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Theorem chtppilimlem2 24963

Description: Lemma for chtppilim 24964. (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 476	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 𝑥 ∈ (2[,)+∞))
2		2re 10967	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
3		elicopnf 12140	. . . . . . . . . 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 207	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥 ∈ ℝ ∧ 2 ≤ 𝑥))
6	5	simpld 474	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 𝑥 ∈ ℝ)
7		0red 9920	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 0 ∈ ℝ)
8	2	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 2 ∈ ℝ)
9		2pos 10989	. . . . . . . . 9 ⊢ 0 < 2
10	9	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 0 < 2)
11	5	simprd 478	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 2 ≤ 𝑥)
12	7, 8, 6, 10, 11	ltletrd 10076	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 0 < 𝑥)
13	6, 12	elrpd 11745	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 𝑥 ∈ ℝ⁺)
14		chtppilim.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
15	14	rpred 11748	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
16	15	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 𝐴 ∈ ℝ)
17	13, 16	rpcxpcld 24276	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐𝐴) ∈ ℝ⁺)
18		ppinncl 24700	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ ∧ 2 ≤ 𝑥) → (π‘𝑥) ∈ ℕ)
19	5, 18	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (π‘𝑥) ∈ ℕ)
20	19	nnrpd 11746	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (π‘𝑥) ∈ ℝ⁺)
21	17, 20	rpdivcld 11765	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐𝐴) / (π‘𝑥)) ∈ ℝ⁺)
22	21	ralrimiva 2949	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (2[,)+∞)((𝑥↑_𝑐𝐴) / (π‘𝑥)) ∈ ℝ⁺)
23		chtppilim.2	. . . 4 ⊢ (𝜑 → 𝐴 < 1)
24		1re 9918	. . . . 5 ⊢ 1 ∈ ℝ
25		difrp 11744	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐴 < 1 ↔ (1 − 𝐴) ∈ ℝ⁺))
26	15, 24, 25	sylancl 693	. . . 4 ⊢ (𝜑 → (𝐴 < 1 ↔ (1 − 𝐴) ∈ ℝ⁺))
27	23, 26	mpbid 221	. . 3 ⊢ (𝜑 → (1 − 𝐴) ∈ ℝ⁺)
28		ovex 6577	. . . . . . 7 ⊢ (2[,)+∞) ∈ V
29	28	a1i 11	. . . . . 6 ⊢ (𝜑 → (2[,)+∞) ∈ V)
30	24	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 1 ∈ ℝ)
31		1lt2 11071	. . . . . . . . . . 11 ⊢ 1 < 2
32	31	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 1 < 2)
33	30, 8, 6, 32, 11	ltletrd 10076	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 1 < 𝑥)
34	6, 33	rplogcld 24179	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (log‘𝑥) ∈ ℝ⁺)
35	13, 34	rpdivcld 11765	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥 / (log‘𝑥)) ∈ ℝ⁺)
36	35, 20	rpdivcld 11765	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥 / (log‘𝑥)) / (π‘𝑥)) ∈ ℝ⁺)
37	27	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (1 − 𝐴) ∈ ℝ⁺)
38	37	rpred 11748	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (1 − 𝐴) ∈ ℝ)
39	13, 38	rpcxpcld 24276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐(1 − 𝐴)) ∈ ℝ⁺)
40	34, 39	rpdivcld 11765	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))) ∈ ℝ⁺)
41		eqidd 2611	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (2[,)+∞) ↦ ((𝑥 / (log‘𝑥)) / (π‘𝑥))) = (𝑥 ∈ (2[,)+∞) ↦ ((𝑥 / (log‘𝑥)) / (π‘𝑥))))
42		eqidd 2611	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (2[,)+∞) ↦ ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) = (𝑥 ∈ (2[,)+∞) ↦ ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))))
43	29, 36, 40, 41, 42	offval2 6812	. . . . 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))))))
44	35	rpcnd 11750	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥 / (log‘𝑥)) ∈ ℂ)
45	40	rpcnd 11750	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))) ∈ ℂ)
46	20	rpcnne0d 11757	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((π‘𝑥) ∈ ℂ ∧ (π‘𝑥) ≠ 0))
47		div23 10583	. . . . . . . 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)))))
