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Theorem cxploglim 27100

Description: The logarithm grows slower than any positive power. (Contributed by Mario Carneiro, 18-Sep-2014.)

**Assertion**
Ref	Expression
cxploglim	⊢ (𝐴 ∈ ℝ⁺ → (𝑛 ∈ ℝ⁺ ↦ ((log‘𝑛) / (𝑛↑_𝑐𝐴))) ⇝_𝑟 0)

Distinct variable group: 𝐴,𝑛

Proof of Theorem cxploglim Dummy variables 𝑚 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpre 13016	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		reefcl 16131	. . . 4 ⊢ (𝐴 ∈ ℝ → (exp‘𝐴) ∈ ℝ)
3	1, 2	syl 18	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘𝐴) ∈ ℝ)
4		efgt1 16162	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → 1 < (exp‘𝐴))
5		cxp2limlem 27098	. . 3 ⊢ (((exp‘𝐴) ∈ ℝ ∧ 1 < (exp‘𝐴)) → (𝑚 ∈ ℝ⁺ ↦ (𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) ⇝_𝑟 0)
6	3, 4, 5	syl2anc 595	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (𝑚 ∈ ℝ⁺ ↦ (𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) ⇝_𝑟 0)
7		reefcl 16131	. . . . . . . 8 ⊢ (𝑧 ∈ ℝ → (exp‘𝑧) ∈ ℝ)
8	7	adantl 486	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) → (exp‘𝑧) ∈ ℝ)
9		1re 11196	. . . . . . 7 ⊢ 1 ∈ ℝ
10		ifcl 4529	. . . . . . 7 ⊢ (((exp‘𝑧) ∈ ℝ ∧ 1 ∈ ℝ) → if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) ∈ ℝ)
11	8, 9, 10	sylancl 597	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) → if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) ∈ ℝ)
12		rpre 13016	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℝ)
13		maxlt 13210	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ (exp‘𝑧) ∈ ℝ ∧ 𝑛 ∈ ℝ) → (if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) < 𝑛 ↔ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛)))
14	9, 8, 12, 13	mp3an3an 1491	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ 𝑛 ∈ ℝ⁺) → (if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) < 𝑛 ↔ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛)))
15		simprrr 793	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘𝑧) < 𝑛)
16		reeflog 26703	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℝ⁺ → (exp‘(log‘𝑛)) = 𝑛)
17	16	ad2antrl 740	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘(log‘𝑛)) = 𝑛)
18	15, 17	breqtrrd 5133	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘𝑧) < (exp‘(log‘𝑛)))
19		simplr 780	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝑧 ∈ ℝ)
20	12	ad2antrl 740	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝑛 ∈ ℝ)
21		simprrl 792	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 1 < 𝑛)
22	20, 21	rplogcld 26752	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (log‘𝑛) ∈ ℝ⁺)
23	22	rpred 13051	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (log‘𝑛) ∈ ℝ)
24		eflt 16163	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℝ ∧ (log‘𝑛) ∈ ℝ) → (𝑧 < (log‘𝑛) ↔ (exp‘𝑧) < (exp‘(log‘𝑛))))
25	19, 23, 24	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (𝑧 < (log‘𝑛) ↔ (exp‘𝑧) < (exp‘(log‘𝑛))))
26	18, 25	mpbird 260	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝑧 < (log‘𝑛))
27		breq2 5109	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (log‘𝑛) → (𝑧 < 𝑚 ↔ 𝑧 < (log‘𝑛)))
28		id 23	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = (log‘𝑛) → 𝑚 = (log‘𝑛))
29		oveq2 7408	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = (log‘𝑛) → ((exp‘𝐴)↑_𝑐𝑚) = ((exp‘𝐴)↑_𝑐(log‘𝑛)))
30	28, 29	oveq12d 7418	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = (log‘𝑛) → (𝑚 / ((exp‘𝐴)↑_𝑐𝑚)) = ((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛))))
31	30	fveq2d 6875	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (log‘𝑛) → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) = (abs‘((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛)))))
32	31	breq1d 5115	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (log‘𝑛) → ((abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥 ↔ (abs‘((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛)))) < 𝑥))
33	27, 32	imbi12d 347	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (log‘𝑛) → ((𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) ↔ (𝑧 < (log‘𝑛) → (abs‘((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛)))) < 𝑥)))
34	33	rspcv 3580	. . . . . . . . . . . . 13 ⊢ ((log‘𝑛) ∈ ℝ⁺ → (∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) → (𝑧 < (log‘𝑛) → (abs‘((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛)))) < 𝑥)))
35	22, 34	syl 18	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) → (𝑧 < (log‘𝑛) → (abs‘((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛)))) < 𝑥)))
36	26, 35	mpid 45	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) → (abs‘((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛)))) < 𝑥))
37	1	ad2antrr 738	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝐴 ∈ ℝ)
38	37	relogefd 26751	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (log‘(exp‘𝐴)) = 𝐴)
39	38	oveq2d 7416	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((log‘𝑛) · (log‘(exp‘𝐴))) = ((log‘𝑛) · 𝐴))
40	22	rpcnd 13053	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (log‘𝑛) ∈ ℂ)
41		rpcn 13018	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
42	41	ad2antrr 738	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝐴 ∈ ℂ)
43	40, 42	mulcomd 11218	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((log‘𝑛) · 𝐴) = (𝐴 · (log‘𝑛)))
44	39, 43	eqtrd 2800	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((log‘𝑛) · (log‘(exp‘𝐴))) = (𝐴 · (log‘𝑛)))
45	44	fveq2d 6875	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘((log‘𝑛) · (log‘(exp‘𝐴)))) = (exp‘(𝐴 · (log‘𝑛))))
46	3	ad2antrr 738	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘𝐴) ∈ ℝ)
47	46	recnd 11225	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘𝐴) ∈ ℂ)
48		efne0 16142	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ≠ 0)
49	42, 48	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘𝐴) ≠ 0)
