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Theorem qredeu 16591

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 12575	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ)
2		gcddvds 16440	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ∧ (𝑧 gcd 𝑛) ∥ 𝑛))
3	2	simpld 495	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∥ 𝑧)
4	1, 3	sylan2 593	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∥ 𝑧)
5		gcdcl 16443	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∈ ℕ₀)
6	1, 5	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℕ₀)
7	6	nn0zd 12580	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℤ)
8		simpl 483	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℤ)
9	1	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
10		nnne0 12242	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → 𝑛 ≠ 0)
11	10	neneqd 2945	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → ¬ 𝑛 = 0)
12	11	intnand 489	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → ¬ (𝑧 = 0 ∧ 𝑛 = 0))
13	12	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ¬ (𝑧 = 0 ∧ 𝑛 = 0))
14		gcdn0cl 16439	. . . . . . . . . . . 12 ⊢ (((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ (𝑧 = 0 ∧ 𝑛 = 0)) → (𝑧 gcd 𝑛) ∈ ℕ)
15	8, 9, 13, 14	syl21anc 836	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℕ)
16		nnne0 12242	. . . . . . . . . . 11 ⊢ ((𝑧 gcd 𝑛) ∈ ℕ → (𝑧 gcd 𝑛) ≠ 0)
17	15, 16	syl 17	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ≠ 0)
18		dvdsval2 16196	. . . . . . . . . 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 1371	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) ∥ 𝑧 ↔ (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ))
20	4, 19	mpbid 231	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ)
21	20	3adant3 1132	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑧 / (𝑧 gcd 𝑛)) ∈ ℤ)
22	2	simprd 496	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 gcd 𝑛) ∥ 𝑛)
23	1, 22	sylan2 593	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∥ 𝑛)
24		dvdsval2 16196	. . . . . . . . . . . 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 1371	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) ∥ 𝑛 ↔ (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ))
26	23, 25	mpbid 231	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ)
27		nnre 12215	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
28	27	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ)
29	6	nn0red 12529	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℝ)
30		nngt0 12239	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 < 𝑛)
31	30	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < 𝑛)
32		nngt0 12239	. . . . . . . . . . . 12 ⊢ ((𝑧 gcd 𝑛) ∈ ℕ → 0 < (𝑧 gcd 𝑛))
33	15, 32	syl 17	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < (𝑧 gcd 𝑛))
34	28, 29, 31, 33	divgt0d 12145	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 0 < (𝑛 / (𝑧 gcd 𝑛)))
35	26, 34	jca 512	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
36	35	3adant3 1132	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
37		elnnz 12564	. . . . . . . 8 ⊢ ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ ↔ ((𝑛 / (𝑧 gcd 𝑛)) ∈ ℤ ∧ 0 < (𝑛 / (𝑧 gcd 𝑛))))
38	36, 37	sylibr 233	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → (𝑛 / (𝑧 gcd 𝑛)) ∈ ℕ)
39	21, 38	opelxpd 5713	. . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ ∈ (ℤ × ℕ))
40	20, 26	gcdcld 16445	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) ∈ ℕ₀)
41	40	nn0cnd 12530	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) ∈ ℂ)
42		1cnd 11205	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 1 ∈ ℂ)
43	6	nn0cnd 12530	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑧 gcd 𝑛) ∈ ℂ)
44	43	mulridd 11227	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · 1) = (𝑧 gcd 𝑛))
45		zcn 12559	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℂ)
46	45	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑧 ∈ ℂ)
47	46, 43, 17	divcan2d 11988	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · (𝑧 / (𝑧 gcd 𝑛))) = 𝑧)
48		nncn 12216	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℂ)
49	48	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℂ)
50	49, 43, 17	divcan2d 11988	. . . . . . . . . 10 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 gcd 𝑛) · (𝑛 / (𝑧 gcd 𝑛))) = 𝑛)
51	47, 50	oveq12d 7423	. . . . . . . . 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 16486	. . . . . . . . . 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 1371	. . . . . . . . 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 2779	. . . . . . . 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 11845	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1)
56	55	3adant3 1132	. . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → ((𝑧 / (𝑧 gcd 𝑛)) gcd (𝑛 / (𝑧 gcd 𝑛))) = 1)
57	10	adantl 482	. . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → 𝑛 ≠ 0)
58	46, 49, 43, 57, 17	divcan7d 12014	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))) = (𝑧 / 𝑛))
