Definition df-sub 8132

Description: Define subtraction. Theorem subval 8151 shows its value (and describes how this definition works), Theorem subaddi 8246 relates it to addition, and Theorems subcli 8235 and resubcli 8222 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 8130	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 7811	. . 3 class ℂ
5	3	cv 1352	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1352	. . . . . 6 class 𝑧
8		caddc 7816	. . . . . 6 class +
9	5, 7, 8	co 5877	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1352	. . . . 5 class 𝑥
11	9, 10	wceq 1353	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 5832	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 5879	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1353	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  8151  subf  8161