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Theorem xaddf 11524
Description: The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
xaddf  |-  +e : ( RR*  X.  RR* )
--> RR*

Proof of Theorem xaddf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0xr 9694 . . . . . 6  |-  0  e.  RR*
2 pnfxr 11419 . . . . . 6  |- +oo  e.  RR*
31, 2keepel 3978 . . . . 5  |-  if ( y  = -oo , 
0 , +oo )  e.  RR*
43a1i 11 . . . 4  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  x  = +oo )  ->  if ( y  = -oo ,  0 , +oo )  e. 
RR* )
5 mnfxr 11421 . . . . . . 7  |- -oo  e.  RR*
61, 5keepel 3978 . . . . . 6  |-  if ( y  = +oo , 
0 , -oo )  e.  RR*
76a1i 11 . . . . 5  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  x  = -oo )  ->  if ( y  = +oo ,  0 , -oo )  e.  RR* )
82a1i 11 . . . . . . . 8  |-  ( ( ( ( x  e. 
RR*  /\  ( -.  x  = +oo  /\  -.  x  = -oo )
)  /\  y  e.  RR* )  /\  y  = +oo )  -> +oo  e.  RR* )
95a1i 11 . . . . . . . . 9  |-  ( ( ( ( ( x  e.  RR*  /\  ( -.  x  = +oo  /\ 
-.  x  = -oo ) )  /\  y  e.  RR* )  /\  -.  y  = +oo )  /\  y  = -oo )  -> -oo  e.  RR* )
10 ioran 492 . . . . . . . . . . . . . 14  |-  ( -.  ( x  = +oo  \/  x  = -oo ) 
<->  ( -.  x  = +oo  /\  -.  x  = -oo ) )
11 elxr 11423 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  RR*  <->  ( x  e.  RR  \/  x  = +oo  \/  x  = -oo ) )
12 3orass 985 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  \/  x  = +oo  \/  x  = -oo )  <->  ( x  e.  RR  \/  ( x  = +oo  \/  x  = -oo ) ) )
1311, 12sylbb 200 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  RR*  ->  ( x  e.  RR  \/  (
x  = +oo  \/  x  = -oo )
) )
1413ord 378 . . . . . . . . . . . . . . . 16  |-  ( x  e.  RR*  ->  ( -.  x  e.  RR  ->  ( x  = +oo  \/  x  = -oo )
) )
1514con1d 127 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR*  ->  ( -.  ( x  = +oo  \/  x  = -oo )  ->  x  e.  RR ) )
1615imp 430 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR*  /\  -.  ( x  = +oo  \/  x  = -oo ) )  ->  x  e.  RR )
1710, 16sylan2br 478 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR*  /\  ( -.  x  = +oo  /\ 
-.  x  = -oo ) )  ->  x  e.  RR )
18 ioran 492 . . . . . . . . . . . . . 14  |-  ( -.  ( y  = +oo  \/  y  = -oo ) 
<->  ( -.  y  = +oo  /\  -.  y  = -oo ) )
19 elxr 11423 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  RR*  <->  ( y  e.  RR  \/  y  = +oo  \/  y  = -oo ) )
20 3orass 985 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  RR  \/  y  = +oo  \/  y  = -oo )  <->  ( y  e.  RR  \/  ( y  = +oo  \/  y  = -oo ) ) )
2119, 20sylbb 200 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  RR*  ->  ( y  e.  RR  \/  (
y  = +oo  \/  y  = -oo )
) )
2221ord 378 . . . . . . . . . . . . . . . 16  |-  ( y  e.  RR*  ->  ( -.  y  e.  RR  ->  ( y  = +oo  \/  y  = -oo )
) )
2322con1d 127 . . . . . . . . . . . . . . 15  |-  ( y  e.  RR*  ->  ( -.  ( y  = +oo  \/  y  = -oo )  ->  y  e.  RR ) )
2423imp 430 . . . . . . . . . . . . . 14  |-  ( ( y  e.  RR*  /\  -.  ( y  = +oo  \/  y  = -oo ) )  ->  y  e.  RR )
2518, 24sylan2br 478 . . . . . . . . . . . . 13  |-  ( ( y  e.  RR*  /\  ( -.  y  = +oo  /\ 
-.  y  = -oo ) )  ->  y  e.  RR )
26 readdcl 9629 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
2717, 25, 26syl2an 479 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR*  /\  ( -.  x  = +oo  /\  -.  x  = -oo ) )  /\  ( y  e.  RR*  /\  ( -.  y  = +oo  /\  -.  y  = -oo ) ) )  ->  ( x  +  y )  e.  RR )
2827rexrd 9697 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR*  /\  ( -.  x  = +oo  /\  -.  x  = -oo ) )  /\  ( y  e.  RR*  /\  ( -.  y  = +oo  /\  -.  y  = -oo ) ) )  ->  ( x  +  y )  e.  RR* )
2928anassrs 652 . . . . . . . . . 10  |-  ( ( ( ( x  e. 
