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Theorem chtppilimlem2 26966

Description: Lemma for chtppilim 26967. (Contributed by Mario Carneiro, 22-Sep-2014.)

**Hypotheses**
Ref	Expression
chtppilim.1	⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
chtppilim.2	⊢ (𝜑 → 𝐴 < 1)

**Assertion**
Ref	Expression
chtppilimlem2	⊢ (𝜑 → ∃𝑧 ∈ ℝ ∀𝑥 ∈ (2[,)+∞)(𝑧 ≤ 𝑥 → ((𝐴↑2) · ((π‘𝑥) · (log‘𝑥))) < (θ‘𝑥)))

Distinct variable groups: 𝑥,𝑧,𝐴 𝜑,𝑥,𝑧

**Proof of Theorem chtppilimlem2**
Step	Hyp	Ref	Expression
1		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 𝑥 ∈ (2[,)+∞))
2		2re 12282	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
3		elicopnf 13418	. . . . . . . . . 10 ⊢ (2 ∈ ℝ → (𝑥 ∈ (2[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥)))
4	2, 3	ax-mp 5	. . . . . . . . 9 ⊢ (𝑥 ∈ (2[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥))
5	1, 4	sylib 217	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥))
6	5	simpld 495	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 𝑥 ∈ ℝ)
7		0red 11213	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 0 ∈ ℝ)
8	2	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 2 ∈ ℝ)
9		2pos 12311	. . . . . . . . 9 ⊢ 0 < 2
10	9	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 0 < 2)
11	5	simprd 496	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 2 ≤ 𝑥)
12	7, 8, 6, 10, 11	ltletrd 11370	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 0 < 𝑥)
13	6, 12	elrpd 13009	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 𝑥 ∈ ℝ⁺)
14		chtppilim.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
15	14	rpred 13012	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
16	15	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 𝐴 ∈ ℝ)
17	13, 16	rpcxpcld 26231	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐𝐴) ∈ ℝ⁺)
18		ppinncl 26667	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 2 ≤ 𝑥) → (π‘𝑥) ∈ ℕ)
19	5, 18	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (π‘𝑥) ∈ ℕ)
20	19	nnrpd 13010	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (π‘𝑥) ∈ ℝ⁺)
21	17, 20	rpdivcld 13029	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐𝐴) / (π‘𝑥)) ∈ ℝ⁺)
22	21	ralrimiva 3146	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (2[,)+∞)((𝑥↑_𝑐𝐴) / (π‘𝑥)) ∈ ℝ⁺)
23		chtppilim.2	. . . 4 ⊢ (𝜑 → 𝐴 < 1)
24		1re 11210	. . . . 5 ⊢ 1 ∈ ℝ
25		difrp 13008	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐴 < 1 ↔ (1 − 𝐴) ∈ ℝ⁺))
26	15, 24, 25	sylancl 586	. . . 4 ⊢ (𝜑 → (𝐴 < 1 ↔ (1 − 𝐴) ∈ ℝ⁺))
27	23, 26	mpbid 231	. . 3 ⊢ (𝜑 → (1 − 𝐴) ∈ ℝ⁺)
28		ovexd 7440	. . . . . 6 ⊢ (𝜑 → (2[,)+∞) ∈ V)
29	24	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 1 ∈ ℝ)
30		1lt2 12379	. . . . . . . . . . 11 ⊢ 1 < 2
31	30	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 1 < 2)
32	29, 8, 6, 31, 11	ltletrd 11370	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 1 < 𝑥)
33	6, 32	rplogcld 26128	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (log‘𝑥) ∈ ℝ⁺)
34	13, 33	rpdivcld 13029	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥 / (log‘𝑥)) ∈ ℝ⁺)
35	34, 20	rpdivcld 13029	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥 / (log‘𝑥)) / (π‘𝑥)) ∈ ℝ⁺)
36	27	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (1 − 𝐴) ∈ ℝ⁺)
37	36	rpred 13012	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (1 − 𝐴) ∈ ℝ)
38	13, 37	rpcxpcld 26231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐(1 − 𝐴)) ∈ ℝ⁺)
39	33, 38	rpdivcld 13029	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))) ∈ ℝ⁺)
