Theorem pnfaddmnf 10269

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 8384	. . 3 ⊢ +∞ ∈ ℝ*
2		mnfxr 8388	. . 3 ⊢ -∞ ∈ ℝ*
3		xaddval 10264	. . 3 ⊢ ((+∞ ∈ ℝ* ∧ -∞ ∈ ℝ*) → (+∞ +e -∞) = if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞))))))
4	1, 2, 3	mp2an 430	. 2 ⊢ (+∞ +e -∞) = if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞)))))
5		eqid 2238	. . 3 ⊢ +∞ = +∞
6	5	iftruei 3646	. 2 ⊢ if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞))))) = if(-∞ = -∞, 0, +∞)
7		eqid 2238	. . 3 ⊢ -∞ = -∞
8	7	iftruei 3646	. 2 ⊢ if(-∞ = -∞, 0, +∞) = 0
9	4, 6, 8	3eqtri 2263	1 ⊢ (+∞ +e -∞) = 0

This proof depends on syntax axioms:   = wceq 1402   ∈ wcel 2209  ifcif 3638  (class class class)co 6088  0cc0 8185   + caddc 8188  +∞cpnf 8363  -∞cmnf 8364  ℝ^*cxr 8365   +_e cxad 10189

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1re 8279  ax-addrcl 8282  ax-rnegex 8294

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-iota 5340  df-fun 5382  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-pnf 8368  df-mnf 8369  df-xr 8370  df-xadd 10192

This theorem is used by:  xnegid  10278  xaddcom  10280  xnegdi  10287  xsubge0  10300  xposdif  10301  xlesubadd  10302  xrmaxadd  12052  xblss2  15631