50	47, 49, 40	cxpefd 26835	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((exp‘𝐴)↑_𝑐(log‘𝑛)) = (exp‘((log‘𝑛) · (log‘(exp‘𝐴)))))
51		rpcn 13018	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℂ)
52	51	ad2antrl 740	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝑛 ∈ ℂ)
53		rpne0 13024	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ≠ 0)
54	53	ad2antrl 740	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝑛 ≠ 0)
55	52, 54, 42	cxpefd 26835	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (𝑛↑_𝑐𝐴) = (exp‘(𝐴 · (log‘𝑛))))
56	45, 50, 55	3eqtr4d 2810	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((exp‘𝐴)↑_𝑐(log‘𝑛)) = (𝑛↑_𝑐𝐴))
57	56	oveq2d 7416	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛))) = ((log‘𝑛) / (𝑛↑_𝑐𝐴)))
58	57	fveq2d 6875	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (abs‘((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛)))) = (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))))
59	58	breq1d 5115	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((abs‘((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛)))) < 𝑥 ↔ (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥))
60	36, 59	sylibd 242	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥))
61	60	expr 461	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ 𝑛 ∈ ℝ⁺) → ((1 < 𝑛 ∧ (exp‘𝑧) < 𝑛) → (∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥)))
62	14, 61	sylbid 243	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ 𝑛 ∈ ℝ⁺) → (if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) < 𝑛 → (∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥)))
63	62	com23 87	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ 𝑛 ∈ ℝ⁺) → (∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) → (if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) < 𝑛 → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥)))
64	63	ralrimdva 3165	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) → (∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) → ∀𝑛 ∈ ℝ⁺ (if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) < 𝑛 → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥)))
65		breq1 5108	. . . . . . 7 ⊢ (𝑦 = if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) → (𝑦 < 𝑛 ↔ if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) < 𝑛))
66	65	rspceaimv 3590	. . . . . 6 ⊢ ((if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) ∈ ℝ ∧ ∀𝑛 ∈ ℝ⁺ (if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) < 𝑛 → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥)) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥))
67	11, 64, 66	syl6an 696	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) → (∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥)))
68	67	rexlimdva 3166	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (∃𝑧 ∈ ℝ ∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥)))
69	68	ralimdv 3179	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (∀𝑥 ∈ ℝ⁺ ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥) → ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥)))
70		simpr 489	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → 𝑚 ∈ ℝ⁺)
71	1	adantr 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
72	71	rpefcld 16151	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → (exp‘𝐴) ∈ ℝ⁺)
73		rpre 13016	. . . . . . . . 9 ⊢ (𝑚 ∈ ℝ⁺ → 𝑚 ∈ ℝ)
74	73	adantl 486	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → 𝑚 ∈ ℝ)
75	72, 74	rpcxpcld 26856	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → ((exp‘𝐴)↑_𝑐𝑚) ∈ ℝ⁺)
76	70, 75	rpdivcld 13068	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → (𝑚 / ((exp‘𝐴)↑_𝑐𝑚)) ∈ ℝ⁺)
77	76	rpcnd 13053	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → (𝑚 / ((exp‘𝐴)↑_𝑐𝑚)) ∈ ℂ)
78	77	ralrimiva 3157	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ∀𝑚 ∈ ℝ⁺ (𝑚 / ((exp‘𝐴)↑_𝑐𝑚)) ∈ ℂ)
79		rpssre 13015	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
80	79	a1i 11	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ℝ⁺ ⊆ ℝ)
81	78, 80	rlim0lt 15550	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((𝑚 ∈ ℝ⁺ ↦ (𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) ⇝_𝑟 0 ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℝ⁺ (𝑧 < 𝑚 → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥)))
82		relogcl 26698	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ⁺ → (log‘𝑛) ∈ ℝ)
83	82	adantl 486	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (log‘𝑛) ∈ ℝ)
84		simpr 489	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → 𝑛 ∈ ℝ⁺)
85	1	adantr 485	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
86	84, 85	rpcxpcld 26856	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑛↑_𝑐𝐴) ∈ ℝ⁺)
87	83, 86	rerpdivcld 13082	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → ((log‘𝑛) / (𝑛↑_𝑐𝐴)) ∈ ℝ)
88	87	recnd 11225	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → ((log‘𝑛) / (𝑛↑_𝑐𝐴)) ∈ ℂ)
89	88	ralrimiva 3157	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ∀𝑛 ∈ ℝ⁺ ((log‘𝑛) / (𝑛↑_𝑐𝐴)) ∈ ℂ)
90	89, 80	rlim0lt 15550	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((𝑛 ∈ ℝ⁺ ↦ ((log‘𝑛) / (𝑛↑_𝑐𝐴))) ⇝_𝑟 0 ↔ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ ∀𝑛 ∈ ℝ⁺ (𝑦 < 𝑛 → (abs‘((log‘𝑛) / (𝑛↑_𝑐𝐴))) < 𝑥)))
91	69, 81, 90	3imtr4d 297	. 2 ⊢ (𝐴 ∈ ℝ⁺ → ((𝑚 ∈ ℝ⁺ ↦ (𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) ⇝_𝑟 0 → (𝑛 ∈ ℝ⁺ ↦ ((log‘𝑛) / (𝑛↑_𝑐𝐴))) ⇝_𝑟 0))
92	6, 91	mpd 16	1 ⊢ (𝐴 ∈ ℝ⁺ → (𝑛 ∈ ℝ⁺ ↦ ((log‘𝑛) / (𝑛↑_𝑐𝐴))) ⇝_𝑟 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ≠ wne 2960 ∀wral 3079 ∃wrex 3089 ⊆ wss 3907 ifcif 4483 class class class wbr 5105 ↦ cmpt 5186 ‘cfv 6525 (class class class)co 7400 ℂcc 11086 ℝcr 11087 0cc0 11088 1c1 11089 · cmul 11093 < clt 11231 ≤ cle 11232 / cdiv 11859 ℝ⁺crp 13007 abscabs 15275 ⇝_𝑟 crli 15526 expce 16105 logclog 26677 ↑_𝑐ccxp 26678

