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Theorem qredeu 16838

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 12717	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
2		gcddvds 16678	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ∧ (𝑧 gcd 𝑛) ∥ 𝑛))
3	2	simpld 500	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∥ 𝑧)
4	1, 3	sylan2 605	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∥ 𝑧)
5		gcdcl 16681	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∈ ℕ₀)
6	1, 5	sylan2 605	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℕ₀)
7	6	nn0zd 12721	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℤ)
8		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℤ)
9	1	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
10		nnne0 12375	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
11	10	neneqd 2960	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
12	11	intnand 494	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → ¬ (𝑧 = 0 ∧ 𝑛 = 0))
13	12	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ¬ (𝑧 = 0 ∧ 𝑛 = 0))
14		gcdn0cl 16677	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ (𝑧 = 0 ∧ 𝑛 = 0)) → (𝑧 gcd 𝑛) ∈ ℕ)
15	8, 9, 13, 14	syl21anc 851	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℕ)
16		nnne0 12375	. . . . . . . . . . 11 ⊢ ((𝑧 gcd 𝑛) ∈ ℕ → (𝑧 gcd 𝑛) ≠ 0)
17	15, 16	syl 18	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ≠ 0)
18		dvdsval2 16430	. . . . . . . . . 10 ⊢ (((𝑧 gcd 𝑛) ∈ ℤ ∧ (𝑧 gcd 𝑛) ≠ 0 ∧ 𝑧 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ↔ (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ))
19	7, 17, 8, 18	syl3anc 1398	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ↔ (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ))
20	4, 19	mpbid 235	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ)
21	20	3adant3 1150	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ)
22	2	simprd 501	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∥ 𝑛)
23	1, 22	sylan2 605	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∥ 𝑛)
24		dvdsval2 16430	. . . . . . . . . . . 12 ⊢ (((𝑧 gcd 𝑛) ∈ ℤ ∧ (𝑧 gcd 𝑛) ≠ 0 ∧ 𝑛 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑛 ↔ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ))
25	7, 17, 9, 24	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) ∥ 𝑛 ↔ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ))
26	23, 25	mpbid 235	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ)
27		nnre 12345	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
28	27	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
29	6	nn0red 12671	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℝ)
30		nngt0 12372	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 < 𝑛)
31	30	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < 𝑛)
32		nngt0 12372	. . . . . . . . . . . 12 ⊢ ((𝑧 gcd 𝑛) ∈ ℕ → 0 < (𝑧 gcd 𝑛))
33	15, 32	syl 18	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < (𝑧 gcd 𝑛))
34	28, 29, 31, 33	divgt0d 12255	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < (𝑛 / (𝑧 gcd 𝑛)))
35	26, 34	jca 521	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
36	35	3adant3 1150	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
37		elnnz 12706	. . . . . . . 8 ⊢ ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ ↔ ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
38	36, 37	sylibr 237	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ)
39	21, 38	opelxpd 5689	. . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ ∈ (ℤ × ℕ))
40	20, 26	gcdcld 16683	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) ∈ ℕ₀)
41	40	nn0cnd 12672	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) ∈ ℂ)
42		1cnd 11305	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
43	6	nn0cnd 12672	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℂ)
44	43	mulridd 11329	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · 1) = (𝑧 gcd 𝑛))
45		zcn 12701	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℂ)
46	45	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℂ)
47	46, 43, 17	divcan2d 12098	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · (𝑧 / (𝑧 gcd 𝑛))) = 𝑧)
48		nncn 12346	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
49	48	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
50	49, 43, 17	divcan2d 12098	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · (𝑛 / (𝑧 gcd 𝑛))) = 𝑛)
51	47, 50	oveq12d 7438	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (((𝑧 gcd 𝑛) · (𝑧 / (𝑧 gcd 𝑛))) gcd ((𝑧 gcd 𝑛) · (𝑛 / (𝑧 gcd 𝑛)))) = (𝑧 gcd 𝑛))
52		mulgcd 16726	. . . . . . . . . 10 ⊢ (((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)))))
53	6, 20, 26, 52	syl3anc 1398	. . . . . . . . 9 ⊢ ((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)))))
54	44, 51, 53	3eqtr2rd 2802	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛)))) = ((𝑧 gcd 𝑛) · 1))
55	41, 42, 43, 17, 54	mulcanad 11954	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1)
56	55	3adant3 1150	. . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1)
57	10	adantl 487	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
58	46, 49, 43, 57, 17	divcan7d 12124	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))) = (𝑧 / 𝑛))
59	58	eqeq2d 2771	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))) ↔ 𝐴 = (𝑧 / 𝑛)))
60	59	biimp3ar 1499	. . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))))
61		ovex 7453	. . . . . . . . . . 11 ⊢ (𝑧 / (𝑧 gcd 𝑛)) ∈ V
62		ovex 7453	. . . . . . . . . . 11 ⊢ (𝑛 / (𝑧 gcd 𝑛)) ∈ V
63	61, 62	op1std 8011	. . . . . . . . . 10 ⊢ (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → (1^st ‘𝑥) = (𝑧 / (𝑧 gcd 𝑛)))
64	61, 62	op2ndd 8012	. . . . . . . . . 10 ⊢ (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → (2^nd ‘𝑥) = (𝑛 / (𝑧 gcd 𝑛)))
