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Mirrors > Home > ILE Home > Th. List > ltmul12a | Unicode version |
Description: Comparison of product of two positive numbers. (Contributed by NM, 30-Dec-2005.) |
Ref | Expression |
---|---|
ltmul12a |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplll 507 | . . . 4 | |
2 | simpllr 508 | . . . 4 | |
3 | simpll 503 | . . . . . 6 | |
4 | simprl 505 | . . . . . 6 | |
5 | 3, 4 | jca 304 | . . . . 5 |
6 | 5 | ad2ant2l 499 | . . . 4 |
7 | ltle 7819 | . . . . . . 7 | |
8 | 7 | imp 123 | . . . . . 6 |
9 | 8 | adantrl 469 | . . . . 5 |
10 | 9 | ad2ant2r 500 | . . . 4 |
11 | lemul1a 8584 | . . . 4 | |
12 | 1, 2, 6, 10, 11 | syl31anc 1204 | . . 3 |
13 | simplrl 509 | . . . . . . 7 | |
14 | simplrr 510 | . . . . . . 7 | |
15 | simpllr 508 | . . . . . . 7 | |
16 | 0re 7734 | . . . . . . . . . 10 | |
17 | lelttr 7820 | . . . . . . . . . 10 | |
18 | 16, 17 | mp3an1 1287 | . . . . . . . . 9 |
19 | 18 | imp 123 | . . . . . . . 8 |
20 | 19 | adantlr 468 | . . . . . . 7 |
21 | ltmul2 8582 | . . . . . . 7 | |
22 | 13, 14, 15, 20, 21 | syl112anc 1205 | . . . . . 6 |
23 | 22 | biimpa 294 | . . . . 5 |
24 | 23 | anasss 396 | . . . 4 |
25 | 24 | adantrrl 477 | . . 3 |
26 | remulcl 7716 | . . . . . 6 | |
27 | 26 | ad2ant2r 500 | . . . . 5 |
28 | remulcl 7716 | . . . . . 6 | |
29 | 28 | ad2ant2lr 501 | . . . . 5 |
30 | remulcl 7716 | . . . . . 6 | |
31 | 30 | ad2ant2l 499 | . . . . 5 |
32 | lelttr 7820 | . . . . 5 | |
33 | 27, 29, 31, 32 | syl3anc 1201 | . . . 4 |
34 | 33 | adantr 274 | . . 3 |
35 | 12, 25, 34 | mp2and 429 | . 2 |
36 | 35 | an4s 562 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 1465 class class class wbr 3899 (class class class)co 5742 cr 7587 cc0 7588 cmul 7593 clt 7768 cle 7769 |
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-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
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-po 4188 df-iso 4189 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-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 |
This theorem is referenced by: ltmul12ad 8667 |
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