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

StepHypRef Expression
1 mnfxr 7869 . . . 4 -∞ ∈ ℝ*
2 xaddval 9681 . . . 4 ((𝐴 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (𝐴 +𝑒 -∞) = if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))))
31, 2mpan2 422 . . 3 (𝐴 ∈ ℝ* → (𝐴 +𝑒 -∞) = if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))))
43adantr 274 . 2 ((𝐴 ∈ ℝ*𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))))
5 ifnefalse 3491 . . 3 (𝐴 ≠ +∞ → if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))) = if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))))
6 mnfnepnf 7868 . . . . . 6 -∞ ≠ +∞
7 ifnefalse 3491 . . . . . 6 (-∞ ≠ +∞ → if(-∞ = +∞, 0, -∞) = -∞)
86, 7ax-mp 5 . . . . 5 if(-∞ = +∞, 0, -∞) = -∞
9 ifnefalse 3491 . . . . . . 7 (-∞ ≠ +∞ → if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = if(-∞ = -∞, -∞, (𝐴 + -∞)))
106, 9ax-mp 5 . . . . . 6 if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = if(-∞ = -∞, -∞, (𝐴 + -∞))
11 eqid 2140 . . . . . . 7 -∞ = -∞
1211iftruei 3486 . . . . . 6 if(-∞ = -∞, -∞, (𝐴 + -∞)) = -∞
1310, 12eqtri 2161 . . . . 5 if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = -∞
14 ifeq12 3494 . . . . 5 ((if(-∞ = +∞, 0, -∞) = -∞ ∧ if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = -∞) → if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))) = if(𝐴 = -∞, -∞, -∞))
158, 13, 14mp2an 423 . . . 4 if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))) = if(𝐴 = -∞, -∞, -∞)
16 xrmnfdc 9679 . . . . 5 (𝐴 ∈ ℝ*DECID 𝐴 = -∞)
17 ifiddc 3511 . . . . 5 (DECID 𝐴 = -∞ → if(𝐴 = -∞, -∞, -∞) = -∞)
1816, 17syl 14 . . . 4 (𝐴 ∈ ℝ* → if(𝐴 = -∞, -∞, -∞) = -∞)
1915, 18syl5eq 2185 . . 3 (𝐴 ∈ ℝ* → if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))) = -∞)
205, 19sylan9eqr 2195 . 2 ((𝐴 ∈ ℝ*𝐴 ≠ +∞) → if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))) = -∞)
214, 20eqtrd 2173 1 ((𝐴 ∈ ℝ*𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = -∞)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  DECID wdc 820   = wceq 1332   ∈ wcel 1481   ≠ wne 2309  ifcif 3480  (class class class)co 5784  0cc0 7667   + caddc 7670  +∞cpnf 7844  -∞cmnf 7845  ℝ*cxr 7846   +𝑒 cxad 9610 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4140  ax-un 4364  ax-setind 4461  ax-cnex 7758  ax-resscn 7759  ax-1re 7761  ax-addrcl 7764  ax-rnegex 7776 This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2692  df-sbc 2915  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-if 3481  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-br 3939  df-opab 3999  df-id 4224  df-xp 4555  df-rel 4556  df-cnv 4557  df-co 4558  df-dm 4559  df-iota 5098  df-fun 5135  df-fv 5141  df-ov 5787  df-oprab 5788  df-mpo 5789  df-pnf 7849  df-mnf 7850  df-xr 7851  df-xadd 9613 This theorem is referenced by:  xaddnepnf  9694  xaddcom  9697  xnegdi  9704  xleadd1a  9709  xsubge0  9717  xposdif  9718  xlesubadd  9719  xleaddadd  9723  xblss2ps  12635  xblss2  12636
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