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Mirrors > Home > ILE Home > Th. List > prodgt0 | Unicode version |
Description: Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 24-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
prodgt0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpllr 524 | . . . . . . 7 | |
2 | 1 | renegcld 8270 | . . . . . 6 |
3 | simplll 523 | . . . . . . 7 | |
4 | 3 | renegcld 8270 | . . . . . 6 |
5 | simplr 520 | . . . . . . . 8 | |
6 | 5 | lt0neg1d 8405 | . . . . . . 7 |
7 | 6 | biimpa 294 | . . . . . 6 |
8 | simprr 522 | . . . . . . . . 9 | |
9 | simpll 519 | . . . . . . . . . . 11 | |
10 | 9 | recnd 7919 | . . . . . . . . . 10 |
11 | 5 | recnd 7919 | . . . . . . . . . 10 |
12 | 10, 11 | mul2negd 8303 | . . . . . . . . 9 |
13 | 8, 12 | breqtrrd 4005 | . . . . . . . 8 |
14 | 10 | negcld 8188 | . . . . . . . . 9 |
15 | 11 | negcld 8188 | . . . . . . . . 9 |
16 | 14, 15 | mulcomd 7912 | . . . . . . . 8 |
17 | 13, 16 | breqtrd 4003 | . . . . . . 7 |
18 | 17 | adantr 274 | . . . . . 6 |
19 | prodgt0gt0 8738 | . . . . . 6 | |
20 | 2, 4, 7, 18, 19 | syl22anc 1228 | . . . . 5 |
21 | 3 | lt0neg1d 8405 | . . . . 5 |
22 | 20, 21 | mpbird 166 | . . . 4 |
23 | simplrl 525 | . . . . 5 | |
24 | 0red 7892 | . . . . . 6 | |
25 | 24, 3 | lenltd 8008 | . . . . 5 |
26 | 23, 25 | mpbid 146 | . . . 4 |
27 | 22, 26 | pm2.65da 651 | . . 3 |
28 | 0red 7892 | . . . 4 | |
29 | 28, 5 | lenltd 8008 | . . 3 |
30 | 27, 29 | mpbird 166 | . 2 |
31 | 9, 5 | remulcld 7921 | . . . . 5 |
32 | 31, 8 | gt0ap0d 8519 | . . . 4 # |
33 | 10, 11, 32 | mulap0bbd 8549 | . . 3 # |
34 | 0cnd 7884 | . . . 4 | |
35 | apsym 8496 | . . . 4 # # | |
36 | 11, 34, 35 | syl2anc 409 | . . 3 # # |
37 | 33, 36 | mpbid 146 | . 2 # |
38 | ltleap 8522 | . . 3 # | |
39 | 28, 5, 38 | syl2anc 409 | . 2 # |
40 | 30, 37, 39 | mpbir2and 933 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wcel 2135 class class class wbr 3977 (class class class)co 5837 cc 7743 cr 7744 cc0 7745 cmul 7750 clt 7925 cle 7926 cneg 8062 # cap 8471 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-cnex 7836 ax-resscn 7837 ax-1cn 7838 ax-1re 7839 ax-icn 7840 ax-addcl 7841 ax-addrcl 7842 ax-mulcl 7843 ax-mulrcl 7844 ax-addcom 7845 ax-mulcom 7846 ax-addass 7847 ax-mulass 7848 ax-distr 7849 ax-i2m1 7850 ax-0lt1 7851 ax-1rid 7852 ax-0id 7853 ax-rnegex 7854 ax-precex 7855 ax-cnre 7856 ax-pre-ltirr 7857 ax-pre-ltwlin 7858 ax-pre-lttrn 7859 ax-pre-apti 7860 ax-pre-ltadd 7861 ax-pre-mulgt0 7862 ax-pre-mulext 7863 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2724 df-sbc 2948 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-br 3978 df-opab 4039 df-id 4266 df-po 4269 df-iso 4270 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-iota 5148 df-fun 5185 df-fv 5191 df-riota 5793 df-ov 5840 df-oprab 5841 df-mpo 5842 df-pnf 7927 df-mnf 7928 df-xr 7929 df-ltxr 7930 df-le 7931 df-sub 8063 df-neg 8064 df-reap 8465 df-ap 8472 df-div 8561 |
This theorem is referenced by: prodgt02 8740 prodgt0i 8795 evennn2n 11806 |
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