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Theorem xaddpnf1 12616
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
StepHypRef Expression
1 pnfxr 10693 . . 3 +∞ ∈ ℝ*
2 xaddval 12613 . . 3 ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 +𝑒 +∞) = if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))))
31, 2mpan2 690 . 2 (𝐴 ∈ ℝ* → (𝐴 +𝑒 +∞) = if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))))
4 pnfnemnf 10694 . . . . 5 +∞ ≠ -∞
5 ifnefalse 4462 . . . . 5 (+∞ ≠ -∞ → if(+∞ = -∞, 0, +∞) = +∞)
64, 5mp1i 13 . . . 4 (𝐴 ≠ -∞ → if(+∞ = -∞, 0, +∞) = +∞)
7 ifnefalse 4462 . . . . 5 (𝐴 ≠ -∞ → if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) = if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))
8 eqid 2824 . . . . . 6 +∞ = +∞
98iftruei 4457 . . . . 5 if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))) = +∞
107, 9syl6eq 2875 . . . 4 (𝐴 ≠ -∞ → if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) = +∞)
116, 10ifeq12d 4470 . . 3 (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = if(𝐴 = +∞, +∞, +∞))
12 ifid 4489 . . 3 if(𝐴 = +∞, +∞, +∞) = +∞
1311, 12syl6eq 2875 . 2 (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = +∞)
143, 13sylan9eq 2879 1 ((𝐴 ∈ ℝ*𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wne 3014  ifcif 4450  (class class class)co 7149  0cc0 10535   + caddc 10538  +∞cpnf 10670  -∞cmnf 10671  *cxr 10672   +𝑒 cxad 12502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-mulcl 10597  ax-i2m1 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-pnf 10675  df-mnf 10676  df-xr 10677  df-xadd 12505
This theorem is referenced by:  xnn0xaddcl  12625  xaddnemnf  12626  xaddcom  12630  xnn0xadd0  12637  xnegdi  12638  xaddass  12639  xleadd1a  12643  xlt2add  12650  xsubge0  12651  xlesubadd  12653  xadddilem  12684  xrsdsreclblem  20144  isxmet2d  22940  xrge0iifhom  31240  esumpr2  31386  hasheuni  31404  carsgclctunlem2  31637  ovolsplit  42560  sge0pr  42963  sge0split  42978  sge0xadd  43004
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