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Mirrors > Home > MPE Home > Th. List > df-sub | Structured version Visualization version GIF version |
Description: Define subtraction. Theorem subval 10675 shows its value (and describes how this definition works), theorem subaddi 10772 relates it to addition, and theorems subcli 10761 and resubcli 10747 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 10668 | . 2 class − | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vy | . . 3 setvar 𝑦 | |
4 | cc 10331 | . . 3 class ℂ | |
5 | 3 | cv 1506 | . . . . . 6 class 𝑦 |
6 | vz | . . . . . . 7 setvar 𝑧 | |
7 | 6 | cv 1506 | . . . . . 6 class 𝑧 |
8 | caddc 10336 | . . . . . 6 class + | |
9 | 5, 7, 8 | co 6974 | . . . . 5 class (𝑦 + 𝑧) |
10 | 2 | cv 1506 | . . . . 5 class 𝑥 |
11 | 9, 10 | wceq 1507 | . . . 4 wff (𝑦 + 𝑧) = 𝑥 |
12 | 11, 6, 4 | crio 6934 | . . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥) |
13 | 2, 3, 4, 4, 12 | cmpo 6976 | . 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) |
14 | 1, 13 | wceq 1507 | 1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) |
Colors of variables: wff setvar class |
This definition is referenced by: subval 10675 subf 10686 |
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