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Theorem xmulf 11565
Description: The extended real multiplication operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
xmulf  |-  xe : ( RR*  X.  RR* )
--> RR*

Proof of Theorem xmulf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0xr 9694 . . . . 5  |-  0  e.  RR*
21a1i 11 . . . 4  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  ( x  =  0  \/  y  =  0 ) )  ->  0  e.  RR* )
3 pnfxr 11419 . . . . . 6  |- +oo  e.  RR*
43a1i 11 . . . . 5  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  -.  (
x  =  0  \/  y  =  0 ) )  /\  ( ( ( 0  <  y  /\  x  = +oo )  \/  ( y  <  0  /\  x  = -oo ) )  \/  ( ( 0  < 
x  /\  y  = +oo )  \/  (
x  <  0  /\  y  = -oo )
) ) )  -> +oo  e.  RR* )
5 mnfxr 11421 . . . . . . 7  |- -oo  e.  RR*
65a1i 11 . . . . . 6  |-  ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  ( x  =  0  \/  y  =  0
) )  /\  -.  ( ( ( 0  <  y  /\  x  = +oo )  \/  (
y  <  0  /\  x  = -oo )
)  \/  ( ( 0  <  x  /\  y  = +oo )  \/  ( x  <  0  /\  y  = -oo ) ) ) )  /\  ( ( ( 0  <  y  /\  x  = -oo )  \/  ( y  <  0  /\  x  = +oo ) )  \/  (
( 0  <  x  /\  y  = -oo )  \/  ( x  <  0  /\  y  = +oo ) ) ) )  -> -oo  e.  RR* )
7 xmullem 11557 . . . . . . . 8  |-  ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  ( x  =  0  \/  y  =  0
) )  /\  -.  ( ( ( 0  <  y  /\  x  = +oo )  \/  (
y  <  0  /\  x  = -oo )
)  \/  ( ( 0  <  x  /\  y  = +oo )  \/  ( x  <  0  /\  y  = -oo ) ) ) )  /\  -.  ( ( ( 0  <  y  /\  x  = -oo )  \/  ( y  <  0  /\  x  = +oo ) )  \/  ( ( 0  < 
x  /\  y  = -oo )  \/  (
x  <  0  /\  y  = +oo )
) ) )  ->  x  e.  RR )
8 ancom 451 . . . . . . . . . . 11  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  <->  ( y  e.  RR*  /\  x  e. 
RR* ) )
9 orcom 388 . . . . . . . . . . . 12  |-  ( ( x  =  0  \/  y  =  0 )  <-> 
( y  =  0  \/  x  =  0 ) )
109notbii 297 . . . . . . . . . . 11  |-  ( -.  ( x  =  0  \/  y  =  0 )  <->  -.  ( y  =  0  \/  x  =  0 ) )
118, 10anbi12i 701 . . . . . . . . . 10  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  ( x  =  0  \/  y  =  0 ) )  <->  ( (
y  e.  RR*  /\  x  e.  RR* )  /\  -.  ( y  =  0  \/  x  =  0 ) ) )
12 orcom 388 . . . . . . . . . . 11  |-  ( ( ( ( 0  < 
y  /\  x  = +oo )  \/  (
y  <  0  /\  x  = -oo )
)  \/  ( ( 0  <  x  /\  y  = +oo )  \/  ( x  <  0  /\  y  = -oo ) ) )  <->  ( (
( 0  <  x  /\  y  = +oo )  \/  ( x  <  0  /\  y  = -oo ) )  \/  ( ( 0  < 
y  /\  x  = +oo )  \/  (
y  <  0  /\  x  = -oo )
) ) )
1312notbii 297 . . . . . . . . . 10  |-  ( -.  ( ( ( 0  <  y  /\  x  = +oo )  \/  (
y  <  0  /\  x  = -oo )
)  \/  ( ( 0  <  x  /\  y  = +oo )  \/  ( x  <  0  /\  y  = -oo ) ) )  <->  -.  (
( ( 0  < 
x  /\  y  = +oo )  \/  (
x  <  0  /\  y  = -oo )
)  \/  ( ( 0  <  y  /\  x  = +oo )  \/  ( y  <  0  /\  x  = -oo ) ) ) )
1411, 13anbi12i 701 . . . . . . . . 9  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  -.  (
x  =  0  \/  y  =  0 ) )  /\  -.  (
( ( 0  < 
y  /\  x  = +oo )  \/  (
y  <  0  /\  x  = -oo )
)  \/  ( ( 0  <  x  /\  y  = +oo )  \/  ( x  <  0  /\  y  = -oo ) ) ) )  <-> 
( ( ( y  e.  RR*  /\  x  e.  RR* )  /\  -.  ( y  =  0  \/  x  =  0 ) )  /\  -.  ( ( ( 0  <  x  /\  y  = +oo )  \/  (
x  <  0  /\  y  = -oo )
)  \/  ( ( 0  <  y  /\  x  = +oo )  \/  ( y  <  0  /\  x  = -oo ) ) ) ) )
15 orcom 388 . . . . . . . . . 10  |-  ( ( ( ( 0  < 
y  /\  x  = -oo )  \/  (
y  <  0  /\  x  = +oo )
)  \/  ( ( 0  <  x  /\  y  = -oo )  \/  ( x  <  0  /\  y  = +oo ) ) )  <->  ( (
( 0  <  x  /\  y  = -oo )  \/  ( x  <  0  /\  y  = +oo ) )  \/  ( ( 0  < 
y  /\  x  = -oo )  \/  (
y  <  0  /\  x  = +oo )
) ) )
1615notbii 297 . . . . . . . . 9  |-  ( -.  ( ( ( 0  <  y  /\  x  = -oo )  \/  (