48	44, 45, 46, 47	syl3anc 1318	. . . . . . 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)))))
49	34	rpcnne0d 11757	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((log‘𝑥) ∈ ℂ ∧ (log‘𝑥) ≠ 0))
50	39	rpcnne0d 11757	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐(1 − 𝐴)) ∈ ℂ ∧ (𝑥↑_𝑐(1 − 𝐴)) ≠ 0))
51	6	recnd 9947	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 𝑥 ∈ ℂ)
52		dmdcan 10614	. . . . . . . . . 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))))
53	49, 50, 51, 52	syl3anc 1318	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))) · (𝑥 / (log‘𝑥))) = (𝑥 / (𝑥↑_𝑐(1 − 𝐴))))
54	44, 45	mulcomd 9940	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥 / (log‘𝑥)) · ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) = (((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))) · (𝑥 / (log‘𝑥))))
55	13	rpcnne0d 11757	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
56		ax-1cn 9873	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
57	56	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 1 ∈ ℂ)
58	37	rpcnd 11750	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (1 − 𝐴) ∈ ℂ)
59		cxpsub 24228	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ 1 ∈ ℂ ∧ (1 − 𝐴) ∈ ℂ) → (𝑥↑_𝑐(1 − (1 − 𝐴))) = ((𝑥↑_𝑐1) / (𝑥↑_𝑐(1 − 𝐴))))
60	55, 57, 58, 59	syl3anc 1318	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐(1 − (1 − 𝐴))) = ((𝑥↑_𝑐1) / (𝑥↑_𝑐(1 − 𝐴))))
61	16	recnd 9947	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 𝐴 ∈ ℂ)
62		nncan 10189	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − (1 − 𝐴)) = 𝐴)
63	56, 61, 62	sylancr 694	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (1 − (1 − 𝐴)) = 𝐴)
64	63	oveq2d 6565	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐(1 − (1 − 𝐴))) = (𝑥↑_𝑐𝐴))
65	60, 64	eqtr3d 2646	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐1) / (𝑥↑_𝑐(1 − 𝐴))) = (𝑥↑_𝑐𝐴))
66	51	cxp1d 24252	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐1) = 𝑥)
67	66	oveq1d 6564	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐1) / (𝑥↑_𝑐(1 − 𝐴))) = (𝑥 / (𝑥↑_𝑐(1 − 𝐴))))
68	65, 67	eqtr3d 2646	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (𝑥↑_𝑐𝐴) = (𝑥 / (𝑥↑_𝑐(1 − 𝐴))))
69	53, 54, 68	3eqtr4d 2654	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥 / (log‘𝑥)) · ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) = (𝑥↑_𝑐𝐴))
70	69	oveq1d 6564	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (((𝑥 / (log‘𝑥)) · ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) / (π‘𝑥)) = ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
71	48, 70	eqtr3d 2646	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (((𝑥 / (log‘𝑥)) / (π‘𝑥)) · ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) = ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
72	71	mpteq2dva 4672	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (2[,)+∞) ↦ (((𝑥 / (log‘𝑥)) / (π‘𝑥)) · ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))))) = (𝑥 ∈ (2[,)+∞) ↦ ((𝑥↑_𝑐𝐴) / (π‘𝑥))))
73	43, 72	eqtrd 2644	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ (2[,)+∞) ↦ ((𝑥 / (log‘𝑥)) / (π‘𝑥))) ∘_𝑓 · (𝑥 ∈ (2[,)+∞) ↦ ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))))) = (𝑥 ∈ (2[,)+∞) ↦ ((𝑥↑_𝑐𝐴) / (π‘𝑥))))
74		chebbnd1 24961	. . . . 5 ⊢ (𝑥 ∈ (2[,)+∞) ↦ ((𝑥 / (log‘𝑥)) / (π‘𝑥))) ∈ 𝑂(1)
75	13	ex 449	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (2[,)+∞) → 𝑥 ∈ ℝ⁺))
76	75	ssrdv 3574	. . . . . 6 ⊢ (𝜑 → (2[,)+∞) ⊆ ℝ⁺)
77		cxploglim 24504	. . . . . . 7 ⊢ ((1 − 𝐴) ∈ ℝ⁺ → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) ⇝_𝑟 0)
78	27, 77	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) ⇝_𝑟 0)
79	76, 78	rlimres2 14140	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (2[,)+∞) ↦ ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴)))) ⇝_𝑟 0)
80		o1rlimmul 14197	. . . . 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)
