Definition df-sub 11522

Description: Define subtraction. Theorem subval 11527 shows its value (and describes how this definition works), Theorem subaddi 11623 relates it to addition, and Theorems subcli 11612 and resubcli 11598 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 11520	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 11182	. . 3 class ℂ
5	3	cv 1536	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1536	. . . . . 6 class 𝑧
8		caddc 11187	. . . . . 6 class +
9	5, 7, 8	co 7448	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1536	. . . . 5 class 𝑥
11	9, 10	wceq 1537	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 7403	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 7450	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1537	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  11527  subf  11538  sn-subf  42404