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Theorem prodge0 8369
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.)
Assertion
Ref Expression
prodge0  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <_  ( A  x.  B ) ) )  ->  0  <_  B )

Proof of Theorem prodge0
StepHypRef Expression
1 simpll 497 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  A  e.  RR )
2 simplr 498 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  B  e.  RR )
32renegcld 7912 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  -u B  e.  RR )
4 simprl 499 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
0  <  A )
5 simprr 500 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
0  <  -u B )
61, 3, 4, 5mulgt0d 7660 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
0  <  ( A  x.  -u B ) )
71recnd 7570 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  A  e.  CC )
82recnd 7570 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  B  e.  CC )
97, 8mulneg2d 7944 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
( A  x.  -u B
)  =  -u ( A  x.  B )
)
106, 9breqtrd 3875 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
0  <  -u ( A  x.  B ) )
1110expr 368 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( 0  <  -u B  ->  0  <  -u ( A  x.  B ) ) )
12 simplr 498 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  B  e.  RR )
1312lt0neg1d 8047 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( B  <  0  <->  0  <  -u B
) )
14 simpll 497 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  A  e.  RR )
1514, 12remulcld 7572 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( A  x.  B )  e.  RR )
1615lt0neg1d 8047 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( ( A  x.  B )  <  0  <->  0  <  -u ( A  x.  B )
) )
1711, 13, 163imtr4d 202 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( B  <  0  ->  ( A  x.  B )  <  0
) )
1817con3d 597 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( -.  ( A  x.  B
)  <  0  ->  -.  B  <  0 ) )
19 0red 7543 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  0  e.  RR )
2019, 15lenltd 7655 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( 0  <_  ( A  x.  B )  <->  -.  ( A  x.  B )  <  0 ) )
2119, 12lenltd 7655 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( 0  <_  B  <->  -.  B  <  0 ) )
2218, 20, 213imtr4d 202 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( 0  <_  ( A  x.  B )  ->  0  <_  B ) )
2322impr 372 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <_  ( A  x.  B ) ) )  ->  0  <_  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    e. wcel 1439   class class class wbr 3851  (class class class)co 5666   RRcr 7403   0cc0 7404    x. cmul 7409    < clt 7576    <_ cle 7577   -ucneg 7708
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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