qredeu - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > qredeu		GIF version

Theorem qredeu 12662

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 9491	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
2		gcddvds 12527	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ∧ (𝑧 gcd 𝑛) ∥ 𝑛))
3	2	simpld 112	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∥ 𝑧)
4	1, 3	sylan2 286	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∥ 𝑧)
5		gcdcl 12530	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∈ ℕ₀)
6	1, 5	sylan2 286	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℕ₀)
7	6	nn0zd 9593	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℤ)
8		simpl 109	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℤ)
9	1	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
10		nnne0 9164	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
11	10	neneqd 2421	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
12	11	intnand 936	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → ¬ (𝑧 = 0 ∧ 𝑛 = 0))
13	12	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ¬ (𝑧 = 0 ∧ 𝑛 = 0))
14		gcdn0cl 12526	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ (𝑧 = 0 ∧ 𝑛 = 0)) → (𝑧 gcd 𝑛) ∈ ℕ)
15	8, 9, 13, 14	syl21anc 1270	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℕ)
16	15	nnne0d 9181	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ≠ 0)
17		dvdsval2 12344	. . . . . . . . . 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 1271	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ↔ (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ))
19	4, 18	mpbid 147	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ)
20	19	3adant3 1041	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ)
21	2	simprd 114	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∥ 𝑛)
22	1, 21	sylan2 286	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∥ 𝑛)
23		dvdsval2 12344	. . . . . . . . . . . 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 1271	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) ∥ 𝑛 ↔ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ))
25	22, 24	mpbid 147	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ)
26		nnre 9143	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
27	26	adantl 277	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
28	6	nn0red 9449	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℝ)
29		nngt0 9161	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 < 𝑛)
30	29	adantl 277	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < 𝑛)
31	15	nngt0d 9180	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < (𝑧 gcd 𝑛))
32	27, 28, 30, 31	divgt0d 9108	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < (𝑛 / (𝑧 gcd 𝑛)))
33	25, 32	jca 306	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
34	33	3adant3 1041	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
35		elnnz 9482	. . . . . . . 8 ⊢ ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ ↔ ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
36	34, 35	sylibr 134	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ)
37		opelxpi 4755	. . . . . . 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 411	. . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ ∈ (ℤ × ℕ))
39		fveq2 5635	. . . . . . . . . 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 1021	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → 𝑧 ∈ ℤ)
41	15	3adant3 1041	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑧 gcd 𝑛) ∈ ℕ)
42		znq 9851	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ (𝑧 gcd 𝑛) ∈ ℕ) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℚ)
43	40, 41, 42	syl2anc 411	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℚ)
44	9	3adant3 1041	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → 𝑛 ∈ ℤ)
45		znq 9851	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℤ ∧ (𝑧 gcd 𝑛) ∈ ℕ) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℚ)
46	44, 41, 45	syl2anc 411	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℚ)
47		op1stg 6308	. . . . . . . . . . 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 411	. . . . . . . . . 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 2284	. . . . . . . . 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 5635	. . . . . . . . . 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 6309	. . . . . . . . . . 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 411	. . . . . . . . . 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 2284	. . . . . . . . 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 6031	. . . . . . . 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 2238	. . . . . . 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 6031	. . . . . . . 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 2241	. . . . . . 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 473	. . . . . 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 12532	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) ∈ ℕ₀)
60	59	nn0cnd 9450	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) ∈ ℂ)
61		1cnd 8188	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
62	6	nn0cnd 9450	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℂ)
63	15	nnap0d 9182	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) # 0)
64	62	mulridd 8189	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · 1) = (𝑧 gcd 𝑛))
65		zcn 9477	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℂ)
66	65	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℂ)
67	66, 62, 63	divcanap2d 8965	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · (𝑧 / (𝑧 gcd 𝑛))) = 𝑧)
68		nncn 9144	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
69	68	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
70	69, 62, 63	divcanap2d 8965	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · (𝑛 / (𝑧 gcd 𝑛))) = 𝑛)
71	67, 70	oveq12d 6031	. . . . . . . . . 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 12580	. . . . . . . . . . 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 1271	. . . . . . . . . 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 2269	. . . . . . . . 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 8836	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1)
76	75	3adant3 1041	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1)
77		nnap0 9165	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → 𝑛 # 0)
78	77	adantl 277	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 # 0)
79	66, 69, 62, 78, 63	divcanap7d 8992	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))) = (𝑧 / 𝑛))
80	79	eqeq2d 2241	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))) ↔ 𝐴 = (𝑧 / 𝑛)))
81	80	biimp3ar 1380	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))))
82	76, 81	jca 306	. . . . . 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 2914	. . . . 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 6327	. . . . . . 7 ⊢ (𝑥 ∈ (ℤ × ℕ) ↔ (𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)))
85		elxp6 6327	. . . . . . 7 ⊢ (𝑦 ∈ (ℤ × ℕ) ↔ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ)))
86		simprl 529	. . . . . . . . . . . 12 ⊢ ((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (1^st ‘𝑥) ∈ ℤ)
87	86	ad2antrr 488	. . . . . . . . . . 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 531	. . . . . . . . . . . 12 ⊢ ((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (2^nd ‘𝑥) ∈ ℕ)
89	88	ad2antrr 488	. . . . . . . . . . 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 537	. . . . . . . . . . 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 529	. . . . . . . . . . . 12 ⊢ ((𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ)) → (1^st ‘𝑦) ∈ ℤ)
92	91	ad2antlr 489	. . . . . . . . . . 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 531	. . . . . . . . . . . 12 ⊢ ((𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ)) → (2^nd ‘𝑦) ∈ ℕ)
94	93	ad2antlr 489	. . . . . . . . . . 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 539	. . . . . . . . . . 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 538	. . . . . . . . . . . 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 540	. . . . . . . . . . . 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 2264	. . . . . . . . . . 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 12661	. . . . . . . . . . 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 1296	. . . . . . . . . 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 2803	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
102		1stexg 6325	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ V → (1^st ‘𝑥) ∈ V)
103	101, 102	ax-mp 5	. . . . . . . . . . 11 ⊢ (1^st ‘𝑥) ∈ V
104		2ndexg 6326	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ V → (2^nd ‘𝑥) ∈ V)
105	101, 104	ax-mp 5	. . . . . . . . . . 11 ⊢ (2^nd ‘𝑥) ∈ V
106	103, 105	opth 4327	. . . . . . . . . 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 134	. . . . . . . . 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 533	. . . . . . . . 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 535	. . . . . . . . 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 2272	. . . . . . . 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 115	. . . . . . 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 291	. . . . . 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 2584	. . . . 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 313	. . . 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 1229	. . 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 2654	. 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 9849	. 2 ⊢ (𝐴 ∈ ℚ ↔ ∃𝑧 ∈ ℤ ∃𝑛 ∈ ℕ 𝐴 = (𝑧 / 𝑛))
118		fveq2 5635	. . . . . 6 ⊢ (𝑥 = 𝑦 → (1^st ‘𝑥) = (1^st ‘𝑦))
119		fveq2 5635	. . . . . 6 ⊢ (𝑥 = 𝑦 → (2^nd ‘𝑥) = (2^nd ‘𝑦))
120	118, 119	oveq12d 6031	. . . . 5 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = ((1^st ‘𝑦) gcd (2^nd ‘𝑦)))
121	120	eqeq1d 2238	. . . 4 ⊢ (𝑥 = 𝑦 → (((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ↔ ((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1))
122	118, 119	oveq12d 6031	. . . . 5 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝑥) / (2^nd ‘𝑥)) = ((1^st ‘𝑦) / (2^nd ‘𝑦)))
123	122	eqeq2d 2241	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥)) ↔ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦))))
124	121, 123	anbi12d 473	. . 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 2998	. 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 201	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 104 ↔ wb 105 ∧ w3a 1002 = wceq 1395 ∈ wcel 2200 ≠ wne 2400 ∀wral 2508 ∃wrex 2509 ∃!wreu 2510 Vcvv 2800 ⟨cop 3670 class class class wbr 4086 × cxp 4721 ‘cfv 5324 (class class class)co 6013 1^st c1st 6296 2^nd c2nd 6297 ℂcc 8023 ℝcr 8024 0cc0 8025 1c1 8026 · cmul 8030 < clt 8207 # cap 8754 / cdiv 8845 ℕcn 9136 ℕ₀cn0 9395 ℤcz 9472 ℚcq 9846 ∥ cdvds 12341 gcd cgcd 12517

