Definition df-sub 10265

Description: Define subtraction. Theorem subval 10269 shows its value (and describes how this definition works), theorem subaddi 10365 relates it to addition, and theorems subcli 10354 and resubcli 10340 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 10263	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 9931	. . 3 class ℂ
5	3	cv 1481	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1481	. . . . . 6 class 𝑧
8		caddc 9936	. . . . . 6 class +
9	5, 7, 8	co 6647	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1481	. . . . 5 class 𝑥
11	9, 10	wceq 1482	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 6607	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpt2 6649	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1482	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  10269  subf  10280