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Mirrors > Home > ILE Home > Th. List > df-sub | GIF version |
Description: Define subtraction. Theorem subval 8147 shows its value (and describes how this definition works), Theorem subaddi 8242 relates it to addition, and Theorems subcli 8231 and resubcli 8218 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 8126 | . 2 class − | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vy | . . 3 setvar 𝑦 | |
4 | cc 7808 | . . 3 class ℂ | |
5 | 3 | cv 1352 | . . . . . 6 class 𝑦 |
6 | vz | . . . . . . 7 setvar 𝑧 | |
7 | 6 | cv 1352 | . . . . . 6 class 𝑧 |
8 | caddc 7813 | . . . . . 6 class + | |
9 | 5, 7, 8 | co 5874 | . . . . 5 class (𝑦 + 𝑧) |
10 | 2 | cv 1352 | . . . . 5 class 𝑥 |
11 | 9, 10 | wceq 1353 | . . . 4 wff (𝑦 + 𝑧) = 𝑥 |
12 | 11, 6, 4 | crio 5829 | . . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥) |
13 | 2, 3, 4, 4, 12 | cmpo 5876 | . 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) |
14 | 1, 13 | wceq 1353 | 1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) |
Colors of variables: wff set class |
This definition is referenced by: subval 8147 subf 8157 |
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