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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 8166 | . . . . . 6 |
3 | simplll 523 | . . . . . . 7 | |
4 | 3 | renegcld 8166 | . . . . . 6 |
5 | simplr 520 | . . . . . . . 8 | |
6 | 5 | lt0neg1d 8301 | . . . . . . 7 |
7 | 6 | biimpa 294 | . . . . . 6 |
8 | simprr 522 | . . . . . . . . 9 | |
9 | simpll 519 | . . . . . . . . . . 11 | |
10 | 9 | recnd 7818 | . . . . . . . . . 10 |
11 | 5 | recnd 7818 | . . . . . . . . . 10 |
12 | 10, 11 | mul2negd 8199 | . . . . . . . . 9 |
13 | 8, 12 | breqtrrd 3964 | . . . . . . . 8 |
14 | 10 | negcld 8084 | . . . . . . . . 9 |
15 | 11 | negcld 8084 | . . . . . . . . 9 |
16 | 14, 15 | mulcomd 7811 | . . . . . . . 8 |
17 | 13, 16 | breqtrd 3962 | . . . . . . 7 |
18 | 17 | adantr 274 | . . . . . 6 |
19 | prodgt0gt0 8633 | . . . . . 6 | |
20 | 2, 4, 7, 18, 19 | syl22anc 1218 | . . . . 5 |
21 | 3 | lt0neg1d 8301 | . . . . 5 |
22 | 20, 21 | mpbird 166 | . . . 4 |
23 | simplrl 525 | . . . . 5 | |
24 | 0red 7791 | . . . . . 6 | |
25 | 24, 3 | lenltd 7904 | . . . . 5 |
26 | 23, 25 | mpbid 146 | . . . 4 |
27 | 22, 26 | pm2.65da 651 | . . 3 |
28 | 0red 7791 | . . . 4 | |
29 | 28, 5 | lenltd 7904 | . . 3 |
30 | 27, 29 | mpbird 166 | . 2 |
31 | 9, 5 | remulcld 7820 | . . . . 5 |
32 | 31, 8 | gt0ap0d 8415 | . . . 4 # |
33 | 10, 11, 32 | mulap0bbd 8445 | . . 3 # |
34 | 0cnd 7783 | . . . 4 | |
35 | apsym 8392 | . . . 4 # # | |
36 | 11, 34, 35 | syl2anc 409 | . . 3 # # |
37 | 33, 36 | mpbid 146 | . 2 # |
38 | ltleap 8418 | . . 3 # | |
39 | 28, 5, 38 | syl2anc 409 | . 2 # |
40 | 30, 37, 39 | mpbir2and 929 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wcel 1481 class class class wbr 3937 (class class class)co 5782 cc 7642 cr 7643 cc0 7644 cmul 7649 clt 7824 cle 7825 cneg 7958 # cap 8367 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-po 4226 df-iso 4227 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 |
This theorem is referenced by: prodgt02 8635 prodgt0i 8690 evennn2n 11616 |
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