Definition df-sub 8499

Description: Define subtraction. Theorem subval 8518 shows its value (and describes how this definition works), Theorem subaddi 8613 relates it to addition, and Theorems subcli 8602 and resubcli 8589 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 8497	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 8177	. . 3 class ℂ
5	3	cv 1401	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1401	. . . . . 6 class 𝑧
8		caddc 8182	. . . . . 6 class +
9	5, 7, 8	co 6085	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1401	. . . . 5 class 𝑥
11	9, 10	wceq 1402	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 6037	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 6087	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1402	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is used by:  subval  8518  subf  8528  cndsex  14890