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Theorem xaddpnf1 13245
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 11306 . . 3 +∞ ∈ ℝ*
2 xaddval 13242 . . 3 ((𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝐴 +𝑒 +∞) = if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))))
31, 2mpan2 689 . 2 (𝐴 ∈ ℝ* → (𝐴 +𝑒 +∞) = if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))))
4 pnfnemnf 11307 . . . . 5 +∞ ≠ -∞
5 ifnefalse 4544 . . . . 5 (+∞ ≠ -∞ → if(+∞ = -∞, 0, +∞) = +∞)
64, 5mp1i 13 . . . 4 (𝐴 ≠ -∞ → if(+∞ = -∞, 0, +∞) = +∞)
7 ifnefalse 4544 . . . . 5 (𝐴 ≠ -∞ → if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) = if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))
8 eqid 2728 . . . . . 6 +∞ = +∞
98iftruei 4539 . . . . 5 if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))) = +∞
107, 9eqtrdi 2784 . . . 4 (𝐴 ≠ -∞ → if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) = +∞)
116, 10ifeq12d 4553 . . 3 (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = if(𝐴 = +∞, +∞, +∞))
12 ifid 4572 . . 3 if(𝐴 = +∞, +∞, +∞) = +∞
1311, 12eqtrdi 2784 . 2 (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = +∞)
143, 13sylan9eq 2788 1 ((𝐴 ∈ ℝ*𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wne 2937  ifcif 4532  (class class class)co 7426  0cc0 11146   + caddc 11149  +∞cpnf 11283  -∞cmnf 11284  *cxr 11285   +𝑒 cxad 13130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-mulcl 11208  ax-i2m1 11214
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-pnf 11288  df-mnf 11289  df-xr 11290  df-xadd 13133
This theorem is referenced by:  xnn0xaddcl  13254  xaddnemnf  13255  xaddcom  13259  xnn0xadd0  13266  xnegdi  13267  xaddass  13268  xleadd1a  13272  xlt2add  13279  xsubge0  13280  xlesubadd  13282  xadddilem  13313  xrsdsreclblem  21352  isxmet2d  24253  xrge0iifhom  33571  esumpr2  33719  hasheuni  33737  carsgclctunlem2  33972  ovolsplit  45405  sge0pr  45811  sge0split  45826  sge0xadd  45852
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