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4202 ax-sep 4205 ax-nul 4213 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-iinf 4684 ax-cnex 8116 ax-resscn 8117 ax-1cn 8118 ax-1re 8119 ax-icn 8120 ax-addcl 8121 ax-addrcl 8122 ax-mulcl 8123 ax-mulrcl 8124 ax-addcom 8125 ax-mulcom 8126 ax-addass 8127 ax-mulass 8128 ax-distr 8129 ax-i2m1 8130 ax-0lt1 8131 ax-1rid 8132 ax-0id 8133 ax-rnegex 8134 ax-precex 8135 ax-cnre 8136 ax-pre-ltirr 8137 ax-pre-ltwlin 8138 ax-pre-lttrn 8139 ax-pre-apti 8140 ax-pre-ltadd 8141 ax-pre-mulgt0 8142 ax-pre-mulext 8143 ax-arch 8144 ax-caucvg 8145

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2802 df-sbc 3030 df-csb 3126 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-nul 3493 df-if 3604 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-int 3927 df-iun 3970 df-br 4087 df-opab 4149 df-mpt 4150 df-tr 4186 df-id 4388 df-po 4391 df-iso 4392 df-iord 4461 df-on 4463 df-ilim 4464 df-suc 4466 df-iom 4687 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-rn 4734 df-res 4735 df-ima 4736 df-iota 5284 df-fun 5326 df-fn 5327 df-f 5328 df-f1 5329 df-fo 5330 df-f1o 5331 df-fv 5332 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-1st 6298 df-2nd 6299 df-recs 6466 df-frec 6552 df-sup 7177 df-pnf 8209 df-mnf 8210 df-xr 8211 df-ltxr 8212 df-le 8213 df-sub 8345 df-neg 8346 df-reap 8748 df-ap 8755 df-div 8846 df-inn 9137 df-2 9195 df-3 9196 df-4 9197 df-n0 9396 df-z 9473 df-uz 9749 df-q 9847 df-rp 9882 df-fz 10237 df-fzo 10371 df-fl 10523 df-mod 10578 df-seqfrec 10703 df-exp 10794 df-cj 11396 df-re 11397 df-im 11398 df-rsqrt 11552 df-abs 11553 df-dvds 12342 df-gcd 12518

This theorem is referenced by: qnumdencl 12752 qnumdenbi 12757

W3C validator