Definition df-sub 11376

Description: Define subtraction. Theorem subval 11381 shows its value (and describes how this definition works), Theorem subaddi 11478 relates it to addition, and Theorems subcli 11467 and resubcli 11453 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 11374	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 11033	. . 3 class ℂ
5	3	cv 1541	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1541	. . . . . 6 class 𝑧
8		caddc 11038	. . . . . 6 class +
9	5, 7, 8	co 7364	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1541	. . . . 5 class 𝑥
11	9, 10	wceq 1542	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 7320	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 7366	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1542	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  11381  subf  11392  sn-subf  42858