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Theorem xaddf 11412
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 9629 . . . . . 6  |-  0  e.  RR*
2 pnfxr 11310 . . . . . 6  |- +oo  e.  RR*
31, 2keepel 4000 . . . . 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 11312 . . . . . . 7  |- -oo  e.  RR*
61, 5keepel 4000 . . . . . 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 490 . . . . . . . . . . . . . 14  |-  ( -.  ( x  = +oo  \/  x  = -oo ) 
<->  ( -.  x  = +oo  /\  -.  x  = -oo ) )
11 elxr 11314 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  RR*  <->  ( x  e.  RR  \/  x  = +oo  \/  x  = -oo ) )
12 3orass 971 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  \/  x  = +oo  \/  x  = -oo )  <->  ( x  e.  RR  \/  ( x  = +oo  \/  x  = -oo ) ) )
1311, 12sylbb 197 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  RR*  ->  ( x  e.  RR  \/  (
x  = +oo  \/  x  = -oo )
) )
1413ord 377 . . . . . . . . . . . . . . . 16  |-  ( x  e.  RR*  ->  ( -.  x  e.  RR  ->  ( x  = +oo  \/  x  = -oo )
) )
1514con1d 124 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR*  ->  ( -.  ( x  = +oo  \/  x  = -oo )  ->  x  e.  RR ) )
1615imp 429 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR*  /\  -.  ( x  = +oo  \/  x  = -oo ) )  ->  x  e.  RR )
1710, 16sylan2br 476 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR*  /\  ( -.  x  = +oo  /\ 
-.  x  = -oo ) )  ->  x  e.  RR )
18 ioran 490 . . . . . . . . . . . . . 14  |-  ( -.  ( y  = +oo  \/  y  = -oo ) 
<->  ( -.  y  = +oo  /\  -.  y  = -oo ) )
19 elxr 11314 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  RR*  <->  ( y  e.  RR  \/  y  = +oo  \/  y  = -oo ) )
20 3orass 971 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  e.  RR  \/  y  = +oo  \/  y  = -oo )  <->  ( y  e.  RR  \/  ( y  = +oo  \/  y  = -oo ) ) )
2119, 20sylbb 197 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  RR*  ->  ( y  e.  RR  \/  (
y  = +oo  \/  y  = -oo )
) )
2221ord 377 . . . . . . . . . . . . . . . 16  |-  ( y  e.  RR*  ->  ( -.  y  e.  RR  ->  ( y  = +oo  \/  y  = -oo )
) )
2322con1d 124 . . . . . . . . . . . . . . 15  |-  ( y  e.  RR*  ->  ( -.  ( y  = +oo  \/  y  = -oo )  ->  y  e.  RR ) )
2423imp 429 . . . . . . . . . . . . . 14  |-  ( ( y  e.  RR*  /\  -.  ( y  = +oo  \/  y  = -oo ) )  ->  y  e.  RR )
2518, 24sylan2br 476 . . . . . . . . . . . . 13  |-  ( ( y  e.  RR*  /\  ( -.  y  = +oo  /\ 
-.  y  = -oo ) )  ->  y  e.  RR )
26 readdcl 9564 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
2717, 25, 26syl2an 477 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR*  /\  ( -.  x  = +oo  /\  -.  x  = -oo ) )  /\  ( y  e.  RR*  /\  ( -.  y  = +oo  /\  -.  y  = -oo ) ) )  ->  ( x  +  y )  e.  RR )
2827rexrd 9632 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR*  /\  ( -.  x  = +oo  /\  -.  x  = -oo ) )  /\  ( y  e.  RR*  /\  ( -.  y  = +oo  /\  -.  y  = -oo ) ) )  ->  ( x  +  y )  e.  RR* )
2928anassrs 648 . . . . . . . . . 10  |-  ( ( ( ( x  e. 
RR*  /\  ( -.  x  = +oo  /\  -.  x  = -oo )
)  /\  y  e.  RR* )  /\  ( -.  y  = +oo  /\  -.  y  = -oo ) )  ->  (
x  +  y )  e.  RR* )
3029anassrs 648 . . . . . . . . 9  |-  ( ( ( ( ( x  e.  RR*  /\  ( -.  x  = +oo  /\ 
-.  x  = -oo ) )  /\  y  e.  RR* )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  ( x  +  y )  e.  RR* )
319, 30ifclda 3964 . . . . . . . 8  |-  ( ( ( ( x  e. 
RR*  /\  ( -.  x  = +oo  /\  -.  x  = -oo )
)  /\  y  e.  RR* )  /\  -.  y  = +oo )  ->  if ( y  = -oo , -oo ,  ( x  +  y ) )  e.  RR* )
328, 31ifclda 3964 . . . . . . 7  |-  ( ( ( x  e.  RR*  /\  ( -.  x  = +oo  /\  -.  x  = -oo ) )  /\  y  e.  RR* )  ->  if ( y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) )  e.  RR* )
3332an32s 802 . . . . . 6  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  ( -.  x  = +oo  /\  -.  x  = -oo ) )  ->  if ( y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) )  e.  RR* )
3433anassrs 648 . . . . 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 3964 . . . 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 3964 . . 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 2884 . 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 11308 . . 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 6841 . 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 208 1  |-  +e : ( RR*  X.  RR* )
--> RR*
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 368    /\ wa 369    \/ w3o 967    = wceq 1374    e. wcel 1762   A.wral 2807   ifcif 3932    X. cxp 4990   -->wf 5575  (class class class)co 6275   RRcr 9480   0cc0 9481    + caddc 9484   +oocpnf 9614   -oocmnf 9615   RR*cxr 9616   +ecxad 11305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-i2m1 9549  ax-1ne0 9550  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-pnf 9619  df-mnf 9620  df-xr 9621  df-xadd 11308
This theorem is referenced by:  xaddcl  11425  xrsadd  18199  xrofsup  27236  xrge0pluscn  27544  xrge0tmdOLD  27549
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