Theorem pnfaddmnf 10129

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 8274	. . 3 ⊢ +∞ ∈ ℝ*
2		mnfxr 8278	. . 3 ⊢ -∞ ∈ ℝ*
3		xaddval 10124	. . 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 2231	. . 3 ⊢ +∞ = +∞
6	5	iftruei 3615	. 2 ⊢ if(+∞ = +∞, if(-∞ = -∞, 0, +∞), if(+∞ = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (+∞ + -∞))))) = if(-∞ = -∞, 0, +∞)
7		eqid 2231	. . 3 ⊢ -∞ = -∞
8	7	iftruei 3615	. 2 ⊢ if(-∞ = -∞, 0, +∞) = 0
9	4, 6, 8	3eqtri 2256	1 ⊢ (+∞ +𝑒 -∞) = 0

Syntax hints:   = wceq 1398   ∈ wcel 2202  ifcif 3607  (class class class)co 6028  0cc0 8075   + caddc 8078  +∞cpnf 8253  -∞cmnf 8254  ℝ^*cxr 8255   +_𝑒 cxad 10049

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172  ax-rnegex 8184

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8258  df-mnf 8259  df-xr 8260  df-xadd 10052

This theorem is referenced by:  xnegid  10138  xaddcom  10140  xnegdi  10147  xsubge0  10160  xposdif  10161  xlesubadd  10162  xrmaxadd  11884  xblss2  15199