Definition df-sub 11370

Description: Define subtraction. Theorem subval 11375 shows its value (and describes how this definition works), Theorem subaddi 11472 relates it to addition, and Theorems subcli 11461 and resubcli 11447 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 11368	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 11028	. . 3 class ℂ
5	3	cv 1541	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1541	. . . . . 6 class 𝑧
8		caddc 11033	. . . . . 6 class +
9	5, 7, 8	co 7360	. . . . 5 class (𝑦 + 𝑧)
10	2	cv 1541	. . . . 5 class 𝑥
11	9, 10	wceq 1542	. . . 4 wff (𝑦 + 𝑧) = 𝑥
12	11, 6, 4	crio 7316	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 7362	. 2 class (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))
14	1, 13	wceq 1542	1 wff − = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥))

This definition is referenced by:  subval  11375  subf  11386  sn-subf  42751