Definition df-sub 8067

Description: Define subtraction. Theorem subval 8086 shows its value (and describes how this definition works), Theorem subaddi 8181 relates it to addition, and Theorems subcli 8170 and resubcli 8157 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 8065	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 7747	. . 3 class ℂ
5	3	cv 1342	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1342	. . . . . 6 class 𝑧
8		caddc 7752	. . . . . 6 class +
9	5, 7, 8	co 5841	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1342	. . . . 5 class 𝑥
11	9, 10	wceq 1343	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 5796	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 5843	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1343	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  8086  subf  8096