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Mirrors > Home > MPE Home > Th. List > df-sub | Structured version Visualization version GIF version |
Description: Define subtraction. Theorem subval 10151 shows its value (and describes how this definition works), theorem subaddi 10247 relates it to addition, and theorems subcli 10236 and resubcli 10222 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 10145 | . 2 class − | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vy | . . 3 setvar 𝑦 | |
4 | cc 9813 | . . 3 class ℂ | |
5 | 3 | cv 1474 | . . . . . 6 class 𝑦 |
6 | vz | . . . . . . 7 setvar 𝑧 | |
7 | 6 | cv 1474 | . . . . . 6 class 𝑧 |
8 | caddc 9818 | . . . . . 6 class + | |
9 | 5, 7, 8 | co 6549 | . . . . 5 class (𝑦 + 𝑧) |
10 | 2 | cv 1474 | . . . . 5 class 𝑥 |
11 | 9, 10 | wceq 1475 | . . . 4 wff (𝑦 + 𝑧) = 𝑥 |
12 | 11, 6, 4 | crio 6510 | . . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥) |
13 | 2, 3, 4, 4, 12 | cmpt2 6551 | . 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) |
14 | 1, 13 | wceq 1475 | 1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) |
Colors of variables: wff setvar class |
This definition is referenced by: subval 10151 subf 10162 |
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