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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 8666 |
. . 3
| |
| 2 | 1 | ex 115 |
. 2
|
| 3 | recexgt0 8655 |
. . . 4
  | |
| 4 | 3 | 3ad2ant3 1023 |
. . 3
|
| 5 | simpl1 1003 |
. . . . . . . . . 10
| |
| 6 | simpl3l 1055 |
. . . . . . . . . 10
| |
| 7 | 5, 6 | remulcld 8105 |
. . . . . . . . 9
|
| 8 | simpl2 1004 |
. . . . . . . . . 10
| |
| 9 | 8, 6 | remulcld 8105 |
. . . . . . . . 9
|
| 10 | simprl 529 |
. . . . . . . . . 10
| |
| 11 | simprrl 539 |
. . . . . . . . . 10
| |
| 12 | 10, 11 | jca 306 |
. . . . . . . . 9
|
| 13 | 7, 9, 12 | 3jca 1180 |
. . . . . . . 8
|
| 14 | ltmul1a 8666 |
. . . . . . . 8
| |
| 15 | 13, 14 | sylan 283 |
. . . . . . 7
|
| 16 | 5 | recnd 8103 |
. . . . . . . . 9
 |
| 17 | 16 | adantr 276 |
. . . . . . . 8
|
| 18 | 6 | recnd 8103 |
. . . . . . . . 9
|
| 19 | 18 | adantr 276 |
. . . . . . . 8
|
| 20 | 10 | recnd 8103 |
. . . . . . . . 9
|
| 21 | 20 | adantr 276 |
. . . . . . . 8
|
| 22 | 17, 19, 21 | mulassd 8098 |
. . . . . . 7
|
| 23 | 8 | recnd 8103 |
. . . . . . . . 9
|
| 24 | 23 | adantr 276 |
. . . . . . . 8
|
| 25 | 24, 19, 21 | mulassd 8098 |
. . . . . . 7
|
| 26 | 15, 22, 25 | 3brtr3d 4076 |
. . . . . 6
|
| 27 | simprrr 540 |
. . . . . . . 8
| |
| 28 | 27 | adantr 276 |
. . . . . . 7
|
| 29 | 28 | oveq2d 5962 |
. . . . . 6
|
| 30 | 28 | oveq2d 5962 |
. . . . . 6
|
| 31 | 26, 29, 30 | 3brtr3d 4076 |
. . . . 5
|
| 32 | 17 | mulridd 8091 |
. . . . 5
|
| 33 | 24 | mulridd 8091 |
. . . . 5
|
| 34 | 31, 32, 33 | 3brtr3d 4076 |
. . . 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 4045
(class class class)co 5946 cc 7925 cr 7926 cc0 7927 c1 7928 cmul 7932 clt 8109 |
| 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 4163 ax-pow 4219 ax-pr 4254 ax-un 4481 ax-setind 4586 ax-cnex 8018 ax-resscn 8019 ax-1cn 8020 ax-1re 8021 ax-icn 8022 ax-addcl 8023 ax-addrcl 8024 ax-mulcl 8025 ax-mulrcl 8026 ax-addcom 8027 ax-mulcom 8028 ax-addass 8029 ax-mulass 8030 ax-distr 8031 ax-i2m1 8032 ax-1rid 8034 ax-0id 8035 ax-rnegex 8036 ax-precex 8037 ax-cnre 8038 ax-pre-ltadd 8043 ax-pre-mulgt0 8044 |
| 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 4046 df-opab 4107 df-id 4341 df-xp 4682 df-rel 4683 df-cnv 4684 df-co 4685 df-dm 4686 df-iota 5233 df-fun 5274 df-fv 5280 df-riota 5901 df-ov 5949 df-oprab 5950 df-mpo 5951 df-pnf 8111 df-mnf 8112 df-ltxr 8114 df-sub 8247 df-neg 8248 |
| This theorem is referenced by: lemul1 8668 reapmul1lem 8669 ltmul2 8931 ltdiv1 8943 ltdiv23 8967 recp1lt1 8974 ltmul1i 8995 ltmul1d 9862 mertenslemi1 11879 flodddiv4t2lthalf 12283 qnumgt0 12553 4sqlem12 12758 tangtx 15343 lgsquadlem1 15587 lgsquadlem2 15588 |
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