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Theorem pnfaddmnf 12820
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 10887 . . 3 +∞ ∈ ℝ*
2 mnfxr 10890 . . 3 -∞ ∈ ℝ*
3 xaddval 12813 . . 3 ((+∞ ∈ ℝ* ∧ -∞ ∈ ℝ*) → (+∞ +𝑒 -∞) = if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞))))))
41, 2, 3mp2an 692 . 2 (+∞ +𝑒 -∞) = if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞)))))
5 eqid 2737 . . 3 +∞ = +∞
65iftruei 4446 . 2 if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞))))) = if(-∞ = -∞, 0, +∞)
7 eqid 2737 . . 3 -∞ = -∞
87iftruei 4446 . 2 if(-∞ = -∞, 0, +∞) = 0
94, 6, 83eqtri 2769 1 (+∞ +𝑒 -∞) = 0
Colors of variables: wff setvar class
Syntax hints:   = wceq 1543  wcel 2110  ifcif 4439  (class class class)co 7213  0cc0 10729   + caddc 10732  +∞cpnf 10864  -∞cmnf 10865  *cxr 10866   +𝑒 cxad 12702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-mulcl 10791  ax-i2m1 10797
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-iota 6338  df-fun 6382  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-pnf 10869  df-mnf 10870  df-xr 10871  df-xadd 12705
This theorem is referenced by:  xnegid  12828  xaddcom  12830  xnegdi  12838  xsubge0  12851  xlesubadd  12853  xadddilem  12884  xblss2  23300
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