Definition df-sub 10670

Description: Define subtraction. Theorem subval 10675 shows its value (and describes how this definition works), theorem subaddi 10772 relates it to addition, and theorems subcli 10761 and resubcli 10747 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 10668	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 10331	. . 3 class ℂ
5	3	cv 1506	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1506	. . . . . 6 class 𝑧
8		caddc 10336	. . . . . 6 class +
9	5, 7, 8	co 6974	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1506	. . . . 5 class 𝑥
11	9, 10	wceq 1507	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 6934	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 6976	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1507	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  10675  subf  10686