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Theorem qredeu 12051

Description: Every rational number has a unique reduced form. (Contributed by Jeff Hankins, 29-Sep-2013.)

**Assertion**
Ref	Expression
qredeu	⊢ (𝐴 ∈ ℚ → ∃!𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))

Distinct variable group: 𝑥,𝐴

Proof of Theorem qredeu Dummy variables 𝑛 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnz 9231	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
2		gcddvds 11918	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ∧ (𝑧 gcd 𝑛) ∥ 𝑛))
3	2	simpld 111	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∥ 𝑧)
4	1, 3	sylan2 284	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∥ 𝑧)
5		gcdcl 11921	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∈ ℕ₀)
6	1, 5	sylan2 284	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℕ₀)
7	6	nn0zd 9332	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℤ)
8		simpl 108	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℤ)
9	1	adantl 275	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
10		nnne0 8906	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
11	10	neneqd 2361	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
12	11	intnand 926	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → ¬ (𝑧 = 0 ∧ 𝑛 = 0))
13	12	adantl 275	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ¬ (𝑧 = 0 ∧ 𝑛 = 0))
14		gcdn0cl 11917	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ (𝑧 = 0 ∧ 𝑛 = 0)) → (𝑧 gcd 𝑛) ∈ ℕ)
15	8, 9, 13, 14	syl21anc 1232	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℕ)
16	15	nnne0d 8923	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ≠ 0)
17		dvdsval2 11752	. . . . . . . . . 10 ⊢ (((𝑧 gcd 𝑛) ∈ ℤ ∧ (𝑧 gcd 𝑛) ≠ 0 ∧ 𝑧 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ↔ (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ))
18	7, 16, 8, 17	syl3anc 1233	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ↔ (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ))
19	4, 18	mpbid 146	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ)
20	19	3adant3 1012	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ)
21	2	simprd 113	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∥ 𝑛)
22	1, 21	sylan2 284	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∥ 𝑛)
23		dvdsval2 11752	. . . . . . . . . . . 12 ⊢ (((𝑧 gcd 𝑛) ∈ ℤ ∧ (𝑧 gcd 𝑛) ≠ 0 ∧ 𝑛 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑛 ↔ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ))
24	7, 16, 9, 23	syl3anc 1233	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) ∥ 𝑛 ↔ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ))
25	22, 24	mpbid 146	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ)
26		nnre 8885	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
27	26	adantl 275	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
28	6	nn0red 9189	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℝ)
29		nngt0 8903	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 < 𝑛)
30	29	adantl 275	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < 𝑛)
31	15	nngt0d 8922	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < (𝑧 gcd 𝑛))
32	27, 28, 30, 31	divgt0d 8851	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < (𝑛 / (𝑧 gcd 𝑛)))
33	25, 32	jca 304	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
34	33	3adant3 1012	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
35		elnnz 9222	. . . . . . . 8 ⊢ ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ ↔ ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
36	34, 35	sylibr 133	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ)
37		opelxpi 4643	. . . . . . 7 ⊢ (((𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ) → ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ ∈ (ℤ × ℕ))
38	20, 36, 37	syl2anc 409	. . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ ∈ (ℤ × ℕ))
39		fveq2 5496	. . . . . . . . . 10 ⊢ (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → (1^st ‘𝑥) = (1^st ‘⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩))
40		simp1 992	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → 𝑧 ∈ ℤ)
41	15	3adant3 1012	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑧 gcd 𝑛) ∈ ℕ)
42		znq 9583	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ (𝑧 gcd 𝑛) ∈ ℕ) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℚ)
43	40, 41, 42	syl2anc 409	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℚ)
44	9	3adant3 1012	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → 𝑛 ∈ ℤ)
45		znq 9583	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℤ ∧ (𝑧 gcd 𝑛) ∈ ℕ) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℚ)
46	44, 41, 45	syl2anc 409	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℚ)
47		op1stg 6129	. . . . . . . . . . 11 ⊢ (((𝑧 / (𝑧 gcd 𝑛)) ∈ ℚ ∧ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℚ) → (1^st ‘⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) = (𝑧 / (𝑧 gcd 𝑛)))
