Definition df-sub 10119

Description: Define subtraction. Theorem subval 10123 shows its value (and describes how this definition works), theorem subaddi 10219 relates it to addition, and theorems subcli 10208 and resubcli 10194 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 10117	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 9790	. . 3 class ℂ
5	3	cv 1473	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1473	. . . . . 6 class 𝑧
8		caddc 9795	. . . . . 6 class +
9	5, 7, 8	co 6526	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1473	. . . . 5 class 𝑥
11	9, 10	wceq 1474	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 6487	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpt2 6528	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1474	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  10123  subf  10134