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Theorem axmulgt0 8894
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 8811 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 8811 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  <RR  A  /\  0  <RR  B )  ->  0  <RR  ( A  x.  B ) ) )
2 0re 8835 . . . 4  |-  0  e.  RR
3 ltxrlt 8890 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  <->  0 
<RR  A ) )
42, 3mpan 653 . . 3  |-  ( A  e.  RR  ->  (
0  <  A  <->  0  <RR  A ) )
5 ltxrlt 8890 . . . 4  |-  ( ( 0  e.  RR  /\  B  e.  RR )  ->  ( 0  <  B  <->  0 
<RR  B ) )
62, 5mpan 653 . . 3  |-  ( B  e.  RR  ->  (
0  <  B  <->  0  <RR  B ) )
74, 6bi2anan9 845 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  < 
A  /\  0  <  B )  <->  ( 0  <RR  A  /\  0  <RR  B ) ) )
8 remulcl 8819 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
9 ltxrlt 8890 . . 3  |-  ( ( 0  e.  RR  /\  ( A  x.  B
)  e.  RR )  ->  ( 0  < 
( A  x.  B
)  <->  0  <RR  ( A  x.  B ) ) )
102, 8, 9sylancr 646 . 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 1687   class class class wbr 4026  (class class class)co 5821   RRcr 8733   0cc0 8734    <RR cltrr 8738    x. cmul 8739    < clt 8864
This theorem is referenced by:  mulgt0  8897  mulgt0i  8948  sin02gt0  12468  sinq12gt0  19871
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-resscn 8791  ax-1cn 8792  ax-icn 8793  ax-addcl 8794  ax-addrcl 8795  ax-mulcl 8796  ax-mulrcl 8797  ax-i2m1 8802  ax-1ne0 8803  ax-rnegex 8805  ax-rrecex 8806  ax-cnre 8807  ax-pre-mulgt0 8811
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3831  df-br 4027  df-opab 4081  df-mpt 4082  df-id 4310  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-ov 5824  df-er 6657  df-en 6861  df-dom 6862  df-sdom 6863  df-pnf 8866  df-mnf 8867  df-ltxr 8869
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