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Mirrors > Home > MPE Home > Th. List > df-sub | Structured version Visualization version GIF version |
Description: Define subtraction. Theorem subval 10310 shows its value (and describes how this definition works), theorem subaddi 10406 relates it to addition, and theorems subcli 10395 and resubcli 10381 prove its closure laws. (Contributed by NM, 26-Nov-1994.) |
Ref | Expression |
---|---|
df-sub | ⊢ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmin 10304 | . 2 class − | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vy | . . 3 setvar 𝑦 | |
4 | cc 9972 | . . 3 class ℂ | |
5 | 3 | cv 1522 | . . . . . 6 class 𝑦 |
6 | vz | . . . . . . 7 setvar 𝑧 | |
7 | 6 | cv 1522 | . . . . . 6 class 𝑧 |
8 | caddc 9977 | . . . . . 6 class + | |
9 | 5, 7, 8 | co 6690 | . . . . 5 class (𝑦 + 𝑧) |
10 | 2 | cv 1522 | . . . . 5 class 𝑥 |
11 | 9, 10 | wceq 1523 | . . . 4 wff (𝑦 + 𝑧) = 𝑥 |
12 | 11, 6, 4 | crio 6650 | . . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥) |
13 | 2, 3, 4, 4, 12 | cmpt2 6692 | . 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) |
14 | 1, 13 | wceq 1523 | 1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) |
Colors of variables: wff setvar class |
This definition is referenced by: subval 10310 subf 10321 |
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