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Theorem xaddval 12436
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 475 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 = 𝐴)
21eqeq1d 2780 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = +∞ ↔ 𝐴 = +∞))
3 simpr 477 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
43eqeq1d 2780 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 = -∞ ↔ 𝐵 = -∞))
54ifbid 4373 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = -∞, 0, +∞) = if(𝐵 = -∞, 0, +∞))
61eqeq1d 2780 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = -∞ ↔ 𝐴 = -∞))
73eqeq1d 2780 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 = +∞ ↔ 𝐵 = +∞))
87ifbid 4373 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = +∞, 0, -∞) = if(𝐵 = +∞, 0, -∞))
9 oveq12 6987 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 + 𝑦) = (𝐴 + 𝐵))
104, 9ifbieq2d 4376 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = -∞, -∞, (𝑥 + 𝑦)) = if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))
117, 10ifbieq2d 4376 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))
126, 8, 11ifbieq12d 4378 . . 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 4378 . 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 12328 . 2 +𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
15 c0ex 10435 . . . 4 0 ∈ V
16 pnfex 10495 . . . 4 +∞ ∈ V
1715, 16ifex 4399 . . 3 if(𝐵 = -∞, 0, +∞) ∈ V
18 mnfxr 10500 . . . . . 6 -∞ ∈ ℝ*
1918elexi 3434 . . . . 5 -∞ ∈ V
2015, 19ifex 4399 . . . 4 if(𝐵 = +∞, 0, -∞) ∈ V
21 ovex 7010 . . . . . 6 (𝐴 + 𝐵) ∈ V
2219, 21ifex 4399 . . . . 5 if(𝐵 = -∞, -∞, (𝐴 + 𝐵)) ∈ V
2316, 22ifex 4399 . . . 4 if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) ∈ V
2420, 23ifex 4399 . . 3 if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) ∈ V
2517, 24ifex 4399 . 2 if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) ∈ V
2613, 14, 25ovmpoa 7123 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 387   = wceq 1507  wcel 2050  ifcif 4351  (class class class)co 6978  0cc0 10337   + caddc 10340  +∞cpnf 10473  -∞cmnf 10474  *cxr 10475   +𝑒 cxad 12325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281  ax-cnex 10393  ax-1cn 10395  ax-icn 10396  ax-addcl 10397  ax-mulcl 10399  ax-i2m1 10405
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3684  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-opab 4993  df-id 5313  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-iota 6154  df-fun 6192  df-fv 6198  df-ov 6981  df-oprab 6982  df-mpo 6983  df-pnf 10478  df-mnf 10479  df-xr 10480  df-xadd 12328
This theorem is referenced by:  xaddpnf1  12439  xaddpnf2  12440  xaddmnf1  12441  xaddmnf2  12442  pnfaddmnf  12443  mnfaddpnf  12444  rexadd  12445
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