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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 7101 | . . . . 5 | |
2 | 1 | ad2ant2r 492 | . . . 4 |
3 | 0re 7119 | . . . 4 | |
4 | ltnsym2 7201 | . . . 4 | |
5 | 2, 3, 4 | sylancl 404 | . . 3 |
6 | orc 665 | . . . . . 6 | |
7 | reaplt 7688 | . . . . . . 7 # | |
8 | 2, 3, 7 | sylancl 404 | . . . . . 6 # |
9 | 6, 8 | syl5ibr 154 | . . . . 5 # |
10 | simplll 499 | . . . . . . 7 # | |
11 | simplrl 501 | . . . . . . 7 # | |
12 | recn 7106 | . . . . . . . . . . . . . 14 | |
13 | recn 7106 | . . . . . . . . . . . . . . 15 | |
14 | mulap0r 7715 | . . . . . . . . . . . . . . 15 # # # | |
15 | 13, 14 | syl3an1 1202 | . . . . . . . . . . . . . 14 # # # |
16 | 12, 15 | syl3an2 1203 | . . . . . . . . . . . . 13 # # # |
17 | 16 | 3expia 1140 | . . . . . . . . . . . 12 # # # |
18 | 17 | ad2ant2r 492 | . . . . . . . . . . 11 # # # |
19 | 18 | imp 122 | . . . . . . . . . 10 # # # |
20 | 19 | simpld 110 | . . . . . . . . 9 # # |
21 | reaplt 7688 | . . . . . . . . . . 11 # | |
22 | 3, 21 | mpan2 415 | . . . . . . . . . 10 # |
23 | 22 | ad3antrrr 475 | . . . . . . . . 9 # # |
24 | 20, 23 | mpbid 145 | . . . . . . . 8 # |
25 | lenlt 7187 | . . . . . . . . . . . 12 | |
26 | 3, 25 | mpan 414 | . . . . . . . . . . 11 |
27 | 26 | biimpa 290 | . . . . . . . . . 10 |
28 | 27 | ad2antrr 471 | . . . . . . . . 9 # |
29 | biorf 695 | . . . . . . . . 9 | |
30 | 28, 29 | syl 14 | . . . . . . . 8 # |
31 | 24, 30 | mpbird 165 | . . . . . . 7 # |
32 | 19 | simprd 112 | . . . . . . . . 9 # # |
33 | reaplt 7688 | . . . . . . . . . . . 12 # | |
34 | 3, 33 | mpan2 415 | . . . . . . . . . . 11 # |
35 | 34 | ad2antrl 473 | . . . . . . . . . 10 # |
36 | 35 | adantr 270 | . . . . . . . . 9 # # |
37 | 32, 36 | mpbid 145 | . . . . . . . 8 # |
38 | lenlt 7187 | . . . . . . . . . . . 12 | |
39 | 3, 38 | mpan 414 | . . . . . . . . . . 11 |
40 | 39 | biimpa 290 | . . . . . . . . . 10 |
41 | 40 | ad2antlr 472 | . . . . . . . . 9 # |
42 | biorf 695 | . . . . . . . . 9 | |
43 | 41, 42 | syl 14 | . . . . . . . 8 # |
44 | 37, 43 | mpbird 165 | . . . . . . 7 # |
45 | 10, 11, 31, 44 | mulgt0d 7232 | . . . . . 6 # |
46 | 45 | ex 113 | . . . . 5 # |
47 | 9, 46 | syld 44 | . . . 4 |
48 | 47 | ancld 318 | . . 3 |
49 | 5, 48 | mtod 621 | . 2 |
50 | lenlt 7187 | . . 3 | |
51 | 3, 2, 50 | sylancr 405 | . 2 |
52 | 49, 51 | mpbird 165 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wo 661 wcel 1433 class class class wbr 3785 (class class class)co 5532 cc 6979 cr 6980 cc0 6981 cmul 6986 clt 7153 cle 7154 # cap 7681 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 |
This theorem is referenced by: mulge0i 7720 mulge0d 7721 ge0mulcl 9005 expge0 9512 bernneq 9593 sqrtmul 9921 amgm2 10004 |
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