Definition df-sub 11346

Description: Define subtraction. Theorem subval 11351 shows its value (and describes how this definition works), Theorem subaddi 11448 relates it to addition, and Theorems subcli 11437 and resubcli 11423 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 11344	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 11004	. . 3 class ℂ
5	3	cv 1540	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1540	. . . . . 6 class 𝑧
8		caddc 11009	. . . . . 6 class +
9	5, 7, 8	co 7346	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1540	. . . . 5 class 𝑥
11	9, 10	wceq 1541	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 7302	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 7348	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1541	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  11351  subf  11362  sn-subf  42470