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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 8497 |
. . 3
| |
| 2 | 1 | ex 114 |
. 2
|
| 3 | recexgt0 8486 |
. . . 4
  | |
| 4 | 3 | 3ad2ant3 1015 |
. . 3
|
| 5 | simpl1 995 |
. . . . . . . . . 10
| |
| 6 | simpl3l 1047 |
. . . . . . . . . 10
| |
| 7 | 5, 6 | remulcld 7937 |
. . . . . . . . 9
|
| 8 | simpl2 996 |
. . . . . . . . . 10
| |
| 9 | 8, 6 | remulcld 7937 |
. . . . . . . . 9
|
| 10 | simprl 526 |
. . . . . . . . . 10
| |
| 11 | simprrl 534 |
. . . . . . . . . 10
| |
| 12 | 10, 11 | jca 304 |
. . . . . . . . 9
|
| 13 | 7, 9, 12 | 3jca 1172 |
. . . . . . . 8
|
| 14 | ltmul1a 8497 |
. . . . . . . 8
| |
| 15 | 13, 14 | sylan 281 |
. . . . . . 7
|
| 16 | 5 | recnd 7935 |
. . . . . . . . 9
 |
| 17 | 16 | adantr 274 |
. . . . . . . 8
|
| 18 | 6 | recnd 7935 |
. . . . . . . . 9
|
| 19 | 18 | adantr 274 |
. . . . . . . 8
|
| 20 | 10 | recnd 7935 |
. . . . . . . . 9
|
| 21 | 20 | adantr 274 |
. . . . . . . 8
|
| 22 | 17, 19, 21 | mulassd 7930 |
. . . . . . 7
|
| 23 | 8 | recnd 7935 |
. . . . . . . . 9
|
| 24 | 23 | adantr 274 |
. . . . . . . 8
|
| 25 | 24, 19, 21 | mulassd 7930 |
. . . . . . 7
|
| 26 | 15, 22, 25 | 3brtr3d 4018 |
. . . . . 6
|
| 27 | simprrr 535 |
. . . . . . . 8
| |
| 28 | 27 | adantr 274 |
. . . . . . 7
|
| 29 | 28 | oveq2d 5866 |
. . . . . 6
|
| 30 | 28 | oveq2d 5866 |
. . . . . 6
|
| 31 | 26, 29, 30 | 3brtr3d 4018 |
. . . . 5
|
| 32 | 17 | mulid1d 7924 |
. . . . 5
|
| 33 | 24 | mulid1d 7924 |
. . . . 5
|
| 34 | 31, 32, 33 | 3brtr3d 4018 |
. . . 4
|
| 35 | 34 | ex 114 |
. . 3
|
| 36 | 4, 35 | rexlimddv 2592 |
. 2
|
| 37 | 2, 36 | impbid 128 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 w3a 973 wceq 1348 wcel 2141 wrex 2449 class class class wbr 3987
(class class class)co 5850 cc 7759 cr 7760 cc0 7761 c1 7762 cmul 7766 clt 7941 |
| 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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-mulrcl 7860 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-precex 7871 ax-cnre 7872 ax-pre-ltadd 7877 ax-pre-mulgt0 7878 |
| This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-pnf 7943 df-mnf 7944 df-ltxr 7946 df-sub 8079 df-neg 8080 |
| This theorem is referenced by: lemul1 8499 reapmul1lem 8500 ltmul2 8759 ltdiv1 8771 ltdiv23 8795 recp1lt1 8802 ltmul1i 8823 ltmul1d 9682 mertenslemi1 11485 flodddiv4t2lthalf 11883 qnumgt0 12139 tangtx 13512 |
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