MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xaddpnf1 Structured version   Visualization version   GIF version

Theorem xaddpnf1 13154
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 11217 . . 3 +∞ ∈ ℝ*
2 xaddval 13151 . . 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 11218 . . . . 5 +∞ ≠ -∞
5 ifnefalse 4502 . . . . 5 (+∞ ≠ -∞ → if(+∞ = -∞, 0, +∞) = +∞)
64, 5mp1i 13 . . . 4 (𝐴 ≠ -∞ → if(+∞ = -∞, 0, +∞) = +∞)
7 ifnefalse 4502 . . . . 5 (𝐴 ≠ -∞ → if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) = if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))
8 eqid 2733 . . . . . 6 +∞ = +∞
98iftruei 4497 . . . . 5 if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))) = +∞
107, 9eqtrdi 2789 . . . 4 (𝐴 ≠ -∞ → if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞)))) = +∞)
116, 10ifeq12d 4511 . . 3 (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = if(𝐴 = +∞, +∞, +∞))
12 ifid 4530 . . 3 if(𝐴 = +∞, +∞, +∞) = +∞
1311, 12eqtrdi 2789 . 2 (𝐴 ≠ -∞ → if(𝐴 = +∞, if(+∞ = -∞, 0, +∞), if(𝐴 = -∞, if(+∞ = +∞, 0, -∞), if(+∞ = +∞, +∞, if(+∞ = -∞, -∞, (𝐴 + +∞))))) = +∞)
143, 13sylan9eq 2793 1 ((𝐴 ∈ ℝ*𝐴 ≠ -∞) → (𝐴 +𝑒 +∞) = +∞)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2940  ifcif 4490  (class class class)co 7361  0cc0 11059   + caddc 11062  +∞cpnf 11194  -∞cmnf 11195  *cxr 11196   +𝑒 cxad 13039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-mulcl 11121  ax-i2m1 11127
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-pnf 11199  df-mnf 11200  df-xr 11201  df-xadd 13042
This theorem is referenced by:  xnn0xaddcl  13163  xaddnemnf  13164  xaddcom  13168  xnn0xadd0  13175  xnegdi  13176  xaddass  13177  xleadd1a  13181  xlt2add  13188  xsubge0  13189  xlesubadd  13191  xadddilem  13222  xrsdsreclblem  20866  isxmet2d  23703  xrge0iifhom  32582  esumpr2  32730  hasheuni  32748  carsgclctunlem2  32983  ovolsplit  44319  sge0pr  44725  sge0split  44740  sge0xadd  44766
  Copyright terms: Public domain W3C validator