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Mirrors > Home > MPE Home > Th. List > df-sub | Structured version Visualization version GIF version |
Description: Define subtraction. Theorem subval 11483 shows its value (and describes how this definition works), Theorem subaddi 11579 relates it to addition, and Theorems subcli 11568 and resubcli 11554 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 11476 | . 2 class − | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vy | . . 3 setvar 𝑦 | |
4 | cc 11138 | . . 3 class ℂ | |
5 | 3 | cv 1532 | . . . . . 6 class 𝑦 |
6 | vz | . . . . . . 7 setvar 𝑧 | |
7 | 6 | cv 1532 | . . . . . 6 class 𝑧 |
8 | caddc 11143 | . . . . . 6 class + | |
9 | 5, 7, 8 | co 7419 | . . . . 5 class (𝑦 + 𝑧) |
10 | 2 | cv 1532 | . . . . 5 class 𝑥 |
11 | 9, 10 | wceq 1533 | . . . 4 wff (𝑦 + 𝑧) = 𝑥 |
12 | 11, 6, 4 | crio 7374 | . . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥) |
13 | 2, 3, 4, 4, 12 | cmpo 7421 | . 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) |
14 | 1, 13 | wceq 1533 | 1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) |
Colors of variables: wff setvar class |
This definition is referenced by: subval 11483 subf 11494 sn-subf 42118 |
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