Definition df-sub 8092

Description: Define subtraction. Theorem subval 8111 shows its value (and describes how this definition works), Theorem subaddi 8206 relates it to addition, and Theorems subcli 8195 and resubcli 8182 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 8090	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 7772	. . 3 class ℂ
5	3	cv 1347	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1347	. . . . . 6 class 𝑧
8		caddc 7777	. . . . . 6 class +
9	5, 7, 8	co 5853	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1347	. . . . 5 class 𝑥
11	9, 10	wceq 1348	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 5808	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 5855	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1348	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  8111  subf  8121