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Theorem xaddpnf1 12607
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 10683 . . 3 +∞ ∈ ℝ*
2 xaddval 12604 . . 3 ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 +𝑒 +∞) = if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))))
31, 2mpan2 687 . 2 (𝐴 ∈ ℝ* → (𝐴 +𝑒 +∞) = if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))))
4 pnfnemnf 10684 . . . . 5 +∞ ≠ -∞
5 ifnefalse 4475 . . . . 5 (+∞ ≠ -∞ → if(+∞ = -∞, 0, +∞) = +∞)
64, 5mp1i 13 . . . 4 (𝐴 ≠ -∞ → if(+∞ = -∞, 0, +∞) = +∞)
7 ifnefalse 4475 . . . . 5 (𝐴 ≠ -∞ → if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) = if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))
8 eqid 2818 . . . . . 6 +∞ = +∞
98iftruei 4470 . . . . 5 if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))) = +∞
107, 9syl6eq 2869 . . . 4 (𝐴 ≠ -∞ → if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) = +∞)
116, 10ifeq12d 4483 . . 3 (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = if(𝐴 = +∞, +∞, +∞))
12 ifid 4502 . . 3 if(𝐴 = +∞, +∞, +∞) = +∞
1311, 12syl6eq 2869 . 2 (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = +∞)
143, 13sylan9eq 2873 1 ((𝐴 ∈ ℝ*𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wne 3013  ifcif 4463  (class class class)co 7145  0cc0 10525   + caddc 10528  +∞cpnf 10660  -∞cmnf 10661  *cxr 10662   +𝑒 cxad 12493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-mulcl 10587  ax-i2m1 10593
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-pnf 10665  df-mnf 10666  df-xr 10667  df-xadd 12496
This theorem is referenced by:  xnn0xaddcl  12616  xaddnemnf  12617  xaddcom  12621  xnn0xadd0  12628  xnegdi  12629  xaddass  12630  xleadd1a  12634  xlt2add  12641  xsubge0  12642  xlesubadd  12644  xadddilem  12675  xrsdsreclblem  20519  isxmet2d  22864  xrge0iifhom  31079  esumpr2  31225  hasheuni  31243  carsgclctunlem2  31476  ovolsplit  42150  sge0pr  42553  sge0split  42568  sge0xadd  42594
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