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| Mirrors > Home > MPE Home > Th. List > Mathboxes > df-resub | Structured version Visualization version GIF version | ||
| Description: Define subtraction between real numbers. This operator saves a few axioms over df-sub 11494 in certain situations. Theorem resubval 42397 shows its value, resubadd 42409 relates it to addition, and rersubcl 42408 proves its closure. It is the restriction of df-sub 11494 to the reals: subresre 42460. (Contributed by Steven Nguyen, 7-Jan-2023.) |
| Ref | Expression |
|---|---|
| df-resub | ⊢ −ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (℩𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cresub 42395 | . 2 class −ℝ | |
| 2 | vx | . . 3 setvar 𝑥 | |
| 3 | vy | . . 3 setvar 𝑦 | |
| 4 | cr 11154 | . . 3 class ℝ | |
| 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 ∈ ℝ (𝑦 + 𝑧) = 𝑥) |
| 13 | 2, 3, 4, 4, 12 | cmpo 7433 | . 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (℩𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥)) |
| 14 | 1, 13 | wceq 1540 | 1 wff −ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (℩𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥)) |
| Colors of variables: wff setvar class |
| This definition is referenced by: resubval 42397 resubf 42411 |
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