Definition df-sub 8500

Description: Define subtraction. Theorem subval 8519 shows its value (and describes how this definition works), Theorem subaddi 8614 relates it to addition, and Theorems subcli 8603 and resubcli 8590 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 8498	. 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  8519  subf  8529  cndsex  14939