Definition df-sub 11112

Description: Define subtraction. Theorem subval 11117 shows its value (and describes how this definition works), Theorem subaddi 11213 relates it to addition, and Theorems subcli 11202 and resubcli 11188 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 11110	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 10775	. . 3 class ℂ
5	3	cv 1542	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1542	. . . . . 6 class 𝑧
8		caddc 10780	. . . . . 6 class +
9	5, 7, 8	co 7252	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1542	. . . . 5 class 𝑥
11	9, 10	wceq 1543	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 7208	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 7254	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1543	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  11117  subf  11128  sn-subf  40303