Definition df-sub 10556

Description: Define subtraction. Theorem subval 10560 shows its value (and describes how this definition works), theorem subaddi 10656 relates it to addition, and theorems subcli 10645 and resubcli 10631 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 10554	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 10222	. . 3 class ℂ
5	3	cv 1636	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1636	. . . . . 6 class 𝑧
8		caddc 10227	. . . . . 6 class +
9	5, 7, 8	co 6877	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1636	. . . . 5 class 𝑥
11	9, 10	wceq 1637	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 6837	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpt2 6879	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1637	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  10560  subf  10571