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Theorem chtppilimlem2 27437

Description: Lemma for chtppilim 27438. (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 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 𝑥 ∈ (2[,)+∞))
2		2re 12255	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
3		elicopnf 13398	. . . . . . . . . 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 218	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥))
6	5	simpld 494	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 𝑥 ∈ ℝ)
7		0red 11147	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 0 ∈ ℝ)
8	2	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 2 ∈ ℝ)
9		2pos 12284	. . . . . . . . 9 ⊢ 0 < 2
10	9	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 0 < 2)
11	5	simprd 495	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 2 ≤ 𝑥)
12	7, 8, 6, 10, 11	ltletrd 11306	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 0 < 𝑥)
13	6, 12	elrpd 12983	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 𝑥 ∈ ℝ⁺)
14		chtppilim.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
15	14	rpred 12986	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
16	15	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 𝐴 ∈ ℝ)
17	13, 16	rpcxpcld 26697	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐𝐴) ∈ ℝ⁺)
18		ppinncl 27137	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 2 ≤ 𝑥) → (π‘𝑥) ∈ ℕ)
19	5, 18	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (π‘𝑥) ∈ ℕ)
20	19	nnrpd 12984	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (π‘𝑥) ∈ ℝ⁺)
21	17, 20	rpdivcld 13003	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐𝐴) / (π‘𝑥)) ∈ ℝ⁺)
22	21	ralrimiva 3129	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (2[,)+∞)((𝑥↑_𝑐𝐴) / (π‘𝑥)) ∈ ℝ⁺)
23		chtppilim.2	. . . 4 ⊢ (𝜑 → 𝐴 < 1)
24		1re 11144	. . . . 5 ⊢ 1 ∈ ℝ
25		difrp 12982	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐴 < 1 ↔ (1 − 𝐴) ∈ ℝ⁺))
26	15, 24, 25	sylancl 587	. . . 4 ⊢ (𝜑 → (𝐴 < 1 ↔ (1 − 𝐴) ∈ ℝ⁺))
27	23, 26	mpbid 232	. . 3 ⊢ (𝜑 → (1 − 𝐴) ∈ ℝ⁺)
28		ovexd 7402	. . . . . 6 ⊢ (𝜑 → (2[,)+∞) ∈ V)
29	24	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 1 ∈ ℝ)
30		1lt2 12347	. . . . . . . . . . 11 ⊢ 1 < 2
31	30	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 1 < 2)
32	29, 8, 6, 31, 11	ltletrd 11306	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 1 < 𝑥)
33	6, 32	rplogcld 26593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (log‘𝑥) ∈ ℝ⁺)
34	13, 33	rpdivcld 13003	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥 / (log‘𝑥)) ∈ ℝ⁺)
35	34, 20	rpdivcld 13003	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥 / (log‘𝑥)) / (π‘𝑥)) ∈ ℝ⁺)
36	27	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (1 − 𝐴) ∈ ℝ⁺)
37	36	rpred 12986	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (1 − 𝐴) ∈ ℝ)
38	13, 37	rpcxpcld 26697	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐(1 − 𝐴)) ∈ ℝ⁺)
39	33, 38	rpdivcld 13003	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))) ∈ ℝ⁺)
40		eqidd 2737	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (2[,)+∞) ↦ ((𝑥 / (log‘𝑥)) / (π‘𝑥))) = (𝑥 ∈ (2[,)+∞) ↦ ((𝑥 / (log‘𝑥)) / (π‘𝑥))))
41		eqidd 2737	. . . . . 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 7651	. . . . 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 12988	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥 / (log‘𝑥)) ∈ ℂ)
44	39	rpcnd 12988	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))) ∈ ℂ)
45	20	rpcnne0d 12995	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((π‘𝑥) ∈ ℂ ∧ (π‘𝑥) ≠ 0))
46		div23 11828	. . . . . . . 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 1374	. . . . . . 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 12995	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((log‘𝑥) ∈ ℂ ∧ (log‘𝑥) ≠ 0))
49	38	rpcnne0d 12995	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐(1 − 𝐴)) ∈ ℂ ∧ (𝑥↑_𝑐(1 − 𝐴)) ≠ 0))
50	6	recnd 11173	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 𝑥 ∈ ℂ)
51		dmdcan 11865	. . . . . . . . . 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 1374	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))) · (𝑥 / (log‘𝑥))) = (𝑥 / (𝑥↑_𝑐(1 − 𝐴))))
53	43, 44	mulcomd 11166	. . . . . . . . 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 12995	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
55		ax-1cn 11096	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
56	55	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 1 ∈ ℂ)
57	36	rpcnd 12988	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (1 − 𝐴) ∈ ℂ)
58		cxpsub 26646	. . . . . . . . . . . 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 1374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐(1 − (1 − 𝐴))) = ((𝑥↑_𝑐1) / (𝑥↑_𝑐(1 − 𝐴))))
60	16	recnd 11173	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 𝐴 ∈ ℂ)
61		nncan 11423	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − (1 − 𝐴)) = 𝐴)
62	55, 60, 61	sylancr 588	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (1 − (1 − 𝐴)) = 𝐴)
63	62	oveq2d 7383	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐(1 − (1 − 𝐴))) = (𝑥↑_𝑐𝐴))
