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Theorem aaliou2b 25835

Description: Liouville's approximation theorem extended to complex 𝐴. (Contributed by Stefan O'Rear, 20-Nov-2014.)

**Assertion**
Ref	Expression
aaliou2b	⊢ (𝐴 ∈ 𝔸 → ∃𝑘 ∈ ℕ ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))

Distinct variable group: 𝐴,𝑘,𝑥,𝑝,𝑞

**Proof of Theorem aaliou2b**
Step	Hyp	Ref	Expression
1		elin 3962	. . 3 ⊢ (𝐴 ∈ (𝔸 ∩ ℝ) ↔ (𝐴 ∈ 𝔸 ∧ 𝐴 ∈ ℝ))
2		aaliou2 25834	. . 3 ⊢ (𝐴 ∈ (𝔸 ∩ ℝ) → ∃𝑘 ∈ ℕ ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))
3	1, 2	sylbir 234	. 2 ⊢ ((𝐴 ∈ 𝔸 ∧ 𝐴 ∈ ℝ) → ∃𝑘 ∈ ℕ ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))
4		1nn 12218	. . 3 ⊢ 1 ∈ ℕ
5		aacn 25811	. . . . . . . 8 ⊢ (𝐴 ∈ 𝔸 → 𝐴 ∈ ℂ)
6	5	adantr 482	. . . . . . 7 ⊢ ((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) → 𝐴 ∈ ℂ)
7	6	imcld 15137	. . . . . 6 ⊢ ((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) → (ℑ‘𝐴) ∈ ℝ)
8	7	recnd 11237	. . . . 5 ⊢ ((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) → (ℑ‘𝐴) ∈ ℂ)
9		reim0b 15061	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0))
10	5, 9	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ 𝔸 → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0))
11	10	necon3bbid 2979	. . . . . 6 ⊢ (𝐴 ∈ 𝔸 → (¬ 𝐴 ∈ ℝ ↔ (ℑ‘𝐴) ≠ 0))
12	11	biimpa 478	. . . . 5 ⊢ ((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) → (ℑ‘𝐴) ≠ 0)
13	8, 12	absrpcld 15390	. . . 4 ⊢ ((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) → (abs‘(ℑ‘𝐴)) ∈ ℝ⁺)
14	13	rphalfcld 13023	. . 3 ⊢ ((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) → ((abs‘(ℑ‘𝐴)) / 2) ∈ ℝ⁺)
15	14	adantr 482	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((abs‘(ℑ‘𝐴)) / 2) ∈ ℝ⁺)
16		1nn0 12483	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ₀
17		nnexpcl 14035	. . . . . . . . . . 11 ⊢ ((𝑞 ∈ ℕ ∧ 1 ∈ ℕ₀) → (𝑞↑1) ∈ ℕ)
18	16, 17	mpan2 690	. . . . . . . . . 10 ⊢ (𝑞 ∈ ℕ → (𝑞↑1) ∈ ℕ)
19	18	ad2antll 728	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑞↑1) ∈ ℕ)
20	19	nnrpd 13009	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑞↑1) ∈ ℝ⁺)
21	15, 20	rpdivcld 13028	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (((abs‘(ℑ‘𝐴)) / 2) / (𝑞↑1)) ∈ ℝ⁺)
22	21	rpred 13011	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (((abs‘(ℑ‘𝐴)) / 2) / (𝑞↑1)) ∈ ℝ)
23	15	rpred 13011	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((abs‘(ℑ‘𝐴)) / 2) ∈ ℝ)
24	6	adantr 482	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝐴 ∈ ℂ)
25		znq 12931	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) → (𝑝 / 𝑞) ∈ ℚ)
26	25	adantl 483	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑝 / 𝑞) ∈ ℚ)
27		qre 12932	. . . . . . . . . 10 ⊢ ((𝑝 / 𝑞) ∈ ℚ → (𝑝 / 𝑞) ∈ ℝ)
28	26, 27	syl 17	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑝 / 𝑞) ∈ ℝ)
29	28	recnd 11237	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑝 / 𝑞) ∈ ℂ)
30	24, 29	subcld 11566	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝐴 − (𝑝 / 𝑞)) ∈ ℂ)
