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Theorem mnfaddpnf 11483
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 11376 . . 3  |- -oo  e.  RR*
2 pnfxr 11374 . . 3  |- +oo  e.  RR*
3 xaddval 11475 . . 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 670 . 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 11380 . . . 4  |- -oo  =/= +oo
6 ifnefalse 3897 . . . 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 2402 . . . . 5  |- -oo  = -oo
98iftruei 3892 . . . 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 2402 . . . . 5  |- +oo  = +oo
1110iftruei 3892 . . . 4  |-  if ( +oo  = +oo , 
0 , -oo )  =  0
129, 11eqtri 2431 . . 3  |-  if ( -oo  = -oo ,  if ( +oo  = +oo ,  0 , -oo ) ,  if ( +oo  = +oo , +oo ,  if ( +oo  = -oo , -oo ,  ( -oo  + +oo )
) ) )  =  0
137, 12eqtri 2431 . 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 2431 1  |-  ( -oo +e +oo )  =  0
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405    e. wcel 1842    =/= wne 2598   ifcif 3885  (class class class)co 6278   0cc0 9522    + caddc 9525   +oocpnf 9655   -oocmnf 9656   RR*cxr 9657   +ecxad 11369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-mulcl 9584  ax-i2m1 9590
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-pnf 9660  df-mnf 9661  df-xr 9662  df-xadd 11372
This theorem is referenced by:  xnegid  11488  xaddcom  11490  xnegdi  11493  xsubge0  11506  xadddilem  11539  xrsnsgrp  18774
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