64	59, 63	eqtr3d 2773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐1) / (𝑥↑_𝑐(1 − 𝐴))) = (𝑥↑_𝑐𝐴))
65	50	cxp1d 26670	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐1) = 𝑥)
66	65	oveq1d 7382	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐1) / (𝑥↑_𝑐(1 − 𝐴))) = (𝑥 / (𝑥↑_𝑐(1 − 𝐴))))
67	64, 66	eqtr3d 2773	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐𝐴) = (𝑥 / (𝑥↑_𝑐(1 − 𝐴))))
68	52, 53, 67	3eqtr4d 2781	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥 / (log‘𝑥)) · ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) = (𝑥↑_𝑐𝐴))
69	68	oveq1d 7382	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (((𝑥 / (log‘𝑥)) · ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) / (π‘𝑥)) = ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
70	47, 69	eqtr3d 2773	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (((𝑥 / (log‘𝑥)) / (π‘𝑥)) · ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) = ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
71	70	mpteq2dva 5178	. . . . 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 2771	. . . 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 27435	. . . . 5 ⊢ (𝑥 ∈ (2[,)+∞) ↦ ((𝑥 / (log‘𝑥)) / (π‘𝑥))) ∈ 𝑂(1)
74	13	ex 412	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℝ⁺))
75	74	ssrdv 3927	. . . . . 6 ⊢ (𝜑 → (2[,)+∞) ⊆ ℝ⁺)
76		cxploglim 26941	. . . . . . 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 15523	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (2[,)+∞) ↦ ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) ⇝_𝑟 0)
79		o1rlimmul 15581	. . . . 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 588	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ (2[,)+∞) ↦ ((𝑥 / (log‘𝑥)) / (π‘𝑥))) ∘_f · (𝑥 ∈ (2[,)+∞) ↦ ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))))) ⇝_𝑟 0)
81	72, 80	eqbrtrrd 5109	. . 3 ⊢ (𝜑 → (𝑥 ∈ (2[,)+∞) ↦ ((𝑥↑_𝑐𝐴) / (π‘𝑥))) ⇝_𝑟 0)
82	22, 27, 81	rlimi 15475	. 2 ⊢ (𝜑 → ∃𝑧 ∈ ℝ ∀𝑥 ∈ (2[,)+∞)(𝑧 ≤ 𝑥 → (abs‘(((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0)) < (1 − 𝐴)))
83	21	rpcnd 12988	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐𝐴) / (π‘𝑥)) ∈ ℂ)
84	83	subid1d 11494	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0) = ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
85	84	fveq2d 6844	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (abs‘(((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0)) = (abs‘((𝑥↑_𝑐𝐴) / (π‘𝑥))))
86	21	rpred 12986	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐𝐴) / (π‘𝑥)) ∈ ℝ)
87	21	rpge0d 12990	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 0 ≤ ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
88	86, 87	absidd 15385	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (abs‘((𝑥↑_𝑐𝐴) / (π‘𝑥))) = ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
89	85, 88	eqtrd 2771	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (abs‘(((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0)) = ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
90	89	breq1d 5095	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((abs‘(((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0)) < (1 − 𝐴) ↔ ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴)))
91	14	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (2[,)+∞) ∧ ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴))) → 𝐴 ∈ ℝ⁺)
92	23	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (2[,)+∞) ∧ ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴))) → 𝐴 < 1)
93		simprl 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (2[,)+∞) ∧ ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴))) → 𝑥 ∈ (2[,)+∞))
94		simprr 773	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (2[,)+∞) ∧ ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴))) → ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴))
95	91, 92, 93, 94	chtppilimlem1 27436	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (2[,)+∞) ∧ ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴))) → ((𝐴↑2) · ((π‘𝑥) · (log‘𝑥))) < (θ‘𝑥))
96	95	expr 456	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴) → ((𝐴↑2) · ((π‘𝑥) · (log‘𝑥))) < (θ‘𝑥)))
97	90, 96	sylbid 240	. . . . 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 3149	. . 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 3152	. 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 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 ∃wrex 3061 Vcvv 3429 class class class wbr 5085 ↦ cmpt 5166 ‘cfv 6498 (class class class)co 7367 ∘_f cof 7629 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 · cmul 11043 +∞cpnf 11176 < clt 11179 ≤ cle 11180 − cmin 11377 / cdiv 11807 ℕcn 12174 2c2 12236 ℝ⁺crp 12942 [,)cico 13300 ↑cexp 14023 abscabs 15196 ⇝_𝑟 crli 15447 𝑂(1)co1 15448 logclog 26518 ↑_𝑐ccxp 26519 θccht 27054 πcppi 27057

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-xnn0 12511 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-o1 15452 df-lo1 15453 df-sum 15649 df-ef 16032 df-e 16033 df-sin 16034 df-cos 16035 df-pi 16037 df-dvds 16222 df-gcd 16464 df-prm 16641 df-pc 16808 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834 df-log 26520 df-cxp 26521 df-cht 27060 df-ppi 27063

This theorem is referenced by: chtppilim 27438

W3C validator