Definition df-sub 11418

Description: Define subtraction. Theorem subval 11423 shows its value (and describes how this definition works), Theorem subaddi 11520 relates it to addition, and Theorems subcli 11509 and resubcli 11495 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 11416	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 11073	. . 3 class ℂ
5	3	cv 1561	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1561	. . . . . 6 class 𝑧
8		caddc 11078	. . . . . 6 class +
9	5, 7, 8	co 7398	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1561	. . . . 5 class 𝑥
11	9, 10	wceq 1562	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 7354	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 7400	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1562	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  11423  subf  11434  sn-subf  43043