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Theorem axmulgt0 9652
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 9560 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 9560 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  <RR  A  /\  0  <RR  B )  ->  0  <RR  ( A  x.  B ) ) )
2 0re 9587 . . . 4  |-  0  e.  RR
3 ltxrlt 9648 . . . 4  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  <  A  <->  0 
<RR  A ) )
42, 3mpan 674 . . 3  |-  ( A  e.  RR  ->  (
0  <  A  <->  0  <RR  A ) )
5 ltxrlt 9648 . . . 4  |-  ( ( 0  e.  RR  /\  B  e.  RR )  ->  ( 0  <  B  <->  0 
<RR  B ) )
62, 5mpan 674 . . 3  |-  ( B  e.  RR  ->  (
0  <  B  <->  0  <RR  B ) )
74, 6bi2anan9 881 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  < 
A  /\  0  <  B )  <->  ( 0  <RR  A  /\  0  <RR  B ) ) )
8 remulcl 9568 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
9 ltxrlt 9648 . . 3  |-  ( ( 0  e.  RR  /\  ( A  x.  B
)  e.  RR )  ->  ( 0  < 
( A  x.  B
)  <->  0  <RR  ( A  x.  B ) ) )
102, 8, 9sylancr 667 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  ( A  x.  B )  <->  0 
<RR  ( A  x.  B
) ) )
111, 7, 103imtr4d 271 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0  < 
A  /\  0  <  B )  ->  0  <  ( A  x.  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    e. wcel 1872   class class class wbr 4359  (class class class)co 6242   RRcr 9482   0cc0 9483    <RR cltrr 9487    x. cmul 9488    < clt 9619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-sep 4482  ax-nul 4491  ax-pow 4538  ax-pr 4596  ax-un 6534  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-i2m1 9551  ax-1ne0 9552  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-nel 2596  df-ral 2713  df-rex 2714  df-rab 2717  df-v 3018  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3698  df-if 3848  df-pw 3919  df-sn 3935  df-pr 3937  df-op 3941  df-uni 4156  df-br 4360  df-opab 4419  df-mpt 4420  df-id 4704  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-rn 4800  df-res 4801  df-ima 4802  df-iota 5501  df-fun 5539  df-fn 5540  df-f 5541  df-f1 5542  df-fo 5543  df-f1o 5544  df-fv 5545  df-ov 6245  df-er 7311  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9621  df-mnf 9622  df-ltxr 9624
This theorem is referenced by:  mulgt0  9655  mulgt0i  9711  sin02gt0  14182  sinq12gt0  23397
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