Definition df-gcd 16532

Description: Define the gcd operator. For example, (-6 gcd 9) = 3 (ex-gcd 30476). For an alternate definition, based on the definition in [ApostolNT] p. 15, see dfgcd2 16583. (Contributed by Paul Chapman, 21-Mar-2011.)

Assertion

Ref	Expression
df-gcd	⊢ gcd = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)}, ℝ, < )))

Distinct variable group:   𝑥,𝑛,𝑦

Detailed syntax breakdown of Definition df-gcd

Step	Hyp	Ref	Expression
1		cgcd 16531	. 2 class gcd
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cz 12613	. . 3 class ℤ
5	2	cv 1539	. . . . . 6 class 𝑥
6		cc0 11155	. . . . . 6 class 0
7	5, 6	wceq 1540	. . . . 5 wff 𝑥 = 0
8	3	cv 1539	. . . . . 6 class 𝑦
9	8, 6	wceq 1540	. . . . 5 wff 𝑦 = 0
10	7, 9	wa 395	. . . 4 wff (𝑥 = 0 ∧ 𝑦 = 0)
11		vn	. . . . . . . . 9 setvar 𝑛
12	11	cv 1539	. . . . . . . 8 class 𝑛
13		cdvds 16290	. . . . . . . 8 class ∥
14	12, 5, 13	wbr 5143	. . . . . . 7 wff 𝑛 ∥ 𝑥
15	12, 8, 13	wbr 5143	. . . . . . 7 wff 𝑛 ∥ 𝑦
16	14, 15	wa 395	. . . . . 6 wff (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)
17	16, 11, 4	crab 3436	. . . . 5 class {𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)}
18		cr 11154	. . . . 5 class ℝ
19		clt 11295	. . . . 5 class <
20	17, 18, 19	csup 9480	. . . 4 class sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)}, ℝ, < )
21	10, 6, 20	cif 4525	. . 3 class if((𝑥 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)}, ℝ, < ))
22	2, 3, 4, 4, 21	cmpo 7433	. 2 class (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)}, ℝ, < )))
23	1, 22	wceq 1540	1 wff gcd = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛 ∥ 𝑥 ∧ 𝑛 ∥ 𝑦)}, ℝ, < )))

This definition is referenced by:  gcdval  16533  gcdf  16549