Definition df-sub 10268

Description: Define subtraction. Theorem subval 10272 shows its value (and describes how this definition works), theorem subaddi 10368 relates it to addition, and theorems subcli 10357 and resubcli 10343 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 10266	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 9934	. . 3 class ℂ
5	3	cv 1482	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1482	. . . . . 6 class 𝑧
8		caddc 9939	. . . . . 6 class +
9	5, 7, 8	co 6650	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1482	. . . . 5 class 𝑥
11	9, 10	wceq 1483	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 6610	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpt2 6652	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1483	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  10272  subf  10283