RR*  /\  ( -.  x  = +oo  /\  -.  x  = -oo )
)  /\  y  e.  RR* )  /\  ( -.  y  = +oo  /\  -.  y  = -oo ) )  ->  (
x  +  y )  e.  RR* )
3029anassrs 652 . . . . . . . . 9  |-  ( ( ( ( ( x  e.  RR*  /\  ( -.  x  = +oo  /\ 
-.  x  = -oo ) )  /\  y  e.  RR* )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  ( x  +  y )  e.  RR* )
319, 30ifclda 3943 . . . . . . . 8  |-  ( ( ( ( x  e. 
RR*  /\  ( -.  x  = +oo  /\  -.  x  = -oo )
)  /\  y  e.  RR* )  /\  -.  y  = +oo )  ->  if ( y  = -oo , -oo ,  ( x  +  y ) )  e.  RR* )
328, 31ifclda 3943 . . . . . . 7  |-  ( ( ( x  e.  RR*  /\  ( -.  x  = +oo  /\  -.  x  = -oo ) )  /\  y  e.  RR* )  ->  if ( y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) )  e.  RR* )
3332an32s 811 . . . . . 6  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  ( -.  x  = +oo  /\  -.  x  = -oo ) )  ->  if ( y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) )  e.  RR* )
3433anassrs 652 . . . . 5  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  /\  -.  x  = -oo )  ->  if ( y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) )  e.  RR* )
357, 34ifclda 3943 . . . 4  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  -.  x  = +oo )  ->  if ( x  = -oo ,  if ( y  = +oo ,  0 , -oo ) ,  if (
y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) ) )  e.  RR* )
364, 35ifclda 3943 . . 3  |-  ( ( x  e.  RR*  /\  y  e.  RR* )  ->  if ( x  = +oo ,  if ( y  = -oo ,  0 , +oo ) ,  if ( x  = -oo ,  if ( y  = +oo ,  0 , -oo ) ,  if ( y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) ) ) )  e.  RR* )
3736rgen2a 2849 . 2  |-  A. x  e.  RR*  A. y  e. 
RR*  if ( x  = +oo ,  if ( y  = -oo , 
0 , +oo ) ,  if ( x  = -oo ,  if ( y  = +oo , 
0 , -oo ) ,  if ( y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) ) ) )  e. 
RR*
38 df-xadd 11417 . . 3  |-  +e 
=  ( x  e. 
RR* ,  y  e.  RR*  |->  if ( x  = +oo ,  if ( y  = -oo , 
0 , +oo ) ,  if ( x  = -oo ,  if ( y  = +oo , 
0 , -oo ) ,  if ( y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) ) ) ) )
3938fmpt2 6874 . 2  |-  ( A. x  e.  RR*  A. y  e.  RR*  if ( x  = +oo ,  if ( y  = -oo ,  0 , +oo ) ,  if (
x  = -oo ,  if ( y  = +oo ,  0 , -oo ) ,  if (
y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) ) ) )  e.  RR*  <->  +e : ( RR*  X.  RR* )
--> RR* )
4037, 39mpbi 211 1  |-  +e : ( RR*  X.  RR* )
--> RR*
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 369    /\ wa 370    \/ w3o 981    = wceq 1437    e. wcel 1872   A.wral 2771   ifcif 3911    X. cxp 4851   -->wf 5597  (class class class)co 6305   RRcr 9545   0cc0 9546    + caddc 9549   +oocpnf 9679   -oocmnf 9680   RR*cxr 9681   +ecxad 11414
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-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
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-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-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-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-pnf 9684  df-mnf 9685  df-xr 9686  df-xadd 11417
This theorem is referenced by:  xaddcl  11537  xrsadd  18984  xrofsup  28359  xrge0pluscn  28754  xrge0tmdOLD  28759
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