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Theorem mnfaddpnf 10558
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 10456 . . 3  |-  -oo  e.  RR*
2 pnfxr 10455 . . 3  |-  +oo  e.  RR*
3 xaddval 10550 . . 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 pnfnemnf 10459 . . . . 5  |-  +oo  =/=  -oo
65necomi 2528 . . . 4  |-  -oo  =/=  +oo
7 ifnefalse 3573 . . . 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 2283 . . . . 5  |-  -oo  =  -oo
10 iftrue 3571 . . . . 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 2283 . . . . 5  |-  +oo  =  +oo
13 iftrue 3571 . . . . 5  |-  (  +oo  =  +oo  ->  if (  +oo  =  +oo ,  0 ,  -oo )  =  0 )
1412, 13ax-mp 8 . . . 4  |-  if ( 
+oo  =  +oo , 
0 ,  -oo )  =  0
1511, 14eqtri 2303 . . 3  |-  if ( 
-oo  =  -oo ,  if (  +oo  =  +oo ,  0 ,  -oo ) ,  if (  +oo  =  +oo ,  +oo ,  if (  +oo  =  -oo ,  -oo ,  (  -oo  +  +oo ) ) ) )  =  0
168, 15eqtri 2303 . 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 2303 1  |-  (  -oo + e  +oo )  =  0
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684    =/= wne 2446   ifcif 3565  (class class class)co 5858   0cc0 8737    + caddc 8740    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866   + ecxad 10450
This theorem is referenced by:  xnegid  10563  xaddcom  10565  xnegdi  10568  xsubge0  10581  xadddilem  10614
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-mulcl 8799  ax-i2m1 8805
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-pnf 8869  df-mnf 8870  df-xr 8871  df-xadd 10453
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