Definition df-sub 11468

Description: Define subtraction. Theorem subval 11473 shows its value (and describes how this definition works), Theorem subaddi 11570 relates it to addition, and Theorems subcli 11559 and resubcli 11545 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 11466	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 11123	. . 3 class ℂ
5	3	cv 1569	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1569	. . . . . 6 class 𝑧
8		caddc 11128	. . . . . 6 class +
9	5, 7, 8	co 7414	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1569	. . . . 5 class 𝑥
11	9, 10	wceq 1570	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 7370	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 7416	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1570	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is used by:  subval  11473  subf  11484  sn-subf  43305