Definition df-sub 10470

Description: Define subtraction. Theorem subval 10474 shows its value (and describes how this definition works), theorem subaddi 10570 relates it to addition, and theorems subcli 10559 and resubcli 10545 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 10468	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 10136	. . 3 class ℂ
5	3	cv 1630	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1630	. . . . . 6 class 𝑧
8		caddc 10141	. . . . . 6 class +
9	5, 7, 8	co 6793	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1630	. . . . 5 class 𝑥
11	9, 10	wceq 1631	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 6753	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpt2 6795	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1631	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  10474  subf  10485