y  <  0  /\  x  = +oo )
)  \/  ( ( 0  <  x  /\  y  = -oo )  \/  ( x  <  0  /\  y  = +oo ) ) )  <->  -.  (
( ( 0  < 
x  /\  y  = -oo )  \/  (
x  <  0  /\  y  = +oo )
)  \/  ( ( 0  <  y  /\  x  = -oo )  \/  ( y  <  0  /\  x  = +oo ) ) ) )
17 xmullem 11557 . . . . . . . . 9  |-  ( ( ( ( ( y  e.  RR*  /\  x  e.  RR* )  /\  -.  ( y  =  0  \/  x  =  0 ) )  /\  -.  ( ( ( 0  <  x  /\  y  = +oo )  \/  (
x  <  0  /\  y  = -oo )
)  \/  ( ( 0  <  y  /\  x  = +oo )  \/  ( y  <  0  /\  x  = -oo ) ) ) )  /\  -.  ( ( ( 0  <  x  /\  y  = -oo )  \/  ( x  <  0  /\  y  = +oo ) )  \/  ( ( 0  < 
y  /\  x  = -oo )  \/  (
y  <  0  /\  x  = +oo )
) ) )  -> 
y  e.  RR )
1814, 16, 17syl2anb 481 . . . . . . . 8  |-  ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  ( x  =  0  \/  y  =  0
) )  /\  -.  ( ( ( 0  <  y  /\  x  = +oo )  \/  (
y  <  0  /\  x  = -oo )
)  \/  ( ( 0  <  x  /\  y  = +oo )  \/  ( x  <  0  /\  y  = -oo ) ) ) )  /\  -.  ( ( ( 0  <  y  /\  x  = -oo )  \/  ( y  <  0  /\  x  = +oo ) )  \/  ( ( 0  < 
x  /\  y  = -oo )  \/  (
x  <  0  /\  y  = +oo )
) ) )  -> 
y  e.  RR )
197, 18remulcld 9678 . . . . . . 7  |-  ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  ( x  =  0  \/  y  =  0
) )  /\  -.  ( ( ( 0  <  y  /\  x  = +oo )  \/  (
y  <  0  /\  x  = -oo )
)  \/  ( ( 0  <  x  /\  y  = +oo )  \/  ( x  <  0  /\  y  = -oo ) ) ) )  /\  -.  ( ( ( 0  <  y  /\  x  = -oo )  \/  ( y  <  0  /\  x  = +oo ) )  \/  ( ( 0  < 
x  /\  y  = -oo )  \/  (
x  <  0  /\  y  = +oo )
) ) )  -> 
( x  x.  y
)  e.  RR )
2019rexrd 9697 . . . . . 6  |-  ( ( ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  ( x  =  0  \/  y  =  0
) )  /\  -.  ( ( ( 0  <  y  /\  x  = +oo )  \/  (
y  <  0  /\  x  = -oo )
)  \/  ( ( 0  <  x  /\  y  = +oo )  \/  ( x  <  0  /\  y  = -oo ) ) ) )  /\  -.  ( ( ( 0  <  y  /\  x  = -oo )  \/  ( y  <  0  /\  x  = +oo ) )  \/  ( ( 0  < 
x  /\  y  = -oo )  \/  (
x  <  0  /\  y  = +oo )
) ) )  -> 
( x  x.  y
)  e.  RR* )
216, 20ifclda 3943 . . . . 5  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  -.  (
x  =  0  \/  y  =  0 ) )  /\  -.  (
( ( 0  < 
y  /\  x  = +oo )  \/  (
y  <  0  /\  x  = -oo )
)  \/  ( ( 0  <  x  /\  y  = +oo )  \/  ( x  <  0  /\  y  = -oo ) ) ) )  ->  if ( ( ( ( 0  < 
y  /\  x  = -oo )  \/  (
y  <  0  /\  x  = +oo )
)  \/  ( ( 0  <  x  /\  y  = -oo )  \/  ( x  <  0  /\  y  = +oo ) ) ) , -oo ,  ( x  x.  y ) )  e.  RR* )
224, 21ifclda 3943 . . . 4  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  ( x  =  0  \/  y  =  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 ) ) )  e.  RR* )
232, 22ifclda 3943 . . 3  |-  ( ( 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 ) ) ) )  e.  RR* )
2423rgen2a 2849 . 2  |-  A. x  e.  RR*  A. 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 ) ) ) )  e.  RR*
25 df-xmul 11418 . . 3  |-  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 ) ) ) ) )
2625fmpt2 6874 . 2  |-  ( A. x  e.  RR*  A. 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 ) ) ) )  e. 
RR* 
<->  xe : (
RR*  X.  RR* ) --> RR* )
2724, 26mpbi 211 1  |-  xe : ( RR*  X.  RR* )
--> RR*
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2771   ifcif 3911   class class class wbr 4423    X. cxp 4851   -->wf 5597  (class class class)co 6305   RRcr 9545   0cc0 9546    x. cmul 9551   +oocpnf 9679   -oocmnf 9680   RR*cxr 9681    < clt 9682   xecxmu 11415
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 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-i2m1 9614  ax-1ne0 9615  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-xmul 11418
This theorem is referenced by:  xmulcl  11566  xrsmul  18985
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