Users' Mathboxes Mathbox for Steven Nguyen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df-resub Structured version   Visualization version   GIF version

Definition df-resub 41542
Description: Define subtraction between real numbers. This operator saves a few axioms over df-sub 11451 in certain situations. Theorem resubval 41543 shows its value, resubadd 41555 relates it to addition, and rersubcl 41554 proves its closure. It is the restriction of df-sub 11451 to the reals: subresre 41606. (Contributed by Steven Nguyen, 7-Jan-2023.)
Assertion
Ref Expression
df-resub = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥))
Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-resub
StepHypRef Expression
1 cresub 41541 . 2 class
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 cr 11112 . . 3 class
53cv 1539 . . . . . 6 class 𝑦
6 vz . . . . . . 7 setvar 𝑧
76cv 1539 . . . . . 6 class 𝑧
8 caddc 11116 . . . . . 6 class +
95, 7, 8co 7412 . . . . 5 class (𝑦 + 𝑧)
102cv 1539 . . . . 5 class 𝑥
119, 10wceq 1540 . . . 4 wff (𝑦 + 𝑧) = 𝑥
1211, 6, 4crio 7367 . . 3 class (𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥)
132, 3, 4, 4, 12cmpo 7414 . 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥))
141, 13wceq 1540 1 wff = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥))
Colors of variables: wff setvar class
This definition is referenced by:  resubval  41543  resubf  41557
  Copyright terms: Public domain W3C validator