15, 22, 25 | 3brtr3d 3924 |
. . . . . 6
|
| 27 | simprrr 512 |
. . . . . . . 8
| |
| 28 | 27 | adantr 272 |
. . . . . . 7
|
| 29 | 28 | oveq2d 5744 |
. . . . . 6
|
| 30 | 28 | oveq2d 5744 |
. . . . . 6
|
| 31 | 26, 29, 30 | 3brtr3d 3924 |
. . . . 5
|
| 32 | 17 | mulid1d 7707 |
. . . . 5
|
| 33 | 24 | mulid1d 7707 |
. . . . 5
|
| 34 | 31, 32, 33 | 3brtr3d 3924 |
. . . 4
|
| 35 | 34 | ex 114 |
. . 3
|
| 36 | 4, 35 | rexlimddv 2528 |
. 2
|
| 37 | 2, 36 | impbid 128 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 w3a 945 wceq 1314 wcel 1463 wrex 2391 class class class wbr 3895
(class class class)co 5728 cc 7545 cr 7546 cc0 7547 c1 7548 cmul 7552 clt 7724 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-cnex 7636 ax-resscn 7637 ax-1cn 7638 ax-1re 7639 ax-icn 7640 ax-addcl 7641 ax-addrcl 7642 ax-mulcl 7643 ax-mulrcl 7644 ax-addcom 7645 ax-mulcom 7646 ax-addass 7647 ax-mulass 7648 ax-distr 7649 ax-i2m1 7650 ax-1rid 7652 ax-0id 7653 ax-rnegex 7654 ax-precex 7655 ax-cnre 7656 ax-pre-ltadd 7661 ax-pre-mulgt0 7662 |
| This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-iota 5046 df-fun 5083 df-fv 5089 df-riota 5684 df-ov 5731 df-oprab 5732 df-mpo 5733 df-pnf 7726 df-mnf 7727 df-ltxr 7729 df-sub 7858 df-neg 7859 |
| This theorem is referenced by: lemul1 8273 reapmul1lem 8274 ltmul2 8524 ltdiv1 8536 ltdiv23 8560 recp1lt1 8567 ltmul1i 8588 ltmul1d 9424 mertenslemi1 11196 flodddiv4t2lthalf 11482 qnumgt0 11721 |
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