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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 8618 |
. . 3
| |
| 2 | 1 | ex 115 |
. 2
|
| 3 | recexgt0 8607 |
. . . 4
  | |
| 4 | 3 | 3ad2ant3 1022 |
. . 3
|
| 5 | simpl1 1002 |
. . . . . . . . . 10
| |
| 6 | simpl3l 1054 |
. . . . . . . . . 10
| |
| 7 | 5, 6 | remulcld 8057 |
. . . . . . . . 9
|
| 8 | simpl2 1003 |
. . . . . . . . . 10
| |
| 9 | 8, 6 | remulcld 8057 |
. . . . . . . . 9
|
| 10 | simprl 529 |
. . . . . . . . . 10
| |
| 11 | simprrl 539 |
. . . . . . . . . 10
| |
| 12 | 10, 11 | jca 306 |
. . . . . . . . 9
|
| 13 | 7, 9, 12 | 3jca 1179 |
. . . . . . . 8
|
| 14 | ltmul1a 8618 |
. . . . . . . 8
| |
| 15 | 13, 14 | sylan 283 |
. . . . . . 7
|
| 16 | 5 | recnd 8055 |
. . . . . . . . 9
 |
| 17 | 16 | adantr 276 |
. . . . . . . 8
|
| 18 | 6 | recnd 8055 |
. . . . . . . . 9
|
| 19 | 18 | adantr 276 |
. . . . . . . 8
|
| 20 | 10 | recnd 8055 |
. . . . . . . . 9
|
| 21 | 20 | adantr 276 |
. . . . . . . 8
|
| 22 | 17, 19, 21 | mulassd 8050 |
. . . . . . 7
|
| 23 | 8 | recnd 8055 |
. . . . . . . . 9
|
| 24 | 23 | adantr 276 |
. . . . . . . 8
|
| 25 | 24, 19, 21 | mulassd 8050 |
. . . . . . 7
|
| 26 | 15, 22, 25 | 3brtr3d 4064 |
. . . . . 6
|
| 27 | simprrr 540 |
. . . . . . . 8
| |
| 28 | 27 | adantr 276 |
. . . . . . 7
|
| 29 | 28 | oveq2d 5938 |
. . . . . 6
|
| 30 | 28 | oveq2d 5938 |
. . . . . 6
|
| 31 | 26, 29, 30 | 3brtr3d 4064 |
. . . . 5
|
| 32 | 17 | mulridd 8043 |
. . . . 5
|
| 33 | 24 | mulridd 8043 |
. . . . 5
|
| 34 | 31, 32, 33 | 3brtr3d 4064 |
. . . 4
|
| 35 | 34 | ex 115 |
. . 3
|
| 36 | 4, 35 | rexlimddv 2619 |
. 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 2167 wrex 2476 class class class wbr 4033
(class class class)co 5922 cc 7877 cr 7878 cc0 7879 c1 7880 cmul 7884 clt 8061 |
| 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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-br 4034 df-opab 4095 df-id 4328 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-iota 5219 df-fun 5260 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-pnf 8063 df-mnf 8064 df-ltxr 8066 df-sub 8199 df-neg 8200 |
| This theorem is referenced by: lemul1 8620 reapmul1lem 8621 ltmul2 8883 ltdiv1 8895 ltdiv23 8919 recp1lt1 8926 ltmul1i 8947 ltmul1d 9813 mertenslemi1 11700 flodddiv4t2lthalf 12104 qnumgt0 12366 4sqlem12 12571 tangtx 15074 lgsquadlem1 15318 lgsquadlem2 15319 |
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