Definition df-sub 11470

Description: Define subtraction. Theorem subval 11475 shows its value (and describes how this definition works), Theorem subaddi 11572 relates it to addition, and Theorems subcli 11561 and resubcli 11547 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 11468	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 11125	. . 3 class ℂ
5	3	cv 1569	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1569	. . . . . 6 class 𝑧
8		caddc 11130	. . . . . 6 class +
9	5, 7, 8	co 7416	. . . . 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 7372	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 7418	. 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  11475  subf  11486  sn-subf  43291