Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > df-div | Structured version Visualization version GIF version |
Description: Define division. Theorem divmuli 11738 relates it to multiplication, and divcli 11726 and redivcli 11751 prove its closure laws. (Contributed by NM, 2-Feb-1995.) Use divval 11644 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 11641 | . 2 class / | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vy | . . 3 setvar 𝑦 | |
4 | cc 10878 | . . 3 class ℂ | |
5 | cc0 10880 | . . . . 5 class 0 | |
6 | 5 | csn 4562 | . . . 4 class {0} |
7 | 4, 6 | cdif 3885 | . . 3 class (ℂ ∖ {0}) |
8 | 3 | cv 1538 | . . . . . 6 class 𝑦 |
9 | vz | . . . . . . 7 setvar 𝑧 | |
10 | 9 | cv 1538 | . . . . . 6 class 𝑧 |
11 | cmul 10885 | . . . . . 6 class · | |
12 | 8, 10, 11 | co 7284 | . . . . 5 class (𝑦 · 𝑧) |
13 | 2 | cv 1538 | . . . . 5 class 𝑥 |
14 | 12, 13 | wceq 1539 | . . . 4 wff (𝑦 · 𝑧) = 𝑥 |
15 | 14, 9, 4 | crio 7240 | . . 3 class (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) |
16 | 2, 3, 4, 7, 15 | cmpo 7286 | . 2 class (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) |
17 | 1, 16 | wceq 1539 | 1 wff / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥)) |
Colors of variables: wff setvar class |
This definition is referenced by: 1div0 11643 divval 11644 elq 12699 cnflddiv 20637 divcn 24040 1div0apr 28841 |
Copyright terms: Public domain | W3C validator |