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Theorem prodge0 9128
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 527 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> A e. RR )
2 simplr 529 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> B e. RR )
32renegcld 8653 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> -u B e. RR )
4 simprl 531 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> 0 < A )
5 simprr 533 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> 0 < -u B )
61, 3, 4, 5mulgt0d 8396 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> 0 < ( A x. -u B ) )
71recnd 8302 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> A e. CC )
82recnd 8302 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> B e. CC )
97, 8mulneg2d 8685 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> ( A x. -u B ) = -u ( A x. B ) )
106, 9breqtrd 4135 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> 0 < -u ( A x. B ) )
1110expr 375 . . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> ( 0 < -u B -> 0 < -u ( A x. B ) ) )
12 simplr 529 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> B e. RR )
1312lt0neg1d 8789 . . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> ( B < 0 <-> 0 < -u B ) )
14 simpll 527 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> A e. RR )
1514, 12remulcld 8304 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> ( A x. B ) e. RR )
1615lt0neg1d 8789 . . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> ( ( A x. B ) < 0 <-> 0 < -u ( A x. B ) ) )
1711, 13, 163imtr4d 203 . . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> ( B < 0 -> ( A x. B ) < 0 ) )
1817con3d 636 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> ( -. ( A x. B ) < 0 -> -. B < 0 ) )
19 0red 8275 . . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> 0 e. RR )
2019, 15lenltd 8391 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> ( 0 <_ ( A x. B ) <-> -. ( A x. B ) < 0 ) )
2119, 12lenltd 8391 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> ( 0 <_ B <-> -. B < 0 ) )
2218, 20, 213imtr4d 203 . 2 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> ( 0 <_ ( A x. B ) -> 0 <_ B ) )
2322impr 379 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 104   e. wcel 2203   class class class wbr 4109  (class class class)co 6050   RRcr 8126   0cc0 8127   x. cmul 8132   < clt 8308   <_ cle 8309   -ucneg 8445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltadd 8243  ax-pre-mulgt0 8244
This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447
This theorem is referenced by:  prodge02  9129  prodge0i  9183  oexpneg  12563  evennn02n  12568
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