Definition df-sub 11478

Description: Define subtraction. Theorem subval 11483 shows its value (and describes how this definition works), Theorem subaddi 11579 relates it to addition, and Theorems subcli 11568 and resubcli 11554 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 11476	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 11138	. . 3 class ℂ
5	3	cv 1532	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1532	. . . . . 6 class 𝑧
8		caddc 11143	. . . . . 6 class +
9	5, 7, 8	co 7419	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1532	. . . . 5 class 𝑥
11	9, 10	wceq 1533	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 7374	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 7421	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1533	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  11483  subf  11494  sn-subf  42118