Definition df-denom 12116

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)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑦 = ((1st ‘𝑥) / (2nd ‘𝑥))))))

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-denom

Step	Hyp	Ref	Expression
1		cdenom 12114	. 2 class denom
2		vy	. . 3 setvar 𝑦
3		cq 9557	. . 3 class ℚ
4		vx	. . . . . . . . . 10 setvar 𝑥
5	4	cv 1342	. . . . . . . . 9 class 𝑥
6		c1st 6106	. . . . . . . . 9 class 1st
7	5, 6	cfv 5188	. . . . . . . 8 class (1st ‘𝑥)
8		c2nd 6107	. . . . . . . . 9 class 2nd
9	5, 8	cfv 5188	. . . . . . . 8 class (2nd ‘𝑥)
10		cgcd 11875	. . . . . . . 8 class gcd
11	7, 9, 10	co 5842	. . . . . . 7 class ((1st ‘𝑥) gcd (2nd ‘𝑥))
12		c1 7754	. . . . . . 7 class 1
13	11, 12	wceq 1343	. . . . . 6 wff ((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1
14	2	cv 1342	. . . . . . 7 class 𝑦
15		cdiv 8568	. . . . . . . 8 class /
16	7, 9, 15	co 5842	. . . . . . 7 class ((1st ‘𝑥) / (2nd ‘𝑥))
17	14, 16	wceq 1343	. . . . . 6 wff 𝑦 = ((1st ‘𝑥) / (2nd ‘𝑥))
18	13, 17	wa 103	. . . . 5 wff (((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑦 = ((1st ‘𝑥) / (2nd ‘𝑥)))
19		cz 9191	. . . . . 6 class ℤ
20		cn 8857	. . . . . 6 class ℕ
21	19, 20	cxp 4602	. . . . 5 class (ℤ × ℕ)
22	18, 4, 21	crio 5797	. . . 4 class (℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑦 = ((1st ‘𝑥) / (2nd ‘𝑥))))
23	22, 8	cfv 5188	. . 3 class (2nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑦 = ((1st ‘𝑥) / (2nd ‘𝑥)))))
24	2, 3, 23	cmpt 4043	. 2 class (𝑦 ∈ ℚ ↦ (2nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑦 = ((1st ‘𝑥) / (2nd ‘𝑥))))))
25	1, 24	wceq 1343	1 wff denom = (𝑦 ∈ ℚ ↦ (2nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑦 = ((1st ‘𝑥) / (2nd ‘𝑥))))))

This definition is referenced by:  qdenval  12118  fden  12123