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Theorem mnfaddpnf 10809
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 10706 . . 3  |-  -oo  e.  RR*
2 pnfxr 10705 . . 3  |-  +oo  e.  RR*
3 xaddval 10801 . . 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 pnfnemnf 10709 . . . . 5  |-  +oo  =/=  -oo
65necomi 2680 . . . 4  |-  -oo  =/=  +oo
7 ifnefalse 3739 . . . 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 ) ) ) ) )
86, 7ax-mp 8 . . 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 ) ) ) )
9 eqid 2435 . . . . 5  |-  -oo  =  -oo
10 iftrue 3737 . . . . 5  |-  (  -oo  =  -oo  ->  if (  -oo  =  -oo ,  if (  +oo  =  +oo , 
0 ,  -oo ) ,  if (  +oo  =  +oo ,  +oo ,  if (  +oo  =  -oo ,  -oo ,  (  -oo  +  +oo ) ) ) )  =  if (  +oo  =  +oo ,  0 , 
-oo ) )
119, 10ax-mp 8 . . . 4  |-  if ( 
-oo  =  -oo ,  if (  +oo  =  +oo ,  0 ,  -oo ) ,  if (  +oo  =  +oo ,  +oo ,  if (  +oo  =  -oo ,  -oo ,  (  -oo  +  +oo ) ) ) )  =  if (  +oo  =  +oo ,  0 , 
-oo )
12 eqid 2435 . . . . 5  |-  +oo  =  +oo
13 iftrue 3737 . . . . 5  |-  (  +oo  =  +oo  ->  if (  +oo  =  +oo ,  0 ,  -oo )  =  0 )
1412, 13ax-mp 8 . . . 4  |-  if ( 
+oo  =  +oo , 
0 ,  -oo )  =  0
1511, 14eqtri 2455 . . 3  |-  if ( 
-oo  =  -oo ,  if (  +oo  =  +oo ,  0 ,  -oo ) ,  if (  +oo  =  +oo ,  +oo ,  if (  +oo  =  -oo ,  -oo ,  (  -oo  +  +oo ) ) ) )  =  0
168, 15eqtri 2455 . 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
174, 16eqtri 2455 1  |-  (  -oo + e  +oo )  =  0
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725    =/= wne 2598   ifcif 3731  (class class class)co 6073   0cc0 8982    + caddc 8985    +oocpnf 9109    -oocmnf 9110   RR*cxr 9111   + ecxad 10700
This theorem is referenced by:  xnegid  10814  xaddcom  10816  xnegdi  10819  xsubge0  10832  xadddilem  10865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-mulcl 9044  ax-i2m1 9050
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-pnf 9114  df-mnf 9115  df-xr 9116  df-xadd 10703
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