MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xaddf Unicode version

Theorem xaddf 10703
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 9025 . . . . . 6  |-  0  e.  RR*
2 pnfxr 10606 . . . . . 6  |-  +oo  e.  RR*
31, 2keepel 3711 . . . . 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 10607 . . . . . . 7  |-  -oo  e.  RR*
61, 5keepel 3711 . . . . . 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 10609 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  RR*  <->  ( x  e.  RR  \/  x  = 
+oo  \/  x  =  -oo ) )
12 3orass 938 . . . . . . . . . . . . . . . . . . 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 10609 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  RR*  <->  ( y  e.  RR  \/  y  = 
+oo  \/  y  =  -oo ) )
21 3orass 938 . . . . . . . . . . . . . . . . . . 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 8967 . . . . . . . . . . . . 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 9028 . . . . . . . . . . 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 3681 . . . . . . . 8  |-  ( ( ( ( x  e. 
RR*  /\  ( -.  x  =  +oo  /\  -.  x  =  -oo ) )  /\  y  e.  RR* )  /\  -.  y  = 
+oo )  ->  if ( y  =  -oo , 
-oo ,  ( x  +  y ) )  e.  RR* )
348, 33ifclda 3681 . . . . . . 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 3681 . . . 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 3681 . . 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 2694 . 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 10604 . . 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 6318 . 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 934    = wceq 1647    e. wcel 1715   A.wral 2628   ifcif 3654    X. cxp 4790   -->wf 5354  (class class class)co 5981   RRcr 8883   0cc0 8884    + caddc 8887    +oocpnf 9011    -oocmnf 9012   RR*cxr 9013   + ecxad 10601
This theorem is referenced by:  xaddcl  10716  xrsadd  16608  xrofsup  23526  xrge0pluscn  23681  xrge0tmdALT  23686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-i2m1 8952  ax-1ne0 8953  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-pnf 9016  df-mnf 9017  df-xr 9018  df-xadd 10604
  Copyright terms: Public domain W3C validator