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Theorem prodge0 8612
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 518 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  A  e.  RR )
2 simplr 519 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  B  e.  RR )
32renegcld 8142 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  -u B  e.  RR )
4 simprl 520 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
0  <  A )
5 simprr 521 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
0  <  -u B )
61, 3, 4, 5mulgt0d 7885 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
0  <  ( A  x.  -u B ) )
71recnd 7794 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  A  e.  CC )
82recnd 7794 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  B  e.  CC )
97, 8mulneg2d 8174 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
( A  x.  -u B
)  =  -u ( A  x.  B )
)
106, 9breqtrd 3954 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
0  <  -u ( A  x.  B ) )
1110expr 372 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( 0  <  -u B  ->  0  <  -u ( A  x.  B ) ) )
12 simplr 519 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  B  e.  RR )
1312lt0neg1d 8277 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( B  <  0  <->  0  <  -u B
) )
14 simpll 518 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  A  e.  RR )
1514, 12remulcld 7796 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( A  x.  B )  e.  RR )
1615lt0neg1d 8277 . . . . 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 620 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( -.  ( A  x.  B
)  <  0  ->  -.  B  <  0 ) )
19 0red 7767 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  0  e.  RR )
2019, 15lenltd 7880 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( 0  <_  ( A  x.  B )  <->  -.  ( A  x.  B )  <  0 ) )
2119, 12lenltd 7880 . . 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 376 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 1480   class class class wbr 3929  (class class class)co 5774   RRcr 7619   0cc0 7620    x. cmul 7625    < clt 7800    <_ cle 7801   -ucneg 7934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltadd 7736  ax-pre-mulgt0 7737
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936
This theorem is referenced by:  prodge02  8613  prodge0i  8667  oexpneg  11574  evennn02n  11579
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