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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 7741 | . . . . 5 | |
2 | 1 | ad2ant2r 500 | . . . 4 |
3 | 0re 7759 | . . . 4 | |
4 | ltnsym2 7847 | . . . 4 | |
5 | 2, 3, 4 | sylancl 409 | . . 3 |
6 | orc 701 | . . . . . 6 | |
7 | reaplt 8343 | . . . . . . 7 # | |
8 | 2, 3, 7 | sylancl 409 | . . . . . 6 # |
9 | 6, 8 | syl5ibr 155 | . . . . 5 # |
10 | simplll 522 | . . . . . . 7 # | |
11 | simplrl 524 | . . . . . . 7 # | |
12 | recn 7746 | . . . . . . . . . . . . . 14 | |
13 | recn 7746 | . . . . . . . . . . . . . . 15 | |
14 | mulap0r 8370 | . . . . . . . . . . . . . . 15 # # # | |
15 | 13, 14 | syl3an1 1249 | . . . . . . . . . . . . . 14 # # # |
16 | 12, 15 | syl3an2 1250 | . . . . . . . . . . . . 13 # # # |
17 | 16 | 3expia 1183 | . . . . . . . . . . . 12 # # # |
18 | 17 | ad2ant2r 500 | . . . . . . . . . . 11 # # # |
19 | 18 | imp 123 | . . . . . . . . . 10 # # # |
20 | 19 | simpld 111 | . . . . . . . . 9 # # |
21 | reaplt 8343 | . . . . . . . . . . 11 # | |
22 | 3, 21 | mpan2 421 | . . . . . . . . . 10 # |
23 | 22 | ad3antrrr 483 | . . . . . . . . 9 # # |
24 | 20, 23 | mpbid 146 | . . . . . . . 8 # |
25 | lenlt 7833 | . . . . . . . . . . . 12 | |
26 | 3, 25 | mpan 420 | . . . . . . . . . . 11 |
27 | 26 | biimpa 294 | . . . . . . . . . 10 |
28 | 27 | ad2antrr 479 | . . . . . . . . 9 # |
29 | biorf 733 | . . . . . . . . 9 | |
30 | 28, 29 | syl 14 | . . . . . . . 8 # |
31 | 24, 30 | mpbird 166 | . . . . . . 7 # |
32 | 19 | simprd 113 | . . . . . . . . 9 # # |
33 | reaplt 8343 | . . . . . . . . . . . 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 7833 | . . . . . . . . . . . 12 | |
39 | 3, 38 | mpan 420 | . . . . . . . . . . 11 |
40 | 39 | biimpa 294 | . . . . . . . . . 10 |
41 | 40 | ad2antlr 480 | . . . . . . . . 9 # |
42 | biorf 733 | . . . . . . . . 9 | |
43 | 41, 42 | syl 14 | . . . . . . . 8 # |
44 | 37, 43 | mpbird 166 | . . . . . . 7 # |
45 | 10, 11, 31, 44 | mulgt0d 7878 | . . . . . 6 # |
46 | 45 | ex 114 | . . . . 5 # |
47 | 9, 46 | syld 45 | . . . 4 |
48 | 47 | ancld 323 | . . 3 |
49 | 5, 48 | mtod 652 | . 2 |
50 | lenlt 7833 | . . 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 697 wcel 1480 class class class wbr 3924 (class class class)co 5767 cc 7611 cr 7612 cc0 7613 cmul 7618 clt 7793 cle 7794 # cap 8336 |
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-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 |
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-po 4213 df-iso 4214 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-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 |
This theorem is referenced by: mulge0i 8375 mulge0d 8376 ge0mulcl 9758 expge0 10322 bernneq 10405 sqrtmul 10800 amgm2 10883 |
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