Definition df-sub 11567

Description: Define subtraction. Theorem subval 11572 shows its value (and describes how this definition works), Theorem subaddi 11669 relates it to addition, and Theorems subcli 11658 and resubcli 11644 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 11565	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 11222	. . 3 class ℂ
5	3	cv 1569	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1569	. . . . . 6 class 𝑧
8		caddc 11227	. . . . . 6 class +
9	5, 7, 8	co 7440	. . . . 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 7396	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 7442	. 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  11572  subf  11583  sn-subf  43671