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Theorem aaliou2b 25701

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 3926	. . 3 ⊢ (𝐴 ∈ (𝔸 ∩ ℝ) ↔ (𝐴 ∈ 𝔸 ∧ 𝐴 ∈ ℝ))
2		aaliou2 25700	. . 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 12164	. . 3 ⊢ 1 ∈ ℕ
5		aacn 25677	. . . . . . . 8 ⊢ (𝐴 ∈ 𝔸 → 𝐴 ∈ ℂ)
6	5	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) → 𝐴 ∈ ℂ)
7	6	imcld 15080	. . . . . 6 ⊢ ((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) → (ℑ‘𝐴) ∈ ℝ)
8	7	recnd 11183	. . . . 5 ⊢ ((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) → (ℑ‘𝐴) ∈ ℂ)
9		reim0b 15004	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0))
10	5, 9	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ 𝔸 → (𝐴 ∈ ℝ ↔ (ℑ‘𝐴) = 0))
11	10	necon3bbid 2981	. . . . . 6 ⊢ (𝐴 ∈ 𝔸 → (¬ 𝐴 ∈ ℝ ↔ (ℑ‘𝐴) ≠ 0))
12	11	biimpa 477	. . . . 5 ⊢ ((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) → (ℑ‘𝐴) ≠ 0)
13	8, 12	absrpcld 15333	. . . 4 ⊢ ((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) → (abs‘(ℑ‘𝐴)) ∈ ℝ⁺)
14	13	rphalfcld 12969	. . 3 ⊢ ((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) → ((abs‘(ℑ‘𝐴)) / 2) ∈ ℝ⁺)
15	14	adantr 481	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((abs‘(ℑ‘𝐴)) / 2) ∈ ℝ⁺)
16		1nn0 12429	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ₀
17		nnexpcl 13980	. . . . . . . . . . 11 ⊢ ((𝑞 ∈ ℕ ∧ 1 ∈ ℕ₀) → (𝑞↑1) ∈ ℕ)
18	16, 17	mpan2 689	. . . . . . . . . 10 ⊢ (𝑞 ∈ ℕ → (𝑞↑1) ∈ ℕ)
19	18	ad2antll 727	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑞↑1) ∈ ℕ)
20	19	nnrpd 12955	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑞↑1) ∈ ℝ⁺)
21	15, 20	rpdivcld 12974	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (((abs‘(ℑ‘𝐴)) / 2) / (𝑞↑1)) ∈ ℝ⁺)
22	21	rpred 12957	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (((abs‘(ℑ‘𝐴)) / 2) / (𝑞↑1)) ∈ ℝ)
23	15	rpred 12957	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((abs‘(ℑ‘𝐴)) / 2) ∈ ℝ)
24	6	adantr 481	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 𝐴 ∈ ℂ)
25		znq 12877	. . . . . . . . . . 11 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) → (𝑝 / 𝑞) ∈ ℚ)
26	25	adantl 482	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑝 / 𝑞) ∈ ℚ)
27		qre 12878	. . . . . . . . . 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 11183	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝑝 / 𝑞) ∈ ℂ)
30	24, 29	subcld 11512	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (𝐴 − (𝑝 / 𝑞)) ∈ ℂ)
31	30	abscld 15321	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (abs‘(𝐴 − (𝑝 / 𝑞))) ∈ ℝ)
32	19	nnge1d 12201	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → 1 ≤ (𝑞↑1))
33		1rp 12919	. . . . . . . . . 10 ⊢ 1 ∈ ℝ⁺
34		rpregt0 12929	. . . . . . . . . 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 12963	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((𝑞↑1) ∈ ℝ ∧ 0 < (𝑞↑1)))
37	15	rpregt0d 12963	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (((abs‘(ℑ‘𝐴)) / 2) ∈ ℝ ∧ 0 < ((abs‘(ℑ‘𝐴)) / 2)))
38		lediv2 12045	. . . . . . . . 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 1371	. . . . . . . 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 12959	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((abs‘(ℑ‘𝐴)) / 2) ∈ ℂ)
42	41	div1d 11923	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (((abs‘(ℑ‘𝐴)) / 2) / 1) = ((abs‘(ℑ‘𝐴)) / 2))
43	40, 42	breqtrd 5131	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (((abs‘(ℑ‘𝐴)) / 2) / (𝑞↑1)) ≤ ((abs‘(ℑ‘𝐴)) / 2))
44	13	adantr 481	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (abs‘(ℑ‘𝐴)) ∈ ℝ⁺)
