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

Step	Hyp	Ref	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(-∞ = -∞, -∞, (+∞ + -∞))))))
4	1, 2, 3	mp2an 692	. 2 ⊢ (+∞ +𝑒 -∞) = if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞)))))
5		eqid 2737	. . 3 ⊢ +∞ = +∞
6	5	iftruei 4446	. 2 ⊢ if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞))))) = if(-∞ = -∞, 0, +∞)
7		eqid 2737	. . 3 ⊢ -∞ = -∞
8	7	iftruei 4446	. 2 ⊢ if(-∞ = -∞, 0, +∞) = 0
9	4, 6, 8	3eqtri 2769	1 ⊢ (+∞ +𝑒 -∞) = 0

Syntax hints:   = wceq 1543   ∈ wcel 2110  ifcif 4439  (class class class)co 7213  0cc0 10729   + caddc 10732  +∞cpnf 10864  -∞cmnf 10865  ℝ^*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