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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 8738 |
. . 3
| |
| 2 | 1 | ex 115 |
. 2
|
| 3 | recexgt0 8727 |
. . . 4
  | |
| 4 | 3 | 3ad2ant3 1044 |
. . 3
|
| 5 | simpl1 1024 |
. . . . . . . . . 10
| |
| 6 | simpl3l 1076 |
. . . . . . . . . 10
| |
| 7 | 5, 6 | remulcld 8177 |
. . . . . . . . 9
|
| 8 | simpl2 1025 |
. . . . . . . . . 10
| |
| 9 | 8, 6 | remulcld 8177 |
. . . . . . . . 9
|
| 10 | simprl 529 |
. . . . . . . . . 10
| |
| 11 | simprrl 539 |
. . . . . . . . . 10
| |
| 12 | 10, 11 | jca 306 |
. . . . . . . . 9
|
| 13 | 7, 9, 12 | 3jca 1201 |
. . . . . . . 8
|
| 14 | ltmul1a 8738 |
. . . . . . . 8
| |
| 15 | 13, 14 | sylan 283 |
. . . . . . 7
|
| 16 | 5 | recnd 8175 |
. . . . . . . . 9
 |
| 17 | 16 | adantr 276 |
. . . . . . . 8
|
| 18 | 6 | recnd 8175 |
. . . . . . . . 9
|
| 19 | 18 | adantr 276 |
. . . . . . . 8
|
| 20 | 10 | recnd 8175 |
. . . . . . . . 9
|
| 21 | 20 | adantr 276 |
. . . . . . . 8
|
| 22 | 17, 19, 21 | mulassd 8170 |
. . . . . . 7
|
| 23 | 8 | recnd 8175 |
. . . . . . . . 9
|
| 24 | 23 | adantr 276 |
. . . . . . . 8
|
| 25 | 24, 19, 21 | mulassd 8170 |
. . . . . . 7
|
| 26 | 15, 22, 25 | 3brtr3d 4114 |
. . . . . 6
|
| 27 | simprrr 540 |
. . . . . . . 8
| |
| 28 | 27 | adantr 276 |
. . . . . . 7
|
| 29 | 28 | oveq2d 6017 |
. . . . . 6
|
| 30 | 28 | oveq2d 6017 |
. . . . . 6
|
| 31 | 26, 29, 30 | 3brtr3d 4114 |
. . . . 5
|
| 32 | 17 | mulridd 8163 |
. . . . 5
|
| 33 | 24 | mulridd 8163 |
. . . . 5
|
| 34 | 31, 32, 33 | 3brtr3d 4114 |
. . . 4
|
| 35 | 34 | ex 115 |
. . 3
|
| 36 | 4, 35 | rexlimddv 2653 |
. 2
|
| 37 | 2, 36 | impbid 129 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 w3a 1002 wceq 1395 wcel 2200 wrex 2509 class class class wbr 4083
(class class class)co 6001 cc 7997 cr 7998 cc0 7999 c1 8000 cmul 8004 clt 8181 |
| 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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-cnex 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-mulrcl 8098 ax-addcom 8099 ax-mulcom 8100 ax-addass 8101 ax-mulass 8102 ax-distr 8103 ax-i2m1 8104 ax-1rid 8106 ax-0id 8107 ax-rnegex 8108 ax-precex 8109 ax-cnre 8110 ax-pre-ltadd 8115 ax-pre-mulgt0 8116 |
| This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-br 4084 df-opab 4146 df-id 4384 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-iota 5278 df-fun 5320 df-fv 5326 df-riota 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-pnf 8183 df-mnf 8184 df-ltxr 8186 df-sub 8319 df-neg 8320 |
| This theorem is referenced by: lemul1 8740 reapmul1lem 8741 ltmul2 9003 ltdiv1 9015 ltdiv23 9039 recp1lt1 9046 ltmul1i 9067 ltmul1d 9934 mertenslemi1 12046 flodddiv4t2lthalf 12450 qnumgt0 12720 4sqlem12 12925 tangtx 15512 lgsquadlem1 15756 lgsquadlem2 15757 |
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