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Theorem xaddf 10553
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 8880 . . . . . 6  |-  0  e.  RR*
2 pnfxr 10457 . . . . . 6  |-  +oo  e.  RR*
31, 2keepel 3624 . . . . 5  |-  if ( y  =  -oo , 
0 ,  +oo )  e.  RR*
43a1i 10 . . . 4  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  x  =  +oo )  ->  if ( y  =  -oo ,  0 ,  +oo )  e. 
RR* )
5 mnfxr 10458 . . . . . . 7  |-  -oo  e.  RR*
61, 5keepel 3624 . . . . . 6  |-  if ( y  =  +oo , 
0 ,  -oo )  e.  RR*
76a1i 10 . . . . 5  |-  ( ( ( ( x  e. 
RR*  /\  y  e.  RR* )  /\  -.  x  =  +oo )  /\  x  =  -oo )  ->  if ( y  =  +oo ,  0 ,  -oo )  e.  RR* )
82a1i 10 . . . . . . . 8  |-  ( ( ( ( x  e. 
RR*  /\  ( -.  x  =  +oo  /\  -.  x  =  -oo ) )  /\  y  e.  RR* )  /\  y  =  +oo )  ->  +oo  e.  RR* )
95a1i 10 . . . . . . . . 9  |-  ( ( ( ( ( x  e.  RR*  /\  ( -.  x  =  +oo  /\ 
-.  x  =  -oo ) )  /\  y  e.  RR* )  /\  -.  y  =  +oo )  /\  y  =  -oo )  ->  -oo  e.  RR* )
10 ioran 476 . . . . . . . . . . . . . 14  |-  ( -.  ( x  =  +oo  \/  x  =  -oo )  <->  ( -.  x  =  +oo  /\ 
-.  x  =  -oo ) )
11 elxr 10460 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  RR*  <->  ( x  e.  RR  \/  x  = 
+oo  \/  x  =  -oo ) )
12 3orass 937 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  e.  RR  \/  x  =  +oo  \/  x  =  -oo )  <->  ( x  e.  RR  \/  ( x  =  +oo  \/  x  =  -oo ) ) )
1311, 12bitri 240 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  RR*  <->  ( x  e.  RR  \/  ( x  =  +oo  \/  x  =  -oo ) ) )
1413biimpi 186 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  RR*  ->  ( x  e.  RR  \/  (
x  =  +oo  \/  x  =  -oo ) ) )
1514ord 366 . . . . . . . . . . . . . . . 16  |-  ( x  e.  RR*  ->  ( -.  x  e.  RR  ->  ( x  =  +oo  \/  x  =  -oo ) ) )
1615con1d 116 . . . . . . . . . . . . . . 15  |-  ( x  e.  RR*  ->  ( -.  ( x  =  +oo  \/  x  =  -oo )  ->  x  e.  RR ) )
1716imp 418 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR*  /\  -.  ( x  =  +oo  \/  x  =  -oo )
)  ->  x  e.  RR )
1810, 17sylan2br 462 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR*  /\  ( -.  x  =  +oo  /\ 
-.  x  =  -oo ) )  ->  x  e.  RR )
19 ioran 476 . . . . . . . . . . . . . 14  |-  ( -.  ( y  =  +oo  \/  y  =  -oo )  <->  ( -.  y  =  +oo  /\ 
-.  y  =  -oo ) )
20 elxr 10460 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  RR*  <->  ( y  e.  RR  \/  y  = 
+oo  \/  y  =  -oo ) )
21 3orass 937 . . . . . . . . . . . . . . . . . . 19  |-  ( ( y  e.  RR  \/  y  =  +oo  \/  y  =  -oo )  <->  ( y  e.  RR  \/  ( y  =  +oo  \/  y  =  -oo ) ) )
2220, 21bitri 240 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  RR*  <->  ( y  e.  RR  \/  ( y  =  +oo  \/  y  =  -oo ) ) )
2322biimpi 186 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  RR*  ->  ( y  e.  RR  \/  (
y  =  +oo  \/  y  =  -oo ) ) )
2423ord 366 . . . . . . . . . . . . . . . 16  |-  ( y  e.  RR*  ->  ( -.  y  e.  RR  ->  ( y  =  +oo  \/  y  =  -oo ) ) )
