Theorem mnfaddpnf 13258

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 11267	. . 3 ⊢ -∞ ∈ ℝ*
2		pnfxr 11264	. . 3 ⊢ +∞ ∈ ℝ*
3		xaddval 13250	. . 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 11266	. . . 4 ⊢ -∞ ≠ +∞
6		ifnefalse 4500	. . . 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 2763	. . . . 5 ⊢ -∞ = -∞
9	8	iftruei 4495	. . . 4 ⊢ if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞)))) = if(+∞ = +∞, 0, -∞)
10		eqid 2763	. . . . 5 ⊢ +∞ = +∞
11	10	iftruei 4495	. . . 4 ⊢ if(+∞ = +∞, 0, -∞) = 0
12	9, 11	eqtri 2786	. . 3 ⊢ if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞)))) = 0
13	7, 12	eqtri 2786	. 2 ⊢ if(-∞ = +∞, if(+∞ = -∞, 0, +∞), if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞))))) = 0
14	4, 13	eqtri 2786	1 ⊢ (-∞ +𝑒 +∞) = 0

Syntax hints:   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ifcif 4488  (class class class)co 7412  0cc0 11101   + caddc 11104  +∞cpnf 11241  -∞cmnf 11242  ℝ^*cxr 11243   +_𝑒 cxad 13136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-mulcl 11163  ax-i2m1 11169

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-pnf 11246  df-mnf 11247  df-xr 11248  df-xadd 13139

This theorem is referenced by:  xnegid  13265  xaddcom  13267  xnegdi  13275  xsubge0  13288  xadddilem  13321  xrsnsgrp  21539