MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xaddmnf2 Structured version   Visualization version   GIF version

Theorem xaddmnf2 12625
Description: Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xaddmnf2 ((𝐴 ∈ ℝ*𝐴 ≠ +∞) → (-∞ +𝑒 𝐴) = -∞)

Proof of Theorem xaddmnf2
StepHypRef Expression
1 mnfxr 10701 . . 3 -∞ ∈ ℝ*
2 xaddval 12619 . . 3 ((-∞ ∈ ℝ*𝐴 ∈ ℝ*) → (-∞ +𝑒 𝐴) = if(-∞ = +∞, if(𝐴 = -∞, 0, +∞), if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴))))))
31, 2mpan 688 . 2 (𝐴 ∈ ℝ* → (-∞ +𝑒 𝐴) = if(-∞ = +∞, if(𝐴 = -∞, 0, +∞), if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴))))))
4 mnfnepnf 10700 . . . . 5 -∞ ≠ +∞
5 ifnefalse 4482 . . . . 5 (-∞ ≠ +∞ → if(-∞ = +∞, if(𝐴 = -∞, 0, +∞), if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴))))) = if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴)))))
64, 5ax-mp 5 . . . 4 if(-∞ = +∞, if(𝐴 = -∞, 0, +∞), if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴))))) = if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴))))
7 eqid 2824 . . . . 5 -∞ = -∞
87iftruei 4477 . . . 4 if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴)))) = if(𝐴 = +∞, 0, -∞)
96, 8eqtri 2847 . . 3 if(-∞ = +∞, if(𝐴 = -∞, 0, +∞), if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴))))) = if(𝐴 = +∞, 0, -∞)
10 ifnefalse 4482 . . 3 (𝐴 ≠ +∞ → if(𝐴 = +∞, 0, -∞) = -∞)
119, 10syl5eq 2871 . 2 (𝐴 ≠ +∞ → if(-∞ = +∞, if(𝐴 = -∞, 0, +∞), if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴))))) = -∞)
123, 11sylan9eq 2879 1 ((𝐴 ∈ ℝ*𝐴 ≠ +∞) → (-∞ +𝑒 𝐴) = -∞)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1536  wcel 2113  wne 3019  ifcif 4470  (class class class)co 7159  0cc0 10540   + caddc 10543  +∞cpnf 10675  -∞cmnf 10676  *cxr 10677   +𝑒 cxad 12508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-mulcl 10602  ax-i2m1 10608
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-iota 6317  df-fun 6360  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-pnf 10680  df-mnf 10681  df-xr 10682  df-xadd 12511
This theorem is referenced by:  xaddnepnf  12633  xaddcom  12636  xaddid1  12637  xnegdi  12644  xpncan  12647  xleadd1a  12649  xlt2add  12656  xadddilem  12690  xadddi2  12693  xrsnsgrp  20584  xaddeq0  30480  supxrgelem  41611  supxrge  41612  xrlexaddrp  41626  infleinflem2  41645
  Copyright terms: Public domain W3C validator