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Theorem prodgt0 8739
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 524 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  B  e.  RR )
21renegcld 8270 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  -u B  e.  RR )
3 simplll 523 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  A  e.  RR )
43renegcld 8270 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  -u A  e.  RR )
5 simplr 520 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  B  e.  RR )
65lt0neg1d 8405 . . . . . . 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 522 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  <  ( A  x.  B
) )
9 simpll 519 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  A  e.  RR )
109recnd 7919 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  A  e.  CC )
115recnd 7919 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  B  e.  CC )
1210, 11mul2negd 8303 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  ( -u A  x.  -u B
)  =  ( A  x.  B ) )
138, 12breqtrrd 4005 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  <  ( -u A  x.  -u B ) )
1410negcld 8188 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  -u A  e.  CC )
1511negcld 8188 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  -u B  e.  CC )
1614, 15mulcomd 7912 . . . . . . . 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 4003 . . . . . . 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 8738 . . . . . 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 1228 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  0  <  -u A )
213lt0neg1d 8405 . . . . 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 525 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  0  <_  A )
24 0red 7892 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  0  <  ( A  x.  B ) ) )  /\  B  <  0
)  ->  0  e.  RR )
2524, 3lenltd 8008 . . . . 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 651 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  -.  B  <  0 )
28 0red 7892 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  e.  RR )
2928, 5lenltd 8008 . . 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 7921 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  ( A  x.  B )  e.  RR )
3231, 8gt0ap0d 8519 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  ( A  x.  B ) #  0 )
3310, 11, 32mulap0bbd 8549 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  B #  0 )
34 0cnd 7884 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  0  e.  CC )
35 apsym 8496 . . . 4  |-  ( ( B  e.  CC  /\  0  e.  CC )  ->  ( B #  0  <->  0 #  B ) )
3611, 34, 35syl2anc 409 . . 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 8522 . . 3  |-  ( ( 0  e.  RR  /\  B  e.  RR )  ->  ( 0  <  B  <->  ( 0  <_  B  /\  0 #  B ) ) )
3928, 5, 38syl2anc 409 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  0  <  ( A  x.  B )
) )  ->  (
0  <  B  <->  ( 0  <_  B  /\  0 #  B ) ) )
4030, 37, 39mpbir2and 933 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 2135   class class class wbr 3977  (class class class)co 5837   CCcc 7743   RRcr 7744   0cc0 7745    x. cmul 7750    < clt 7925    <_ cle 7926   -ucneg 8062   # cap 8471
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182  ax-un 4406  ax-setind 4509  ax-cnex 7836  ax-resscn 7837  ax-1cn 7838  ax-1re 7839  ax-icn 7840  ax-addcl 7841  ax-addrcl 7842  ax-mulcl 7843  ax-mulrcl 7844  ax-addcom 7845  ax-mulcom 7846  ax-addass 7847  ax-mulass 7848  ax-distr 7849  ax-i2m1 7850  ax-0lt1 7851  ax-1rid 7852  ax-0id 7853  ax-rnegex 7854  ax-precex 7855  ax-cnre 7856  ax-pre-ltirr 7857  ax-pre-ltwlin 7858  ax-pre-lttrn 7859  ax-pre-apti 7860  ax-pre-ltadd 7861  ax-pre-mulgt0 7862  ax-pre-mulext 7863
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2724  df-sbc 2948  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-br 3978  df-opab 4039  df-id 4266  df-po 4269  df-iso 4270  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-iota 5148  df-fun 5185  df-fv 5191  df-riota 5793  df-ov 5840  df-oprab 5841  df-mpo 5842  df-pnf 7927  df-mnf 7928  df-xr 7929  df-ltxr 7930  df-le 7931  df-sub 8063  df-neg 8064  df-reap 8465  df-ap 8472  df-div 8561
This theorem is referenced by:  prodgt02  8740  prodgt0i  8795  evennn2n  11806
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