48	43, 46, 47	syl2anc 409	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (1^st ‘⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) = (𝑧 / (𝑧 gcd 𝑛)))
49	39, 48	sylan9eqr 2225	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) ∧ 𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) → (1^st ‘𝑥) = (𝑧 / (𝑧 gcd 𝑛)))
50		fveq2 5496	. . . . . . . . . 10 ⊢ (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → (2^nd ‘𝑥) = (2^nd ‘⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩))
51		op2ndg 6130	. . . . . . . . . . 11 ⊢ (((𝑧 / (𝑧 gcd 𝑛)) ∈ ℚ ∧ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℚ) → (2^nd ‘⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) = (𝑛 / (𝑧 gcd 𝑛)))
52	43, 46, 51	syl2anc 409	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (2^nd ‘⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) = (𝑛 / (𝑧 gcd 𝑛)))
53	50, 52	sylan9eqr 2225	. . . . . . . . 9 ⊢ (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) ∧ 𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) → (2^nd ‘𝑥) = (𝑛 / (𝑧 gcd 𝑛)))
54	49, 53	oveq12d 5871	. . . . . . . 8 ⊢ (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) ∧ 𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) → ((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))))
55	54	eqeq1d 2179	. . . . . . 7 ⊢ (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) ∧ 𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) → (((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ↔ ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1))
56	49, 53	oveq12d 5871	. . . . . . . 8 ⊢ (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) ∧ 𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) → ((1^st ‘𝑥) / (2^nd ‘𝑥)) = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))))
57	56	eqeq2d 2182	. . . . . . 7 ⊢ (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) ∧ 𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) → (𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥)) ↔ 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛)))))
58	55, 57	anbi12d 470	. . . . . 6 ⊢ (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) ∧ 𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩) → ((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ↔ (((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1 ∧ 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))))))
59	19, 25	gcdcld 11923	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) ∈ ℕ₀)
60	59	nn0cnd 9190	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) ∈ ℂ)
61		1cnd 7936	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
62	6	nn0cnd 9190	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℂ)
63	15	nnap0d 8924	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) # 0)
64	62	mulid1d 7937	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · 1) = (𝑧 gcd 𝑛))
65		zcn 9217	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℂ)
66	65	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℂ)
67	66, 62, 63	divcanap2d 8709	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · (𝑧 / (𝑧 gcd 𝑛))) = 𝑧)
68		nncn 8886	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
69	68	adantl 275	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
70	69, 62, 63	divcanap2d 8709	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · (𝑛 / (𝑧 gcd 𝑛))) = 𝑛)
71	67, 70	oveq12d 5871	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (((𝑧 gcd 𝑛) · (𝑧 / (𝑧 gcd 𝑛))) gcd ((𝑧 gcd 𝑛) · (𝑛 / (𝑧 gcd 𝑛)))) = (𝑧 gcd 𝑛))
72		mulgcd 11971	. . . . . . . . . . 11 ⊢ (((𝑧 gcd 𝑛) ∈ ℕ₀ ∧ (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ) → (((𝑧 gcd 𝑛) · (𝑧 / (𝑧 gcd 𝑛))) gcd ((𝑧 gcd 𝑛) · (𝑛 / (𝑧 gcd 𝑛)))) = ((𝑧 gcd 𝑛) · ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛)))))
73	6, 19, 25, 72	syl3anc 1233	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (((𝑧 gcd 𝑛) · (𝑧 / (𝑧 gcd 𝑛))) gcd ((𝑧 gcd 𝑛) · (𝑛 / (𝑧 gcd 𝑛)))) = ((𝑧 gcd 𝑛) · ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛)))))
74	64, 71, 73	3eqtr2rd 2210	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛)))) = ((𝑧 gcd 𝑛) · 1))
75	60, 61, 62, 63, 74	mulcanapad 8581	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1)
76	75	3adant3 1012	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1)
77		nnap0 8907	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 𝑛 # 0)
78	77	adantl 275	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 # 0)
79	66, 69, 62, 78, 63	divcanap7d 8736	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))) = (𝑧 / 𝑛))
80	79	eqeq2d 2182	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))) ↔ 𝐴 = (𝑧 / 𝑛)))
81	80	biimp3ar 1341	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))))
82	76, 81	jca 304	. . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1 ∧ 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛)))))
83	38, 58, 82	rspcedvd 2840	. . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ∃𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))
84		elxp6 6148	. . . . . . 7 ⊢ (𝑥 ∈ (ℤ × ℕ) ↔ (𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)))
