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Definition df-lcm 10821
Description: Define the lcm operator. For example, (6 lcm 9) = 18. (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 ∣ (𝑥𝑛𝑦𝑛)}, ℝ, < )))
Distinct variable group:   𝑥,𝑛,𝑦

Detailed syntax breakdown of Definition df-lcm
StepHypRef Expression
1 clcm 10820 . 2 class lcm
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 cz 8644 . . 3 class
52cv 1284 . . . . . 6 class 𝑥
6 cc0 7251 . . . . . 6 class 0
75, 6wceq 1285 . . . . 5 wff 𝑥 = 0
83cv 1284 . . . . . 6 class 𝑦
98, 6wceq 1285 . . . . 5 wff 𝑦 = 0
107, 9wo 662 . . . 4 wff (𝑥 = 0 ∨ 𝑦 = 0)
11 vn . . . . . . . . 9 setvar 𝑛
1211cv 1284 . . . . . . . 8 class 𝑛
13 cdvds 10574 . . . . . . . 8 class
145, 12, 13wbr 3811 . . . . . . 7 wff 𝑥𝑛
158, 12, 13wbr 3811 . . . . . . 7 wff 𝑦𝑛
1614, 15wa 102 . . . . . 6 wff (𝑥𝑛𝑦𝑛)
17 cn 8314 . . . . . 6 class
1816, 11, 17crab 2357 . . . . 5 class {𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)}
19 cr 7250 . . . . 5 class
20 clt 7423 . . . . 5 class <
2118, 19, 20cinf 6583 . . . 4 class inf({𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)}, ℝ, < )
2210, 6, 21cif 3373 . . 3 class if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)}, ℝ, < ))
232, 3, 4, 4, 22cmpt2 5591 . 2 class (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)}, ℝ, < )))
241, 23wceq 1285 1 wff lcm = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∨ 𝑦 = 0), 0, inf({𝑛 ∈ ℕ ∣ (𝑥𝑛𝑦𝑛)}, ℝ, < )))
Colors of variables: wff set class
This definition is referenced by:  lcmval  10823
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