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Definition df-lcmf 15285
Description: Define the lcm function on a set of integers. (Contributed by AV, 21-Aug-2020.) (Revised by AV, 16-Sep-2020.)
Assertion
Ref Expression
df-lcmf lcm = (𝑧 ∈ 𝒫 ℤ ↦ if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛}, ℝ, < )))
Distinct variable group:   𝑚,𝑛,𝑧

Detailed syntax breakdown of Definition df-lcmf
StepHypRef Expression
1 clcmf 15283 . 2 class lcm
2 vz . . 3 setvar 𝑧
3 cz 11362 . . . 4 class
43cpw 4149 . . 3 class 𝒫 ℤ
5 cc0 9921 . . . . 5 class 0
62cv 1480 . . . . 5 class 𝑧
75, 6wcel 1988 . . . 4 wff 0 ∈ 𝑧
8 vm . . . . . . . . 9 setvar 𝑚
98cv 1480 . . . . . . . 8 class 𝑚
10 vn . . . . . . . . 9 setvar 𝑛
1110cv 1480 . . . . . . . 8 class 𝑛
12 cdvds 14964 . . . . . . . 8 class
139, 11, 12wbr 4644 . . . . . . 7 wff 𝑚𝑛
1413, 8, 6wral 2909 . . . . . 6 wff 𝑚𝑧 𝑚𝑛
15 cn 11005 . . . . . 6 class
1614, 10, 15crab 2913 . . . . 5 class {𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛}
17 cr 9920 . . . . 5 class
18 clt 10059 . . . . 5 class <
1916, 17, 18cinf 8332 . . . 4 class inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛}, ℝ, < )
207, 5, 19cif 4077 . . 3 class if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛}, ℝ, < ))
212, 4, 20cmpt 4720 . 2 class (𝑧 ∈ 𝒫 ℤ ↦ if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛}, ℝ, < )))
221, 21wceq 1481 1 wff lcm = (𝑧 ∈ 𝒫 ℤ ↦ if(0 ∈ 𝑧, 0, inf({𝑛 ∈ ℕ ∣ ∀𝑚𝑧 𝑚𝑛}, ℝ, < )))
Colors of variables: wff setvar class
This definition is referenced by:  lcmfval  15315  lcmf0val  15316
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