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Theorem mnfaddpnf 10749
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 10646 . . 3  |-  -oo  e.  RR*
2 pnfxr 10645 . . 3  |-  +oo  e.  RR*
3 xaddval 10741 . . 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 10649 . . . . 5  |-  +oo  =/=  -oo
65necomi 2632 . . . 4  |-  -oo  =/=  +oo
7 ifnefalse 3690 . . . 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 2387 . . . . 5  |-  -oo  =  -oo
10 iftrue 3688 . . . . 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 2387 . . . . 5  |-  +oo  =  +oo
13 iftrue 3688 . . . . 5  |-  (  +oo  =  +oo  ->  if (  +oo  =  +oo ,  0 ,  -oo )  =  0 )
1412, 13ax-mp 8 . . . 4  |-  if ( 
+oo  =  +oo , 
0 ,  -oo )  =  0
1511, 14eqtri 2407 . . 3  |-  if ( 
-oo  =  -oo ,  if (  +oo  =  +oo ,  0 ,  -oo ) ,  if (  +oo  =  +oo ,  +oo ,  if (  +oo  =  -oo ,  -oo ,  (  -oo  +  +oo ) ) ) )  =  0
168, 15eqtri 2407 . 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 2407 1  |-  (  -oo + e  +oo )  =  0
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1717    =/= wne 2550   ifcif 3682  (class class class)co 6020   0cc0 8923    + caddc 8926    +oocpnf 9050    -oocmnf 9051   RR*cxr 9052   + ecxad 10640
This theorem is referenced by:  xnegid  10754  xaddcom  10756  xnegdi  10759  xsubge0  10772  xadddilem  10805
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-mulcl 8985  ax-i2m1 8991
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-pnf 9055  df-mnf 9056  df-xr 9057  df-xadd 10643
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