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Theorem prodgt0gt0 8827
Description: Infer that a multiplicand is positive from a positive multiplier and positive product. See prodgt0 8828 for the same theorem with  0  < 
A replaced by the weaker condition 
0  <_  A. (Contributed by Jim Kingdon, 29-Feb-2020.)
Assertion
Ref Expression
prodgt0gt0  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  B )

Proof of Theorem prodgt0gt0
StepHypRef Expression
1 simpll 527 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  A  e.  RR )
2 simplr 528 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  B  e.  RR )
31, 2remulcld 8007 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  ( A  x.  B )  e.  RR )
4 simprl 529 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  A )
51, 4gt0ap0d 8605 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  A #  0 )
61, 5rerecclapd 8810 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  (
1  /  A )  e.  RR )
7 simprr 531 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  ( A  x.  B
) )
8 recgt0 8826 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
0  <  ( 1  /  A ) )
98ad2ant2r 509 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  ( 1  /  A
) )
103, 6, 7, 9mulgt0d 8099 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  ( ( A  x.  B )  x.  (
1  /  A ) ) )
113recnd 8005 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  ( A  x.  B )  e.  CC )
121recnd 8005 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  A  e.  CC )
1311, 12, 5divrecapd 8769 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  (
( A  x.  B
)  /  A )  =  ( ( A  x.  B )  x.  ( 1  /  A
) ) )
14 simpr 110 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  RR )
1514recnd 8005 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  e.  CC )
1615adantr 276 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  B  e.  CC )
1716, 12, 5divcanap3d 8771 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  (
( A  x.  B
)  /  A )  =  B )
1813, 17eqtr3d 2224 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  (
( A  x.  B
)  x.  ( 1  /  A ) )  =  B )
1910, 18breqtrd 4044 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  < 
A  /\  0  <  ( A  x.  B ) ) )  ->  0  <  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2160   class class class wbr 4018  (class class class)co 5891   CCcc 7828   RRcr 7829   0cc0 7830   1c1 7831    x. cmul 7835    < clt 8011    / cdiv 8648
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 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-mulrcl 7929  ax-addcom 7930  ax-mulcom 7931  ax-addass 7932  ax-mulass 7933  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-1rid 7937  ax-0id 7938  ax-rnegex 7939  ax-precex 7940  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-apti 7945  ax-pre-ltadd 7946  ax-pre-mulgt0 7947  ax-pre-mulext 7948
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 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-reap 8551  df-ap 8558  df-div 8649
This theorem is referenced by:  prodgt0  8828
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