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Theorem pnfaddmnf 10649
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 10547 . . 3  |-  +oo  e.  RR*
2 mnfxr 10548 . . 3  |-  -oo  e.  RR*
3 xaddval 10642 . . 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 653 . 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 2358 . . 3  |-  +oo  =  +oo
6 iftrue 3647 . . 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 2358 . . 3  |-  -oo  =  -oo
9 iftrue 3647 . . 3  |-  (  -oo  =  -oo  ->  if (  -oo  =  -oo ,  0 ,  +oo )  =  0 )
108, 9ax-mp 8 . 2  |-  if ( 
-oo  =  -oo , 
0 ,  +oo )  =  0
114, 7, 103eqtri 2382 1  |-  (  +oo + e  -oo )  =  0
Colors of variables: wff set class
Syntax hints:    = wceq 1642    e. wcel 1710   ifcif 3641  (class class class)co 5945   0cc0 8827    + caddc 8830    +oocpnf 8954    -oocmnf 8955   RR*cxr 8956   + ecxad 10542
This theorem is referenced by:  xnegid  10655  xaddcom  10657  xnegdi  10660  xsubge0  10673  xlesubadd  10675  xadddilem  10706  xblss2  18060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-mulcl 8889  ax-i2m1 8895
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-pnf 8959  df-mnf 8960  df-xr 8961  df-xadd 10545
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