Definition df-sub 11446

Description: Define subtraction. Theorem subval 11451 shows its value (and describes how this definition works), Theorem subaddi 11547 relates it to addition, and Theorems subcli 11536 and resubcli 11522 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 11444	. 2 class −
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cc 11108	. . 3 class ℂ
5	3	cv 1541	. . . . . 6 class 𝑦
6		vz	. . . . . . 7 setvar 𝑧
7	6	cv 1541	. . . . . 6 class 𝑧
8		caddc 11113	. . . . . 6 class +
9	5, 7, 8	co 7409	. . . . 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 7364	. . 3 class (℩𝑧 ∈ ℂ (𝑦 + 𝑧) = 𝑥)
13	2, 3, 4, 4, 12	cmpo 7411	. 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  11451  subf  11462  sn-subf  41349