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Theorem xaddf 11394
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 9590 . . . . . 6  |-  0  e.  RR*
2 pnfxr 11292 . . . . . 6  |- +oo  e.  RR*
31, 2keepel 3951 . . . . 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 11294 . . . . . . 7  |- -oo  e.  RR*
61, 5keepel 3951 . . . . . 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 488 . . . . . . . . . . . . . 14  |-  ( -.  ( x  = +oo  \/  x  = -oo ) 
<->  ( -.  x  = +oo  /\  -.  x  = -oo ) )
11 elxr 11296 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  RR*  <->  ( x  e.  RR  \/  x  = +oo  \/  x  = -oo ) )
12 3orass 977 . . . . . . . . . . . . . . . . . 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 375 . . . . . . . . . . . . . . . 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 427 . . . . . . . . . . . . . 14  |-  ( ( x  e.  RR*  /\  -.  ( x  = +oo  \/  x  = -oo ) )  ->  x  e.  RR )
1710, 16sylan2br 474 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR*  /\  ( -.  x  = +oo  /\ 
-.  x  = -oo ) )  ->  x  e.  RR )
18 ioran 488 . . . . . . . . . . . . . 14  |-  ( -.  ( y  = +oo  \/  y  = -oo ) 
<->  ( -.  y  = +oo  /\  -.  y  = -oo ) )
19 elxr 11296 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  RR*  <->  ( y  e.  RR  \/  y  = +oo  \/  y  = -oo ) )
20 3orass 977 . . . . . . . . . . . . . . . . . 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 375 . . . . . . . . . . . . . . . 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 427 . . . . . . . . . . . . . 14  |-  ( ( y  e.  RR*  /\  -.  ( y  = +oo  \/  y  = -oo ) )  ->  y  e.  RR )
2518, 24sylan2br 474 . . . . . . . . . . . . 13  |-  ( ( y  e.  RR*  /\  ( -.  y  = +oo  /\ 
-.  y  = -oo ) )  ->  y  e.  RR )
26 readdcl 9525 . . . . . . . . . . . . 13  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  e.  RR )
2717, 25, 26syl2an 475 . . . . . . . . . . . 12  |-  ( ( ( x  e.  RR*  /\  ( -.  x  = +oo  /\  -.  x  = -oo ) )  /\  ( y  e.  RR*  /\  ( -.  y  = +oo  /\  -.  y  = -oo ) ) )  ->  ( x  +  y )  e.  RR )
2827rexrd 9593 . . . . . . . . . . 11  |-  ( ( ( x  e.  RR*  /\  ( -.  x  = +oo  /\  -.  x  = -oo ) )  /\  ( y  e.  RR*  /\  ( -.  y  = +oo  /\  -.  y  = -oo ) ) )  ->  ( x  +  y )  e.  RR* )
2928anassrs 646 . . . . . . . . . 10  |-  ( ( ( ( x  e. 
RR*  /\  ( -.  x  = +oo  /\  -.  x  = -oo )
)  /\  y  e.  RR* )  /\  ( -.  y  = +oo  /\  -.  y  = -oo ) )  ->  (
x  +  y )  e.  RR* )
3029anassrs 646 . . . . . . . . 9  |-  ( ( ( ( ( x  e.  RR*  /\  ( -.  x  = +oo  /\ 
-.  x  = -oo ) )  /\  y  e.  RR* )  /\  -.  y  = +oo )  /\  -.  y  = -oo )  ->  ( x  +  y )  e.  RR* )
319, 30ifclda 3916 . . . . . . . 8  |-  ( ( ( ( x  e. 
RR*  /\  ( -.  x  = +oo  /\  -.  x  = -oo )
)  /\  y  e.  RR* )  /\  -.  y  = +oo )  ->  if ( y  = -oo , -oo ,  ( x  +  y ) )  e.  RR* )
328, 31ifclda 3916 . . . . . . 7  |-  ( ( ( x  e.  RR*  /\  ( -.  x  = +oo  /\  -.  x  = -oo ) )  /\  y  e.  RR* )  ->  if ( y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) )  e.  RR* )
3332an32s 805 . . . . . 6  |-  ( ( ( x  e.  RR*  /\  y  e.  RR* )  /\  ( -.  x  = +oo  /\  -.  x  = -oo ) )  ->  if ( y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) )  e.  RR* )
3433anassrs 646 . . . . 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 3916 . . . 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 3916 . . 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 2830 . 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 11290 . . 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 6805 . 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 366    /\ wa 367    \/ w3o 973    = wceq 1405    e. wcel 1842   A.wral 2753   ifcif 3884    X. cxp 4940   -->wf 5521  (class class class)co 6234   RRcr 9441   0cc0 9442    + caddc 9445   +oocpnf 9575   -oocmnf 9576   RR*cxr 9577   +ecxad 11287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-i2m1 9510  ax-1ne0 9511  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-1st 6738  df-2nd 6739  df-pnf 9580  df-mnf 9581  df-xr 9582  df-xadd 11290
This theorem is referenced by:  xaddcl  11407  xrsadd  18647  xrofsup  27910  xrge0pluscn  28255  xrge0tmdOLD  28260
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