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Definition df-gcd 15211
Description: Define the gcd operator. For example, (-6 gcd 9) = 3 (ex-gcd 27298). For an alternate definition, based on the definition in [ApostolNT] p. 15, see dfgcd2 15257. (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 ∣ (𝑛𝑥𝑛𝑦)}, ℝ, < )))
Distinct variable group:   𝑥,𝑛,𝑦

Detailed syntax breakdown of Definition df-gcd
StepHypRef Expression
1 cgcd 15210 . 2 class gcd
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 cz 11374 . . 3 class
52cv 1481 . . . . . 6 class 𝑥
6 cc0 9933 . . . . . 6 class 0
75, 6wceq 1482 . . . . 5 wff 𝑥 = 0
83cv 1481 . . . . . 6 class 𝑦
98, 6wceq 1482 . . . . 5 wff 𝑦 = 0
107, 9wa 384 . . . 4 wff (𝑥 = 0 ∧ 𝑦 = 0)
11 vn . . . . . . . . 9 setvar 𝑛
1211cv 1481 . . . . . . . 8 class 𝑛
13 cdvds 14977 . . . . . . . 8 class
1412, 5, 13wbr 4651 . . . . . . 7 wff 𝑛𝑥
1512, 8, 13wbr 4651 . . . . . . 7 wff 𝑛𝑦
1614, 15wa 384 . . . . . 6 wff (𝑛𝑥𝑛𝑦)
1716, 11, 4crab 2915 . . . . 5 class {𝑛 ∈ ℤ ∣ (𝑛𝑥𝑛𝑦)}
18 cr 9932 . . . . 5 class
19 clt 10071 . . . . 5 class <
2017, 18, 19csup 8343 . . . 4 class sup({𝑛 ∈ ℤ ∣ (𝑛𝑥𝑛𝑦)}, ℝ, < )
2110, 6, 20cif 4084 . . 3 class if((𝑥 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛𝑥𝑛𝑦)}, ℝ, < ))
222, 3, 4, 4, 21cmpt2 6649 . 2 class (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛𝑥𝑛𝑦)}, ℝ, < )))
231, 22wceq 1482 1 wff gcd = (𝑥 ∈ ℤ, 𝑦 ∈ ℤ ↦ if((𝑥 = 0 ∧ 𝑦 = 0), 0, sup({𝑛 ∈ ℤ ∣ (𝑛𝑥𝑛𝑦)}, ℝ, < )))
Colors of variables: wff setvar class
This definition is referenced by:  gcdval  15212  gcdf  15228
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