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Theorem mnfaddpnf 9475
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 7694 . . 3  |- -oo  e.  RR*
2 pnfxr 7690 . . 3  |- +oo  e.  RR*
3 xaddval 9469 . . 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 420 . 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 7693 . . . 4  |- -oo  =/= +oo
6 ifnefalse 3432 . . . 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 7 . . 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 2100 . . . . 5  |- -oo  = -oo
98iftruei 3427 . . . 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 2100 . . . . 5  |- +oo  = +oo
1110iftruei 3427 . . . 4  |-  if ( +oo  = +oo , 
0 , -oo )  =  0
129, 11eqtri 2120 . . 3  |-  if ( -oo  = -oo ,  if ( +oo  = +oo ,  0 , -oo ) ,  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo ,  ( -oo  + +oo )
) ) )  =  0
137, 12eqtri 2120 . 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 2120 1  |-  ( -oo +e +oo )  =  0
Colors of variables: wff set class
Syntax hints:    = wceq 1299    e. wcel 1448    =/= wne 2267   ifcif 3421  (class class class)co 5706   0cc0 7500    + caddc 7503   +oocpnf 7669   -oocmnf 7670   RR*cxr 7671   +ecxad 9398
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1re 7589  ax-addrcl 7592  ax-rnegex 7604
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-pnf 7674  df-mnf 7675  df-xr 7676  df-xadd 9401
This theorem is referenced by:  xnegid  9483  xaddcom  9485  xnegdi  9492  xsubge0  9505  xposdif  9506  xrmaxadd  10869
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