Definition df-sub 11543

Description: Define subtraction. Theorem subval 11548 shows its value (and describes how this definition works), Theorem subaddi 11645 relates it to addition, and Theorems subcli 11634 and resubcli 11620 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 11541	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 11198	. . 3 class ℂ
5	3	cv 1568	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1568	. . . . . 6 class 𝑧
8		caddc 11203	. . . . . 6 class +
9	5, 7, 8	co 7420	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1568	. . . . 5 class 𝑥
11	9, 10	wceq 1569	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 7376	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 7422	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1569	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is used by:  subval  11548  subf  11559  sn-subf  43523