Definition df-sub 8199

Description: Define subtraction. Theorem subval 8218 shows its value (and describes how this definition works), Theorem subaddi 8313 relates it to addition, and Theorems subcli 8302 and resubcli 8289 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 8197	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 7877	. . 3 class ℂ
5	3	cv 1363	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1363	. . . . . 6 class 𝑧
8		caddc 7882	. . . . . 6 class +
9	5, 7, 8	co 5922	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1363	. . . . 5 class 𝑥
11	9, 10	wceq 1364	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 5876	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 5924	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1364	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  8218  subf  8228  cndsex  14109