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Theorem xaddval 11418
Description: Value of the extended real addition operation. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xaddval  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e B )  =  if ( A  = +oo ,  if ( B  = -oo ,  0 , +oo ) ,  if ( A  = -oo ,  if ( B  = +oo ,  0 , -oo ) ,  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo , 
( A  +  B
) ) ) ) ) )

Proof of Theorem xaddval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 457 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  x  =  A )
21eqeq1d 2469 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  = +oo  <->  A  = +oo ) )
3 simpr 461 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
43eqeq1d 2469 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  = -oo  <->  B  = -oo ) )
54ifbid 3961 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  = -oo ,  0 , +oo )  =  if ( B  = -oo ,  0 , +oo ) )
61eqeq1d 2469 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  = -oo  <->  A  = -oo ) )
73eqeq1d 2469 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( y  = +oo  <->  B  = +oo ) )
87ifbid 3961 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  = +oo ,  0 , -oo )  =  if ( B  = +oo ,  0 , -oo ) )
9 oveq12 6291 . . . . . 6  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  +  y )  =  ( A  +  B ) )
104, 9ifbieq2d 3964 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  = -oo , -oo , 
( x  +  y ) )  =  if ( B  = -oo , -oo ,  ( A  +  B ) ) )
117, 10ifbieq2d 3964 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) )  =  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo ,  ( A  +  B ) ) ) )
126, 8, 11ifbieq12d 3966 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  if ( x  = -oo ,  if ( y  = +oo , 
0 , -oo ) ,  if ( y  = +oo , +oo ,  if ( y  = -oo , -oo ,  ( x  +  y ) ) ) )  =  if ( A  = -oo ,  if ( B  = +oo ,  0 , -oo ) ,  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo ,  ( A  +  B ) ) ) ) )
132, 5, 12ifbieq12d 3966 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  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 ) ) ) ) )  =  if ( A  = +oo ,  if ( B  = -oo , 
0 , +oo ) ,  if ( A  = -oo ,  if ( B  = +oo , 
0 , -oo ) ,  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo ,  ( A  +  B ) ) ) ) ) )
14 df-xadd 11315 . 2  |-  +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 ) ) ) ) ) )
15 c0ex 9586 . . . 4  |-  0  e.  _V
16 pnfex 11318 . . . 4  |- +oo  e.  _V
1715, 16ifex 4008 . . 3  |-  if ( B  = -oo , 
0 , +oo )  e.  _V
18 mnfxr 11319 . . . . . 6  |- -oo  e.  RR*
1918elexi 3123 . . . . 5  |- -oo  e.  _V
2015, 19ifex 4008 . . . 4  |-  if ( B  = +oo , 
0 , -oo )  e.  _V
21 ovex 6307 . . . . . 6  |-  ( A  +  B )  e. 
_V
2219, 21ifex 4008 . . . . 5  |-  if ( B  = -oo , -oo ,  ( A  +  B ) )  e. 
_V
2316, 22ifex 4008 . . . 4  |-  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo ,  ( A  +  B ) ) )  e.  _V
2420, 23ifex 4008 . . 3  |-  if ( A  = -oo ,  if ( B  = +oo ,  0 , -oo ) ,  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo , 
( A  +  B
) ) ) )  e.  _V
2517, 24ifex 4008 . 2  |-  if ( A  = +oo ,  if ( B  = -oo ,  0 , +oo ) ,  if ( A  = -oo ,  if ( B  = +oo ,  0 , -oo ) ,  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo , 
( A  +  B
) ) ) ) )  e.  _V
2613, 14, 25ovmpt2a 6415 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e B )  =  if ( A  = +oo ,  if ( B  = -oo ,  0 , +oo ) ,  if ( A  = -oo ,  if ( B  = +oo ,  0 , -oo ) ,  if ( B  = +oo , +oo ,  if ( B  = -oo , -oo , 
( A  +  B
) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   ifcif 3939  (class class class)co 6282   0cc0 9488    + caddc 9491   +oocpnf 9621   -oocmnf 9622   RR*cxr 9623   +ecxad 11312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-mulcl 9550  ax-i2m1 9556
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-pnf 9626  df-mnf 9627  df-xr 9628  df-xadd 11315
This theorem is referenced by:  xaddpnf1  11421  xaddpnf2  11422  xaddmnf1  11423  xaddmnf2  11424  pnfaddmnf  11425  mnfaddpnf  11426  rexadd  11427
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