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Theorem prodge0 8035
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 496 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  A  e.  RR )
2 simplr 497 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  B  e.  RR )
32renegcld 7587 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  -u B  e.  RR )
4 simprl 498 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
0  <  A )
5 simprr 499 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
0  <  -u B )
61, 3, 4, 5mulgt0d 7335 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
0  <  ( A  x.  -u B ) )
71recnd 7245 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  A  e.  CC )
82recnd 7245 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  B  e.  CC )
97, 8mulneg2d 7619 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
( A  x.  -u B
)  =  -u ( A  x.  B )
)
106, 9breqtrd 3830 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
0  <  -u ( A  x.  B ) )
1110expr 367 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( 0  <  -u B  ->  0  <  -u ( A  x.  B ) ) )
12 simplr 497 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  B  e.  RR )
1312lt0neg1d 7719 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( B  <  0  <->  0  <  -u B
) )
14 simpll 496 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  A  e.  RR )
1514, 12remulcld 7247 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( A  x.  B )  e.  RR )
1615lt0neg1d 7719 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( ( A  x.  B )  <  0  <->  0  <  -u ( A  x.  B )
) )
1711, 13, 163imtr4d 201 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( B  <  0  ->  ( A  x.  B )  <  0
) )
1817con3d 594 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( -.  ( A  x.  B
)  <  0  ->  -.  B  <  0 ) )
19 0red 7218 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  0  e.  RR )
2019, 15lenltd 7330 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( 0  <_  ( A  x.  B )  <->  -.  ( A  x.  B )  <  0 ) )
2119, 12lenltd 7330 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( 0  <_  B  <->  -.  B  <  0 ) )
2218, 20, 213imtr4d 201 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( 0  <_  ( A  x.  B )  ->  0  <_  B ) )
2322impr 371 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 102    e. wcel 1434   class class class wbr 3806  (class class class)co 5564   RRcr 7078   0cc0 7079    x. cmul 7084    < clt 7251    <_ cle 7252   -ucneg 7383
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-cnex 7165  ax-resscn 7166  ax-1cn 7167  ax-1re 7168  ax-icn 7169  ax-addcl 7170  ax-addrcl 7171  ax-mulcl 7172  ax-mulrcl 7173  ax-addcom 7174  ax-mulcom 7175  ax-addass 7176  ax-distr 7178  ax-i2m1 7179  ax-0id 7182  ax-rnegex 7183  ax-cnre 7185  ax-pre-ltadd 7190  ax-pre-mulgt0 7191
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2826  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-opab 3861  df-id 4077  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-iota 4918  df-fun 4955  df-fv 4961  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-pnf 7253  df-mnf 7254  df-xr 7255  df-ltxr 7256  df-le 7257  df-sub 7384  df-neg 7385
This theorem is referenced by:  prodge02  8036  prodge0i  8090  oexpneg  10468  evennn02n  10473
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