Theorem xaddpnf1 12345

Description: Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xaddpnf1	⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞)

Proof of Theorem xaddpnf1

Step	Hyp	Ref	Expression
1		pnfxr 10410	. . 3 ⊢ +∞ ∈ ℝ*
2		xaddval 12342	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 +𝑒 +∞) = if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))))
3	1, 2	mpan2 682	. 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 +∞) = if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))))
4		pnfnemnf 10412	. . . . 5 ⊢ +∞ ≠ -∞
5		ifnefalse 4318	. . . . 5 ⊢ (+∞ ≠ -∞ → if(+∞ = -∞, 0, +∞) = +∞)
6	4, 5	mp1i 13	. . . 4 ⊢ (𝐴 ≠ -∞ → if(+∞ = -∞, 0, +∞) = +∞)
7		ifnefalse 4318	. . . . 5 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) = if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))
8		eqid 2825	. . . . . 6 ⊢ +∞ = +∞
9	8	iftruei 4313	. . . . 5 ⊢ if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))) = +∞
10	7, 9	syl6eq 2877	. . . 4 ⊢ (𝐴 ≠ -∞ → if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) = +∞)
11	6, 10	ifeq12d 4326	. . 3 ⊢ (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = if(𝐴 = +∞, +∞, +∞))
12		ifid 4345	. . 3 ⊢ if(𝐴 = +∞, +∞, +∞) = +∞
13	11, 12	syl6eq 2877	. 2 ⊢ (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = +∞)
14	3, 13	sylan9eq 2881	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞)

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1656   ∈ wcel 2164   ≠ wne 2999  ifcif 4306  (class class class)co 6905  0cc0 10252   + caddc 10255  +∞cpnf 10388  -∞cmnf 10389  ℝ^*cxr 10390   +_𝑒 cxad 12230

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-mulcl 10314  ax-i2m1 10320

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-iota 6086  df-fun 6125  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-pnf 10393  df-mnf 10394  df-xr 10395  df-xadd 12233

This theorem is referenced by:  xnn0xaddcl  12354  xaddnemnf  12355  xaddcom  12359  xnn0xadd0  12365  xnegdi  12366  xaddass  12367  xleadd1a  12371  xlt2add  12378  xsubge0  12379  xlesubadd  12381  xadddilem  12412  xrsdsreclblem  20152  isxmet2d  22502  xrge0iifhom  30517  esumpr2  30663  hasheuni  30681  carsgclctunlem2  30915  ovolsplit  40992  sge0pr  41395  sge0split  41410  sge0xadd  41436