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Mirrors > Home > ILE Home > Th. List > prodge0 | Unicode version |
Description: Infer that a multiplicand is nonnegative from a positive multiplier and nonnegative product. (Contributed by NM, 2-Jul-2005.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
prodge0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 503 | . . . . . . . 8 | |
2 | simplr 504 | . . . . . . . . 9 | |
3 | 2 | renegcld 8110 | . . . . . . . 8 |
4 | simprl 505 | . . . . . . . 8 | |
5 | simprr 506 | . . . . . . . 8 | |
6 | 1, 3, 4, 5 | mulgt0d 7853 | . . . . . . 7 |
7 | 1 | recnd 7762 | . . . . . . . 8 |
8 | 2 | recnd 7762 | . . . . . . . 8 |
9 | 7, 8 | mulneg2d 8142 | . . . . . . 7 |
10 | 6, 9 | breqtrd 3924 | . . . . . 6 |
11 | 10 | expr 372 | . . . . 5 |
12 | simplr 504 | . . . . . 6 | |
13 | 12 | lt0neg1d 8245 | . . . . 5 |
14 | simpll 503 | . . . . . . 7 | |
15 | 14, 12 | remulcld 7764 | . . . . . 6 |
16 | 15 | lt0neg1d 8245 | . . . . 5 |
17 | 11, 13, 16 | 3imtr4d 202 | . . . 4 |
18 | 17 | con3d 605 | . . 3 |
19 | 0red 7735 | . . . 4 | |
20 | 19, 15 | lenltd 7848 | . . 3 |
21 | 19, 12 | lenltd 7848 | . . 3 |
22 | 18, 20, 21 | 3imtr4d 202 | . 2 |
23 | 22 | impr 376 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wcel 1465 class class class wbr 3899 (class class class)co 5742 cr 7587 cc0 7588 cmul 7593 clt 7768 cle 7769 cneg 7902 |
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-distr 7692 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 |
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-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 |
This theorem is referenced by: prodge02 8581 prodge0i 8635 oexpneg 11501 evennn02n 11506 |
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