Definition df-sub 8394

Description: Define subtraction. Theorem subval 8413 shows its value (and describes how this definition works), Theorem subaddi 8508 relates it to addition, and Theorems subcli 8497 and resubcli 8484 prove its closure laws. (Contributed by NM, 26-Nov-1994.)

Assertion

Ref	Expression
df-sub	⊢ − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-sub

Step	Hyp	Ref	Expression
1		cmin 8392	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 8073	. . 3 class ℂ
5	3	cv 1397	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1397	. . . . . 6 class 𝑧
8		caddc 8078	. . . . . 6 class +
9	5, 7, 8	co 6028	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1397	. . . . 5 class 𝑥
11	9, 10	wceq 1398	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 5980	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 6030	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1398	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  8413  subf  8423  cndsex  14632