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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 534 |
. . . . . . 7
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2 | 1 | renegcld 8355 |
. . . . . 6
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3 | simplll 533 |
. . . . . . 7
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4 | 3 | renegcld 8355 |
. . . . . 6
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5 | simplr 528 |
. . . . . . . 8
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6 | 5 | lt0neg1d 8490 |
. . . . . . 7
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7 | 6 | biimpa 296 |
. . . . . 6
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8 | simprr 531 |
. . . . . . . . 9
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9 | simpll 527 |
. . . . . . . . . . 11
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10 | 9 | recnd 8004 |
. . . . . . . . . 10
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11 | 5 | recnd 8004 |
. . . . . . . . . 10
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12 | 10, 11 | mul2negd 8388 |
. . . . . . . . 9
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13 | 8, 12 | breqtrrd 4046 |
. . . . . . . 8
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14 | 10 | negcld 8273 |
. . . . . . . . 9
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15 | 11 | negcld 8273 |
. . . . . . . . 9
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16 | 14, 15 | mulcomd 7997 |
. . . . . . . 8
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17 | 13, 16 | breqtrd 4044 |
. . . . . . 7
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18 | 17 | adantr 276 |
. . . . . 6
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19 | prodgt0gt0 8826 |
. . . . . 6
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20 | 2, 4, 7, 18, 19 | syl22anc 1250 |
. . . . 5
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21 | 3 | lt0neg1d 8490 |
. . . . 5
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22 | 20, 21 | mpbird 167 |
. . . 4
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23 | simplrl 535 |
. . . . 5
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24 | 0red 7976 |
. . . . . 6
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25 | 24, 3 | lenltd 8093 |
. . . . 5
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26 | 23, 25 | mpbid 147 |
. . . 4
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27 | 22, 26 | pm2.65da 662 |
. . 3
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28 | 0red 7976 |
. . . 4
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29 | 28, 5 | lenltd 8093 |
. . 3
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30 | 27, 29 | mpbird 167 |
. 2
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31 | 9, 5 | remulcld 8006 |
. . . . 5
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32 | 31, 8 | gt0ap0d 8604 |
. . . 4
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33 | 10, 11, 32 | mulap0bbd 8635 |
. . 3
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34 | 0cnd 7968 |
. . . 4
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35 | apsym 8581 |
. . . 4
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36 | 11, 34, 35 | syl2anc 411 |
. . 3
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37 | 33, 36 | mpbid 147 |
. 2
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38 | ltleap 8607 |
. . 3
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39 | 28, 5, 38 | syl2anc 411 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
40 | 30, 37, 39 | mpbir2and 946 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-mulrcl 7928 ax-addcom 7929 ax-mulcom 7930 ax-addass 7931 ax-mulass 7932 ax-distr 7933 ax-i2m1 7934 ax-0lt1 7935 ax-1rid 7936 ax-0id 7937 ax-rnegex 7938 ax-precex 7939 ax-cnre 7940 ax-pre-ltirr 7941 ax-pre-ltwlin 7942 ax-pre-lttrn 7943 ax-pre-apti 7944 ax-pre-ltadd 7945 ax-pre-mulgt0 7946 ax-pre-mulext 7947 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4308 df-po 4311 df-iso 4312 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-iota 5193 df-fun 5233 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-pnf 8012 df-mnf 8013 df-xr 8014 df-ltxr 8015 df-le 8016 df-sub 8148 df-neg 8149 df-reap 8550 df-ap 8557 df-div 8648 |
This theorem is referenced by: prodgt02 8828 prodgt0i 8883 evennn2n 11906 |
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