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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 8577 |
. . 3
| |
| 2 | 1 | ex 115 |
. 2
|
| 3 | recexgt0 8566 |
. . . 4
  | |
| 4 | 3 | 3ad2ant3 1022 |
. . 3
|
| 5 | simpl1 1002 |
. . . . . . . . . 10
| |
| 6 | simpl3l 1054 |
. . . . . . . . . 10
| |
| 7 | 5, 6 | remulcld 8017 |
. . . . . . . . 9
|
| 8 | simpl2 1003 |
. . . . . . . . . 10
| |
| 9 | 8, 6 | remulcld 8017 |
. . . . . . . . 9
|
| 10 | simprl 529 |
. . . . . . . . . 10
| |
| 11 | simprrl 539 |
. . . . . . . . . 10
| |
| 12 | 10, 11 | jca 306 |
. . . . . . . . 9
|
| 13 | 7, 9, 12 | 3jca 1179 |
. . . . . . . 8
|
| 14 | ltmul1a 8577 |
. . . . . . . 8
| |
| 15 | 13, 14 | sylan 283 |
. . . . . . 7
|
| 16 | 5 | recnd 8015 |
. . . . . . . . 9
 |
| 17 | 16 | adantr 276 |
. . . . . . . 8
|
| 18 | 6 | recnd 8015 |
. . . . . . . . 9
|
| 19 | 18 | adantr 276 |
. . . . . . . 8
|
| 20 | 10 | recnd 8015 |
. . . . . . . . 9
|
| 21 | 20 | adantr 276 |
. . . . . . . 8
|
| 22 | 17, 19, 21 | mulassd 8010 |
. . . . . . 7
|
| 23 | 8 | recnd 8015 |
. . . . . . . . 9
|
| 24 | 23 | adantr 276 |
. . . . . . . 8
|
| 25 | 24, 19, 21 | mulassd 8010 |
. . . . . . 7
|
| 26 | 15, 22, 25 | 3brtr3d 4049 |
. . . . . 6
|
| 27 | simprrr 540 |
. . . . . . . 8
| |
| 28 | 27 | adantr 276 |
. . . . . . 7
|
| 29 | 28 | oveq2d 5911 |
. . . . . 6
|
| 30 | 28 | oveq2d 5911 |
. . . . . 6
|
| 31 | 26, 29, 30 | 3brtr3d 4049 |
. . . . 5
|
| 32 | 17 | mulridd 8003 |
. . . . 5
|
| 33 | 24 | mulridd 8003 |
. . . . 5
|
| 34 | 31, 32, 33 | 3brtr3d 4049 |
. . . 4
|
| 35 | 34 | ex 115 |
. . 3
|
| 36 | 4, 35 | rexlimddv 2612 |
. 2
|
| 37 | 2, 36 | impbid 129 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 w3a 980 wceq 1364 wcel 2160 wrex 2469 class class class wbr 4018
(class class class)co 5895 cc 7838 cr 7839 cc0 7840 c1 7841 cmul 7845 clt 8021 |
| 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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-mulrcl 7939 ax-addcom 7940 ax-mulcom 7941 ax-addass 7942 ax-mulass 7943 ax-distr 7944 ax-i2m1 7945 ax-1rid 7947 ax-0id 7948 ax-rnegex 7949 ax-precex 7950 ax-cnre 7951 ax-pre-ltadd 7956 ax-pre-mulgt0 7957 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-pnf 8023 df-mnf 8024 df-ltxr 8026 df-sub 8159 df-neg 8160 |
| This theorem is referenced by: lemul1 8579 reapmul1lem 8580 ltmul2 8842 ltdiv1 8854 ltdiv23 8878 recp1lt1 8885 ltmul1i 8906 ltmul1d 9767 mertenslemi1 11574 flodddiv4t2lthalf 11973 qnumgt0 12229 4sqlem12 12433 tangtx 14711 |
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