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Mirrors > Home > ILE Home > Th. List > df-div | GIF version |
Description: Define division. Theorem divmulap 8585 relates it to multiplication, and divclap 8588 and redivclap 8641 prove its closure laws. (Contributed by NM, 2-Feb-1995.) Use divvalap 8584 instead. (Revised by Mario Carneiro, 1-Apr-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
df-div | ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdiv 8582 | . 2 class / | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vy | . . 3 setvar 𝑦 | |
4 | cc 7765 | . . 3 class ℂ | |
5 | cc0 7767 | . . . . 5 class 0 | |
6 | 5 | csn 3581 | . . . 4 class {0} |
7 | 4, 6 | cdif 3118 | . . 3 class (ℂ ∖ {0}) |
8 | 3 | cv 1347 | . . . . . 6 class 𝑦 |
9 | vz | . . . . . . 7 setvar 𝑧 | |
10 | 9 | cv 1347 | . . . . . 6 class 𝑧 |
11 | cmul 7772 | . . . . . 6 class · | |
12 | 8, 10, 11 | co 5851 | . . . . 5 class (𝑦 · 𝑧) |
13 | 2 | cv 1347 | . . . . 5 class 𝑥 |
14 | 12, 13 | wceq 1348 | . . . 4 wff (𝑦 · 𝑧) = 𝑥 |
15 | 14, 9, 4 | crio 5806 | . . 3 class (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) |
16 | 2, 3, 4, 7, 15 | cmpo 5853 | . 2 class (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) |
17 | 1, 16 | wceq 1348 | 1 wff / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) |
Colors of variables: wff set class |
This definition is referenced by: divvalap 8584 divfnzn 9573 |
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