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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 8548 |
. . 3
| |
| 2 | 1 | ex 115 |
. 2
|
| 3 | recexgt0 8537 |
. . . 4
  | |
| 4 | 3 | 3ad2ant3 1020 |
. . 3
|
| 5 | simpl1 1000 |
. . . . . . . . . 10
| |
| 6 | simpl3l 1052 |
. . . . . . . . . 10
| |
| 7 | 5, 6 | remulcld 7988 |
. . . . . . . . 9
|
| 8 | simpl2 1001 |
. . . . . . . . . 10
| |
| 9 | 8, 6 | remulcld 7988 |
. . . . . . . . 9
|
| 10 | simprl 529 |
. . . . . . . . . 10
| |
| 11 | simprrl 539 |
. . . . . . . . . 10
| |
| 12 | 10, 11 | jca 306 |
. . . . . . . . 9
|
| 13 | 7, 9, 12 | 3jca 1177 |
. . . . . . . 8
|
| 14 | ltmul1a 8548 |
. . . . . . . 8
| |
| 15 | 13, 14 | sylan 283 |
. . . . . . 7
|
| 16 | 5 | recnd 7986 |
. . . . . . . . 9
 |
| 17 | 16 | adantr 276 |
. . . . . . . 8
|
| 18 | 6 | recnd 7986 |
. . . . . . . . 9
|
| 19 | 18 | adantr 276 |
. . . . . . . 8
|
| 20 | 10 | recnd 7986 |
. . . . . . . . 9
|
| 21 | 20 | adantr 276 |
. . . . . . . 8
|
| 22 | 17, 19, 21 | mulassd 7981 |
. . . . . . 7
|
| 23 | 8 | recnd 7986 |
. . . . . . . . 9
|
| 24 | 23 | adantr 276 |
. . . . . . . 8
|
| 25 | 24, 19, 21 | mulassd 7981 |
. . . . . . 7
|
| 26 | 15, 22, 25 | 3brtr3d 4035 |
. . . . . 6
|
| 27 | simprrr 540 |
. . . . . . . 8
| |
| 28 | 27 | adantr 276 |
. . . . . . 7
|
| 29 | 28 | oveq2d 5891 |
. . . . . 6
|
| 30 | 28 | oveq2d 5891 |
. . . . . 6
|
| 31 | 26, 29, 30 | 3brtr3d 4035 |
. . . . 5
|
| 32 | 17 | mulridd 7974 |
. . . . 5
|
| 33 | 24 | mulridd 7974 |
. . . . 5
|
| 34 | 31, 32, 33 | 3brtr3d 4035 |
. . . 4
|
| 35 | 34 | ex 115 |
. . 3
|
| 36 | 4, 35 | rexlimddv 2599 |
. 2
|
| 37 | 2, 36 | impbid 129 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 w3a 978 wceq 1353 wcel 2148 wrex 2456 class class class wbr 4004
(class class class)co 5875 cc 7809 cr 7810 cc0 7811 c1 7812 cmul 7816 clt 7992 |
| 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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-mulrcl 7910 ax-addcom 7911 ax-mulcom 7912 ax-addass 7913 ax-mulass 7914 ax-distr 7915 ax-i2m1 7916 ax-1rid 7918 ax-0id 7919 ax-rnegex 7920 ax-precex 7921 ax-cnre 7922 ax-pre-ltadd 7927 ax-pre-mulgt0 7928 |
| This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2740 df-sbc 2964 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-br 4005 df-opab 4066 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-iota 5179 df-fun 5219 df-fv 5225 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-pnf 7994 df-mnf 7995 df-ltxr 7997 df-sub 8130 df-neg 8131 |
| This theorem is referenced by: lemul1 8550 reapmul1lem 8551 ltmul2 8813 ltdiv1 8825 ltdiv23 8849 recp1lt1 8856 ltmul1i 8877 ltmul1d 9738 mertenslemi1 11543 flodddiv4t2lthalf 11942 qnumgt0 12198 tangtx 14262 |
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