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Theorem xaddmnf1 12018
Description: Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xaddmnf1 ((𝐴 ∈ ℝ*𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = -∞)

Proof of Theorem xaddmnf1
StepHypRef Expression
1 mnfxr 10056 . . 3 -∞ ∈ ℝ*
2 xaddval 12013 . . 3 ((𝐴 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (𝐴 +𝑒 -∞) = if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))))
31, 2mpan2 706 . 2 (𝐴 ∈ ℝ* → (𝐴 +𝑒 -∞) = if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))))
4 ifnefalse 4076 . . 3 (𝐴 ≠ +∞ → if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))) = if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))))
5 mnfnepnf 10055 . . . . . 6 -∞ ≠ +∞
6 ifnefalse 4076 . . . . . 6 (-∞ ≠ +∞ → if(-∞ = +∞, 0, -∞) = -∞)
75, 6ax-mp 5 . . . . 5 if(-∞ = +∞, 0, -∞) = -∞
8 ifnefalse 4076 . . . . . . 7 (-∞ ≠ +∞ → if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = if(-∞ = -∞, -∞, (𝐴 + -∞)))
95, 8ax-mp 5 . . . . . 6 if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = if(-∞ = -∞, -∞, (𝐴 + -∞))
10 eqid 2621 . . . . . . 7 -∞ = -∞
1110iftruei 4071 . . . . . 6 if(-∞ = -∞, -∞, (𝐴 + -∞)) = -∞
129, 11eqtri 2643 . . . . 5 if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = -∞
13 ifeq12 4081 . . . . 5 ((if(-∞ = +∞, 0, -∞) = -∞ ∧ if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = -∞) → if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))) = if(𝐴 = -∞, -∞, -∞))
147, 12, 13mp2an 707 . . . 4 if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))) = if(𝐴 = -∞, -∞, -∞)
15 ifid 4103 . . . 4 if(𝐴 = -∞, -∞, -∞) = -∞
1614, 15eqtri 2643 . . 3 if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))) = -∞
174, 16syl6eq 2671 . 2 (𝐴 ≠ +∞ → if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))) = -∞)
183, 17sylan9eq 2675 1 ((𝐴 ∈ ℝ*𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = -∞)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  ifcif 4064  (class class class)co 6615  0cc0 9896   + caddc 9899  +∞cpnf 10031  -∞cmnf 10032  *cxr 10033   +𝑒 cxad 11904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-mulcl 9958  ax-i2m1 9964
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-pnf 10036  df-mnf 10037  df-xr 10038  df-xadd 11907
This theorem is referenced by:  xaddnepnf  12027  xaddcom  12030  xnegdi  12037  xleadd1a  12042  xsubge0  12050  xlesubadd  12052  xadddilem  12083  xblss2ps  22146  xblss2  22147
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