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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 497 |
. . . . . . . 8
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2 | simplr 498 |
. . . . . . . . 9
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3 | 2 | renegcld 7912 |
. . . . . . . 8
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4 | simprl 499 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | simprr 500 |
. . . . . . . 8
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6 | 1, 3, 4, 5 | mulgt0d 7660 |
. . . . . . 7
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7 | 1 | recnd 7570 |
. . . . . . . 8
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8 | 2 | recnd 7570 |
. . . . . . . 8
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9 | 7, 8 | mulneg2d 7944 |
. . . . . . 7
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10 | 6, 9 | breqtrd 3875 |
. . . . . 6
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11 | 10 | expr 368 |
. . . . 5
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12 | simplr 498 |
. . . . . 6
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13 | 12 | lt0neg1d 8047 |
. . . . 5
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14 | simpll 497 |
. . . . . . 7
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15 | 14, 12 | remulcld 7572 |
. . . . . 6
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16 | 15 | lt0neg1d 8047 |
. . . . 5
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17 | 11, 13, 16 | 3imtr4d 202 |
. . . 4
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18 | 17 | con3d 597 |
. . 3
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19 | 0red 7543 |
. . . 4
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20 | 19, 15 | lenltd 7655 |
. . 3
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21 | 19, 12 | lenltd 7655 |
. . 3
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22 | 18, 20, 21 | 3imtr4d 202 |
. 2
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23 | 22 | impr 372 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7490 ax-resscn 7491 ax-1cn 7492 ax-1re 7493 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-mulrcl 7498 ax-addcom 7499 ax-mulcom 7500 ax-addass 7501 ax-distr 7503 ax-i2m1 7504 ax-0id 7507 ax-rnegex 7508 ax-cnre 7510 ax-pre-ltadd 7515 ax-pre-mulgt0 7516 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-id 4129 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-pnf 7578 df-mnf 7579 df-xr 7580 df-ltxr 7581 df-le 7582 df-sub 7709 df-neg 7710 |
This theorem is referenced by: prodge02 8370 prodge0i 8424 oexpneg 11209 evennn02n 11214 |
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