Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df-xdiv Structured version   Visualization version   GIF version

Definition df-xdiv 29960
Description: Define division over extended real numbers. (Contributed by Thierry Arnoux, 17-Dec-2016.)
Assertion
Ref Expression
df-xdiv /𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ (ℝ ∖ {0}) ↦ (𝑧 ∈ ℝ* (𝑦 ·e 𝑧) = 𝑥))
Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-xdiv
StepHypRef Expression
1 cxdiv 29959 . 2 class /𝑒
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 cxr 10274 . . 3 class *
5 cr 10136 . . . 4 class
6 cc0 10137 . . . . 5 class 0
76csn 4314 . . . 4 class {0}
85, 7cdif 3718 . . 3 class (ℝ ∖ {0})
93cv 1629 . . . . . 6 class 𝑦
10 vz . . . . . . 7 setvar 𝑧
1110cv 1629 . . . . . 6 class 𝑧
12 cxmu 12149 . . . . . 6 class ·e
139, 11, 12co 6792 . . . . 5 class (𝑦 ·e 𝑧)
142cv 1629 . . . . 5 class 𝑥
1513, 14wceq 1630 . . . 4 wff (𝑦 ·e 𝑧) = 𝑥
1615, 10, 4crio 6752 . . 3 class (𝑧 ∈ ℝ* (𝑦 ·e 𝑧) = 𝑥)
172, 3, 4, 8, 16cmpt2 6794 . 2 class (𝑥 ∈ ℝ*, 𝑦 ∈ (ℝ ∖ {0}) ↦ (𝑧 ∈ ℝ* (𝑦 ·e 𝑧) = 𝑥))
181, 17wceq 1630 1 wff /𝑒 = (𝑥 ∈ ℝ*, 𝑦 ∈ (ℝ ∖ {0}) ↦ (𝑧 ∈ ℝ* (𝑦 ·e 𝑧) = 𝑥))
Colors of variables: wff setvar class
This definition is referenced by:  xdivval  29961
  Copyright terms: Public domain W3C validator