Definition df-sub 8327

Description: Define subtraction. Theorem subval 8346 shows its value (and describes how this definition works), Theorem subaddi 8441 relates it to addition, and Theorems subcli 8430 and resubcli 8417 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 8325	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 8005	. . 3 class ℂ
5	3	cv 1394	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1394	. . . . . 6 class 𝑧
8		caddc 8010	. . . . . 6 class +
9	5, 7, 8	co 6007	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1394	. . . . 5 class 𝑥
11	9, 10	wceq 1395	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 5959	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 6009	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1395	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  8346  subf  8356  cndsex  14525