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Theorem prodge0 8580
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 503 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  A  e.  RR )
2 simplr 504 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  B  e.  RR )
32renegcld 8110 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  -u B  e.  RR )
4 simprl 505 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
0  <  A )
5 simprr 506 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
0  <  -u B )
61, 3, 4, 5mulgt0d 7853 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
0  <  ( A  x.  -u B ) )
71recnd 7762 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  A  e.  CC )
82recnd 7762 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  B  e.  CC )
97, 8mulneg2d 8142 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
( A  x.  -u B
)  =  -u ( A  x.  B )
)
106, 9breqtrd 3924 . . . . . 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 504 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  B  e.  RR )
1312lt0neg1d 8245 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( B  <  0  <->  0  <  -u B
) )
14 simpll 503 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  A  e.  RR )
1514, 12remulcld 7764 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( A  x.  B )  e.  RR )
1615lt0neg1d 8245 . . . . 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 605 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( -.  ( A  x.  B
)  <  0  ->  -.  B  <  0 ) )
19 0red 7735 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  0  e.  RR )
2019, 15lenltd 7848 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( 0  <_  ( A  x.  B )  <->  -.  ( A  x.  B )  <  0 ) )
2119, 12lenltd 7848 . . 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 1465   class class class wbr 3899  (class class class)co 5742   RRcr 7587   0cc0 7588    x. cmul 7593    < clt 7768    <_ cle 7769   -ucneg 7902
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltadd 7704  ax-pre-mulgt0 7705
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904
This theorem is referenced by:  prodge02  8581  prodge0i  8635  oexpneg  11501  evennn02n  11506
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