40		eqidd 2733	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (2[,)+∞) ↦ ((𝑥 / (log‘𝑥)) / (π‘𝑥))) = (𝑥 ∈ (2[,)+∞) ↦ ((𝑥 / (log‘𝑥)) / (π‘𝑥))))
41		eqidd 2733	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (2[,)+∞) ↦ ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) = (𝑥 ∈ (2[,)+∞) ↦ ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))))
42	28, 35, 39, 40, 41	offval2 7686	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ (2[,)+∞) ↦ ((𝑥 / (log‘𝑥)) / (π‘𝑥))) ∘_f · (𝑥 ∈ (2[,)+∞) ↦ ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))))) = (𝑥 ∈ (2[,)+∞) ↦ (((𝑥 / (log‘𝑥)) / (π‘𝑥)) · ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))))))
43	34	rpcnd 13014	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥 / (log‘𝑥)) ∈ ℂ)
44	39	rpcnd 13014	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))) ∈ ℂ)
45	20	rpcnne0d 13021	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((π‘𝑥) ∈ ℂ ∧ (π‘𝑥) ≠ 0))
46		div23 11887	. . . . . . . 8 ⊢ (((𝑥 / (log‘𝑥)) ∈ ℂ ∧ ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))) ∈ ℂ ∧ ((π‘𝑥) ∈ ℂ ∧ (π‘𝑥) ≠ 0)) → (((𝑥 / (log‘𝑥)) · ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) / (π‘𝑥)) = (((𝑥 / (log‘𝑥)) / (π‘𝑥)) · ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))))
47	43, 44, 45, 46	syl3anc 1371	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (((𝑥 / (log‘𝑥)) · ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) / (π‘𝑥)) = (((𝑥 / (log‘𝑥)) / (π‘𝑥)) · ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))))
48	33	rpcnne0d 13021	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((log‘𝑥) ∈ ℂ ∧ (log‘𝑥) ≠ 0))
49	38	rpcnne0d 13021	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐(1 − 𝐴)) ∈ ℂ ∧ (𝑥↑_𝑐(1 − 𝐴)) ≠ 0))
50	6	recnd 11238	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 𝑥 ∈ ℂ)
51		dmdcan 11920	. . . . . . . . . 10 ⊢ ((((log‘𝑥) ∈ ℂ ∧ (log‘𝑥) ≠ 0) ∧ ((𝑥↑_𝑐(1 − 𝐴)) ∈ ℂ ∧ (𝑥↑_𝑐(1 − 𝐴)) ≠ 0) ∧ 𝑥 ∈ ℂ) → (((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))) · (𝑥 / (log‘𝑥))) = (𝑥 / (𝑥↑_𝑐(1 − 𝐴))))
52	48, 49, 50, 51	syl3anc 1371	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))) · (𝑥 / (log‘𝑥))) = (𝑥 / (𝑥↑_𝑐(1 − 𝐴))))
53	43, 44	mulcomd 11231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥 / (log‘𝑥)) · ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) = (((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))) · (𝑥 / (log‘𝑥))))
54	13	rpcnne0d 13021	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
55		ax-1cn 11164	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
56	55	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 1 ∈ ℂ)
57	36	rpcnd 13014	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (1 − 𝐴) ∈ ℂ)
58		cxpsub 26181	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ 1 ∈ ℂ ∧ (1 − 𝐴) ∈ ℂ) → (𝑥↑_𝑐(1 − (1 − 𝐴))) = ((𝑥↑_𝑐1) / (𝑥↑_𝑐(1 − 𝐴))))
59	54, 56, 57, 58	syl3anc 1371	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐(1 − (1 − 𝐴))) = ((𝑥↑_𝑐1) / (𝑥↑_𝑐(1 − 𝐴))))
60	16	recnd 11238	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 𝐴 ∈ ℂ)
61		nncan 11485	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − (1 − 𝐴)) = 𝐴)
62	55, 60, 61	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (1 − (1 − 𝐴)) = 𝐴)
63	62	oveq2d 7421	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐(1 − (1 − 𝐴))) = (𝑥↑_𝑐𝐴))
64	59, 63	eqtr3d 2774	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐1) / (𝑥↑_𝑐(1 − 𝐴))) = (𝑥↑_𝑐𝐴))
