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Theorem xmul01 11380
Description: Extended real version of mul01 9670. (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 9551 . . 3  |-  0  e.  RR*
2 xmulval 11345 . . 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 669 . 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 2382 . . . 4  |-  0  =  0
54olci 389 . . 3  |-  ( A  =  0  \/  0  =  0 )
65iftruei 3864 . 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 2439 1  |-  ( A  e.  RR*  ->  ( A xe 0 )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1399    e. wcel 1826   ifcif 3857   class class class wbr 4367  (class class class)co 6196   0cc0 9403    x. cmul 9408   +oocpnf 9536   -oocmnf 9537   RR*cxr 9538    < clt 9539   xecxmu 11238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-i2m1 9471  ax-1ne0 9472  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-pnf 9541  df-mnf 9542  df-xr 9543  df-xmul 11241
This theorem is referenced by:  xmul02  11381  xmulge0  11397  xmulass  11400  xlemul1a  11401  xadddilem  11407  xadddi2  11410  psmetge0  20901  xmetge0  20932  nmoix  21321  xrge0mulc1cn  28077  esumcst  28211
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