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Theorem axmulgt0 8851
Description: The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-mulgt0 8768 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 8768 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  <RR  A  /\  0  <RR  B )  ->  0  <RR  ( A  x.  B ) ) )
2 0re 8792 . . . 4  |-  0  e.  RR
3 ltxrlt 8847 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  <->  0 
<RR  A ) )
42, 3mpan 654 . . 3  |-  ( A  e.  RR  ->  (
0  <  A  <->  0  <RR  A ) )
5 ltxrlt 8847 . . . 4  |-  ( ( 0  e.  RR  /\  B  e.  RR )  ->  ( 0  <  B  <->  0 
<RR  B ) )
62, 5mpan 654 . . 3  |-  ( B  e.  RR  ->  (
0  <  B  <->  0  <RR  B ) )
74, 6bi2anan9 848 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  < 
A  /\  0  <  B )  <->  ( 0  <RR  A  /\  0  <RR  B ) ) )
8 remulcl 8776 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
9 ltxrlt 8847 . . 3  |-  ( ( 0  e.  RR  /\  ( A  x.  B
)  e.  RR )  ->  ( 0  < 
( A  x.  B
)  <->  0  <RR  ( A  x.  B ) ) )
102, 8, 9sylancr 647 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  ( A  x.  B )  <->  0 
<RR  ( A  x.  B
) ) )
111, 7, 103imtr4d 261 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  < 
A  /\  0  <  B )  ->  0  <  ( A  x.  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    e. wcel 1621   class class class wbr 3983  (class class class)co 5778   RRcr 8690   0cc0 8691    <RR cltrr 8695    x. cmul 8696    < clt 8821
This theorem is referenced by:  mulgt0  8854  mulgt0i  8905  sin02gt0  12420  sinq12gt0  19823
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-i2m1 8759  ax-1ne0 8760  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-mulgt0 8768
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-pnf 8823  df-mnf 8824  df-ltxr 8826
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