Theorem mnfaddpnf 13248

Description: Addition of negative and positive 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
mnfaddpnf	⊢ (-∞ +𝑒 +∞) = 0

Proof of Theorem mnfaddpnf

Step	Hyp	Ref	Expression
1		mnfxr 11254	. . 3 ⊢ -∞ ∈ ℝ*
2		pnfxr 11251	. . 3 ⊢ +∞ ∈ ℝ*
3		xaddval 13240	. . 3 ⊢ ((-∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞ +𝑒 +∞) = if(-∞ = +∞, if(+∞ = -∞, 0, +∞), if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞))))))
4	1, 2, 3	mp2an 704	. 2 ⊢ (-∞ +𝑒 +∞) = if(-∞ = +∞, if(+∞ = -∞, 0, +∞), if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞)))))
5		mnfnepnf 11253	. . . 4 ⊢ -∞ ≠ +∞
6		ifnefalse 4495	. . . 4 ⊢ (-∞ ≠ +∞ → if(-∞ = +∞, if(+∞ = -∞, 0, +∞), if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞))))) = if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞)))))
7	5, 6	ax-mp 5	. . 3 ⊢ if(-∞ = +∞, if(+∞ = -∞, 0, +∞), if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞))))) = if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞))))
8		eqid 2765	. . . . 5 ⊢ -∞ = -∞
9	8	iftruei 4490	. . . 4 ⊢ if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞)))) = if(+∞ = +∞, 0, -∞)
10		eqid 2765	. . . . 5 ⊢ +∞ = +∞
11	10	iftruei 4490	. . . 4 ⊢ if(+∞ = +∞, 0, -∞) = 0
12	9, 11	eqtri 2788	. . 3 ⊢ if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞)))) = 0
13	7, 12	eqtri 2788	. 2 ⊢ if(-∞ = +∞, if(+∞ = -∞, 0, +∞), if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞))))) = 0
14	4, 13	eqtri 2788	1 ⊢ (-∞ +𝑒 +∞) = 0

Syntax hints:   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ifcif 4483  (class class class)co 7400  0cc0 11088   + caddc 11091  +∞cpnf 11228  -∞cmnf 11229  ℝ^*cxr 11230   +_𝑒 cxad 13126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-mulcl 11150  ax-i2m1 11156

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-pnf 11233  df-mnf 11234  df-xr 11235  df-xadd 13129

This theorem is referenced by:  xnegid  13255  xaddcom  13257  xnegdi  13265  xsubge0  13278  xadddilem  13311  xrsnsgrp  21518