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Theorem chtppilimlem2 27410

Description: Lemma for chtppilim 27411. (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 12196	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
3		elicopnf 13342	. . . . . . . . . 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 11112	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 0 ∈ ℝ)
8	2	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 2 ∈ ℝ)
9		2pos 12225	. . . . . . . . 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 11270	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 0 < 𝑥)
13	6, 12	elrpd 12928	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 𝑥 ∈ ℝ⁺)
14		chtppilim.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
15	14	rpred 12931	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
16	15	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 𝐴 ∈ ℝ)
17	13, 16	rpcxpcld 26667	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐𝐴) ∈ ℝ⁺)
18		ppinncl 27109	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 2 ≤ 𝑥) → (π‘𝑥) ∈ ℕ)
19	5, 18	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (π‘𝑥) ∈ ℕ)
20	19	nnrpd 12929	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (π‘𝑥) ∈ ℝ⁺)
21	17, 20	rpdivcld 12948	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐𝐴) / (π‘𝑥)) ∈ ℝ⁺)
22	21	ralrimiva 3124	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (2[,)+∞)((𝑥↑_𝑐𝐴) / (π‘𝑥)) ∈ ℝ⁺)
23		chtppilim.2	. . . 4 ⊢ (𝜑 → 𝐴 < 1)
24		1re 11109	. . . . 5 ⊢ 1 ∈ ℝ
25		difrp 12927	. . . . 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 232	. . 3 ⊢ (𝜑 → (1 − 𝐴) ∈ ℝ⁺)
28		ovexd 7381	. . . . . 6 ⊢ (𝜑 → (2[,)+∞) ∈ V)
29	24	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 1 ∈ ℝ)
30		1lt2 12288	. . . . . . . . . . 11 ⊢ 1 < 2
31	30	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 1 < 2)
32	29, 8, 6, 31, 11	ltletrd 11270	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 1 < 𝑥)
33	6, 32	rplogcld 26563	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (log‘𝑥) ∈ ℝ⁺)
34	13, 33	rpdivcld 12948	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥 / (log‘𝑥)) ∈ ℝ⁺)
35	34, 20	rpdivcld 12948	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥 / (log‘𝑥)) / (π‘𝑥)) ∈ ℝ⁺)
36	27	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (1 − 𝐴) ∈ ℝ⁺)
37	36	rpred 12931	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (1 − 𝐴) ∈ ℝ)
38	13, 37	rpcxpcld 26667	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐(1 − 𝐴)) ∈ ℝ⁺)
39	33, 38	rpdivcld 12948	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))) ∈ ℝ⁺)
40		eqidd 2732	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (2[,)+∞) ↦ ((𝑥 / (log‘𝑥)) / (π‘𝑥))) = (𝑥 ∈ (2[,)+∞) ↦ ((𝑥 / (log‘𝑥)) / (π‘𝑥))))
41		eqidd 2732	. . . . . 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 7630	. . . . 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 12933	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥 / (log‘𝑥)) ∈ ℂ)
44	39	rpcnd 12933	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))) ∈ ℂ)
45	20	rpcnne0d 12940	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((π‘𝑥) ∈ ℂ ∧ (π‘𝑥) ≠ 0))
46		div23 11792	. . . . . . . 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 1373	. . . . . . 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 12940	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((log‘𝑥) ∈ ℂ ∧ (log‘𝑥) ≠ 0))
49	38	rpcnne0d 12940	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐(1 − 𝐴)) ∈ ℂ ∧ (𝑥↑_𝑐(1 − 𝐴)) ≠ 0))
50	6	recnd 11137	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 𝑥 ∈ ℂ)
51		dmdcan 11828	. . . . . . . . . 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 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))) · (𝑥 / (log‘𝑥))) = (𝑥 / (𝑥↑_𝑐(1 − 𝐴))))
53	43, 44	mulcomd 11130	. . . . . . . . 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 12940	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
55		ax-1cn 11061	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
56	55	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 1 ∈ ℂ)
57	36	rpcnd 12933	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (1 − 𝐴) ∈ ℂ)
58		cxpsub 26616	. . . . . . . . . . . 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 1373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐(1 − (1 − 𝐴))) = ((𝑥↑_𝑐1) / (𝑥↑_𝑐(1 − 𝐴))))
60	16	recnd 11137	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 𝐴 ∈ ℂ)
61		nncan 11387	. . . . . . . . . . . . 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 7362	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐(1 − (1 − 𝐴))) = (𝑥↑_𝑐𝐴))
64	59, 63	eqtr3d 2768	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐1) / (𝑥↑_𝑐(1 − 𝐴))) = (𝑥↑_𝑐𝐴))
65	50	cxp1d 26640	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐1) = 𝑥)
66	65	oveq1d 7361	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐1) / (𝑥↑_𝑐(1 − 𝐴))) = (𝑥 / (𝑥↑_𝑐(1 − 𝐴))))
67	64, 66	eqtr3d 2768	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐𝐴) = (𝑥 / (𝑥↑_𝑐(1 − 𝐴))))
68	52, 53, 67	3eqtr4d 2776	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥 / (log‘𝑥)) · ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) = (𝑥↑_𝑐𝐴))
69	68	oveq1d 7361	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (((𝑥 / (log‘𝑥)) · ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) / (π‘𝑥)) = ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
70	47, 69	eqtr3d 2768	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (((𝑥 / (log‘𝑥)) / (π‘𝑥)) · ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) = ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
71	70	mpteq2dva 5184	. . . . 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 2766	. . . 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 27408	. . . . 5 ⊢ (𝑥 ∈ (2[,)+∞) ↦ ((𝑥 / (log‘𝑥)) / (π‘𝑥))) ∈ 𝑂(1)
74	13	ex 412	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℝ⁺))
75	74	ssrdv 3940	. . . . . 6 ⊢ (𝜑 → (2[,)+∞) ⊆ ℝ⁺)
76		cxploglim 26913	. . . . . . 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 15465	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (2[,)+∞) ↦ ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) ⇝_𝑟 0)
79		o1rlimmul 15523	. . . . 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 5115	. . 3 ⊢ (𝜑 → (𝑥 ∈ (2[,)+∞) ↦ ((𝑥↑_𝑐𝐴) / (π‘𝑥))) ⇝_𝑟 0)
82	22, 27, 81	rlimi 15417	. 2 ⊢ (𝜑 → ∃𝑧 ∈ ℝ ∀𝑥 ∈ (2[,)+∞)(𝑧 ≤ 𝑥 → (abs‘(((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0)) < (1 − 𝐴)))
83	21	rpcnd 12933	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐𝐴) / (π‘𝑥)) ∈ ℂ)
84	83	subid1d 11458	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0) = ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
85	84	fveq2d 6826	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (abs‘(((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0)) = (abs‘((𝑥↑_𝑐𝐴) / (π‘𝑥))))
86	21	rpred 12931	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐𝐴) / (π‘𝑥)) ∈ ℝ)
87	21	rpge0d 12935	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 0 ≤ ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
88	86, 87	absidd 15327	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (abs‘((𝑥↑_𝑐𝐴) / (π‘𝑥))) = ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
89	85, 88	eqtrd 2766	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (abs‘(((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0)) = ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
90	89	breq1d 5101	. . . . . 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 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (2[,)+∞) ∧ ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴))) → 𝑥 ∈ (2[,)+∞))
94		simprr 772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (2[,)+∞) ∧ ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴))) → ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴))
95	91, 92, 93, 94	chtppilimlem1 27409	. . . . . . 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 3144	. . 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 3147	. 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 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 Vcvv 3436 class class class wbr 5091 ↦ cmpt 5172 ‘cfv 6481 (class class class)co 7346 ∘_f cof 7608 ℂcc 11001 ℝcr 11002 0cc0 11003 1c1 11004 · cmul 11008 +∞cpnf 11140 < clt 11143 ≤ cle 11144 − cmin 11341 / cdiv 11771 ℕcn 12122 2c2 12177 ℝ⁺crp 12887 [,)cico 13244 ↑cexp 13965 abscabs 15138 ⇝_𝑟 crli 15389 𝑂(1)co1 15390 logclog 26488 ↑_𝑐ccxp 26489 θccht 27026 πcppi 27029