65	63, 64	oveq12d 7438	. . . . . . . . 9 ⊢ (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))))
66	65	eqeq1d 2762	. . . . . . . 8 ⊢ (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))
67	63, 64	oveq12d 7438	. . . . . . . . 9 ⊢ (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → ((1^st ‘𝑥) / (2^nd ‘𝑥)) = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))))
68	67	eqeq2d 2771	. . . . . . . 8 ⊢ (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)))))
69	66, 68	anbi12d 644	. . . . . . 7 ⊢ (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))))))
70	69	rspcev 3576	. . . . . 6 ⊢ ((⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ ∈ (ℤ × ℕ) ∧ (((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1 ∧ 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))))) → ∃𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))
71	39, 56, 60, 70	syl12anc 850	. . . . 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))))
72		elxp6 8035	. . . . . . 7 ⊢ (𝑥 ∈ (ℤ × ℕ) ↔ (𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)))
73		elxp6 8035	. . . . . . 7 ⊢ (𝑦 ∈ (ℤ × ℕ) ↔ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ)))
74		simprl 783	. . . . . . . . . . . 12 ⊢ ((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (1^st ‘𝑥) ∈ ℤ)
75	74	ad2antrr 739	. . . . . . . . . . 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)
76		simprr 785	. . . . . . . . . . . 12 ⊢ ((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (2^nd ‘𝑥) ∈ ℕ)
77	76	ad2antrr 739	. . . . . . . . . . 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)
78		simprll 791	. . . . . . . . . . 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)
79		simprl 783	. . . . . . . . . . . 12 ⊢ ((𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ)) → (1^st ‘𝑦) ∈ ℤ)
80	79	ad2antlr 740	. . . . . . . . . . 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)
81		simprr 785	. . . . . . . . . . . 12 ⊢ ((𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ)) → (2^nd ‘𝑦) ∈ ℕ)
82	81	ad2antlr 740	. . . . . . . . . . 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)
83		simprrl 793	. . . . . . . . . . 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)
84		simprlr 792	. . . . . . . . . . . 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)))
85		simprrr 794	. . . . . . . . . . . 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)))
86	84, 85	eqtr3d 2797	. . . . . . . . . . 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)))
87		qredeq 16837	. . . . . . . . . . 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)))
88	75, 77, 78, 80, 82, 83, 86, 87	syl331anc 1422	. . . . . . . . . 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)))
89		fvex 6898	. . . . . . . . . . 11 ⊢ (1^st ‘𝑥) ∈ V
90		fvex 6898	. . . . . . . . . . 11 ⊢ (2^nd ‘𝑥) ∈ V
91	89, 90	opth 5444	. . . . . . . . . 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)))
92	88, 91	sylibr 237	. . . . . . . . 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)⟩)
93		simplll 787	. . . . . . . . 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)⟩)
94		simplrl 789	. . . . . . . . 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)⟩)
95	92, 93, 94	3eqtr4d 2805	. . . . . . . 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)
96	95	ex 418	. . . . . . 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))
97	72, 73, 96	syl2anb 610	. . . . . 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))
98	97	rgen2 3202	. . . . 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)
99	71, 98	jctir 530	. . . 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)))
100	99	3expia 1139	. . 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))))
101	100	rexlimivv 3204	. 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)))
102		elq 13080	. 2 ⊢ (𝐴 ∈ ℚ ↔ ∃𝑧 ∈ ℤ ∃𝑛 ∈ ℕ 𝐴 = (𝑧 / 𝑛))
103		fveq2 6885	. . . . . 6 ⊢ (𝑥 = 𝑦 → (1^st ‘𝑥) = (1^st ‘𝑦))
104		fveq2 6885	. . . . . 6 ⊢ (𝑥 = 𝑦 → (2^nd ‘𝑥) = (2^nd ‘𝑦))
105	103, 104	oveq12d 7438	. . . . 5 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = ((1^st ‘𝑦) gcd (2^nd ‘𝑦)))
106	105	eqeq1d 2762	. . . 4 ⊢ (𝑥 = 𝑦 → (((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ↔ ((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1))
107	103, 104	oveq12d 7438	. . . . 5 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝑥) / (2^nd ‘𝑥)) = ((1^st ‘𝑦) / (2^nd ‘𝑦)))
108	107	eqeq2d 2771	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥)) ↔ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦))))
109	106, 108	anbi12d 644	. . 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)))))
110	109	reu4 3688	. 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)))
111	101, 102, 110	3imtr4i 295	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 setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 ∃!wreu 3363 ⟨cop 4834 class class class wbr 5102 × cxp 5648 ‘cfv 6538 (class class class)co 7420 1^st c1st 7999 2^nd c2nd 8000 ℂcc 11201 ℝcr 11202 0cc0 11203 1c1 11204 · cmul 11208 < clt 11346 / cdiv 11976 ℕcn 12338 ℕ₀cn0 12609 ℤcz 12696 ℚcq 13078 ∥ cdvds 16427 gcd cgcd 16669

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280 ax-pre-sup 11281

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-er 8720 df-en 8977 df-dom 8978 df-sdom 8979 df-sup 9437 df-inf 9438 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-div 11977 df-nn 12339 df-2 12408 df-3 12409 df-n0 12610 df-z 12697 df-uz 12969 df-q 13079 df-rp 13124 df-fl 13935 df-mod 14013 df-seq 14148 df-exp 14208 df-cj 15271 df-re 15272 df-im 15273 df-sqrt 15407 df-abs 15408 df-dvds 16428 df-gcd 16670

This theorem is used by: qnumdencl 16920 qnumdenbi 16925

W3C validator