85		elxp6 6148	. . . . . . 7 ⊢ (𝑦 ∈ (ℤ × ℕ) ↔ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ)))
86		simprl 526	. . . . . . . . . . . 12 ⊢ ((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (1^st ‘𝑥) ∈ ℤ)
87	86	ad2antrr 485	. . . . . . . . . . 11 ⊢ ((((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ))) ∧ ((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ (((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1 ∧ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦))))) → (1^st ‘𝑥) ∈ ℤ)
88		simprr 527	. . . . . . . . . . . 12 ⊢ ((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (2^nd ‘𝑥) ∈ ℕ)
89	88	ad2antrr 485	. . . . . . . . . . 11 ⊢ ((((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ))) ∧ ((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ (((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1 ∧ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦))))) → (2^nd ‘𝑥) ∈ ℕ)
90		simprll 532	. . . . . . . . . . 11 ⊢ ((((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ))) ∧ ((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ (((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1 ∧ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦))))) → ((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1)
91		simprl 526	. . . . . . . . . . . 12 ⊢ ((𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ)) → (1^st ‘𝑦) ∈ ℤ)
92	91	ad2antlr 486	. . . . . . . . . . 11 ⊢ ((((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ))) ∧ ((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ (((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1 ∧ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦))))) → (1^st ‘𝑦) ∈ ℤ)
93		simprr 527	. . . . . . . . . . . 12 ⊢ ((𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ)) → (2^nd ‘𝑦) ∈ ℕ)
94	93	ad2antlr 486	. . . . . . . . . . 11 ⊢ ((((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ))) ∧ ((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ (((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1 ∧ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦))))) → (2^nd ‘𝑦) ∈ ℕ)
95		simprrl 534	. . . . . . . . . . 11 ⊢ ((((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ))) ∧ ((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ (((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1 ∧ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦))))) → ((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1)
96		simprlr 533	. . . . . . . . . . . 12 ⊢ ((((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ))) ∧ ((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ (((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1 ∧ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦))))) → 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥)))
97		simprrr 535	. . . . . . . . . . . 12 ⊢ ((((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ))) ∧ ((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ (((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1 ∧ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦))))) → 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦)))
98	96, 97	eqtr3d 2205	. . . . . . . . . . 11 ⊢ ((((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ))) ∧ ((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ (((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1 ∧ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦))))) → ((1^st ‘𝑥) / (2^nd ‘𝑥)) = ((1^st ‘𝑦) / (2^nd ‘𝑦)))
99		qredeq 12050	. . . . . . . . . . 11 ⊢ ((((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ ∧ ((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1) ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ ∧ ((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1) ∧ ((1^st ‘𝑥) / (2^nd ‘𝑥)) = ((1^st ‘𝑦) / (2^nd ‘𝑦))) → ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) = (2^nd ‘𝑦)))
100	87, 89, 90, 92, 94, 95, 98, 99	syl331anc 1258	. . . . . . . . . 10 ⊢ ((((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ))) ∧ ((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ (((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1 ∧ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦))))) → ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) = (2^nd ‘𝑦)))
101		vex 2733	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
102		1stexg 6146	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ V → (1^st ‘𝑥) ∈ V)
103	101, 102	ax-mp 5	. . . . . . . . . . 11 ⊢ (1^st ‘𝑥) ∈ V
104		2ndexg 6147	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ V → (2^nd ‘𝑥) ∈ V)
105	101, 104	ax-mp 5	. . . . . . . . . . 11 ⊢ (2^nd ‘𝑥) ∈ V
106	103, 105	opth 4222	. . . . . . . . . 10 ⊢ (⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ↔ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) = (2^nd ‘𝑦)))
107	100, 106	sylibr 133	. . . . . . . . 9 ⊢ ((((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ))) ∧ ((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ (((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1 ∧ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦))))) → ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
