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| Mirrors > Home > MPE Home > Th. List > df-sub | Structured version Visualization version GIF version | ||
| Description: Define subtraction. Theorem subval 11473 shows its value (and describes how this definition works), Theorem subaddi 11570 relates it to addition, and Theorems subcli 11559 and resubcli 11545 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 11466 | . 2 class − | |
| 2 | vx | . . 3 setvar 𝑥 | |
| 3 | vy | . . 3 setvar 𝑦 | |
| 4 | cc 11123 | . . 3 class ℂ | |
| 5 | 3 | cv 1569 | . . . . . 6 class 𝑦 |
| 6 | vz | . . . . . . 7 setvar 𝑧 | |
| 7 | 6 | cv 1569 | . . . . . 6 class 𝑧 |
| 8 | caddc 11128 | . . . . . 6 class + | |
| 9 | 5, 7, 8 | co 7414 | . . . . 5 class (𝑦 + 𝑧) |
| 10 | 2 | cv 1569 | . . . . 5 class 𝑥 |
| 11 | 9, 10 | wceq 1570 | . . . 4 wff (𝑦 + 𝑧) = 𝑥 |
| 12 | 11, 6, 4 | crio 7370 | . . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥) |
| 13 | 2, 3, 4, 4, 12 | cmpo 7416 | . 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) |
| 14 | 1, 13 | wceq 1570 | 1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)) |
| Colors of variables: wff setvar class |
| This definition is used by: subval 11473 subf 11484 sn-subf 43305 |
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