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Theorem xaddf 11192
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 9428 . . . . . 6  |-  0  e.  RR*
2 pnfxr 11090 . . . . . 6  |- +oo  e.  RR*
31, 2keepel 3855 . . . . 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 11092 . . . . . . 7  |- -oo  e.  RR*
61, 5keepel 3855 . . . . . 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 11094 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  RR*  <->  ( x  e.  RR  \/  x  = +oo  \/  x  = -oo ) )
12 3orass 968 . . . . . . . . . . . . . . . . . 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 11094 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  RR*  <->  ( y  e.  RR  \/  y  = +oo  \/  y  = -oo ) )
20 3orass 968 . . . . . . . . . . . . . . . . . 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 9363 . . . . . . . . . . . . 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 9431 . . . . . . . . . . 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 3819 . . . . . . . 8  |-  ( ( ( ( x  e. 
RR*  /\  ( -.  x  = +oo  /\  -.  x  = -oo )
)  /\  y  e.  RR* )  /\  -.  y  = +oo )  ->  if ( y  = -oo , -oo ,  ( x  +  y ) )  e.  RR* )
328, 31ifclda 3819 . . . . . . 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 3819 . . . 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 3819 . . 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 2780 . 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 11088 . . 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 6639 . 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 964    = wceq 1369    e. wcel 1756   A.wral 2713   ifcif 3789    X. cxp 4836   -->wf 5412  (class class class)co 6089   RRcr 9279   0cc0 9280    + caddc 9283   +oocpnf 9413   -oocmnf 9414   RR*cxr 9415   +ecxad 11085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-i2m1 9348  ax-1ne0 9349  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-pnf 9418  df-mnf 9419  df-xr 9420  df-xadd 11088
This theorem is referenced by:  xaddcl  11205  xrsadd  17831  xrofsup  26053  xrge0pluscn  26368  xrge0tmdOLD  26373
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