45	44	rpred 12957	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (abs‘(ℑ‘𝐴)) ∈ ℝ)
46		rphalflt 12944	. . . . . . . 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 15102	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (ℑ‘(𝐴 − (𝑝 / 𝑞))) = ((ℑ‘𝐴) − (ℑ‘(𝑝 / 𝑞))))
49	28	reim0d 15110	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (ℑ‘(𝑝 / 𝑞)) = 0)
50	49	oveq2d 7373	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((ℑ‘𝐴) − (ℑ‘(𝑝 / 𝑞))) = ((ℑ‘𝐴) − 0))
51	8	adantr 481	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (ℑ‘𝐴) ∈ ℂ)
52	51	subid1d 11501	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((ℑ‘𝐴) − 0) = (ℑ‘𝐴))
53	48, 50, 52	3eqtrd 2780	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (ℑ‘(𝐴 − (𝑝 / 𝑞))) = (ℑ‘𝐴))
54	53	fveq2d 6846	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (abs‘(ℑ‘(𝐴 − (𝑝 / 𝑞)))) = (abs‘(ℑ‘𝐴)))
55		absimle 15194	. . . . . . . . 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 5129	. . . . . . 7 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (abs‘(ℑ‘𝐴)) ≤ (abs‘(𝐴 − (𝑝 / 𝑞))))
58	23, 45, 31, 47, 57	ltletrd 11315	. . . . . 6 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → ((abs‘(ℑ‘𝐴)) / 2) < (abs‘(𝐴 − (𝑝 / 𝑞))))
59	22, 23, 31, 43, 58	lelttrd 11313	. . . . 5 ⊢ (((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) ∧ (𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ)) → (((abs‘(ℑ‘𝐴)) / 2) / (𝑞↑1)) < (abs‘(𝐴 − (𝑝 / 𝑞))))
60	59	olcd 872	. . . 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 3197	. . 3 ⊢ ((𝐴 ∈ 𝔸 ∧ ¬ 𝐴 ∈ ℝ) → ∀𝑝 ∈ ℤ ∀𝑞 ∈ ℕ (𝐴 = (𝑝 / 𝑞) ∨ (((abs‘(ℑ‘𝐴)) / 2) / (𝑞↑1)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))
62		oveq2 7365	. . . . . . . 8 ⊢ (𝑘 = 1 → (𝑞↑𝑘) = (𝑞↑1))
63	62	oveq2d 7373	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑥 / (𝑞↑𝑘)) = (𝑥 / (𝑞↑1)))
64	63	breq1d 5115	. . . . . 6 ⊢ (𝑘 = 1 → ((𝑥 / (𝑞↑𝑘)) < (abs‘(𝐴 − (𝑝 / 𝑞))) ↔ (𝑥 / (𝑞↑1)) < (abs‘(𝐴 − (𝑝 / 𝑞)))))
65	64	orbi2d 914	. . . . 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 3212	. . . 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 7364	. . . . . . 7 ⊢ (𝑥 = ((abs‘(ℑ‘𝐴)) / 2) → (𝑥 / (𝑞↑1)) = (((abs‘(ℑ‘𝐴)) / 2) / (𝑞↑1)))
68	67	breq1d 5115	. . . . . 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 914	. . . . 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 3212	. . . 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 3592	. . 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 1466	. 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 811	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 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 ∩ cin 3909 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 < clt 11189 ≤ cle 11190 − cmin 11385 / cdiv 11812 ℕcn 12153 2c2 12208 ℕ₀cn0 12413 ℤcz 12499 ℚcq 12873 ℝ⁺crp 12915 ↑cexp 13967 ℑcim 14983 abscabs 15119 𝔸caa 25674

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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-xnn0 12486 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-clim 15370 df-rlim 15371 df-sum 15571 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-grp 18751 df-minusg 18752 df-mulg 18873 df-subg 18925 df-cntz 19097 df-cmn 19564 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-subrg 20220 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-0p 25034 df-limc 25230 df-dv 25231 df-dvn 25232 df-cpn 25233 df-ply 25549 df-idp 25550 df-coe 25551 df-dgr 25552 df-quot 25651 df-aa 25675

This theorem is referenced by: aaliou3lem9 25710

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