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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 11029 in certain situations. Theorem resubval 39999 shows its value, resubadd 40011 relates it to addition, and rersubcl 40010 proves its closure. It is the restriction of df-sub 11029 to the reals: subresre 40061. (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 39997 | . 2 class −ℝ | |
2 | vx | . . 3 setvar 𝑥 | |
3 | vy | . . 3 setvar 𝑦 | |
4 | cr 10693 | . . 3 class ℝ | |
5 | 3 | cv 1542 | . . . . . 6 class 𝑦 |
6 | vz | . . . . . . 7 setvar 𝑧 | |
7 | 6 | cv 1542 | . . . . . 6 class 𝑧 |
8 | caddc 10697 | . . . . . 6 class + | |
9 | 5, 7, 8 | co 7191 | . . . . 5 class (𝑦 + 𝑧) |
10 | 2 | cv 1542 | . . . . 5 class 𝑥 |
11 | 9, 10 | wceq 1543 | . . . 4 wff (𝑦 + 𝑧) = 𝑥 |
12 | 11, 6, 4 | crio 7147 | . . 3 class (℩𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥) |
13 | 2, 3, 4, 4, 12 | cmpo 7193 | . 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (℩𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥)) |
14 | 1, 13 | wceq 1543 | 1 wff −ℝ = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (℩𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥)) |
Colors of variables: wff setvar class |
This definition is referenced by: resubval 39999 resubf 40013 |
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