Definition df-sub 8501

Description: Define subtraction. Theorem subval 8520 shows its value (and describes how this definition works), Theorem subaddi 8615 relates it to addition, and Theorems subcli 8604 and resubcli 8591 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 8499	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 8178	. . 3 class ℂ
5	3	cv 1401	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1401	. . . . . 6 class 𝑧
8		caddc 8183	. . . . . 6 class +
9	5, 7, 8	co 6085	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1401	. . . . 5 class 𝑥
11	9, 10	wceq 1402	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 6037	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 6087	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1402	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is used by:  subval  8520  subf  8530  cndsex  14974