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Theorem xaddval 9469
Description: Value of the extended real addition operation. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xaddval ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))))

Proof of Theorem xaddval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0xr 7684 . . . . . 6 0 ∈ ℝ*
21a1i 9 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → 0 ∈ ℝ*)
3 pnfxr 7690 . . . . . 6 +∞ ∈ ℝ*
43a1i 9 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → +∞ ∈ ℝ*)
5 xrmnfdc 9467 . . . . . 6 (𝐵 ∈ ℝ*DECID 𝐵 = -∞)
65adantl 273 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → DECID 𝐵 = -∞)
72, 4, 6ifcldcd 3454 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → if(𝐵 = -∞, 0, +∞) ∈ ℝ*)
87adantr 272 . . 3 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ 𝐴 = +∞) → if(𝐵 = -∞, 0, +∞) ∈ ℝ*)
9 mnfxr 7694 . . . . . . 7 -∞ ∈ ℝ*
109a1i 9 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → -∞ ∈ ℝ*)
11 xrpnfdc 9466 . . . . . . 7 (𝐵 ∈ ℝ*DECID 𝐵 = +∞)
1211adantl 273 . . . . . 6 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → DECID 𝐵 = +∞)
132, 10, 12ifcldcd 3454 . . . . 5 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → if(𝐵 = +∞, 0, -∞) ∈ ℝ*)
1413ad2antrr 475 . . . 4 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ 𝐴 = -∞) → if(𝐵 = +∞, 0, -∞) ∈ ℝ*)
153a1i 9 . . . . 5 (((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ 𝐵 = +∞) → +∞ ∈ ℝ*)
169a1i 9 . . . . . 6 ((((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ 𝐵 = -∞) → -∞ ∈ ℝ*)
17 simp-4r 512 . . . . . . . . 9 ((((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → ¬ 𝐴 = +∞)
18 simpl 108 . . . . . . . . . . 11 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → 𝐴 ∈ ℝ*)
1918ad4antr 481 . . . . . . . . . 10 ((((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐴 ∈ ℝ*)
20 simpllr 504 . . . . . . . . . . 11 ((((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → ¬ 𝐴 = -∞)
2120neqned 2274 . . . . . . . . . 10 ((((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐴 ≠ -∞)
22 xrnemnf 9405 . . . . . . . . . . 11 ((𝐴 ∈ ℝ*𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
2322biimpi 119 . . . . . . . . . 10 ((𝐴 ∈ ℝ*𝐴 ≠ -∞) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
2419, 21, 23syl2anc 406 . . . . . . . . 9 ((((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
2517, 24ecased 1295 . . . . . . . 8 ((((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐴 ∈ ℝ)
26 simplr 500 . . . . . . . . 9 ((((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → ¬ 𝐵 = +∞)
27 simpr 109 . . . . . . . . . . 11 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → 𝐵 ∈ ℝ*)
2827ad4antr 481 . . . . . . . . . 10 ((((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐵 ∈ ℝ*)
29 simpr 109 . . . . . . . . . . 11 ((((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → ¬ 𝐵 = -∞)
3029neqned 2274 . . . . . . . . . 10 ((((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐵 ≠ -∞)
31 xrnemnf 9405 . . . . . . . . . . 11 ((𝐵 ∈ ℝ*𝐵 ≠ -∞) ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞))
3231biimpi 119 . . . . . . . . . 10 ((𝐵 ∈ ℝ*𝐵 ≠ -∞) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞))
3328, 30, 32syl2anc 406 . . . . . . . . 9 ((((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞))
3426, 33ecased 1295 . . . . . . . 8 ((((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → 𝐵 ∈ ℝ)
3525, 34readdcld 7667 . . . . . . 7 ((((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → (𝐴 + 𝐵) ∈ ℝ)
3635rexrd 7687 . . . . . 6 ((((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) ∧ ¬ 𝐵 = -∞) → (𝐴 + 𝐵) ∈ ℝ*)
376ad3antrrr 479 . . . . . 6 (((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) → DECID 𝐵 = -∞)
3816, 36, 37ifcldadc 3448 . . . . 5 (((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) ∧ ¬ 𝐵 = +∞) → if(𝐵 = -∞, -∞, (𝐴 + 𝐵)) ∈ ℝ*)
3912ad2antrr 475 . . . . 5 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → DECID 𝐵 = +∞)
4015, 38, 39ifcldadc 3448 . . . 4 ((((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) ∧ ¬ 𝐴 = -∞) → if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) ∈ ℝ*)
41 xrmnfdc 9467 . . . . 5 (𝐴 ∈ ℝ*DECID 𝐴 = -∞)
4241ad2antrr 475 . . . 4 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) → DECID 𝐴 = -∞)
4314, 40, 42ifcldadc 3448 . . 3 (((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) ∧ ¬ 𝐴 = +∞) → if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) ∈ ℝ*)
44 xrpnfdc 9466 . . . 4 (𝐴 ∈ ℝ*DECID 𝐴 = +∞)
4544adantr 272 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → DECID 𝐴 = +∞)
468, 43, 45ifcldadc 3448 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) ∈ ℝ*)
47 simpl 108 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 = 𝐴)
4847eqeq1d 2108 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = +∞ ↔ 𝐴 = +∞))
49 simpr 109 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
5049eqeq1d 2108 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 = -∞ ↔ 𝐵 = -∞))
5150ifbid 3440 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = -∞, 0, +∞) = if(𝐵 = -∞, 0, +∞))
5247eqeq1d 2108 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = -∞ ↔ 𝐴 = -∞))
5349eqeq1d 2108 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 = +∞ ↔ 𝐵 = +∞))
5453ifbid 3440 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = +∞, 0, -∞) = if(𝐵 = +∞, 0, -∞))
55 oveq12 5715 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 + 𝑦) = (𝐴 + 𝐵))
5650, 55ifbieq2d 3443 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = -∞, -∞, (𝑥 + 𝑦)) = if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))
5753, 56ifbieq2d 3443 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))
5852, 54, 57ifbieq12d 3445 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦)))) = if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))))
5948, 51, 58ifbieq12d 3445 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))))
60 df-xadd 9401 . . 3 +𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
6159, 60ovmpoga 5832 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) ∈ ℝ*) → (𝐴 +𝑒 𝐵) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))))
6246, 61mpd3an3 1284 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴 +𝑒 𝐵) = if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wo 670  DECID wdc 786   = wceq 1299  wcel 1448  wne 2267  ifcif 3421  (class class class)co 5706  cr 7499  0cc0 7500   + caddc 7503  +∞cpnf 7669  -∞cmnf 7670  *cxr 7671   +𝑒 cxad 9398
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 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1re 7589  ax-addrcl 7592  ax-rnegex 7604
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-pnf 7674  df-mnf 7675  df-xr 7676  df-xadd 9401
This theorem is referenced by:  xaddpnf1  9470  xaddpnf2  9471  xaddmnf1  9472  xaddmnf2  9473  pnfaddmnf  9474  mnfaddpnf  9475  rexadd  9476
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