31	30	abscld 15378	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (abs‘(𝐴 − (𝑝 / 𝑞))) ∈ ℝ)
32	19	nnge1d 12255	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 1 ≤ (𝑞↑1))
33		1rp 12973	. . . . . . . . . 10 ⊢ 1 ∈ ℝ⁺
34		rpregt0 12983	. . . . . . . . . 10 ⊢ (1 ∈ ℝ⁺ → (1 ∈ ℝ ∧ 0 < 1))
35	33, 34	mp1i 13	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (1 ∈ ℝ ∧ 0 < 1))
36	20	rpregt0d 13017	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((𝑞↑1) ∈ ℝ ∧ 0 < (𝑞↑1)))
37	15	rpregt0d 13017	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (((abs‘(ℑ‘𝐴)) / 2) ∈ ℝ ∧ 0 < ((abs‘(ℑ‘𝐴)) / 2)))
38		lediv2 12099	. . . . . . . . 9 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ ((𝑞↑1) ∈ ℝ ∧ 0 < (𝑞↑1)) ∧ (((abs‘(ℑ‘𝐴)) / 2) ∈ ℝ ∧ 0 < ((abs‘(ℑ‘𝐴)) / 2))) → (1 ≤ (𝑞↑1) ↔ (((abs‘(ℑ‘𝐴)) / 2) / (𝑞↑1)) ≤ (((abs‘(ℑ‘𝐴)) / 2) / 1)))
39	35, 36, 37, 38	syl3anc 1372	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (1 ≤ (𝑞↑1) ↔ (((abs‘(ℑ‘𝐴)) / 2) / (𝑞↑1)) ≤ (((abs‘(ℑ‘𝐴)) / 2) / 1)))
40	32, 39	mpbid 231	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (((abs‘(ℑ‘𝐴)) / 2) / (𝑞↑1)) ≤ (((abs‘(ℑ‘𝐴)) / 2) / 1))
41	15	rpcnd 13013	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((abs‘(ℑ‘𝐴)) / 2) ∈ ℂ)
42	41	div1d 11977	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (((abs‘(ℑ‘𝐴)) / 2) / 1) = ((abs‘(ℑ‘𝐴)) / 2))
43	40, 42	breqtrd 5172	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (((abs‘(ℑ‘𝐴)) / 2) / (𝑞↑1)) ≤ ((abs‘(ℑ‘𝐴)) / 2))
44	13	adantr 482	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (abs‘(ℑ‘𝐴)) ∈ ℝ⁺)
45	44	rpred 13011	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (abs‘(ℑ‘𝐴)) ∈ ℝ)
46		rphalflt 12998	. . . . . . . 8 ⊢ ((abs‘(ℑ‘𝐴)) ∈ ℝ⁺ → ((abs‘(ℑ‘𝐴)) / 2) < (abs‘(ℑ‘𝐴)))
47	44, 46	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((abs‘(ℑ‘𝐴)) / 2) < (abs‘(ℑ‘𝐴)))
48	24, 29	imsubd 15159	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (ℑ‘(𝐴 − (𝑝 / 𝑞))) = ((ℑ‘𝐴) − (ℑ‘(𝑝 / 𝑞))))
49	28	reim0d 15167	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (ℑ‘(𝑝 / 𝑞)) = 0)
50	49	oveq2d 7419	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((ℑ‘𝐴) − (ℑ‘(𝑝 / 𝑞))) = ((ℑ‘𝐴) − 0))
51	8	adantr 482	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (ℑ‘𝐴) ∈ ℂ)
52	51	subid1d 11555	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((ℑ‘𝐴) − 0) = (ℑ‘𝐴))
53	48, 50, 52	3eqtrd 2777	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (ℑ‘(𝐴 − (𝑝 / 𝑞))) = (ℑ‘𝐴))
54	53	fveq2d 6891	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (abs‘(ℑ‘(𝐴 − (𝑝 / 𝑞)))) = (abs‘(ℑ‘𝐴)))
55		absimle 15251	. . . . . . . . 9 ⊢ ((𝐴 − (𝑝 / 𝑞)) ∈ ℂ → (abs‘(ℑ‘(𝐴 − (𝑝 / 𝑞)))) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))
56	30, 55	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (abs‘(ℑ‘(𝐴 − (𝑝 / 𝑞)))) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))
57	54, 56	eqbrtrrd 5170	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (abs‘(ℑ‘𝐴)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))
