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Theorem xmulval 11310
Description: Value of the extended real multiplication operation. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xmulval  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A xe B )  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) ) )

Proof of Theorem xmulval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 457 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  x  =  A )
21eqeq1d 2456 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  =  0  <-> 
A  =  0 ) )
3 simpr 461 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
43eqeq1d 2456 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  =  0  <-> 
B  =  0 ) )
52, 4orbi12d 709 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  =  0  \/  y  =  0 )  <->  ( A  =  0  \/  B  =  0 ) ) )
63breq2d 4415 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( 0  <  y  <->  0  <  B ) )
71eqeq1d 2456 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  = +oo  <->  A  = +oo ) )
86, 7anbi12d 710 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( 0  < 
y  /\  x  = +oo )  <->  ( 0  < 
B  /\  A  = +oo ) ) )
93breq1d 4413 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  <  0  <->  B  <  0 ) )
101eqeq1d 2456 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  = -oo  <->  A  = -oo ) )
119, 10anbi12d 710 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( y  <  0  /\  x  = -oo )  <->  ( B  <  0  /\  A  = -oo ) ) )
128, 11orbi12d 709 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( 0  <  y  /\  x  = +oo )  \/  (
y  <  0  /\  x  = -oo )
)  <->  ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) ) ) )
131breq2d 4415 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( 0  <  x  <->  0  <  A ) )
143eqeq1d 2456 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  = +oo  <->  B  = +oo ) )
1513, 14anbi12d 710 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( 0  < 
x  /\  y  = +oo )  <->  ( 0  < 
A  /\  B  = +oo ) ) )
161breq1d 4413 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  <  0  <->  A  <  0 ) )
173eqeq1d 2456 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  = -oo  <->  B  = -oo ) )
1816, 17anbi12d 710 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  <  0  /\  y  = -oo )  <->  ( A  <  0  /\  B  = -oo ) ) )
1915, 18orbi12d 709 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( 0  <  x  /\  y  = +oo )  \/  (
x  <  0  /\  y  = -oo )
)  <->  ( ( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) )
2012, 19orbi12d 709 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( ( 0  <  y  /\  x  = +oo )  \/  ( y  <  0  /\  x  = -oo ) )  \/  (
( 0  <  x  /\  y  = +oo )  \/  ( x  <  0  /\  y  = -oo ) ) )  <-> 
( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) ) )
216, 10anbi12d 710 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( 0  < 
y  /\  x  = -oo )  <->  ( 0  < 
B  /\  A  = -oo ) ) )
229, 7anbi12d 710 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( y  <  0  /\  x  = +oo )  <->  ( B  <  0  /\  A  = +oo ) ) )
2321, 22orbi12d 709 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( 0  <  y  /\  x  = -oo )  \/  (
y  <  0  /\  x  = +oo )
)  <->  ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) ) ) )
2413, 17anbi12d 710 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( 0  < 
x  /\  y  = -oo )  <->  ( 0  < 
A  /\  B  = -oo ) ) )
2516, 14anbi12d 710 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( x  <  0  /\  y  = +oo )  <->  ( A  <  0  /\  B  = +oo ) ) )
2624, 25orbi12d 709 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( 0  <  x  /\  y  = -oo )  \/  (
x  <  0  /\  y  = +oo )
)  <->  ( ( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) )
2723, 26orbi12d 709 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( ( ( 0  <  y  /\  x  = -oo )  \/  ( y  <  0  /\  x  = +oo ) )  \/  (
( 0  <  x  /\  y  = -oo )  \/  ( x  <  0  /\  y  = +oo ) ) )  <-> 
( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) ) )
28 oveq12 6212 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  x.  y
)  =  ( A  x.  B ) )
2927, 28ifbieq2d 3925 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( ( ( ( 0  <  y  /\  x  = -oo )  \/  ( y  <  0  /\  x  = +oo ) )  \/  ( ( 0  < 
x  /\  y  = -oo )  \/  (
x  <  0  /\  y  = +oo )
) ) , -oo ,  ( x  x.  y ) )  =  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )
3020, 29ifbieq2d 3925 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( ( ( ( 0  <  y  /\  x  = +oo )  \/  ( y  <  0  /\  x  = -oo ) )  \/  ( ( 0  < 
x  /\  y  = +oo )  \/  (
x  <  0  /\  y  = -oo )
) ) , +oo ,  if ( ( ( ( 0  <  y  /\  x  = -oo )  \/  ( y  <  0  /\  x  = +oo ) )  \/  ( ( 0  < 
x  /\  y  = -oo )  \/  (
x  <  0  /\  y  = +oo )
) ) , -oo ,  ( x  x.  y ) ) )  =  if ( ( ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) )
315, 30ifbieq2d 3925 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( ( x  =  0  \/  y  =  0 ) ,  0 ,  if ( ( ( ( 0  <  y  /\  x  = +oo )  \/  (
y  <  0  /\  x  = -oo )
)  \/  ( ( 0  <  x  /\  y  = +oo )  \/  ( x  <  0  /\  y  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  y  /\  x  = -oo )  \/  (
y  <  0  /\  x  = +oo )
)  \/  ( ( 0  <  x  /\  y  = -oo )  \/  ( x  <  0  /\  y  = +oo ) ) ) , -oo ,  ( x  x.  y ) ) ) )  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) ) )
32 df-xmul 11206 . 2  |-  xe  =  ( x  e. 
RR* ,  y  e.  RR*  |->  if ( ( x  =  0  \/  y  =  0 ) ,  0 ,  if ( ( ( ( 0  <  y  /\  x  = +oo )  \/  (
y  <  0  /\  x  = -oo )
)  \/  ( ( 0  <  x  /\  y  = +oo )  \/  ( x  <  0  /\  y  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  y  /\  x  = -oo )  \/  (
y  <  0  /\  x  = +oo )
)  \/  ( ( 0  <  x  /\  y  = -oo )  \/  ( x  <  0  /\  y  = +oo ) ) ) , -oo ,  ( x  x.  y ) ) ) ) )
33 c0ex 9495 . . 3  |-  0  e.  _V
34 pnfex 11208 . . . 4  |- +oo  e.  _V
35 mnfxr 11209 . . . . . 6  |- -oo  e.  RR*
3635elexi 3088 . . . . 5  |- -oo  e.  _V
37 ovex 6228 . . . . 5  |-  ( A  x.  B )  e. 
_V
3836, 37ifex 3969 . . . 4  |-  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) )  e.  _V
3934, 38ifex 3969 . . 3  |-  if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  e.  _V
4033, 39ifex 3969 . 2  |-  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) )  e. 
_V
4131, 32, 40ovmpt2a 6334 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A xe B )  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758   ifcif 3902   class class class wbr 4403  (class class class)co 6203   0cc0 9397    x. cmul 9402   +oocpnf 9530   -oocmnf 9531   RR*cxr 9532    < clt 9533   xecxmu 11203
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 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-mulcl 9459  ax-i2m1 9465
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-pnf 9535  df-mnf 9536  df-xr 9537  df-xmul 11206
This theorem is referenced by:  xmulcom  11344  xmul01  11345  xmulneg1  11347  rexmul  11349  xmulpnf1  11352
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