Theorem pnfaddmnf 10183

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 8326	. . 3 ⊢ +∞ ∈ ℝ*
2		mnfxr 8330	. . 3 ⊢ -∞ ∈ ℝ*
3		xaddval 10178	. . 3 ⊢ ((+∞ ∈ ℝ* ∧ -∞ ∈ ℝ*) → (+∞ +𝑒 -∞) = if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞))))))
4	1, 2, 3	mp2an 426	. 2 ⊢ (+∞ +𝑒 -∞) = if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞)))))
5		eqid 2232	. . 3 ⊢ +∞ = +∞
6	5	iftruei 3628	. 2 ⊢ if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞))))) = if(-∞ = -∞, 0, +∞)
7		eqid 2232	. . 3 ⊢ -∞ = -∞
8	7	iftruei 3628	. 2 ⊢ if(-∞ = -∞, 0, +∞) = 0
9	4, 6, 8	3eqtri 2257	1 ⊢ (+∞ +𝑒 -∞) = 0

Syntax hints:   = wceq 1398   ∈ wcel 2203  ifcif 3620  (class class class)co 6050  0cc0 8127   + caddc 8130  +∞cpnf 8305  -∞cmnf 8306  ℝ^*cxr 8307   +_𝑒 cxad 10103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224  ax-rnegex 8236

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-xadd 10106

This theorem is referenced by:  xnegid  10192  xaddcom  10194  xnegdi  10201  xsubge0  10214  xposdif  10215  xlesubadd  10216  xrmaxadd  11946  xblss2  15270