Definition df-divs 28082

Description: Define surreal division. This is not the definition used in the literature, but we use it here because it is technically easier to work with. (Contributed by Scott Fenton, 12-Mar-2025.)

Assertion

Ref	Expression
df-divs	⊢ /su = (𝑥 ∈ No , 𝑦 ∈ ( No ∖ { 0s }) ↦ (℩𝑧 ∈ No (𝑦 ·s 𝑧) = 𝑥))

Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-divs

Step	Hyp	Ref	Expression
1		cdivs 28081	. 2 class /su
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		csur 27567	. . 3 class No
5		c0s 27749	. . . . 5 class 0s
6	5	csn 4625	. . . 4 class { 0s }
7	4, 6	cdif 3942	. . 3 class ( No ∖ { 0s })
8	3	cv 1533	. . . . . 6 class 𝑦
9		vz	. . . . . . 7 setvar 𝑧
10	9	cv 1533	. . . . . 6 class 𝑧
11		cmuls 28000	. . . . . 6 class ·s
12	8, 10, 11	co 7415	. . . . 5 class (𝑦 ·s 𝑧)
13	2	cv 1533	. . . . 5 class 𝑥
14	12, 13	wceq 1534	. . . 4 wff (𝑦 ·s 𝑧) = 𝑥
15	14, 9, 4	crio 7370	. . 3 class (℩𝑧 ∈ No (𝑦 ·s 𝑧) = 𝑥)
16	2, 3, 4, 7, 15	cmpo 7417	. 2 class (𝑥 ∈ No , 𝑦 ∈ ( No ∖ { 0s }) ↦ (℩𝑧 ∈ No (𝑦 ·s 𝑧) = 𝑥))
17	1, 16	wceq 1534	1 wff /su = (𝑥 ∈ No , 𝑦 ∈ ( No ∖ { 0s }) ↦ (℩𝑧 ∈ No (𝑦 ·s 𝑧) = 𝑥))

This definition is referenced by:  divsval  28083