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Definition df-denom 11255
Description: The canonical denominator of a rational is the denominator of the rational's reduced fraction representation (no common factors, denominator positive). (Contributed by Stefan O'Rear, 13-Sep-2014.)
Assertion
Ref Expression
df-denom denom = (𝑦 ∈ ℚ ↦ (2nd ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑦 = ((1st𝑥) / (2nd𝑥))))))
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-denom
StepHypRef Expression
1 cdenom 11253 . 2 class denom
2 vy . . 3 setvar 𝑦
3 cq 9073 . . 3 class
4 vx . . . . . . . . . 10 setvar 𝑥
54cv 1288 . . . . . . . . 9 class 𝑥
6 c1st 5891 . . . . . . . . 9 class 1st
75, 6cfv 5002 . . . . . . . 8 class (1st𝑥)
8 c2nd 5892 . . . . . . . . 9 class 2nd
95, 8cfv 5002 . . . . . . . 8 class (2nd𝑥)
10 cgcd 11031 . . . . . . . 8 class gcd
117, 9, 10co 5634 . . . . . . 7 class ((1st𝑥) gcd (2nd𝑥))
12 c1 7330 . . . . . . 7 class 1
1311, 12wceq 1289 . . . . . 6 wff ((1st𝑥) gcd (2nd𝑥)) = 1
142cv 1288 . . . . . . 7 class 𝑦
15 cdiv 8113 . . . . . . . 8 class /
167, 9, 15co 5634 . . . . . . 7 class ((1st𝑥) / (2nd𝑥))
1714, 16wceq 1289 . . . . . 6 wff 𝑦 = ((1st𝑥) / (2nd𝑥))
1813, 17wa 102 . . . . 5 wff (((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑦 = ((1st𝑥) / (2nd𝑥)))
19 cz 8720 . . . . . 6 class
20 cn 8394 . . . . . 6 class
2119, 20cxp 4426 . . . . 5 class (ℤ × ℕ)
2218, 4, 21crio 5589 . . . 4 class (𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑦 = ((1st𝑥) / (2nd𝑥))))
2322, 8cfv 5002 . . 3 class (2nd ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑦 = ((1st𝑥) / (2nd𝑥)))))
242, 3, 23cmpt 3891 . 2 class (𝑦 ∈ ℚ ↦ (2nd ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑦 = ((1st𝑥) / (2nd𝑥))))))
251, 24wceq 1289 1 wff denom = (𝑦 ∈ ℚ ↦ (2nd ‘(𝑥 ∈ (ℤ × ℕ)(((1st𝑥) gcd (2nd𝑥)) = 1 ∧ 𝑦 = ((1st𝑥) / (2nd𝑥))))))
Colors of variables: wff set class
This definition is referenced by:  qdenval  11257  fden  11262
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