Definition df-sub 8128

Description: Define subtraction. Theorem subval 8147 shows its value (and describes how this definition works), Theorem subaddi 8242 relates it to addition, and Theorems subcli 8231 and resubcli 8218 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 8126	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 7808	. . 3 class ℂ
5	3	cv 1352	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1352	. . . . . 6 class 𝑧
8		caddc 7813	. . . . . 6 class +
9	5, 7, 8	co 5874	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1352	. . . . 5 class 𝑥
11	9, 10	wceq 1353	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 5829	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 5876	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1353	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  8147  subf  8157