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Mirrors > Home > ILE Home > Th. List > ltmul2 | Unicode version |
Description: Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.) |
Ref | Expression |
---|---|
ltmul2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltmul1 8322 | . 2 | |
2 | recn 7721 | . . . 4 | |
3 | recn 7721 | . . . . . . 7 | |
4 | mulcom 7717 | . . . . . . 7 | |
5 | 3, 4 | sylan 281 | . . . . . 6 |
6 | 5 | 3adant2 985 | . . . . 5 |
7 | recn 7721 | . . . . . . 7 | |
8 | mulcom 7717 | . . . . . . 7 | |
9 | 7, 8 | sylan 281 | . . . . . 6 |
10 | 9 | 3adant1 984 | . . . . 5 |
11 | 6, 10 | breq12d 3912 | . . . 4 |
12 | 2, 11 | syl3an3 1236 | . . 3 |
13 | 12 | 3adant3r 1198 | . 2 |
14 | 1, 13 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 947 wceq 1316 wcel 1465 class class class wbr 3899 (class class class)co 5742 cc 7586 cr 7587 cc0 7588 cmul 7593 clt 7768 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-ltxr 7773 df-sub 7903 df-neg 7904 |
This theorem is referenced by: ltmul12a 8586 mulgt1 8589 ltmulgt11 8590 lt2msq1 8611 ltdiv2 8613 ltmul2i 8649 ltmul2d 9494 ef01bndlem 11390 cos01gt0 11396 sin4lt0 11400 tangtx 12846 |
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