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Mirrors > Home > ILE Home > Th. List > df-div | GIF version |
Description: Define division. Theorem divmulap 7436 relates it to multiplication, and divclap 7439 and redivclap 7489 prove its closure laws. (Contributed by NM, 2-Feb-1995.) (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 7433 | . 2 class / | |
2 | vx | . . 3 setvar x | |
3 | vy | . . 3 setvar y | |
4 | cc 6709 | . . 3 class ℂ | |
5 | cc0 6711 | . . . . 5 class 0 | |
6 | 5 | csn 3367 | . . . 4 class {0} |
7 | 4, 6 | cdif 2908 | . . 3 class (ℂ ∖ {0}) |
8 | 3 | cv 1241 | . . . . . 6 class y |
9 | vz | . . . . . . 7 setvar z | |
10 | 9 | cv 1241 | . . . . . 6 class z |
11 | cmul 6716 | . . . . . 6 class · | |
12 | 8, 10, 11 | co 5455 | . . . . 5 class (y · z) |
13 | 2 | cv 1241 | . . . . 5 class x |
14 | 12, 13 | wceq 1242 | . . . 4 wff (y · z) = x |
15 | 14, 9, 4 | crio 5410 | . . 3 class (℩z ∈ ℂ (y · z) = x) |
16 | 2, 3, 4, 7, 15 | cmpt2 5457 | . 2 class (x ∈ ℂ, y ∈ (ℂ ∖ {0}) ↦ (℩z ∈ ℂ (y · z) = x)) |
17 | 1, 16 | wceq 1242 | 1 wff / = (x ∈ ℂ, y ∈ (ℂ ∖ {0}) ↦ (℩z ∈ ℂ (y · z) = x)) |
Colors of variables: wff set class |
This definition is referenced by: divvalap 7435 divfnzn 8332 |
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