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Theorem prodge0 8962
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 528 . . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> B e. RR )
32renegcld 8487 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> -u B e. RR )
4 simprl 529 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> 0 < A )
5 simprr 531 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> 0 < -u B )
61, 3, 4, 5mulgt0d 8230 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> 0 < ( A x. -u B ) )
71recnd 8136 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> A e. CC )
82recnd 8136 . . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> B e. CC )
97, 8mulneg2d 8519 . . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < -u B ) ) -> ( A x. -u B ) = -u ( A x. B ) )
106, 9breqtrd 4085 . . . . . 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 528 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> B e. RR )
1312lt0neg1d 8623 . . . . 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 8138 . . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> ( A x. B ) e. RR )
1615lt0neg1d 8623 . . . . 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 632 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> ( -. ( A x. B ) < 0 -> -. B < 0 ) )
19 0red 8108 . . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> 0 e. RR )
2019, 15lenltd 8225 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ 0 < A ) -> ( 0 <_ ( A x. B ) <-> -. ( A x. B ) < 0 ) )
2119, 12lenltd 8225 . . 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 2178   class class class wbr 4059  (class class class)co 5967   RRcr 7959   0cc0 7960   x. cmul 7965   < clt 8142   <_ cle 8143   -ucneg 8279
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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltadd 8076  ax-pre-mulgt0 8077
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281
This theorem is referenced by:  prodge02  8963  prodge0i  9017  oexpneg  12303  evennn02n  12308
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