65	50	cxp1d 26205	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐1) = 𝑥)
66	65	oveq1d 7420	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐1) / (𝑥↑_𝑐(1 − 𝐴))) = (𝑥 / (𝑥↑_𝑐(1 − 𝐴))))
67	64, 66	eqtr3d 2774	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐𝐴) = (𝑥 / (𝑥↑_𝑐(1 − 𝐴))))
68	52, 53, 67	3eqtr4d 2782	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥 / (log‘𝑥)) · ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) = (𝑥↑_𝑐𝐴))
69	68	oveq1d 7420	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (((𝑥 / (log‘𝑥)) · ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) / (π‘𝑥)) = ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
70	47, 69	eqtr3d 2774	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (((𝑥 / (log‘𝑥)) / (π‘𝑥)) · ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) = ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
71	70	mpteq2dva 5247	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (2[,)+∞) ↦ (((𝑥 / (log‘𝑥)) / (π‘𝑥)) · ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))))) = (𝑥 ∈ (2[,)+∞) ↦ ((𝑥↑_𝑐𝐴) / (π‘𝑥))))
72	42, 71	eqtrd 2772	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ (2[,)+∞) ↦ ((𝑥 / (log‘𝑥)) / (π‘𝑥))) ∘_f · (𝑥 ∈ (2[,)+∞) ↦ ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))))) = (𝑥 ∈ (2[,)+∞) ↦ ((𝑥↑_𝑐𝐴) / (π‘𝑥))))
73		chebbnd1 26964	. . . . 5 ⊢ (𝑥 ∈ (2[,)+∞) ↦ ((𝑥 / (log‘𝑥)) / (π‘𝑥))) ∈ 𝑂(1)
74	13	ex 413	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℝ⁺))
75	74	ssrdv 3987	. . . . . 6 ⊢ (𝜑 → (2[,)+∞) ⊆ ℝ⁺)
76		cxploglim 26471	. . . . . . 7 ⊢ ((1 − 𝐴) ∈ ℝ⁺ → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) ⇝_𝑟 0)
77	27, 76	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) ⇝_𝑟 0)
78	75, 77	rlimres2 15501	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (2[,)+∞) ↦ ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) ⇝_𝑟 0)
79		o1rlimmul 15559	. . . . 5 ⊢ (((𝑥 ∈ (2[,)+∞) ↦ ((𝑥 / (log‘𝑥)) / (π‘𝑥))) ∈ 𝑂(1) ∧ (𝑥 ∈ (2[,)+∞) ↦ ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) ⇝_𝑟 0) → ((𝑥 ∈ (2[,)+∞) ↦ ((𝑥 / (log‘𝑥)) / (π‘𝑥))) ∘_f · (𝑥 ∈ (2[,)+∞) ↦ ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))))) ⇝_𝑟 0)
80	73, 78, 79	sylancr 587	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ (2[,)+∞) ↦ ((𝑥 / (log‘𝑥)) / (π‘𝑥))) ∘_f · (𝑥 ∈ (2[,)+∞) ↦ ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))))) ⇝_𝑟 0)
81	72, 80	eqbrtrrd 5171	. . 3 ⊢ (𝜑 → (𝑥 ∈ (2[,)+∞) ↦ ((𝑥↑_𝑐𝐴) / (π‘𝑥))) ⇝_𝑟 0)
82	22, 27, 81	rlimi 15453	. 2 ⊢ (𝜑 → ∃𝑧 ∈ ℝ ∀𝑥 ∈ (2[,)+∞)(𝑧 ≤ 𝑥 → (abs‘(((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0)) < (1 − 𝐴)))
83	21	rpcnd 13014	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐𝐴) / (π‘𝑥)) ∈ ℂ)
84	83	subid1d 11556	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0) = ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
85	84	fveq2d 6892	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (abs‘(((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0)) = (abs‘((𝑥↑_𝑐𝐴) / (π‘𝑥))))
86	21	rpred 13012	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐𝐴) / (π‘𝑥)) ∈ ℝ)
87	21	rpge0d 13016	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 0 ≤ ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
88	86, 87	absidd 15365	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (abs‘((𝑥↑_𝑐𝐴) / (π‘𝑥))) = ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
89	85, 88	eqtrd 2772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (abs‘(((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0)) = ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
