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Theorem pnfaddmnf 10821
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 10718 . . 3  |-  +oo  e.  RR*
2 mnfxr 10719 . . 3  |-  -oo  e.  RR*
3 xaddval 10814 . . 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 655 . 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 2438 . . 3  |-  +oo  =  +oo
6 iftrue 3747 . . 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 5 . 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 2438 . . 3  |-  -oo  =  -oo
9 iftrue 3747 . . 3  |-  (  -oo  =  -oo  ->  if (  -oo  =  -oo ,  0 ,  +oo )  =  0 )
108, 9ax-mp 5 . 2  |-  if ( 
-oo  =  -oo , 
0 ,  +oo )  =  0
114, 7, 103eqtri 2462 1  |-  (  +oo + e  -oo )  =  0
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   ifcif 3741  (class class class)co 6084   0cc0 8995    + caddc 8998    +oocpnf 9122    -oocmnf 9123   RR*cxr 9124   + ecxad 10713
This theorem is referenced by:  xnegid  10827  xaddcom  10829  xnegdi  10832  xsubge0  10845  xlesubadd  10847  xadddilem  10878  xblss2  18437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-mulcl 9057  ax-i2m1 9063
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-pnf 9127  df-mnf 9128  df-xr 9129  df-xadd 10716
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