58	23, 45, 31, 47, 57	ltletrd 11369	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((abs‘(ℑ‘𝐴)) / 2) < (abs‘(𝐴 − (𝑝 / 𝑞))))
59	22, 23, 31, 43, 58	lelttrd 11367	. . . . 5 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (((abs‘(ℑ‘𝐴)) / 2) / (𝑞↑1)) < (abs‘(𝐴 − (𝑝 / 𝑞))))
60	59	olcd 873	. . . 4 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝐴 = (𝑝 / 𝑞) ∨ (((abs‘(ℑ‘𝐴)) / 2) / (𝑞↑1)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))
61	60	ralrimivva 3201	. . 3 ⊢ ((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) → ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (((abs‘(ℑ‘𝐴)) / 2) / (𝑞↑1)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))
62		oveq2 7411	. . . . . . . 8 ⊢ (𝑘 = 1 → (𝑞↑𝑘) = (𝑞↑1))
63	62	oveq2d 7419	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑥 / (𝑞↑𝑘)) = (𝑥 / (𝑞↑1)))
64	63	breq1d 5156	. . . . . 6 ⊢ (𝑘 = 1 → ((𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞))) ↔ (𝑥 / (𝑞↑1)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))
65	64	orbi2d 915	. . . . 5 ⊢ (𝑘 = 1 → ((𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))) ↔ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑1)) < (abs‘(𝐴 − (𝑝 / 𝑞))))))
66	65	2ralbidv 3219	. . . 4 ⊢ (𝑘 = 1 → (∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))) ↔ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑1)) < (abs‘(𝐴 − (𝑝 / 𝑞))))))
67		oveq1 7410	. . . . . . 7 ⊢ (𝑥 = ((abs‘(ℑ‘𝐴)) / 2) → (𝑥 / (𝑞↑1)) = (((abs‘(ℑ‘𝐴)) / 2) / (𝑞↑1)))
68	67	breq1d 5156	. . . . . 6 ⊢ (𝑥 = ((abs‘(ℑ‘𝐴)) / 2) → ((𝑥 / (𝑞↑1)) < (abs‘(𝐴 − (𝑝 / 𝑞))) ↔ (((abs‘(ℑ‘𝐴)) / 2) / (𝑞↑1)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))
69	68	orbi2d 915	. . . . 5 ⊢ (𝑥 = ((abs‘(ℑ‘𝐴)) / 2) → ((𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑1)) < (abs‘(𝐴 − (𝑝 / 𝑞)))) ↔ (𝐴 = (𝑝 / 𝑞) ∨ (((abs‘(ℑ‘𝐴)) / 2) / (𝑞↑1)) < (abs‘(𝐴 − (𝑝 / 𝑞))))))
70	69	2ralbidv 3219	. . . 4 ⊢ (𝑥 = ((abs‘(ℑ‘𝐴)) / 2) → (∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑1)) < (abs‘(𝐴 − (𝑝 / 𝑞)))) ↔ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (((abs‘(ℑ‘𝐴)) / 2) / (𝑞↑1)) < (abs‘(𝐴 − (𝑝 / 𝑞))))))
71	66, 70	rspc2ev 3622	. . 3 ⊢ ((1 ∈ ℕ ∧ ((abs‘(ℑ‘𝐴)) / 2) ∈ ℝ⁺ ∧ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (((abs‘(ℑ‘𝐴)) / 2) / (𝑞↑1)) < (abs‘(𝐴 − (𝑝 / 𝑞))))) → ∃𝑘 ∈ ℕ ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))
72	4, 14, 61, 71	mp3an2i 1467	. 2 ⊢ ((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) → ∃𝑘 ∈ ℕ ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))
73	3, 72	pm2.61dan 812	1 ⊢ (𝐴 ∈ 𝔸 → ∃𝑘 ∈ ℕ ∃𝑥 ∈ ℝ⁺ ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 ∩ cin 3945 class class class wbr 5146 ‘cfv 6539 (class class class)co 7403 ℂcc 11103 ℝcr 11104 0cc0 11105 1c1 11106 < clt 11243 ≤ cle 11244 − cmin 11439 / cdiv 11866 ℕcn 12207 2c2 12262 ℕ₀cn0 12467 ℤcz 12553 ℚcq 12927 ℝ⁺crp 12969 ↑cexp 14022 ℑcim 15040 abscabs 15176 𝔸caa 25808

