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Theorem xaddval 13027
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 483 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑥 = 𝐴)
21eqeq1d 2739 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = +∞ ↔ 𝐴 = +∞))
3 simpr 485 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → 𝑦 = 𝐵)
43eqeq1d 2739 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 = -∞ ↔ 𝐵 = -∞))
54ifbid 4492 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = -∞, 0, +∞) = if(𝐵 = -∞, 0, +∞))
61eqeq1d 2739 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = -∞ ↔ 𝐴 = -∞))
73eqeq1d 2739 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑦 = +∞ ↔ 𝐵 = +∞))
87ifbid 4492 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = +∞, 0, -∞) = if(𝐵 = +∞, 0, -∞))
9 oveq12 7322 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 + 𝑦) = (𝐴 + 𝐵))
104, 9ifbieq2d 4495 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = -∞, -∞, (𝑥 + 𝑦)) = if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))
117, 10ifbieq2d 4495 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))) = if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))
126, 8, 11ifbieq12d 4497 . . 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 4497 . 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 12919 . 2 +𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
15 c0ex 11039 . . . 4 0 ∈ V
16 pnfex 11098 . . . 4 +∞ ∈ V
1715, 16ifex 4519 . . 3 if(𝐵 = -∞, 0, +∞) ∈ V
18 mnfxr 11102 . . . . . 6 -∞ ∈ ℝ*
1918elexi 3460 . . . . 5 -∞ ∈ V
2015, 19ifex 4519 . . . 4 if(𝐵 = +∞, 0, -∞) ∈ V
21 ovex 7346 . . . . . 6 (𝐴 + 𝐵) ∈ V
2219, 21ifex 4519 . . . . 5 if(𝐵 = -∞, -∞, (𝐴 + 𝐵)) ∈ V
2316, 22ifex 4519 . . . 4 if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))) ∈ V
2420, 23ifex 4519 . . 3 if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵)))) ∈ V
2517, 24ifex 4519 . 2 if(𝐴 = +∞, if(𝐵 = -∞, 0, +∞), if(𝐴 = -∞, if(𝐵 = +∞, 0, -∞), if(𝐵 = +∞, +∞, if(𝐵 = -∞, -∞, (𝐴 + 𝐵))))) ∈ V
2613, 14, 25ovmpoa 7466 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 396   = wceq 1540  wcel 2105  ifcif 4469  (class class class)co 7313  0cc0 10941   + caddc 10944  +∞cpnf 11076  -∞cmnf 11077  *cxr 11078   +𝑒 cxad 12916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365  ax-un 7626  ax-cnex 10997  ax-1cn 10999  ax-icn 11000  ax-addcl 11001  ax-mulcl 11003  ax-i2m1 11009
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-sbc 3726  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-br 5086  df-opab 5148  df-id 5505  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-iota 6415  df-fun 6465  df-fv 6471  df-ov 7316  df-oprab 7317  df-mpo 7318  df-pnf 11081  df-mnf 11082  df-xr 11083  df-xadd 12919
This theorem is referenced by:  xaddpnf1  13030  xaddpnf2  13031  xaddmnf1  13032  xaddmnf2  13033  pnfaddmnf  13034  mnfaddpnf  13035  rexadd  13036
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