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Mirrors > Home > MPE Home > Th. List > df-sub | Structured version Visualization version GIF version |
Description: Define subtraction. Theorem subval 10474 shows its value (and describes how this definition works), theorem subaddi 10570 relates it to addition, and theorems subcli 10559 and resubcli 10545 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 10468 | . 2 class − | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vy | . . 3 setvar 𝑦 | |
4 | cc 10136 | . . 3 class ℂ | |
5 | 3 | cv 1630 | . . . . . 6 class 𝑦 |
6 | vz | . . . . . . 7 setvar 𝑧 | |
7 | 6 | cv 1630 | . . . . . 6 class 𝑧 |
8 | caddc 10141 | . . . . . 6 class + | |
9 | 5, 7, 8 | co 6793 | . . . . 5 class (𝑦 + 𝑧) |
10 | 2 | cv 1630 | . . . . 5 class 𝑥 |
11 | 9, 10 | wceq 1631 | . . . 4 wff (𝑦 + 𝑧) = 𝑥 |
12 | 11, 6, 4 | crio 6753 | . . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥) |
13 | 2, 3, 4, 4, 12 | cmpt2 6795 | . 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) |
14 | 1, 13 | wceq 1631 | 1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) |
Colors of variables: wff setvar class |
This definition is referenced by: subval 10474 subf 10485 |
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