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Theorem mulgt0i 8080
Description: The product of two positive numbers is positive. (Contributed by NM, 16-May-1999.)
Hypotheses
Ref Expression
lt.1  |-  A  e.  RR
lt.2  |-  B  e.  RR
Assertion
Ref Expression
mulgt0i  |-  ( ( 0  <  A  /\  0  <  B )  -> 
0  <  ( A  x.  B ) )

Proof of Theorem mulgt0i
StepHypRef Expression
1 lt.1 . 2  |-  A  e.  RR
2 lt.2 . 2  |-  B  e.  RR
3 axmulgt0 8042 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  < 
A  /\  0  <  B )  ->  0  <  ( A  x.  B ) ) )
41, 2, 3mp2an 426 1  |-  ( ( 0  <  A  /\  0  <  B )  -> 
0  <  ( A  x.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2158   class class class wbr 4015  (class class class)co 5888   RRcr 7823   0cc0 7824    x. cmul 7829    < clt 8005
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921  ax-mulrcl 7923  ax-rnegex 7933  ax-pre-mulgt0 7941
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-xp 4644  df-pnf 8007  df-mnf 8008  df-ltxr 8010
This theorem is referenced by:  mulgt0ii  8081
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