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Definition df-div 10670
Description: Define division. Theorem divmuli 10764 relates it to multiplication, and divcli 10752 and redivcli 10777 prove its closure laws. (Contributed by NM, 2-Feb-1995.) Use divval 10672 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 10669 . 2 class /
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 cc 9919 . . 3 class
5 cc0 9921 . . . . 5 class 0
65csn 4168 . . . 4 class {0}
74, 6cdif 3564 . . 3 class (ℂ ∖ {0})
83cv 1480 . . . . . 6 class 𝑦
9 vz . . . . . . 7 setvar 𝑧
109cv 1480 . . . . . 6 class 𝑧
11 cmul 9926 . . . . . 6 class ·
128, 10, 11co 6635 . . . . 5 class (𝑦 · 𝑧)
132cv 1480 . . . . 5 class 𝑥
1412, 13wceq 1481 . . . 4 wff (𝑦 · 𝑧) = 𝑥
1514, 9, 4crio 6595 . . 3 class (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)
162, 3, 4, 7, 15cmpt2 6637 . 2 class (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
171, 16wceq 1481 1 wff / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
Colors of variables: wff setvar class
This definition is referenced by:  1div0  10671  divval  10672  elq  11775  cnflddiv  19757  divcn  22652  1div0apr  27294
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