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Theorem prodge0 9184
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 531 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> A e. RR )
2 simplr 533 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> B e. RR )
32renegcld 8707 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> -u B e. RR )
4 simprl 535 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> 0 < A )
5 simprr 537 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> 0 < -u B )
61, 3, 4, 5mulgt0d 8449 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> 0 < ( A x. -u B ) )
71recnd 8354 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> A e. CC )
82recnd 8354 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> B e. CC )
97, 8mulneg2d 8739 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> ( A x. -u B ) = -u ( A x. B ) )
106, 9breqtrd 4156 . . . . . 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 533 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> B e. RR )
1312lt0neg1d 8843 . . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> ( B < 0 <-> 0 < -u B ) )
14 simpll 531 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> A e. RR )
1514, 12remulcld 8356 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> ( A x. B ) e. RR )
1615lt0neg1d 8843 . . . . 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 640 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> ( -. ( A x. B ) < 0 -> -. B < 0 ) )
19 0red 8327 . . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> 0 e. RR )
2019, 15lenltd 8444 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> ( 0 <_ ( A x. B ) <-> -. ( A x. B ) < 0 ) )
2119, 12lenltd 8444 . . 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
This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   RRcr 8178   0cc0 8179   x. cmul 8184   < clt 8360   <_ cle 8361   -ucneg 8498
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltadd 8295  ax-pre-mulgt0 8296
This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500
This theorem is used by:  prodge02  9185  prodge0i  9239  oexpneg  12644  evennn02n  12649
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