Definition df-sub 11367

Description: Define subtraction. Theorem subval 11372 shows its value (and describes how this definition works), Theorem subaddi 11469 relates it to addition, and Theorems subcli 11458 and resubcli 11444 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 11365	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 11025	. . 3 class ℂ
5	3	cv 1541	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1541	. . . . . 6 class 𝑧
8		caddc 11030	. . . . . 6 class +
9	5, 7, 8	co 7358	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1541	. . . . 5 class 𝑥
11	9, 10	wceq 1542	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 7314	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 7360	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1542	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  11372  subf  11383  sn-subf  42860