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Theorem axmulgt0 7479
Description: The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 7383 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
axmulgt0  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  < 
A  /\  0  <  B )  ->  0  <  ( A  x.  B ) ) )

Proof of Theorem axmulgt0
StepHypRef Expression
1 ax-pre-mulgt0 7383 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  <RR  A  /\  0  <RR  B )  ->  0  <RR  ( A  x.  B ) ) )
2 0re 7409 . . . 4  |-  0  e.  RR
3 ltxrlt 7473 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  <->  0 
<RR  A ) )
42, 3mpan 415 . . 3  |-  ( A  e.  RR  ->  (
0  <  A  <->  0  <RR  A ) )
5 ltxrlt 7473 . . . 4  |-  ( ( 0  e.  RR  /\  B  e.  RR )  ->  ( 0  <  B  <->  0 
<RR  B ) )
62, 5mpan 415 . . 3  |-  ( B  e.  RR  ->  (
0  <  B  <->  0  <RR  B ) )
74, 6bi2anan9 571 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  < 
A  /\  0  <  B )  <->  ( 0  <RR  A  /\  0  <RR  B ) ) )
8 remulcl 7391 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
9 ltxrlt 7473 . . 3  |-  ( ( 0  e.  RR  /\  ( A  x.  B
)  e.  RR )  ->  ( 0  < 
( A  x.  B
)  <->  0  <RR  ( A  x.  B ) ) )
102, 8, 9sylancr 405 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  ( A  x.  B )  <->  0 
<RR  ( A  x.  B
) ) )
111, 7, 103imtr4d 201 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  < 
A  /\  0  <  B )  ->  0  <  ( A  x.  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1436   class class class wbr 3814  (class class class)co 5594   RRcr 7270   0cc0 7271    <RR cltrr 7275    x. cmul 7276    < clt 7443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-cnex 7357  ax-resscn 7358  ax-1re 7360  ax-addrcl 7363  ax-mulrcl 7365  ax-rnegex 7375  ax-pre-mulgt0 7383
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2616  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-br 3815  df-opab 3869  df-xp 4410  df-pnf 7445  df-mnf 7446  df-ltxr 7448
This theorem is referenced by:  mulgt0  7481  mulgt0i  7515
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