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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 481 | . . . . . . . 8 | |
2 | simplr 482 | . . . . . . . . 9 | |
3 | 2 | renegcld 7378 | . . . . . . . 8 |
4 | simprl 483 | . . . . . . . 8 | |
5 | simprr 484 | . . . . . . . 8 | |
6 | 1, 3, 4, 5 | mulgt0d 7137 | . . . . . . 7 |
7 | 1 | recnd 7054 | . . . . . . . 8 |
8 | 2 | recnd 7054 | . . . . . . . 8 |
9 | 7, 8 | mulneg2d 7409 | . . . . . . 7 |
10 | 6, 9 | breqtrd 3788 | . . . . . 6 |
11 | 10 | expr 357 | . . . . 5 |
12 | simplr 482 | . . . . . 6 | |
13 | 12 | lt0neg1d 7507 | . . . . 5 |
14 | simpll 481 | . . . . . . 7 | |
15 | 14, 12 | remulcld 7056 | . . . . . 6 |
16 | 15 | lt0neg1d 7507 | . . . . 5 |
17 | 11, 13, 16 | 3imtr4d 192 | . . . 4 |
18 | 17 | con3d 561 | . . 3 |
19 | 0red 7028 | . . . 4 | |
20 | 19, 15 | lenltd 7134 | . . 3 |
21 | 19, 12 | lenltd 7134 | . . 3 |
22 | 18, 20, 21 | 3imtr4d 192 | . 2 |
23 | 22 | impr 361 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wcel 1393 class class class wbr 3764 (class class class)co 5512 cr 6888 cc0 6889 cmul 6894 clt 7060 cle 7061 cneg 7183 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-mulrcl 6983 ax-addcom 6984 ax-mulcom 6985 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 ax-pre-ltadd 7000 ax-pre-mulgt0 7001 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 |
This theorem is referenced by: prodge02 7821 prodge0i 7875 |
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