Definition df-sub 11396

Description: Define subtraction. Theorem subval 11401 shows its value (and describes how this definition works), Theorem subaddi 11497 relates it to addition, and Theorems subcli 11486 and resubcli 11472 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 11394	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 11058	. . 3 class ℂ
5	3	cv 1540	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1540	. . . . . 6 class 𝑧
8		caddc 11063	. . . . . 6 class +
9	5, 7, 8	co 7362	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1540	. . . . 5 class 𝑥
11	9, 10	wceq 1541	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 7317	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 7364	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1541	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  11401  subf  11412  sn-subf  40955