Definition df-sub 11364

Description: Define subtraction. Theorem subval 11369 shows its value (and describes how this definition works), Theorem subaddi 11466 relates it to addition, and Theorems subcli 11455 and resubcli 11441 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 11362	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 11022	. . 3 class ℂ
5	3	cv 1540	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1540	. . . . . 6 class 𝑧
8		caddc 11027	. . . . . 6 class +
9	5, 7, 8	co 7356	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1540	. . . . 5 class 𝑥
11	9, 10	wceq 1541	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 7312	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 7358	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1541	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  11369  subf  11380  sn-subf  42626