MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-xadd Structured version   Visualization version   GIF version

Definition df-xadd 12500
Description: Define addition over extended real numbers. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
df-xadd +𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-xadd
StepHypRef Expression
1 cxad 12497 . 2 class +𝑒
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 cxr 10667 . . 3 class *
52cv 1537 . . . . 5 class 𝑥
6 cpnf 10665 . . . . 5 class +∞
75, 6wceq 1538 . . . 4 wff 𝑥 = +∞
83cv 1537 . . . . . 6 class 𝑦
9 cmnf 10666 . . . . . 6 class -∞
108, 9wceq 1538 . . . . 5 wff 𝑦 = -∞
11 cc0 10530 . . . . 5 class 0
1210, 11, 6cif 4428 . . . 4 class if(𝑦 = -∞, 0, +∞)
135, 9wceq 1538 . . . . 5 wff 𝑥 = -∞
148, 6wceq 1538 . . . . . 6 wff 𝑦 = +∞
1514, 11, 9cif 4428 . . . . 5 class if(𝑦 = +∞, 0, -∞)
16 caddc 10533 . . . . . . . 8 class +
175, 8, 16co 7139 . . . . . . 7 class (𝑥 + 𝑦)
1810, 9, 17cif 4428 . . . . . 6 class if(𝑦 = -∞, -∞, (𝑥 + 𝑦))
1914, 6, 18cif 4428 . . . . 5 class if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦)))
2013, 15, 19cif 4428 . . . 4 class if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))
217, 12, 20cif 4428 . . 3 class if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦)))))
222, 3, 4, 4, 21cmpo 7141 . 2 class (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
231, 22wceq 1538 1 wff +𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ if(𝑥 = +∞, if(𝑦 = -∞, 0, +∞), if(𝑥 = -∞, if(𝑦 = +∞, 0, -∞), if(𝑦 = +∞, +∞, if(𝑦 = -∞, -∞, (𝑥 + 𝑦))))))
Colors of variables: wff setvar class
This definition is referenced by:  xaddval  12608  xaddf  12609
  Copyright terms: Public domain W3C validator