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

Definition df-div 11287
 Description: Define division. Theorem divmuli 11383 relates it to multiplication, and divcli 11371 and redivcli 11396 prove its closure laws. (Contributed by NM, 2-Feb-1995.) Use divval 11289 instead. (Revised by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.)
Assertion
Ref Expression
df-div / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-div
StepHypRef Expression
1 cdiv 11286 . 2 class /
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 cc 10524 . . 3 class
5 cc0 10526 . . . . 5 class 0
65csn 4525 . . . 4 class {0}
74, 6cdif 3878 . . 3 class (ℂ ∖ {0})
83cv 1537 . . . . . 6 class 𝑦
9 vz . . . . . . 7 setvar 𝑧
109cv 1537 . . . . . 6 class 𝑧
11 cmul 10531 . . . . . 6 class ·
128, 10, 11co 7135 . . . . 5 class (𝑦 · 𝑧)
132cv 1537 . . . . 5 class 𝑥
1412, 13wceq 1538 . . . 4 wff (𝑦 · 𝑧) = 𝑥
1514, 9, 4crio 7092 . . 3 class (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)
162, 3, 4, 7, 15cmpo 7137 . 2 class (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
171, 16wceq 1538 1 wff / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
 Colors of variables: wff setvar class This definition is referenced by:  1div0  11288  divval  11289  elq  12338  cnflddiv  20121  divcn  23473  1div0apr  28253
 Copyright terms: Public domain W3C validator