ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  prodgt0 Unicode version

Theorem prodgt0 8468
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 504 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  B  e.  RR )
21renegcld 8009 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  -u B  e.  RR )
3 simplll 503 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  A  e.  RR )
43renegcld 8009 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  -u A  e.  RR )
5 simplr 500 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  B  e.  RR )
65lt0neg1d 8144 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  ( B  <  0  <->  0  <  -u B ) )
76biimpa 292 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  0  <  -u B )
8 simprr 502 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  <  ( A  x.  B
) )
9 simpll 499 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  A  e.  RR )
109recnd 7666 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  A  e.  CC )
115recnd 7666 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  B  e.  CC )
1210, 11mul2negd 8042 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  ( -u A  x.  -u B
)  =  ( A  x.  B ) )
138, 12breqtrrd 3901 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  <  ( -u A  x.  -u B ) )
1410negcld 7931 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  -u A  e.  CC )
1511negcld 7931 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  -u B  e.  CC )
1614, 15mulcomd 7659 . . . . . . . 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 3899 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  <  ( -u B  x.  -u A ) )
1817adantr 272 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  0  <  (
-u B  x.  -u A
) )
19 prodgt0gt0 8467 . . . . . 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 1185 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  0  <  -u A )
213lt0neg1d 8144 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  ( A  <  0  <->  0  <  -u A
) )
2220, 21mpbird 166 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  A  <  0 )
23 simplrl 505 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  0  <_  A )
24 0red 7639 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  0  e.  RR )
2524, 3lenltd 7751 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  ( 0  <_  A  <->  -.  A  <  0 ) )
2623, 25mpbid 146 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  -.  A  <  0 )
2722, 26pm2.65da 628 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  -.  B  <  0 )
28 0red 7639 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  e.  RR )
2928, 5lenltd 7751 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  (
0  <_  B  <->  -.  B  <  0 ) )
3027, 29mpbird 166 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  <_  B )
319, 5remulcld 7668 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  ( A  x.  B )  e.  RR )
3231, 8gt0ap0d 8257 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  ( A  x.  B ) #  0 )
3310, 11, 32mulap0bbd 8282 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  B #  0 )
34 0cnd 7631 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  e.  CC )
35 apsym 8234 . . . 4  |-  ( ( B  e.  CC  /\  0  e.  CC )  ->  ( B #  0  <->  0 #  B ) )
3611, 34, 35syl2anc 406 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  ( B #  0  <->  0 #  B )
)
3733, 36mpbid 146 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0 #  B )
38 ltleap 8259 . . 3  |-  ( ( 0  e.  RR  /\  B  e.  RR )  ->  ( 0  <  B  <->  ( 0  <_  B  /\  0 #  B ) ) )
3928, 5, 38syl2anc 406 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  (
0  <  B  <->  ( 0  <_  B  /\  0 #  B ) ) )
4030, 37, 39mpbir2and 896 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 103    <-> wb 104    e. wcel 1448   class class class wbr 3875  (class class class)co 5706   CCcc 7498   RRcr 7499   0cc0 7500    x. cmul 7505    < clt 7672    <_ cle 7673   -ucneg 7805   # cap 8209
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-po 4156  df-iso 4157  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294
This theorem is referenced by:  prodgt02  8469  prodgt0i  8524  evennn2n  11375
  Copyright terms: Public domain W3C validator