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Theorem pnfaddmnf 9662
Description: Addition of positive and negative infinity. This is often taken to be a "null" value or out of the domain, but we define it (somewhat arbitrarily) to be zero so that the resulting function is total, which simplifies proofs. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
pnfaddmnf  |-  ( +oo +e -oo )  =  0

Proof of Theorem pnfaddmnf
StepHypRef Expression
1 pnfxr 7841 . . 3  |- +oo  e.  RR*
2 mnfxr 7845 . . 3  |- -oo  e.  RR*
3 xaddval 9657 . . 3  |-  ( ( +oo  e.  RR*  /\ -oo  e.  RR* )  ->  ( +oo +e -oo )  =  if ( +oo  = +oo ,  if ( -oo  = -oo ,  0 , +oo ) ,  if ( +oo  = -oo ,  if ( -oo  = +oo ,  0 , -oo ) ,  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo ,  ( +oo  + -oo )
) ) ) ) )
41, 2, 3mp2an 423 . 2  |-  ( +oo +e -oo )  =  if ( +oo  = +oo ,  if ( -oo  = -oo ,  0 , +oo ) ,  if ( +oo  = -oo ,  if ( -oo  = +oo ,  0 , -oo ) ,  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo ,  ( +oo  + -oo )
) ) ) )
5 eqid 2140 . . 3  |- +oo  = +oo
65iftruei 3484 . 2  |-  if ( +oo  = +oo ,  if ( -oo  = -oo ,  0 , +oo ) ,  if ( +oo  = -oo ,  if ( -oo  = +oo , 
0 , -oo ) ,  if ( -oo  = +oo , +oo ,  if ( -oo  = -oo , -oo ,  ( +oo  + -oo ) ) ) ) )  =  if ( -oo  = -oo , 
0 , +oo )
7 eqid 2140 . . 3  |- -oo  = -oo
87iftruei 3484 . 2  |-  if ( -oo  = -oo , 
0 , +oo )  =  0
94, 6, 83eqtri 2165 1  |-  ( +oo +e -oo )  =  0
Colors of variables: wff set class
Syntax hints:    = wceq 1332    e. wcel 1481   ifcif 3478  (class class class)co 5781   0cc0 7643    + caddc 7646   +oocpnf 7820   -oocmnf 7821   RR*cxr 7822   +ecxad 9586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735  ax-1re 7737  ax-addrcl 7740  ax-rnegex 7752
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-pnf 7825  df-mnf 7826  df-xr 7827  df-xadd 9589
This theorem is referenced by:  xnegid  9671  xaddcom  9673  xnegdi  9680  xsubge0  9693  xposdif  9694  xlesubadd  9695  xrmaxadd  11061  xblss2  12611
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