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-inf2 9531 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-pre-sup 11081 ax-addf 11082

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-se 5570 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-oadd 8389 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-dju 9791 df-card 9829 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-div 11772 df-nn 12123 df-2 12185 df-3 12186 df-4 12187 df-5 12188 df-6 12189 df-7 12190 df-8 12191 df-9 12192 df-n0 12379 df-xnn0 12452 df-z 12466 df-dec 12586 df-uz 12730 df-q 12844 df-rp 12888 df-xneg 13008 df-xadd 13009 df-xmul 13010 df-ioo 13246 df-ioc 13247 df-ico 13248 df-icc 13249 df-fz 13405 df-fzo 13552 df-fl 13693 df-mod 13771 df-seq 13906 df-exp 13966 df-fac 14178 df-bc 14207 df-hash 14235 df-shft 14971 df-cj 15003 df-re 15004 df-im 15005 df-sqrt 15139 df-abs 15140 df-limsup 15375 df-clim 15392 df-rlim 15393 df-o1 15394 df-lo1 15395 df-sum 15591 df-ef 15971 df-e 15972 df-sin 15973 df-cos 15974 df-pi 15976 df-dvds 16161 df-gcd 16403 df-prm 16580 df-pc 16746 df-struct 17055 df-sets 17072 df-slot 17090 df-ndx 17102 df-base 17118 df-ress 17139 df-plusg 17171 df-mulr 17172 df-starv 17173 df-sca 17174 df-vsca 17175 df-ip 17176 df-tset 17177 df-ple 17178 df-ds 17180 df-unif 17181 df-hom 17182 df-cco 17183 df-rest 17323 df-topn 17324 df-0g 17342 df-gsum 17343 df-topgen 17344 df-pt 17345 df-prds 17348 df-xrs 17403 df-qtop 17408 df-imas 17409 df-xps 17411 df-mre 17485 df-mrc 17486 df-acs 17488 df-mgm 18545 df-sgrp 18624 df-mnd 18640 df-submnd 18689 df-mulg 18978 df-cntz 19227 df-cmn 19692 df-psmet 21281 df-xmet 21282 df-met 21283 df-bl 21284 df-mopn 21285 df-fbas 21286 df-fg 21287 df-cnfld 21290 df-top 22807 df-topon 22824 df-topsp 22846 df-bases 22859 df-cld 22932 df-ntr 22933 df-cls 22934 df-nei 23011 df-lp 23049 df-perf 23050 df-cn 23140 df-cnp 23141 df-haus 23228 df-tx 23475 df-hmeo 23668 df-fil 23759 df-fm 23851 df-flim 23852 df-flf 23853 df-xms 24233 df-ms 24234 df-tms 24235 df-cncf 24796 df-limc 25792 df-dv 25793 df-log 26490 df-cxp 26491 df-cht 27032 df-ppi 27035

This theorem is referenced by: chtppilim 27411

W3C validator