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-inf2 9598 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166 ax-addf 11167

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-iin 4955 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-se 5606 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-om 7851 df-1st 7974 df-2nd 7975 df-supp 8145 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-map 8814 df-pm 8815 df-ixp 8884 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13367 df-ioc 13368 df-ico 13369 df-icc 13370 df-fz 13527 df-fzo 13674 df-fl 13816 df-mod 13894 df-seq 14029 df-exp 14089 df-fac 14301 df-bc 14330 df-hash 14358 df-shft 15094 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277 df-limsup 15512 df-clim 15529 df-rlim 15530 df-sum 15728 df-ef 16111 df-sin 16113 df-cos 16114 df-pi 16116 df-struct 17197 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-mulr 17314 df-starv 17315 df-sca 17316 df-vsca 17317 df-ip 17318 df-tset 17319 df-ple 17320 df-ds 17322 df-unif 17323 df-hom 17324 df-cco 17325 df-rest 17465 df-topn 17466 df-0g 17484 df-gsum 17485 df-topgen 17486 df-pt 17487 df-prds 17490 df-xrs 17546 df-qtop 17551 df-imas 17552 df-xps 17554 df-mre 17628 df-mrc 17629 df-acs 17631 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-submnd 18832 df-mulg 19125 df-cntz 19378 df-cmn 19843 df-psmet 21474 df-xmet 21475 df-met 21476 df-bl 21477 df-mopn 21478 df-fbas 21479 df-fg 21480 df-cnfld 21483 df-top 23012 df-topon 23029 df-topsp 23051 df-bases 23064 df-cld 23137 df-ntr 23138 df-cls 23139 df-nei 23216 df-lp 23254 df-perf 23255 df-cn 23345 df-cnp 23346 df-haus 23433 df-tx 23680 df-hmeo 23873 df-fil 23964 df-fm 24056 df-flim 24057 df-flf 24058 df-xms 24438 df-ms 24439 df-tms 24440 df-cncf 24998 df-limc 25986 df-dv 25987 df-log 26679 df-cxp 26680

This theorem is referenced by: cxploglim2 27101 logfacrlim 27346 chtppilimlem2 27596 chpchtlim 27601 dchrvmasumlema 27622 logdivsum 27655

W3C validator