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Theorem prodge0 9093
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 8618 . . . . . . . 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 8361 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> 0 < ( A x. -u B ) )
71recnd 8267 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> A e. CC )
82recnd 8267 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> B e. CC )
97, 8mulneg2d 8650 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> ( A x. -u B ) = -u ( A x. B ) )
106, 9breqtrd 4119 . . . . . 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 8754 . . . . 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 8269 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> ( A x. B ) e. RR )
1615lt0neg1d 8754 . . . . 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 8240 . . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> 0 e. RR )
2019, 15lenltd 8356 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> ( 0 <_ ( A x. B ) <-> -. ( A x. B ) < 0 ) )
2119, 12lenltd 8356 . . 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 2202   class class class wbr 4093  (class class class)co 6028   RRcr 8091   0cc0 8092   x. cmul 8097   < clt 8273   <_ cle 8274   -ucneg 8410
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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltadd 8208  ax-pre-mulgt0 8209
This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412
This theorem is referenced by:  prodge02  9094  prodge0i  9148  oexpneg  12518  evennn02n  12523
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