Definition df-sub 11407

Description: Define subtraction. Theorem subval 11412 shows its value (and describes how this definition works), Theorem subaddi 11509 relates it to addition, and Theorems subcli 11498 and resubcli 11484 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 11405	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 11066	. . 3 class ℂ
5	3	cv 1539	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1539	. . . . . 6 class 𝑧
8		caddc 11071	. . . . . 6 class +
9	5, 7, 8	co 7387	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1539	. . . . 5 class 𝑥
11	9, 10	wceq 1540	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 7343	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 7389	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1540	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  11412  subf  11423  sn-subf  42417