90	89	breq1d 5157	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((abs‘(((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0)) < (1 − 𝐴) ↔ ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴)))
91	14	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (2[,)+∞) ∧ ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴))) → 𝐴 ∈ ℝ⁺)
92	23	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (2[,)+∞) ∧ ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴))) → 𝐴 < 1)
93		simprl 769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (2[,)+∞) ∧ ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴))) → 𝑥 ∈ (2[,)+∞))
94		simprr 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (2[,)+∞) ∧ ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴))) → ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴))
95	91, 92, 93, 94	chtppilimlem1 26965	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (2[,)+∞) ∧ ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴))) → ((𝐴↑2) · ((π‘𝑥) · (log‘𝑥))) < (θ‘𝑥))
96	95	expr 457	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴) → ((𝐴↑2) · ((π‘𝑥) · (log‘𝑥))) < (θ‘𝑥)))
97	90, 96	sylbid 239	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((abs‘(((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0)) < (1 − 𝐴) → ((𝐴↑2) · ((π‘𝑥) · (log‘𝑥))) < (θ‘𝑥)))
98	97	imim2d 57	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑧 ≤ 𝑥 → (abs‘(((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0)) < (1 − 𝐴)) → (𝑧 ≤ 𝑥 → ((𝐴↑2) · ((π‘𝑥) · (log‘𝑥))) < (θ‘𝑥))))
99	98	ralimdva 3167	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ (2[,)+∞)(𝑧 ≤ 𝑥 → (abs‘(((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0)) < (1 − 𝐴)) → ∀𝑥 ∈ (2[,)+∞)(𝑧 ≤ 𝑥 → ((𝐴↑2) · ((π‘𝑥) · (log‘𝑥))) < (θ‘𝑥))))
100	99	reximdv 3170	. 2 ⊢ (𝜑 → (∃𝑧 ∈ ℝ ∀𝑥 ∈ (2[,)+∞)(𝑧 ≤ 𝑥 → (abs‘(((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0)) < (1 − 𝐴)) → ∃𝑧 ∈ ℝ ∀𝑥 ∈ (2[,)+∞)(𝑧 ≤ 𝑥 → ((𝐴↑2) · ((π‘𝑥) · (log‘𝑥))) < (θ‘𝑥))))
101	82, 100	mpd 15	1 ⊢ (𝜑 → ∃𝑧 ∈ ℝ ∀𝑥 ∈ (2[,)+∞)(𝑧 ≤ 𝑥 → ((𝐴↑2) · ((π‘𝑥) · (log‘𝑥))) < (θ‘𝑥)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 Vcvv 3474 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6540 (class class class)co 7405 ∘_f cof 7664 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 · cmul 11111 +∞cpnf 11241 < clt 11244 ≤ cle 11245 − cmin 11440 / cdiv 11867 ℕcn 12208 2c2 12263 ℝ⁺crp 12970 [,)cico 13322 ↑cexp 14023 abscabs 15177 ⇝_𝑟 crli 15425 𝑂(1)co1 15426 logclog 26054 ↑_𝑐ccxp 26055 θccht 26584 πcppi 26587

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-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 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 ax-addf 11185 ax-mulf 11186

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-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 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-se 5631 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-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 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-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-o1 15430 df-lo1 15431 df-sum 15629 df-ef 16007 df-e 16008 df-sin 16009 df-cos 16010 df-pi 16012 df-dvds 16194 df-gcd 16432 df-prm 16605 df-pc 16766 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375 df-log 26056 df-cxp 26057 df-cht 26590 df-ppi 26593

This theorem is referenced by: chtppilim 26967

W3C validator