Definition df-sub 10306

Description: Define subtraction. Theorem subval 10310 shows its value (and describes how this definition works), theorem subaddi 10406 relates it to addition, and theorems subcli 10395 and resubcli 10381 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 10304	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 9972	. . 3 class ℂ
5	3	cv 1522	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1522	. . . . . 6 class 𝑧
8		caddc 9977	. . . . . 6 class +
9	5, 7, 8	co 6690	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1522	. . . . 5 class 𝑥
11	9, 10	wceq 1523	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 6650	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpt2 6692	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1523	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  10310  subf  10321