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Theorem xmul01 11331
Description: Extended real version of mul01 9649. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xmul01  |-  ( A  e.  RR*  ->  ( A xe 0 )  =  0 )

Proof of Theorem xmul01
StepHypRef Expression
1 0xr 9531 . . 3  |-  0  e.  RR*
2 xmulval 11296 . . 3  |-  ( ( A  e.  RR*  /\  0  e.  RR* )  ->  ( A xe 0 )  =  if ( ( A  =  0  \/  0  =  0 ) ,  0 ,  if ( ( ( ( 0  <  0  /\  A  = +oo )  \/  ( 0  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  0  = +oo )  \/  ( A  <  0  /\  0  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  0  /\  A  = -oo )  \/  ( 0  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  0  = -oo )  \/  ( A  <  0  /\  0  = +oo ) ) ) , -oo ,  ( A  x.  0 ) ) ) ) )
31, 2mpan2 671 . 2  |-  ( A  e.  RR*  ->  ( A xe 0 )  =  if ( ( A  =  0  \/  0  =  0 ) ,  0 ,  if ( ( ( ( 0  <  0  /\  A  = +oo )  \/  ( 0  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  0  = +oo )  \/  ( A  <  0  /\  0  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  0  /\  A  = -oo )  \/  ( 0  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  0  = -oo )  \/  ( A  <  0  /\  0  = +oo ) ) ) , -oo ,  ( A  x.  0 ) ) ) ) )
4 eqid 2451 . . . 4  |-  0  =  0
54olci 391 . . 3  |-  ( A  =  0  \/  0  =  0 )
65iftruei 3896 . 2  |-  if ( ( A  =  0  \/  0  =  0 ) ,  0 ,  if ( ( ( ( 0  <  0  /\  A  = +oo )  \/  ( 0  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  0  = +oo )  \/  ( A  <  0  /\  0  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  0  /\  A  = -oo )  \/  ( 0  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  0  = -oo )  \/  ( A  <  0  /\  0  = +oo ) ) ) , -oo ,  ( A  x.  0 ) ) ) )  =  0
73, 6syl6eq 2508 1  |-  ( A  e.  RR*  ->  ( A xe 0 )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758   ifcif 3889   class class class wbr 4390  (class class class)co 6190   0cc0 9383    x. cmul 9388   +oocpnf 9516   -oocmnf 9517   RR*cxr 9518    < clt 9519   xecxmu 11189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-i2m1 9451  ax-1ne0 9452  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-iota 5479  df-fun 5518  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-pnf 9521  df-mnf 9522  df-xr 9523  df-xmul 11192
This theorem is referenced by:  xmul02  11332  xmulge0  11348  xmulass  11351  xlemul1a  11352  xadddilem  11358  xadddi2  11361  psmetge0  20004  xmetge0  20035  nmoix  20424  xrge0mulc1cn  26505  esumcst  26648
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