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Theorem xaddmnf1 9524
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 7746 . . . 4 -∞ ∈ ℝ*
2 xaddval 9521 . . . 4 ((𝐴 ∈ ℝ* ∧ -∞ ∈ ℝ*) → (𝐴 +𝑒 -∞) = if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))))
31, 2mpan2 419 . . 3 (𝐴 ∈ ℝ* → (𝐴 +𝑒 -∞) = if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))))
43adantr 272 . 2 ((𝐴 ∈ ℝ*𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))))
5 ifnefalse 3451 . . 3 (𝐴 ≠ +∞ → if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))) = if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))))
6 mnfnepnf 7745 . . . . . 6 -∞ ≠ +∞
7 ifnefalse 3451 . . . . . 6 (-∞ ≠ +∞ → if(-∞ = +∞, 0, -∞) = -∞)
86, 7ax-mp 7 . . . . 5 if(-∞ = +∞, 0, -∞) = -∞
9 ifnefalse 3451 . . . . . . 7 (-∞ ≠ +∞ → if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = if(-∞ = -∞, -∞, (𝐴 + -∞)))
106, 9ax-mp 7 . . . . . 6 if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = if(-∞ = -∞, -∞, (𝐴 + -∞))
11 eqid 2115 . . . . . . 7 -∞ = -∞
1211iftruei 3446 . . . . . 6 if(-∞ = -∞, -∞, (𝐴 + -∞)) = -∞
1310, 12eqtri 2135 . . . . 5 if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = -∞
14 ifeq12 3454 . . . . 5 ((if(-∞ = +∞, 0, -∞) = -∞ ∧ if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))) = -∞) → if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))) = if(𝐴 = -∞, -∞, -∞))
158, 13, 14mp2an 420 . . . 4 if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))) = if(𝐴 = -∞, -∞, -∞)
16 xrmnfdc 9519 . . . . 5 (𝐴 ∈ ℝ*DECID 𝐴 = -∞)
17 ifiddc 3471 . . . . 5 (DECID 𝐴 = -∞ → if(𝐴 = -∞, -∞, -∞) = -∞)
1816, 17syl 14 . . . 4 (𝐴 ∈ ℝ* → if(𝐴 = -∞, -∞, -∞) = -∞)
1915, 18syl5eq 2159 . . 3 (𝐴 ∈ ℝ* → if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞)))) = -∞)
205, 19sylan9eqr 2169 . 2 ((𝐴 ∈ ℝ*𝐴 ≠ +∞) → if(𝐴 = +∞, if(-∞ = -∞, 0, +∞), if(𝐴 = -∞, if(-∞ = +∞, 0, -∞), if(-∞ = +∞, +∞, if(-∞ = -∞, -∞, (𝐴 + -∞))))) = -∞)
214, 20eqtrd 2147 1 ((𝐴 ∈ ℝ*𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = -∞)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  DECID wdc 802   = wceq 1314  wcel 1463  wne 2282  ifcif 3440  (class class class)co 5728  0cc0 7547   + caddc 7550  +∞cpnf 7721  -∞cmnf 7722  *cxr 7723   +𝑒 cxad 9450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1re 7639  ax-addrcl 7642  ax-rnegex 7654
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-iota 5046  df-fun 5083  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-pnf 7726  df-mnf 7727  df-xr 7728  df-xadd 9453
This theorem is referenced by:  xaddnepnf  9534  xaddcom  9537  xnegdi  9544  xleadd1a  9549  xsubge0  9557  xposdif  9558  xlesubadd  9559  xleaddadd  9563  xblss2ps  12393  xblss2  12394
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