Theorem mnfaddpnf 10035

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 8191	. . 3 ⊢ -∞ ∈ ℝ*
2		pnfxr 8187	. . 3 ⊢ +∞ ∈ ℝ*
3		xaddval 10029	. . 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		mnfnepnf 8190	. . . 4 ⊢ -∞ ≠ +∞
6		ifnefalse 3613	. . . 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 2229	. . . . 5 ⊢ -∞ = -∞
9	8	iftruei 3608	. . . 4 ⊢ if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞)))) = if(+∞ = +∞, 0, -∞)
10		eqid 2229	. . . . 5 ⊢ +∞ = +∞
11	10	iftruei 3608	. . . 4 ⊢ if(+∞ = +∞, 0, -∞) = 0
12	9, 11	eqtri 2250	. . 3 ⊢ if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞)))) = 0
13	7, 12	eqtri 2250	. 2 ⊢ if(-∞ = +∞, if(+∞ = -∞, 0, +∞), if(-∞ = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (-∞ + +∞))))) = 0
14	4, 13	eqtri 2250	1 ⊢ (-∞ +𝑒 +∞) = 0

Syntax hints:   = wceq 1395   ∈ wcel 2200   ≠ wne 2400  ifcif 3602  (class class class)co 5994  0cc0 7987   + caddc 7990  +∞cpnf 8166  -∞cmnf 8167  ℝ^*cxr 8168   +_𝑒 cxad 9954

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1re 8081  ax-addrcl 8084  ax-rnegex 8096

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-iota 5274  df-fun 5316  df-fv 5322  df-ov 5997  df-oprab 5998  df-mpo 5999  df-pnf 8171  df-mnf 8172  df-xr 8173  df-xadd 9957

This theorem is referenced by:  xnegid  10043  xaddcom  10045  xnegdi  10052  xsubge0  10065  xposdif  10066  xrmaxadd  11758