59	58	eqeq2d 2743	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))) ↔ 𝐴 = (𝑧 / 𝑛)))
60	59	biimp3ar 1470	. . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℕ ∧ 𝐴 = (𝑧 / 𝑛)) → 𝐴 = ((𝑧 / (𝑧 gcd 𝑛)) / (𝑛 / (𝑧 gcd 𝑛))))
61		ovex 7438	. . . . . . . . . . 11 ⊢ (𝑧 / (𝑧 gcd 𝑛)) ∈ V
62		ovex 7438	. . . . . . . . . . 11 ⊢ (𝑛 / (𝑧 gcd 𝑛)) ∈ V
63	61, 62	op1std 7981	. . . . . . . . . 10 ⊢ (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → (1^st ‘𝑥) = (𝑧 / (𝑧 gcd 𝑛)))
64	61, 62	op2ndd 7982	. . . . . . . . . 10 ⊢ (𝑥 = ⟨(𝑧 / (𝑧 gcd 𝑛)), (𝑛 / (𝑧 gcd 𝑛))⟩ → (2^nd ‘𝑥) = (𝑛 / (𝑧 gcd 𝑛)))
65	63, 64	oveq12d 7423	. . . . . . . . 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 2734	. . . . . . . 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 7423	. . . . . . . . 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 2743	. . . . . . . 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 631	. . . . . . 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 3612	. . . . . 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 835	. . . . 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 8005	. . . . . . 7 ⊢ (𝑥 ∈ (ℤ × ℕ) ↔ (𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)))
73		elxp6 8005	. . . . . . 7 ⊢ (𝑦 ∈ (ℤ × ℕ) ↔ (𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ)))
74		simprl 769	. . . . . . . . . . . 12 ⊢ ((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (1^st ‘𝑥) ∈ ℤ)
75	74	ad2antrr 724	. . . . . . . . . . 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 771	. . . . . . . . . . . 12 ⊢ ((𝑥 = ⟨(1^st ‘𝑥), (2^nd ‘𝑥)⟩ ∧ ((1^st ‘𝑥) ∈ ℤ ∧ (2^nd ‘𝑥) ∈ ℕ)) → (2^nd ‘𝑥) ∈ ℕ)
77	76	ad2antrr 724	. . . . . . . . . . 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 777	. . . . . . . . . . 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 769	. . . . . . . . . . . 12 ⊢ ((𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ)) → (1^st ‘𝑦) ∈ ℤ)
80	79	ad2antlr 725	. . . . . . . . . . 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 771	. . . . . . . . . . . 12 ⊢ ((𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ ∧ ((1^st ‘𝑦) ∈ ℤ ∧ (2^nd ‘𝑦) ∈ ℕ)) → (2^nd ‘𝑦) ∈ ℕ)
82	81	ad2antlr 725	. . . . . . . . . . 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 779	. . . . . . . . . . 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 778	. . . . . . . . . . . 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 780	. . . . . . . . . . . 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 2774	. . . . . . . . . . 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 16590	. . . . . . . . . . 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 1395	. . . . . . . . . 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 6901	. . . . . . . . . . 11 ⊢ (1^st ‘𝑥) ∈ V
90		fvex 6901	. . . . . . . . . . 11 ⊢ (2^nd ‘𝑥) ∈ V
91	89, 90	opth 5475	. . . . . . . . . 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 233	. . . . . . . . 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 773	. . . . . . . . 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 775	. . . . . . . . 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 2782	. . . . . . . 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 413	. . . . . . 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 598	. . . . . 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 3197	. . . . 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 521	. . . 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 1121	. . 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 3199	. 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 12930	. 2 ⊢ (𝐴 ∈ ℚ ↔ ∃𝑧 ∈ ℤ ∃𝑛 ∈ ℕ 𝐴 = (𝑧 / 𝑛))
103		fveq2 6888	. . . . . 6 ⊢ (𝑥 = 𝑦 → (1^st ‘𝑥) = (1^st ‘𝑦))
104		fveq2 6888	. . . . . 6 ⊢ (𝑥 = 𝑦 → (2^nd ‘𝑥) = (2^nd ‘𝑦))
105	103, 104	oveq12d 7423	. . . . 5 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = ((1^st ‘𝑦) gcd (2^nd ‘𝑦)))
106	105	eqeq1d 2734	. . . 4 ⊢ (𝑥 = 𝑦 → (((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ↔ ((1^st ‘𝑦) gcd (2^nd ‘𝑦)) = 1))
107	103, 104	oveq12d 7423	. . . . 5 ⊢ (𝑥 = 𝑦 → ((1^st ‘𝑥) / (2^nd ‘𝑥)) = ((1^st ‘𝑦) / (2^nd ‘𝑦)))
108	107	eqeq2d 2743	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥)) ↔ 𝐴 = ((1^st ‘𝑦) / (2^nd ‘𝑦))))
109	106, 108	anbi12d 631	. . 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 3726	. 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 291	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

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ∃!wreu 3374 ⟨cop 4633 class class class wbr 5147 × cxp 5673 ‘cfv 6540 (class class class)co 7405 1^st c1st 7969 2^nd c2nd 7970 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 · cmul 11111 < clt 11244 / cdiv 11867 ℕcn 12208 ℕ₀cn0 12468 ℤcz 12554 ℚcq 12928 ∥ cdvds 16193 gcd cgcd 16431

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-dvds 16194 df-gcd 16432

This theorem is referenced by: qnumdencl 16671 qnumdenbi 16676

W3C validator