81	74, 79, 80	sylancr 694	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ (2[,)+∞) ↦ ((𝑥 / (log‘𝑥)) / (π‘𝑥))) ∘_𝑓 · (𝑥 ∈ (2[,)+∞) ↦ ((log‘𝑥) / (𝑥↑_𝑐(1 − 𝐴))))) ⇝_𝑟 0)
82	73, 81	eqbrtrrd 4607	. . 3 ⊢ (𝜑 → (𝑥 ∈ (2[,)+∞) ↦ ((𝑥↑_𝑐𝐴) / (π‘𝑥))) ⇝_𝑟 0)
83	22, 27, 82	rlimi 14092	. 2 ⊢ (𝜑 → ∃𝑧 ∈ ℝ ∀𝑥 ∈ (2[,)+∞)(𝑧 ≤ 𝑥 → (abs‘(((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0)) < (1 − 𝐴)))
84	21	rpcnd 11750	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐𝐴) / (π‘𝑥)) ∈ ℂ)
85	84	subid1d 10260	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0) = ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
86	85	fveq2d 6107	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (abs‘(((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0)) = (abs‘((𝑥↑_𝑐𝐴) / (π‘𝑥))))
87	21	rpred 11748	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑥↑_𝑐𝐴) / (π‘𝑥)) ∈ ℝ)
88	21	rpge0d 11752	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → 0 ≤ ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
89	87, 88	absidd 14009	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (abs‘((𝑥↑_𝑐𝐴) / (π‘𝑥))) = ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
90	86, 89	eqtrd 2644	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (abs‘(((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0)) = ((𝑥↑_𝑐𝐴) / (π‘𝑥)))
91	90	breq1d 4593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((abs‘(((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0)) < (1 − 𝐴) ↔ ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴)))
92	14	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (2[,)+∞) ∧ ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴))) → 𝐴 ∈ ℝ⁺)
93	23	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (2[,)+∞) ∧ ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴))) → 𝐴 < 1)
94		simprl 790	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (2[,)+∞) ∧ ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴))) → 𝑥 ∈ (2[,)+∞))
95		simprr 792	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (2[,)+∞) ∧ ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴))) → ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴))
96	92, 93, 94, 95	chtppilimlem1 24962	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (2[,)+∞) ∧ ((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴))) → ((𝐴↑2) · ((π‘𝑥) · (log‘𝑥))) < (θ‘𝑥))
97	96	expr 641	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → (((𝑥↑_𝑐𝐴) / (π‘𝑥)) < (1 − 𝐴) → ((𝐴↑2) · ((π‘𝑥) · (log‘𝑥))) < (θ‘𝑥)))
98	91, 97	sylbid 229	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((abs‘(((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0)) < (1 − 𝐴) → ((𝐴↑2) · ((π‘𝑥) · (log‘𝑥))) < (θ‘𝑥)))
99	98	imim2d 55	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (2[,)+∞)) → ((𝑧 ≤ 𝑥 → (abs‘(((𝑥↑_𝑐𝐴) / (π‘𝑥)) − 0)) < (1 − 𝐴)) → (𝑧 ≤ 𝑥 → ((𝐴↑2) · ((π‘𝑥) · (log‘𝑥))) < (θ‘𝑥))))
100	99	ralimdva 2945	. . 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))))
101	100	reximdv 2999	. 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))))
102	83, 101	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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 Vcvv 3173 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ∘_𝑓 cof 6793 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 · cmul 9820 +∞cpnf 9950 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℕcn 10897 2c2 10947 ℝ⁺crp 11708 [,)cico 12048 ↑cexp 12722 abscabs 13822 ⇝_𝑟 crli 14064 𝑂(1)co1 14065 logclog 24105 ↑_𝑐ccxp 24106 θccht 24617 πcppi 24620

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-xnn0 11241 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-o1 14069 df-lo1 14070 df-sum 14265 df-ef 14637 df-e 14638 df-sin 14639 df-cos 14640 df-pi 14642 df-dvds 14822 df-gcd 15055 df-prm 15224 df-pc 15380 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-limc 23436 df-dv 23437 df-log 24107 df-cxp 24108 df-cht 24623 df-ppi 24626

This theorem is referenced by: chtppilim 24964

W3C validator