Definition df-sub 11494

Description: Define subtraction. Theorem subval 11499 shows its value (and describes how this definition works), Theorem subaddi 11596 relates it to addition, and Theorems subcli 11585 and resubcli 11571 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 11492	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 11153	. . 3 class ℂ
5	3	cv 1539	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1539	. . . . . 6 class 𝑧
8		caddc 11158	. . . . . 6 class +
9	5, 7, 8	co 7431	. . . . 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 7387	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 7433	. 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  11499  subf  11510  sn-subf  42458