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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 8664 |
. . 3
| |
| 2 | 1 | ex 115 |
. 2
|
| 3 | recexgt0 8653 |
. . . 4
  | |
| 4 | 3 | 3ad2ant3 1023 |
. . 3
|
| 5 | simpl1 1003 |
. . . . . . . . . 10
| |
| 6 | simpl3l 1055 |
. . . . . . . . . 10
| |
| 7 | 5, 6 | remulcld 8103 |
. . . . . . . . 9
|
| 8 | simpl2 1004 |
. . . . . . . . . 10
| |
| 9 | 8, 6 | remulcld 8103 |
. . . . . . . . 9
|
| 10 | simprl 529 |
. . . . . . . . . 10
| |
| 11 | simprrl 539 |
. . . . . . . . . 10
| |
| 12 | 10, 11 | jca 306 |
. . . . . . . . 9
|
| 13 | 7, 9, 12 | 3jca 1180 |
. . . . . . . 8
|
| 14 | ltmul1a 8664 |
. . . . . . . 8
| |
| 15 | 13, 14 | sylan 283 |
. . . . . . 7
|
| 16 | 5 | recnd 8101 |
. . . . . . . . 9
 |
| 17 | 16 | adantr 276 |
. . . . . . . 8
|
| 18 | 6 | recnd 8101 |
. . . . . . . . 9
|
| 19 | 18 | adantr 276 |
. . . . . . . 8
|
| 20 | 10 | recnd 8101 |
. . . . . . . . 9
|
| 21 | 20 | adantr 276 |
. . . . . . . 8
|
| 22 | 17, 19, 21 | mulassd 8096 |
. . . . . . 7
|
| 23 | 8 | recnd 8101 |
. . . . . . . . 9
|
| 24 | 23 | adantr 276 |
. . . . . . . 8
|
| 25 | 24, 19, 21 | mulassd 8096 |
. . . . . . 7
|
| 26 | 15, 22, 25 | 3brtr3d 4075 |
. . . . . 6
|
| 27 | simprrr 540 |
. . . . . . . 8
| |
| 28 | 27 | adantr 276 |
. . . . . . 7
|
| 29 | 28 | oveq2d 5960 |
. . . . . 6
|
| 30 | 28 | oveq2d 5960 |
. . . . . 6
|
| 31 | 26, 29, 30 | 3brtr3d 4075 |
. . . . 5
|
| 32 | 17 | mulridd 8089 |
. . . . 5
|
| 33 | 24 | mulridd 8089 |
. . . . 5
|
| 34 | 31, 32, 33 | 3brtr3d 4075 |
. . . 4
|
| 35 | 34 | ex 115 |
. . 3
|
| 36 | 4, 35 | rexlimddv 2628 |
. 2
|
| 37 | 2, 36 | impbid 129 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 w3a 981 wceq 1373 wcel 2176 wrex 2485 class class class wbr 4044
(class class class)co 5944 cc 7923 cr 7924 cc0 7925 c1 7926 cmul 7930 clt 8107 |
| 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 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltadd 8041 ax-pre-mulgt0 8042 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rab 2493 df-v 2774 df-sbc 2999 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-br 4045 df-opab 4106 df-id 4340 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-iota 5232 df-fun 5273 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-pnf 8109 df-mnf 8110 df-ltxr 8112 df-sub 8245 df-neg 8246 |
| This theorem is referenced by: lemul1 8666 reapmul1lem 8667 ltmul2 8929 ltdiv1 8941 ltdiv23 8965 recp1lt1 8972 ltmul1i 8993 ltmul1d 9860 mertenslemi1 11846 flodddiv4t2lthalf 12250 qnumgt0 12520 4sqlem12 12725 tangtx 15310 lgsquadlem1 15554 lgsquadlem2 15555 |
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