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Theorem xaddmnf2 12306
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 10385 . . 3 -∞ ∈ ℝ*
2 xaddval 12300 . . 3 ((-∞ ∈ ℝ*𝐴 ∈ ℝ*) → (-∞ +𝑒 𝐴) = if(-∞ = +∞, if(𝐴 = -∞, 0, +∞), if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴))))))
31, 2mpan 682 . 2 (𝐴 ∈ ℝ* → (-∞ +𝑒 𝐴) = if(-∞ = +∞, if(𝐴 = -∞, 0, +∞), if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴))))))
4 mnfnepnf 10384 . . . . 5 -∞ ≠ +∞
5 ifnefalse 4288 . . . . 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 2798 . . . . 5 -∞ = -∞
87iftruei 4283 . . . 4 if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴)))) = if(𝐴 = +∞, 0, -∞)
96, 8eqtri 2820 . . 3 if(-∞ = +∞, if(𝐴 = -∞, 0, +∞), if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴))))) = if(𝐴 = +∞, 0, -∞)
10 ifnefalse 4288 . . 3 (𝐴 ≠ +∞ → if(𝐴 = +∞, 0, -∞) = -∞)
119, 10syl5eq 2844 . 2 (𝐴 ≠ +∞ → if(-∞ = +∞, if(𝐴 = -∞, 0, +∞), if(-∞ = -∞, if(𝐴 = +∞, 0, -∞), if(𝐴 = +∞, +∞, if(𝐴 = -∞, -∞, (-∞ + 𝐴))))) = -∞)
123, 11sylan9eq 2852 1 ((𝐴 ∈ ℝ*𝐴 ≠ +∞) → (-∞ +𝑒 𝐴) = -∞)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wne 2970  ifcif 4276  (class class class)co 6877  0cc0 10223   + caddc 10226  +∞cpnf 10359  -∞cmnf 10360  *cxr 10361   +𝑒 cxad 12188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-sep 4974  ax-nul 4982  ax-pow 5034  ax-pr 5096  ax-un 7182  ax-cnex 10279  ax-1cn 10281  ax-icn 10282  ax-addcl 10283  ax-mulcl 10285  ax-i2m1 10291
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-nel 3074  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3386  df-sbc 3633  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4628  df-br 4843  df-opab 4905  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-iota 6063  df-fun 6102  df-fv 6108  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-pnf 10364  df-mnf 10365  df-xr 10366  df-xadd 12191
This theorem is referenced by:  xaddnepnf  12314  xaddcom  12317  xaddid1  12318  xnegdi  12324  xpncan  12327  xleadd1a  12329  xlt2add  12336  xadddilem  12370  xadddi2  12373  xrsnsgrp  20101  xaddeq0  30029  supxrgelem  40286  supxrge  40287  xrlexaddrp  40301  infleinflem2  40320
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