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Theorem prodge0 9817
Description: Infer that a multiplicand is nonnegative from a positive muliplier 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 731 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  A  e.  RR )
2 simplr 732 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  B  e.  RR )
32renegcld 9424 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  -u B  e.  RR )
4 simprl 733 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
0  <  A )
5 simprr 734 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
0  <  -u B )
61, 3, 4, 5mulgt0d 9185 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
0  <  ( A  x.  -u B ) )
71recnd 9074 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  A  e.  CC )
82recnd 9074 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  ->  B  e.  CC )
97, 8mulneg2d 9447 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
( A  x.  -u B
)  =  -u ( A  x.  B )
)
106, 9breqtrd 4200 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  -u B ) )  -> 
0  <  -u ( A  x.  B ) )
1110expr 599 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( 0  <  -u B  ->  0  <  -u ( A  x.  B ) ) )
12 simplr 732 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  B  e.  RR )
1312lt0neg1d 9556 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( B  <  0  <->  0  <  -u B
) )
14 simpll 731 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  A  e.  RR )
1514, 12remulcld 9076 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( A  x.  B )  e.  RR )
1615lt0neg1d 9556 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( ( A  x.  B )  <  0  <->  0  <  -u ( A  x.  B )
) )
1711, 13, 163imtr4d 260 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( B  <  0  ->  ( A  x.  B )  <  0
) )
1817con3d 127 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( -.  ( A  x.  B
)  <  0  ->  -.  B  <  0 ) )
19 0re 9051 . . . . 5  |-  0  e.  RR
2019a1i 11 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  0  e.  RR )
2120, 15lenltd 9179 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( 0  <_  ( A  x.  B )  <->  -.  ( A  x.  B )  <  0 ) )
2220, 12lenltd 9179 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( 0  <_  B  <->  -.  B  <  0 ) )
2318, 21, 223imtr4d 260 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  0  <  A
)  ->  ( 0  <_  ( A  x.  B )  ->  0  <_  B ) )
2423impr 603 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 359    e. wcel 1721   class class class wbr 4176  (class class class)co 6044   RRcr 8949   0cc0 8950    x. cmul 8955    < clt 9080    <_ cle 9081   -ucneg 9252
This theorem is referenced by:  prodge02  9818  prodge0i  9878  oexpneg  12870  nvge0  22120
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-po 4467  df-so 4468  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-riota 6512  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254
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