Theorem pnfaddmnf 13160

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 11219	. . 3 ⊢ +∞ ∈ ℝ*
2		mnfxr 11222	. . 3 ⊢ -∞ ∈ ℝ*
3		xaddval 13153	. . 3 ⊢ ((+∞ ∈ ℝ* ∧ -∞ ∈ ℝ*) → (+∞ +𝑒 -∞) = if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞))))))
4	1, 2, 3	mp2an 691	. 2 ⊢ (+∞ +𝑒 -∞) = if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞)))))
5		eqid 2732	. . 3 ⊢ +∞ = +∞
6	5	iftruei 4499	. 2 ⊢ if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞))))) = if(-∞ = -∞, 0, +∞)
7		eqid 2732	. . 3 ⊢ -∞ = -∞
8	7	iftruei 4499	. 2 ⊢ if(-∞ = -∞, 0, +∞) = 0
9	4, 6, 8	3eqtri 2764	1 ⊢ (+∞ +𝑒 -∞) = 0

Syntax hints:   = wceq 1542   ∈ wcel 2107  ifcif 4492  (class class class)co 7363  0cc0 11061   + caddc 11064  +∞cpnf 11196  -∞cmnf 11197  ℝ^*cxr 11198   +_𝑒 cxad 13041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2703  ax-sep 5262  ax-nul 5269  ax-pow 5326  ax-pr 5390  ax-un 7678  ax-cnex 11117  ax-1cn 11119  ax-icn 11120  ax-addcl 11121  ax-mulcl 11123  ax-i2m1 11129

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4289  df-if 4493  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4872  df-br 5112  df-opab 5174  df-id 5537  df-xp 5645  df-rel 5646  df-cnv 5647  df-co 5648  df-dm 5649  df-iota 6454  df-fun 6504  df-fv 6510  df-ov 7366  df-oprab 7367  df-mpo 7368  df-pnf 11201  df-mnf 11202  df-xr 11203  df-xadd 13044

This theorem is referenced by:  xnegid  13168  xaddcom  13170  xnegdi  13178  xsubge0  13191  xlesubadd  13193  xadddilem  13224  xblss2  23793