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Theorem xaddval 12258
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 simpl 468 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 = 𝐴)
21eqeq1d 2772 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = +∞ ↔ 𝐴 = +∞))
3 simpr 471 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
43eqeq1d 2772 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 = -∞ ↔ 𝐵 = -∞))
54ifbid 4245 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = -∞, 0, +∞) = if(𝐵 = -∞, 0, +∞))
61eqeq1d 2772 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = -∞ ↔ 𝐴 = -∞))
73eqeq1d 2772 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 = +∞ ↔ 𝐵 = +∞))
87ifbid 4245 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = +∞, 0, -∞) = if(𝐵 = +∞, 0, -∞))
9 oveq12 6801 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 + 𝑦) = (𝐴 + 𝐵))
104, 9ifbieq2d 4248 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = -∞, -∞, (𝑥 + 𝑦)) = if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))
117, 10ifbieq2d 4248 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))
126, 8, 11ifbieq12d 4250 . . 3 ((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)))))
132, 5, 12ifbieq12d 4250 . 2 ((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))))))
14 df-xadd 12151 . 2 +𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
15 c0ex 10235 . . . 4 0 ∈ V
16 pnfex 10294 . . . 4 +∞ ∈ V
1715, 16ifex 4293 . . 3 if(𝐵 = -∞, 0, +∞) ∈ V
18 mnfxr 10297 . . . . . 6 -∞ ∈ ℝ*
1918elexi 3362 . . . . 5 -∞ ∈ V
2015, 19ifex 4293 . . . 4 if(𝐵 = +∞, 0, -∞) ∈ V
21 ovex 6822 . . . . . 6 (𝐴 + 𝐵) ∈ V
2219, 21ifex 4293 . . . . 5 if(𝐵 = -∞, -∞, (𝐴 + 𝐵)) ∈ V
2316, 22ifex 4293 . . . 4 if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) ∈ V
2420, 23ifex 4293 . . 3 if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) ∈ V
2517, 24ifex 4293 . 2 if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) ∈ V
2613, 14, 25ovmpt2a 6937 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 setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  ifcif 4223  (class class class)co 6792  0cc0 10137   + caddc 10140  +∞cpnf 10272  -∞cmnf 10273  *cxr 10274   +𝑒 cxad 12148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-mulcl 10199  ax-i2m1 10205
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-pnf 10277  df-mnf 10278  df-xr 10279  df-xadd 12151
This theorem is referenced by:  xaddpnf1  12261  xaddpnf2  12262  xaddmnf1  12263  xaddmnf2  12264  pnfaddmnf  12265  mnfaddpnf  12266  rexadd  12267
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