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Mirrors > Home > ILE Home > Th. List > mulge0 | Unicode version |
Description: The product of two nonnegative numbers is nonnegative. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
mulge0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | remulcl 7716 | . . . . 5 | |
2 | 1 | ad2ant2r 500 | . . . 4 |
3 | 0re 7734 | . . . 4 | |
4 | ltnsym2 7822 | . . . 4 | |
5 | 2, 3, 4 | sylancl 409 | . . 3 |
6 | orc 686 | . . . . . 6 | |
7 | reaplt 8318 | . . . . . . 7 # | |
8 | 2, 3, 7 | sylancl 409 | . . . . . 6 # |
9 | 6, 8 | syl5ibr 155 | . . . . 5 # |
10 | simplll 507 | . . . . . . 7 # | |
11 | simplrl 509 | . . . . . . 7 # | |
12 | recn 7721 | . . . . . . . . . . . . . 14 | |
13 | recn 7721 | . . . . . . . . . . . . . . 15 | |
14 | mulap0r 8345 | . . . . . . . . . . . . . . 15 # # # | |
15 | 13, 14 | syl3an1 1234 | . . . . . . . . . . . . . 14 # # # |
16 | 12, 15 | syl3an2 1235 | . . . . . . . . . . . . 13 # # # |
17 | 16 | 3expia 1168 | . . . . . . . . . . . 12 # # # |
18 | 17 | ad2ant2r 500 | . . . . . . . . . . 11 # # # |
19 | 18 | imp 123 | . . . . . . . . . 10 # # # |
20 | 19 | simpld 111 | . . . . . . . . 9 # # |
21 | reaplt 8318 | . . . . . . . . . . 11 # | |
22 | 3, 21 | mpan2 421 | . . . . . . . . . 10 # |
23 | 22 | ad3antrrr 483 | . . . . . . . . 9 # # |
24 | 20, 23 | mpbid 146 | . . . . . . . 8 # |
25 | lenlt 7808 | . . . . . . . . . . . 12 | |
26 | 3, 25 | mpan 420 | . . . . . . . . . . 11 |
27 | 26 | biimpa 294 | . . . . . . . . . 10 |
28 | 27 | ad2antrr 479 | . . . . . . . . 9 # |
29 | biorf 718 | . . . . . . . . 9 | |
30 | 28, 29 | syl 14 | . . . . . . . 8 # |
31 | 24, 30 | mpbird 166 | . . . . . . 7 # |
32 | 19 | simprd 113 | . . . . . . . . 9 # # |
33 | reaplt 8318 | . . . . . . . . . . . 12 # | |
34 | 3, 33 | mpan2 421 | . . . . . . . . . . 11 # |
35 | 34 | ad2antrl 481 | . . . . . . . . . 10 # |
36 | 35 | adantr 274 | . . . . . . . . 9 # # |
37 | 32, 36 | mpbid 146 | . . . . . . . 8 # |
38 | lenlt 7808 | . . . . . . . . . . . 12 | |
39 | 3, 38 | mpan 420 | . . . . . . . . . . 11 |
40 | 39 | biimpa 294 | . . . . . . . . . 10 |
41 | 40 | ad2antlr 480 | . . . . . . . . 9 # |
42 | biorf 718 | . . . . . . . . 9 | |
43 | 41, 42 | syl 14 | . . . . . . . 8 # |
44 | 37, 43 | mpbird 166 | . . . . . . 7 # |
45 | 10, 11, 31, 44 | mulgt0d 7853 | . . . . . 6 # |
46 | 45 | ex 114 | . . . . 5 # |
47 | 9, 46 | syld 45 | . . . 4 |
48 | 47 | ancld 323 | . . 3 |
49 | 5, 48 | mtod 637 | . 2 |
50 | lenlt 7808 | . . 3 | |
51 | 3, 2, 50 | sylancr 410 | . 2 |
52 | 49, 51 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 682 wcel 1465 class class class wbr 3899 (class class class)co 5742 cc 7586 cr 7587 cc0 7588 cmul 7593 clt 7768 cle 7769 # cap 8311 |
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: mulge0i 8350 mulge0d 8351 ge0mulcl 9733 expge0 10297 bernneq 10380 sqrtmul 10775 amgm2 10858 |
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