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Theorem mnfaddpnf 9648
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 7836 . . 3  |- -oo  e.  RR*
2 pnfxr 7832 . . 3  |- +oo  e.  RR*
3 xaddval 9642 . . 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 7835 . . . 4  |- -oo  =/= +oo
6 ifnefalse 3485 . . . 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 2139 . . . . 5  |- -oo  = -oo
98iftruei 3480 . . . 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 2139 . . . . 5  |- +oo  = +oo
1110iftruei 3480 . . . 4  |-  if ( +oo  = +oo , 
0 , -oo )  =  0
129, 11eqtri 2160 . . 3  |-  if ( -oo  = -oo ,  if ( +oo  = +oo ,  0 , -oo ) ,  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo ,  ( -oo  + +oo )
) ) )  =  0
137, 12eqtri 2160 . 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 2160 1  |-  ( -oo +e +oo )  =  0
Colors of variables: wff set class
Syntax hints:    = wceq 1331    e. wcel 1480    =/= wne 2308   ifcif 3474  (class class class)co 5774   0cc0 7634    + caddc 7637   +oocpnf 7811   -oocmnf 7812   RR*cxr 7813   +ecxad 9571
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7725  ax-resscn 7726  ax-1re 7728  ax-addrcl 7731  ax-rnegex 7743
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7816  df-mnf 7817  df-xr 7818  df-xadd 9574
This theorem is referenced by:  xnegid  9656  xaddcom  9658  xnegdi  9665  xsubge0  9678  xposdif  9679  xrmaxadd  11044
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