Definition df-sub 11491

Description: Define subtraction. Theorem subval 11496 shows its value (and describes how this definition works), Theorem subaddi 11593 relates it to addition, and Theorems subcli 11582 and resubcli 11568 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 11489	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 11150	. . 3 class ℂ
5	3	cv 1535	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1535	. . . . . 6 class 𝑧
8		caddc 11155	. . . . . 6 class +
9	5, 7, 8	co 7430	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1535	. . . . 5 class 𝑥
11	9, 10	wceq 1536	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 7386	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 7432	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1536	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  11496  subf  11507  sn-subf  42434