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Theorem prodgt0 8827
Description: Infer that a multiplicand is positive from a nonnegative multiplier and positive product. (Contributed by NM, 24-Apr-2005.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
prodgt0  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  <  B )

Proof of Theorem prodgt0
StepHypRef Expression
1 simpllr 534 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  B  e.  RR )
21renegcld 8355 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  -u B  e.  RR )
3 simplll 533 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  A  e.  RR )
43renegcld 8355 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  -u A  e.  RR )
5 simplr 528 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  B  e.  RR )
65lt0neg1d 8490 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  ( B  <  0  <->  0  <  -u B ) )
76biimpa 296 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  0  <  -u B )
8 simprr 531 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  <  ( A  x.  B
) )
9 simpll 527 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  A  e.  RR )
109recnd 8004 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  A  e.  CC )
115recnd 8004 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  B  e.  CC )
1210, 11mul2negd 8388 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  ( -u A  x.  -u B
)  =  ( A  x.  B ) )
138, 12breqtrrd 4046 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  <  ( -u A  x.  -u B ) )
1410negcld 8273 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  -u A  e.  CC )
1511negcld 8273 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  -u B  e.  CC )
1614, 15mulcomd 7997 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  ( -u A  x.  -u B
)  =  ( -u B  x.  -u A ) )
1713, 16breqtrd 4044 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  <  ( -u B  x.  -u A ) )
1817adantr 276 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  0  <  (
-u B  x.  -u A
) )
19 prodgt0gt0 8826 . . . . . 6  |-  ( ( ( -u B  e.  RR  /\  -u A  e.  RR )  /\  (
0  <  -u B  /\  0  <  ( -u B  x.  -u A ) ) )  ->  0  <  -u A )
202, 4, 7, 18, 19syl22anc 1250 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  0  <  -u A )
213lt0neg1d 8490 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  ( A  <  0  <->  0  <  -u A
) )
2220, 21mpbird 167 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  A  <  0 )
23 simplrl 535 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  0  <_  A )
24 0red 7976 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  0  e.  RR )
2524, 3lenltd 8093 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  ( 0  <_  A  <->  -.  A  <  0 ) )
2623, 25mpbid 147 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  -.  A  <  0 )
2722, 26pm2.65da 662 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  -.  B  <  0 )
28 0red 7976 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  e.  RR )
2928, 5lenltd 8093 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  (
0  <_  B  <->  -.  B  <  0 ) )
3027, 29mpbird 167 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  <_  B )
319, 5remulcld 8006 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  ( A  x.  B )  e.  RR )
3231, 8gt0ap0d 8604 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  ( A  x.  B ) #  0 )
3310, 11, 32mulap0bbd 8635 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  B #  0 )
34 0cnd 7968 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  e.  CC )
35 apsym 8581 . . . 4  |-  ( ( B  e.  CC  /\  0  e.  CC )  ->  ( B #  0  <->  0 #  B ) )
3611, 34, 35syl2anc 411 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  ( B #  0  <->  0 #  B )
)
3733, 36mpbid 147 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0 #  B )
38 ltleap 8607 . . 3  |-  ( ( 0  e.  RR  /\  B  e.  RR )  ->  ( 0  <  B  <->  ( 0  <_  B  /\  0 #  B ) ) )
3928, 5, 38syl2anc 411 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  (
0  <  B  <->  ( 0  <_  B  /\  0 #  B ) ) )
4030, 37, 39mpbir2and 946 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    <-> wb 105    e. wcel 2160   class class class wbr 4018  (class class class)co 5891   CCcc 7827   RRcr 7828   0cc0 7829    x. cmul 7834    < clt 8010    <_ cle 8011   -ucneg 8147   # cap 8556
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-mulrcl 7928  ax-addcom 7929  ax-mulcom 7930  ax-addass 7931  ax-mulass 7932  ax-distr 7933  ax-i2m1 7934  ax-0lt1 7935  ax-1rid 7936  ax-0id 7937  ax-rnegex 7938  ax-precex 7939  ax-cnre 7940  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943  ax-pre-apti 7944  ax-pre-ltadd 7945  ax-pre-mulgt0 7946  ax-pre-mulext 7947
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4308  df-po 4311  df-iso 4312  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-iota 5193  df-fun 5233  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-sub 8148  df-neg 8149  df-reap 8550  df-ap 8557  df-div 8648
This theorem is referenced by:  prodgt02  8828  prodgt0i  8883  evennn2n  11906
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