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Theorem mnfaddpnf 10574
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 10472 . . 3  |-  -oo  e.  RR*
2 pnfxr 10471 . . 3  |-  +oo  e.  RR*
3 xaddval 10566 . . 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 10475 . . . . 5  |-  +oo  =/=  -oo
65necomi 2541 . . . 4  |-  -oo  =/=  +oo
7 ifnefalse 3586 . . . 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 2296 . . . . 5  |-  -oo  =  -oo
10 iftrue 3584 . . . . 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 2296 . . . . 5  |-  +oo  =  +oo
13 iftrue 3584 . . . . 5  |-  (  +oo  =  +oo  ->  if (  +oo  =  +oo ,  0 ,  -oo )  =  0 )
1412, 13ax-mp 8 . . . 4  |-  if ( 
+oo  =  +oo , 
0 ,  -oo )  =  0
1511, 14eqtri 2316 . . 3  |-  if ( 
-oo  =  -oo ,  if (  +oo  =  +oo ,  0 ,  -oo ) ,  if (  +oo  =  +oo ,  +oo ,  if (  +oo  =  -oo ,  -oo ,  (  -oo  +  +oo ) ) ) )  =  0
168, 15eqtri 2316 . 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 2316 1  |-  (  -oo + e  +oo )  =  0
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696    =/= wne 2459   ifcif 3578  (class class class)co 5874   0cc0 8753    + caddc 8756    +oocpnf 8880    -oocmnf 8881   RR*cxr 8882   + ecxad 10466
This theorem is referenced by:  xnegid  10579  xaddcom  10581  xnegdi  10584  xsubge0  10597  xadddilem  10630
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-mulcl 8815  ax-i2m1 8821
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-pnf 8885  df-mnf 8886  df-xr 8887  df-xadd 10469
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