Theorem pnfaddmnf 13384

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 11387	. . 3 ⊢ +∞ ∈ ℝ*
2		mnfxr 11390	. . 3 ⊢ -∞ ∈ ℝ*
3		xaddval 13377	. . 3 ⊢ ((+∞ ∈ ℝ* ∧ -∞ ∈ ℝ*) → (+∞ +e -∞) = if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞))))))
4	1, 2, 3	mp2an 705	. 2 ⊢ (+∞ +e -∞) = if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞)))))
5		eqid 2763	. . 3 ⊢ +∞ = +∞
6	5	iftruei 4492	. 2 ⊢ if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞))))) = if(-∞ = -∞, 0, +∞)
7		eqid 2763	. . 3 ⊢ -∞ = -∞
8	7	iftruei 4492	. 2 ⊢ if(-∞ = -∞, 0, +∞) = 0
9	4, 6, 8	3eqtri 2790	1 ⊢ (+∞ +e -∞) = 0

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  ifcif 4485  (class class class)co 7440  0cc0 11224   + caddc 11227  +∞cpnf 11364  -∞cmnf 11365  ℝ^*cxr 11366   +_e cxad 13263

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-mulcl 11286  ax-i2m1 11292

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6509  df-fun 6557  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-pnf 11369  df-mnf 11370  df-xr 11371  df-xadd 13266

This theorem is used by:  xnegid  13392  xaddcom  13394  xnegdi  13402  xsubge0  13415  xlesubadd  13417  xadddilem  13448  xblss2  24748