2524con1d 116 . . . . . . . . . . . . . . 15  |-  ( y  e.  RR*  ->  ( -.  ( y  =  +oo  \/  y  =  -oo )  ->  y  e.  RR ) )
2625imp 418 . . . . . . . . . . . . . 14  |-  ( ( y  e.  RR*  /\  -.  ( y  =  +oo  \/  y  =  -oo )
)  ->  y  e.  RR )
2719, 26sylan2br 462 . . . . . . . . . . . . 13  |-  ( ( y  e.  RR*  /\  ( -.  y  =  +oo  /\ 
-.  y  =  -oo ) )  ->  y  e.  RR )
28 readdcl 8822 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
2918, 27, 28syl2an 463 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR*  /\  ( -.  x  = 
+oo  /\  -.  x  =  -oo ) )  /\  ( y  e.  RR*  /\  ( -.  y  = 
+oo  /\  -.  y  =  -oo ) ) )  ->  ( x  +  y )  e.  RR )
3029rexrd 8883 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR*  /\  ( -.  x  = 
+oo  /\  -.  x  =  -oo ) )  /\  ( y  e.  RR*  /\  ( -.  y  = 
+oo  /\  -.  y  =  -oo ) ) )  ->  ( x  +  y )  e.  RR* )
3130anassrs 629 . . . . . . . . . 10  |-  ( ( ( ( x  e. 
RR*  /\  ( -.  x  =  +oo  /\  -.  x  =  -oo ) )  /\  y  e.  RR* )  /\  ( -.  y  =  +oo  /\  -.  y  =  -oo ) )  -> 
( x  +  y )  e.  RR* )
3231anassrs 629 . . . . . . . . 9  |-  ( ( ( ( ( x  e.  RR*  /\  ( -.  x  =  +oo  /\ 
-.  x  =  -oo ) )  /\  y  e.  RR* )  /\  -.  y  =  +oo )  /\  -.  y  =  -oo )  ->  ( x  +  y )  e.  RR* )
339, 32ifclda 3594 . . . . . . . 8  |-  ( ( ( ( x  e. 
RR*  /\  ( -.  x  =  +oo  /\  -.  x  =  -oo ) )  /\  y  e.  RR* )  /\  -.  y  = 
+oo )  ->  if ( y  =  -oo , 
-oo ,  ( x  +  y ) )  e.  RR* )
348, 33ifclda 3594 . . . . . . 7  |-  ( ( ( x  e.  RR*  /\  ( -.  x  = 
+oo  /\  -.  x  =  -oo ) )  /\  y  e.  RR* )  ->  if ( y  =  +oo , 
+oo ,  if (
y  =  -oo ,  -oo ,  ( x  +  y ) ) )  e.  RR* )
3534an32s 779 . . . . . 6  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  ( -.  x  = 
+oo  /\  -.  x  =  -oo ) )  ->  if ( y  =  +oo , 
+oo ,  if (
y  =  -oo ,  -oo ,  ( x  +  y ) ) )  e.  RR* )
3635anassrs 629 . . . . 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 3594 . . . 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 3594 . . 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 2611 . 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 10455 . . 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 6193 . 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 199 1  |-  + e : ( RR*  X.  RR* )
--> RR*
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 357    /\ wa 358    \/ w3o 933    = wceq 1625    e. wcel 1686   A.wral 2545   ifcif 3567    X. cxp 4689   -->wf 5253  (class class class)co 5860   RRcr 8738   0cc0 8739    + caddc 8742    +oocpnf 8866    -oocmnf 8867   RR*cxr 8868   + ecxad 10452
This theorem is referenced by:  xaddcl  10566  xrsadd  16393  xrofsup  23257  xrge0pluscn  23324  xrge0tmdALT  23329
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-i2m1 8807  ax-1ne0 8808  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-pnf 8871  df-mnf 8872  df-xr 8873  df-xadd 10455
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