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Theorem prodgt0 8622
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 523 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  B  e.  RR )
21renegcld 8154 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  -u B  e.  RR )
3 simplll 522 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  A  e.  RR )
43renegcld 8154 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  -u A  e.  RR )
5 simplr 519 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  B  e.  RR )
65lt0neg1d 8289 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  ( B  <  0  <->  0  <  -u B ) )
76biimpa 294 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  0  <  -u B )
8 simprr 521 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  <  ( A  x.  B
) )
9 simpll 518 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  A  e.  RR )
109recnd 7806 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  A  e.  CC )
115recnd 7806 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  B  e.  CC )
1210, 11mul2negd 8187 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  ( -u A  x.  -u B
)  =  ( A  x.  B ) )
138, 12breqtrrd 3956 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  <  ( -u A  x.  -u B ) )
1410negcld 8072 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  -u A  e.  CC )
1511negcld 8072 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  -u B  e.  CC )
1614, 15mulcomd 7799 . . . . . . . 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 3954 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  <  ( -u B  x.  -u A ) )
1817adantr 274 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  0  <  (
-u B  x.  -u A
) )
19 prodgt0gt0 8621 . . . . . 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 1217 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  0  <  -u A )
213lt0neg1d 8289 . . . . 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 524 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  0  <_  A )
24 0red 7779 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  0  e.  RR )
2524, 3lenltd 7892 . . . . 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 650 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  -.  B  <  0 )
28 0red 7779 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  e.  RR )
2928, 5lenltd 7892 . . 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 7808 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  ( A  x.  B )  e.  RR )
3231, 8gt0ap0d 8403 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  ( A  x.  B ) #  0 )
3310, 11, 32mulap0bbd 8433 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  B #  0 )
34 0cnd 7771 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  e.  CC )
35 apsym 8380 . . . 4  |-  ( ( B  e.  CC  /\  0  e.  CC )  ->  ( B #  0  <->  0 #  B ) )
3611, 34, 35syl2anc 408 . . 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 8406 . . 3  |-  ( ( 0  e.  RR  /\  B  e.  RR )  ->  ( 0  <  B  <->  ( 0  <_  B  /\  0 #  B ) ) )
3928, 5, 38syl2anc 408 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  (
0  <  B  <->  ( 0  <_  B  /\  0 #  B ) ) )
4030, 37, 39mpbir2and 928 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 1480   class class class wbr 3929  (class class class)co 5774   CCcc 7630   RRcr 7631   0cc0 7632    x. cmul 7637    < clt 7812    <_ cle 7813   -ucneg 7946   # cap 8355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445
This theorem is referenced by:  prodgt02  8623  prodgt0i  8678  evennn2n  11591
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