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Definition df-div 10973
Description: Define division. Theorem divmuli 11067 relates it to multiplication, and divcli 11055 and redivcli 11080 prove its closure laws. (Contributed by NM, 2-Feb-1995.) Use divval 10975 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 10972 . 2 class /
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 cc 10222 . . 3 class
5 cc0 10224 . . . . 5 class 0
65csn 4377 . . . 4 class {0}
74, 6cdif 3773 . . 3 class (ℂ ∖ {0})
83cv 1636 . . . . . 6 class 𝑦
9 vz . . . . . . 7 setvar 𝑧
109cv 1636 . . . . . 6 class 𝑧
11 cmul 10229 . . . . . 6 class ·
128, 10, 11co 6877 . . . . 5 class (𝑦 · 𝑧)
132cv 1636 . . . . 5 class 𝑥
1412, 13wceq 1637 . . . 4 wff (𝑦 · 𝑧) = 𝑥
1514, 9, 4crio 6837 . . 3 class (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)
162, 3, 4, 7, 15cmpt2 6879 . 2 class (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
171, 16wceq 1637 1 wff / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
Colors of variables: wff setvar class
This definition is referenced by:  1div0  10974  divval  10975  elq  12012  cnflddiv  19987  divcn  22888  1div0apr  27661
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