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-inf2 9631 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 ax-addf 11184 ax-mulf 11185

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4907 df-int 4949 df-iun 4997 df-iin 4998 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8141 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-2o 8461 df-oadd 8464 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-fi 9401 df-sup 9432 df-inf 9433 df-oi 9500 df-dju 9891 df-card 9929 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-div 11867 df-nn 12208 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12468 df-xnn0 12540 df-z 12554 df-dec 12673 df-uz 12818 df-q 12928 df-rp 12970 df-xneg 13087 df-xadd 13088 df-xmul 13089 df-ioo 13323 df-ico 13325 df-icc 13326 df-fz 13480 df-fzo 13623 df-fl 13752 df-seq 13962 df-exp 14023 df-hash 14286 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-clim 15427 df-rlim 15428 df-sum 15628 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17140 df-ress 17169 df-plusg 17205 df-mulr 17206 df-starv 17207 df-sca 17208 df-vsca 17209 df-ip 17210 df-tset 17211 df-ple 17212 df-ds 17214 df-unif 17215 df-hom 17216 df-cco 17217 df-rest 17363 df-topn 17364 df-0g 17382 df-gsum 17383 df-topgen 17384 df-pt 17385 df-prds 17388 df-xrs 17443 df-qtop 17448 df-imas 17449 df-xps 17451 df-mre 17525 df-mrc 17526 df-acs 17528 df-mgm 18556 df-sgrp 18605 df-mnd 18621 df-submnd 18667 df-grp 18817 df-minusg 18818 df-mulg 18944 df-subg 18996 df-cntz 19174 df-cmn 19642 df-mgp 19979 df-ur 19996 df-ring 20048 df-cring 20049 df-subrg 20348 df-psmet 20920 df-xmet 20921 df-met 20922 df-bl 20923 df-mopn 20924 df-fbas 20925 df-fg 20926 df-cnfld 20929 df-top 22377 df-topon 22394 df-topsp 22416 df-bases 22430 df-cld 22504 df-ntr 22505 df-cls 22506 df-nei 22583 df-lp 22621 df-perf 22622 df-cn 22712 df-cnp 22713 df-haus 22800 df-cmp 22872 df-tx 23047 df-hmeo 23240 df-fil 23331 df-fm 23423 df-flim 23424 df-flf 23425 df-xms 23807 df-ms 23808 df-tms 23809 df-cncf 24375 df-0p 25168 df-limc 25364 df-dv 25365 df-dvn 25366 df-cpn 25367 df-ply 25683 df-idp 25684 df-coe 25685 df-dgr 25686 df-quot 25785 df-aa 25809

This theorem is referenced by: aaliou3lem9 25844

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