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Theorem xaddf 10810
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 9131 . . . . . 6  |-  0  e.  RR*
2 pnfxr 10713 . . . . . 6  |-  +oo  e.  RR*
31, 2keepel 3796 . . . . 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 10714 . . . . . . 7  |-  -oo  e.  RR*
61, 5keepel 3796 . . . . . 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 477 . . . . . . . . . . . . . 14  |-  ( -.  ( x  =  +oo  \/  x  =  -oo )  <->  ( -.  x  =  +oo  /\ 
-.  x  =  -oo ) )
11 elxr 10716 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  RR*  <->  ( x  e.  RR  \/  x  = 
+oo  \/  x  =  -oo ) )
12 3orass 939 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  RR  \/  x  =  +oo  \/  x  =  -oo )  <->  ( x  e.  RR  \/  ( x  =  +oo  \/  x  =  -oo ) ) )
1311, 12bitri 241 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  RR*  <->  ( x  e.  RR  \/  ( x  =  +oo  \/  x  =  -oo ) ) )
1413biimpi 187 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  RR*  ->  ( x  e.  RR  \/  (
x  =  +oo  \/  x  =  -oo ) ) )
1514ord 367 . . . . . . . . . . . . . . . 16  |-  ( x  e.  RR*  ->  ( -.  x  e.  RR  ->  ( x  =  +oo  \/  x  =  -oo ) ) )
1615con1d 118 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR*  ->  ( -.  ( x  =  +oo  \/  x  =  -oo )  ->  x  e.  RR ) )
1716imp 419 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR*  /\  -.  ( x  =  +oo  \/  x  =  -oo )
)  ->  x  e.  RR )
1810, 17sylan2br 463 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR*  /\  ( -.  x  =  +oo  /\ 
-.  x  =  -oo ) )  ->  x  e.  RR )
19 ioran 477 . . . . . . . . . . . . . 14  |-  ( -.  ( y  =  +oo  \/  y  =  -oo )  <->  ( -.  y  =  +oo  /\ 
-.  y  =  -oo ) )
20 elxr 10716 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  RR*  <->  ( y  e.  RR  \/  y  = 
+oo  \/  y  =  -oo ) )
21 3orass 939 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  RR  \/  y  =  +oo  \/  y  =  -oo )  <->  ( y  e.  RR  \/  ( y  =  +oo  \/  y  =  -oo ) ) )
2220, 21bitri 241 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  RR*  <->  ( y  e.  RR  \/  ( y  =  +oo  \/  y  =  -oo ) ) )
2322biimpi 187 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  RR*  ->  ( y  e.  RR  \/  (
y  =  +oo  \/  y  =  -oo ) ) )
2423ord 367 . . . . . . . . . . . . . . . 16  |-  ( y  e.  RR*  ->  ( -.  y  e.  RR  ->  ( y  =  +oo  \/  y  =  -oo ) ) )
2524con1d 118 . . . . . . . . . . . . . . 15  |-  ( y  e.  RR*  ->  ( -.  ( y  =  +oo  \/  y  =  -oo )  ->  y  e.  RR ) )
2625imp 419 . . . . . . . . . . . . . 14  |-  ( ( y  e.  RR*  /\  -.  ( y  =  +oo  \/  y  =  -oo )
)  ->  y  e.  RR )
2719, 26sylan2br 463 . . . . . . . . . . . . 13  |-  ( ( y  e.  RR*  /\  ( -.  y  =  +oo  /\ 
-.  y  =  -oo ) )  ->  y  e.  RR )
28 readdcl 9073 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
2918, 27, 28syl2an 464 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR*  /\  ( -.  x  = 
+oo  /\  -.  x  =  -oo ) )  /\  ( y  e.  RR*  /\  ( -.  y  = 
+oo  /\  -.  y  =  -oo ) ) )  ->  ( x  +  y )  e.  RR )
3029rexrd 9134 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR*  /\  ( -.  x  = 
+oo  /\  -.  x  =  -oo ) )  /\  ( y  e.  RR*  /\  ( -.  y  = 
+oo  /\  -.  y  =  -oo ) ) )  ->  ( x  +  y )  e.  RR* )
3130anassrs 630 . . . . . . . . . 10  |-  ( ( ( ( x  e. 
RR*  /\  ( -.  x  =  +oo  /\  -.  x  =  -oo ) )  /\  y  e.  RR* )  /\  ( -.  y  =  +oo  /\  -.  y  =  -oo ) )  -> 
( x  +  y )  e.  RR* )
3231anassrs 630 . . . . . . . . 9  |-  ( ( ( ( ( x  e.  RR*  /\  ( -.  x  =  +oo  /\ 
-.  x  =  -oo ) )  /\  y  e.  RR* )  /\  -.  y  =  +oo )  /\  -.  y  =  -oo )  ->  ( x  +  y )  e.  RR* )
339, 32ifclda 3766 . . . . . . . 8  |-  ( ( ( ( x  e. 
RR*  /\  ( -.  x  =  +oo  /\  -.  x  =  -oo ) )  /\  y  e.  RR* )  /\  -.  y  = 
+oo )  ->  if ( y  =  -oo , 
-oo ,  ( x  +  y ) )  e.  RR* )
348, 33ifclda 3766 . . . . . . 7  |-  ( ( ( x  e.  RR*  /\  ( -.  x  = 
+oo  /\  -.  x  =  -oo ) )  /\  y  e.  RR* )  ->  if ( y  =  +oo , 
+oo ,  if (
y  =  -oo ,  -oo ,  ( x  +  y ) ) )  e.  RR* )
3534an32s 780 . . . . . 6  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  ( -.  x  = 
+oo  /\  -.  x  =  -oo ) )  ->  if ( y  =  +oo , 
+oo ,  if (
y  =  -oo ,  -oo ,  ( x  +  y ) ) )  e.  RR* )
3635anassrs 630 . . . . 5  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  -.  x  =  +oo )  /\  -.  x  =  -oo )  ->  if ( y  =  +oo , 
+oo ,  if (
y  =  -oo ,  -oo ,  ( x  +  y ) ) )  e.  RR* )
377, 36ifclda 3766 . . . 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* )
384, 37ifclda 3766 . . 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* )
3938rgen2a 2772 . 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*
40 df-xadd 10711 . . 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 ) ) ) ) ) )
4140fmpt2 6418 . 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* )
4239, 41mpbi 200 1  |-  + e : ( RR*  X.  RR* )
--> RR*
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 358    /\ wa 359    \/ w3o 935    = wceq 1652    e. wcel 1725   A.wral 2705   ifcif 3739    X. cxp 4876   -->wf 5450  (class class class)co 6081   RRcr 8989   0cc0 8990    + caddc 8993    +oocpnf 9117    -oocmnf 9118   RR*cxr 9119   + ecxad 10708
This theorem is referenced by:  xaddcl  10823  xrsadd  16718  xrofsup  24126  xrge0pluscn  24326  xrge0tmdOLD  24331
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-i2m1 9058  ax-1ne0 9059  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-pnf 9122  df-mnf 9123  df-xr 9124  df-xadd 10711
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