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Theorem pnfaddmnf 12099
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 (+∞ +𝑒 -∞) = 0

Proof of Theorem pnfaddmnf
StepHypRef Expression
1 pnfxr 10130 . . 3 +∞ ∈ ℝ*
2 mnfxr 10134 . . 3 -∞ ∈ ℝ*
3 xaddval 12092 . . 3 ((+∞ ∈ ℝ* ∧ -∞ ∈ ℝ*) → (+∞ +𝑒 -∞) = if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞))))))
41, 2, 3mp2an 708 . 2 (+∞ +𝑒 -∞) = if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞)))))
5 eqid 2651 . . 3 +∞ = +∞
65iftruei 4126 . 2 if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞))))) = if(-∞ = -∞, 0, +∞)
7 eqid 2651 . . 3 -∞ = -∞
87iftruei 4126 . 2 if(-∞ = -∞, 0, +∞) = 0
94, 6, 83eqtri 2677 1 (+∞ +𝑒 -∞) = 0
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  ifcif 4119  (class class class)co 6690  0cc0 9974   + caddc 9977  +∞cpnf 10109  -∞cmnf 10110  *cxr 10111   +𝑒 cxad 11982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-mulcl 10036  ax-i2m1 10042
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-pnf 10114  df-mnf 10115  df-xr 10116  df-xadd 11985
This theorem is referenced by:  xnegid  12107  xaddcom  12109  xnegdi  12116  xsubge0  12129  xlesubadd  12131  xadddilem  12162  xblss2  22254
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