Definition df-lcm 16771

Description: Define the lcm operator. For example, (6 lcm 9) = 18 (ex-lcm 31065). (Contributed by Steve Rodriguez, 20-Jan-2020.) (Revised by AV, 16-Sep-2020.)

Assertion

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

Distinct variable group:   𝑥,𝑛,𝑦

Detailed syntax breakdown of Definition df-lcm

Step	Hyp	Ref	Expression
1		clcm 16769	. 2 class lcm
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		cz 12697	. . 3 class ℤ
5	2	cv 1568	. . . . . 6 class 𝑥
6		cc0 11204	. . . . . 6 class 0
7	5, 6	wceq 1569	. . . . 5 wff 𝑥 = 0
8	3	cv 1568	. . . . . 6 class 𝑦
9	8, 6	wceq 1569	. . . . 5 wff 𝑦 = 0
10	7, 9	wo 861	. . . 4 wff (𝑥 = 0 ∨ 𝑦 = 0)
11		vn	. . . . . . . . 9 setvar 𝑛
12	11	cv 1568	. . . . . . . 8 class 𝑛
13		cdvds 16428	. . . . . . . 8 class ∥
14	5, 12, 13	wbr 5102	. . . . . . 7 wff 𝑥 ∥ 𝑛
15	8, 12, 13	wbr 5102	. . . . . . 7 wff 𝑦 ∥ 𝑛
16	14, 15	wa 401	. . . . . 6 wff (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)
17		cn 12339	. . . . . 6 class ℕ
18	16, 11, 17	crab 3412	. . . . 5 class {𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}
19		cr 11203	. . . . 5 class ℝ
20		clt 11347	. . . . 5 class <
21	18, 19, 20	cinf 9437	. . . 4 class inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )
22	10, 6, 21	cif 4481	. . 3 class if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < ))
23	2, 3, 4, 4, 22	cmpo 7423	. 2 class (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )))
24	1, 23	wceq 1569	1 wff lcm = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥 ∥ 𝑛 ∧ 𝑦 ∥ 𝑛)}, ℝ, < )))

This definition is used by:  lcmval  16773