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Theorem pnfaddmnf 10780
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 10677 . . 3  |-  +oo  e.  RR*
2 mnfxr 10678 . . 3  |-  -oo  e.  RR*
3 xaddval 10773 . . 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 654 . 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 2412 . . 3  |-  +oo  =  +oo
6 iftrue 3713 . . 3  |-  (  +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 , 
0 ,  +oo )
)
75, 6ax-mp 8 . 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 )
8 eqid 2412 . . 3  |-  -oo  =  -oo
9 iftrue 3713 . . 3  |-  (  -oo  =  -oo  ->  if (  -oo  =  -oo ,  0 ,  +oo )  =  0 )
108, 9ax-mp 8 . 2  |-  if ( 
-oo  =  -oo , 
0 ,  +oo )  =  0
114, 7, 103eqtri 2436 1  |-  (  +oo + e  -oo )  =  0
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   ifcif 3707  (class class class)co 6048   0cc0 8954    + caddc 8957    +oocpnf 9081    -oocmnf 9082   RR*cxr 9083   + ecxad 10672
This theorem is referenced by:  xnegid  10786  xaddcom  10788  xnegdi  10791  xsubge0  10804  xlesubadd  10806  xadddilem  10837  xblss2  18393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-mulcl 9016  ax-i2m1 9022
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5385  df-fun 5423  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-pnf 9086  df-mnf 9087  df-xr 9088  df-xadd 10675
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