108		simplll 528	. . . . . . . . 9 ⊢ ((((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ))) ∧ ((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ (((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1 ∧ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦))))) → 𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩)
109		simplrl 530	. . . . . . . . 9 ⊢ ((((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ))) ∧ ((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ (((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1 ∧ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦))))) → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
110	107, 108, 109	3eqtr4d 2213	. . . . . . . 8 ⊢ ((((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ))) ∧ ((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ (((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1 ∧ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦))))) → 𝑥 = 𝑦)
111	110	ex 114	. . . . . . 7 ⊢ (((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) ∧ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ))) → (((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ (((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1 ∧ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦)))) → 𝑥 = 𝑦))
112	84, 85, 111	syl2anb 289	. . . . . 6 ⊢ ((𝑥 ∈ (ℤ × ℕ) ∧ 𝑦 ∈ (ℤ × ℕ)) → (((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ (((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1 ∧ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦)))) → 𝑥 = 𝑦))
113	112	rgen2a 2524	. . . . 5 ⊢ ∀𝑥 ∈ (ℤ × ℕ)∀𝑦 ∈ (ℤ × ℕ)(((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ (((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1 ∧ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦)))) → 𝑥 = 𝑦)
114	83, 113	jctir 311	. . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (∃𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ ∀𝑥 ∈ (ℤ × ℕ)∀𝑦 ∈ (ℤ × ℕ)(((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ (((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1 ∧ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦)))) → 𝑥 = 𝑦)))
115	114	3expia 1200	. . 3 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝐴 = (𝑧 / 𝑛) → (∃𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ ∀𝑥 ∈ (ℤ × ℕ)∀𝑦 ∈ (ℤ × ℕ)(((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ (((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1 ∧ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦)))) → 𝑥 = 𝑦))))
116	115	rexlimivv 2593	. 2 ⊢ (∃𝑧 ∈ ℤ ∃𝑛 ∈ ℕ 𝐴 = (𝑧 / 𝑛) → (∃𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ ∀𝑥 ∈ (ℤ × ℕ)∀𝑦 ∈ (ℤ × ℕ)(((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ (((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1 ∧ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦)))) → 𝑥 = 𝑦)))
117		elq 9581	. 2 ⊢ (𝐴 ∈ ℚ ↔ ∃𝑧 ∈ ℤ ∃𝑛 ∈ ℕ 𝐴 = (𝑧 / 𝑛))
118		fveq2 5496	. . . . . 6 ⊢ (𝑥 = 𝑦 → (1^st ‘𝑥) = (1^st ‘𝑦))
119		fveq2 5496	. . . . . 6 ⊢ (𝑥 = 𝑦 → (2^nd ‘𝑥) = (2^nd ‘𝑦))
120	118, 119	oveq12d 5871	. . . . 5 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = ((1^st ‘𝑦) gcd (2^nd ‘𝑦)))
121	120	eqeq1d 2179	. . . 4 ⊢ (𝑥 = 𝑦 → (((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ↔ ((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1))
122	118, 119	oveq12d 5871	. . . . 5 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝑥) / (2^nd ‘𝑥)) = ((1^st ‘𝑦) / (2^nd ‘𝑦)))
123	122	eqeq2d 2182	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥)) ↔ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦))))
124	121, 123	anbi12d 470	. . 3 ⊢ (𝑥 = 𝑦 → ((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ↔ (((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1 ∧ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦)))))
125	124	reu4 2924	. 2 ⊢ (∃!𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ↔ (∃𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ ∀𝑥 ∈ (ℤ × ℕ)∀𝑦 ∈ (ℤ × ℕ)(((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ∧ (((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1 ∧ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦)))) → 𝑥 = 𝑦)))
126	116, 117, 125	3imtr4i 200	1 ⊢ (𝐴 ∈ ℚ → ∃!𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 ∀wral 2448 ∃wrex 2449 ∃!wreu 2450 Vcvv 2730 ⟨cop 3586 class class class wbr 3989 × cxp 4609 ‘cfv 5198 (class class class)co 5853 1^st c1st 6117 2^nd c2nd 6118 ℂcc 7772 ℝcr 7773 0cc0 7774 1c1 7775 · cmul 7779 < clt 7954 # cap 8500 / cdiv 8589 ℕcn 8878 ℕ₀cn0 9135 ℤcz 9212 ℚcq 9578 ∥ cdvds 11749 gcd cgcd 11897

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-sup 6961 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fzo 10099 df-fl 10226 df-mod 10279 df-seqfrec 10402 df-exp 10476 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-dvds 11750 df-gcd 11898

This theorem is referenced by: qnumdencl 12141 qnumdenbi 12146

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