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Theorem pnfaddmnf 11315
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 11207 . . 3  |- +oo  e.  RR*
2 mnfxr 11209 . . 3  |- -oo  e.  RR*
3 xaddval 11308 . . 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 672 . 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 2454 . . 3  |- +oo  = +oo
65iftruei 3909 . 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 )
7 eqid 2454 . . 3  |- -oo  = -oo
87iftruei 3909 . 2  |-  if ( -oo  = -oo , 
0 , +oo )  =  0
94, 6, 83eqtri 2487 1  |-  ( +oo +e -oo )  =  0
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758   ifcif 3902  (class class class)co 6203   0cc0 9397    + caddc 9400   +oocpnf 9530   -oocmnf 9531   RR*cxr 9532   +ecxad 11202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-mulcl 9459  ax-i2m1 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-iota 5492  df-fun 5531  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-pnf 9535  df-mnf 9536  df-xr 9537  df-xadd 11205
This theorem is referenced by:  xnegid  11321  xaddcom  11323  xnegdi  11326  xsubge0  11339  xlesubadd  11341  xadddilem  11372  xblss2  20119
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