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Theorem mnfaddpnf 9602
Description: Addition of negative and positive 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
mnfaddpnf  |-  ( -oo +e +oo )  =  0

Proof of Theorem mnfaddpnf
StepHypRef Expression
1 mnfxr 7790 . . 3  |- -oo  e.  RR*
2 pnfxr 7786 . . 3  |- +oo  e.  RR*
3 xaddval 9596 . . 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 422 . 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 mnfnepnf 7789 . . . 4  |- -oo  =/= +oo
6 ifnefalse 3455 . . . 4  |-  ( -oo  =/= +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 ) ) ) ) )  =  if ( -oo  = -oo ,  if ( +oo  = +oo ,  0 , -oo ) ,  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo ,  ( -oo  + +oo )
) ) ) )
75, 6ax-mp 5 . . 3  |-  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 ,  if ( +oo  = +oo ,  0 , -oo ) ,  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo ,  ( -oo  + +oo )
) ) )
8 eqid 2117 . . . . 5  |- -oo  = -oo
98iftruei 3450 . . . 4  |-  if ( -oo  = -oo ,  if ( +oo  = +oo ,  0 , -oo ) ,  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo ,  ( -oo  + +oo )
) ) )  =  if ( +oo  = +oo ,  0 , -oo )
10 eqid 2117 . . . . 5  |- +oo  = +oo
1110iftruei 3450 . . . 4  |-  if ( +oo  = +oo , 
0 , -oo )  =  0
129, 11eqtri 2138 . . 3  |-  if ( -oo  = -oo ,  if ( +oo  = +oo ,  0 , -oo ) ,  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo ,  ( -oo  + +oo )
) ) )  =  0
137, 12eqtri 2138 . 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 ) ) ) ) )  =  0
144, 13eqtri 2138 1  |-  ( -oo +e +oo )  =  0
Colors of variables: wff set class
Syntax hints:    = wceq 1316    e. wcel 1465    =/= wne 2285   ifcif 3444  (class class class)co 5742   0cc0 7588    + caddc 7591   +oocpnf 7765   -oocmnf 7766   RR*cxr 7767   +ecxad 9525
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1re 7682  ax-addrcl 7685  ax-rnegex 7697
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-xadd 9528
This theorem is referenced by:  xnegid  9610  xaddcom  9612  xnegdi  9619  xsubge0  9632  xposdif  9633  xrmaxadd  10998
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