Definition df-sub 7184

Description: Define subtraction. Theorem subval 7203 shows its value (and describes how this definition works), theorem subaddi 7298 relates it to addition, and theorems subcli 7287 and resubcli 7274 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 7182	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 6887	. . 3 class ℂ
5	3	cv 1242	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1242	. . . . . 6 class 𝑧
8		caddc 6892	. . . . . 6 class +
9	5, 7, 8	co 5512	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1242	. . . . 5 class 𝑥
11	9, 10	wceq 1243	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 5467	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpt2 5514	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1243	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  7203  subf  7213