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Definition df-resub 39501
Description: Define subtraction between real numbers. This operator saves a few axioms over df-sub 10865 in certain situations. Theorem resubval 39502 shows its value, resubadd 39514 relates it to addition, and rersubcl 39513 proves its closure. It is the restriction of df-sub 10865 to the reals: subresre 39564. (Contributed by Steven Nguyen, 7-Jan-2022.)
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 39500 . 2 class
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 cr 10529 . . 3 class
53cv 1537 . . . . . 6 class 𝑦
6 vz . . . . . . 7 setvar 𝑧
76cv 1537 . . . . . 6 class 𝑧
8 caddc 10533 . . . . . 6 class +
95, 7, 8co 7139 . . . . 5 class (𝑦 + 𝑧)
102cv 1537 . . . . 5 class 𝑥
119, 10wceq 1538 . . . 4 wff (𝑦 + 𝑧) = 𝑥
1211, 6, 4crio 7096 . . 3 class (𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥)
132, 3, 4, 4, 12cmpo 7141 . 2 class (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥))
141, 13wceq 1538 1 wff = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ (𝑧 ∈ ℝ (𝑦 + 𝑧) = 𝑥))
Colors of variables: wff setvar class
This definition is referenced by:  resubval  39502  resubf  39516
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