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| Mirrors > Home > MPE Home > Th. List > df-sub | Structured version Visualization version GIF version | ||
| Description: Define subtraction. Theorem subval 11499 shows its value (and describes how this definition works), Theorem subaddi 11596 relates it to addition, and Theorems subcli 11585 and resubcli 11571 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 11492 | . 2 class − | |
| 2 | vx | . . 3 setvar 𝑥 | |
| 3 | vy | . . 3 setvar 𝑦 | |
| 4 | cc 11153 | . . 3 class ℂ | |
| 5 | 3 | cv 1539 | . . . . . 6 class 𝑦 |
| 6 | vz | . . . . . . 7 setvar 𝑧 | |
| 7 | 6 | cv 1539 | . . . . . 6 class 𝑧 |
| 8 | caddc 11158 | . . . . . 6 class + | |
| 9 | 5, 7, 8 | co 7431 | . . . . 5 class (𝑦 + 𝑧) |
| 10 | 2 | cv 1539 | . . . . 5 class 𝑥 |
| 11 | 9, 10 | wceq 1540 | . . . 4 wff (𝑦 + 𝑧) = 𝑥 |
| 12 | 11, 6, 4 | crio 7387 | . . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥) |
| 13 | 2, 3, 4, 4, 12 | cmpo 7433 | . 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) |
| 14 | 1, 13 | wceq 1540 | 1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) |
| Colors of variables: wff setvar class |
| This definition is referenced by: subval 11499 subf 11510 sn-subf 42458 |
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