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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 8346 |
. . 3
| |
| 2 | 1 | ex 114 |
. 2
|
| 3 | recexgt0 8335 |
. . . 4
  | |
| 4 | 3 | 3ad2ant3 1004 |
. . 3
|
| 5 | simpl1 984 |
. . . . . . . . . 10
| |
| 6 | simpl3l 1036 |
. . . . . . . . . 10
| |
| 7 | 5, 6 | remulcld 7789 |
. . . . . . . . 9
|
| 8 | simpl2 985 |
. . . . . . . . . 10
| |
| 9 | 8, 6 | remulcld 7789 |
. . . . . . . . 9
|
| 10 | simprl 520 |
. . . . . . . . . 10
| |
| 11 | simprrl 528 |
. . . . . . . . . 10
| |
| 12 | 10, 11 | jca 304 |
. . . . . . . . 9
|
| 13 | 7, 9, 12 | 3jca 1161 |
. . . . . . . 8
|
| 14 | ltmul1a 8346 |
. . . . . . . 8
| |
| 15 | 13, 14 | sylan 281 |
. . . . . . 7
|
| 16 | 5 | recnd 7787 |
. . . . . . . . 9
 |
| 17 | 16 | adantr 274 |
. . . . . . . 8
|
| 18 | 6 | recnd 7787 |
. . . . . . . . 9
|
| 19 | 18 | adantr 274 |
. . . . . . . 8
|
| 20 | 10 | recnd 7787 |
. . . . . . . . 9
|
| 21 | 20 | adantr 274 |
. . . . . . . 8
|
| 22 | 17, 19, 21 | mulassd 7782 |
. . . . . . 7
|
| 23 | 8 | recnd 7787 |
. . . . . . . . 9
|
| 24 | 23 | adantr 274 |
. . . . . . . 8
|
| 25 | 24, 19, 21 | mulassd 7782 |
. . . . . . 7
|
| 26 | 15, 22, 25 | 3brtr3d 3954 |
. . . . . 6
|
| 27 | simprrr 529 |
. . . . . . . 8
| |
| 28 | 27 | adantr 274 |
. . . . . . 7
|
| 29 | 28 | oveq2d 5783 |
. . . . . 6
|
| 30 | 28 | oveq2d 5783 |
. . . . . 6
|
| 31 | 26, 29, 30 | 3brtr3d 3954 |
. . . . 5
|
| 32 | 17 | mulid1d 7776 |
. . . . 5
|
| 33 | 24 | mulid1d 7776 |
. . . . 5
|
| 34 | 31, 32, 33 | 3brtr3d 3954 |
. . . 4
|
| 35 | 34 | ex 114 |
. . 3
|
| 36 | 4, 35 | rexlimddv 2552 |
. 2
|
| 37 | 2, 36 | impbid 128 |
1
|
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 w3a 962 wceq 1331 wcel 1480 wrex 2415 class class class wbr 3924
(class class class)co 5767 cc 7611 cr 7612 cc0 7613 c1 7614 cmul 7618 clt 7793 |
| 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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 |
| This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-ltxr 7798 df-sub 7928 df-neg 7929 |
| This theorem is referenced by: lemul1 8348 reapmul1lem 8349 ltmul2 8607 ltdiv1 8619 ltdiv23 8643 recp1lt1 8650 ltmul1i 8671 ltmul1d 9518 mertenslemi1 11297 flodddiv4t2lthalf 11623 qnumgt0 11865 tangtx 12908 |
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