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Theorem cxploglim 26127

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 12738	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		reefcl 15796	. . . 4 ⊢ (𝐴 ∈ ℝ → (exp‘𝐴) ∈ ℝ)
3	1, 2	syl 17	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘𝐴) ∈ ℝ)
4		efgt1 15825	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → 1 < (exp‘𝐴))
5		cxp2limlem 26125	. . 3 ⊢ (((exp‘𝐴) ∈ ℝ ∧ 1 < (exp‘𝐴)) → (𝑚 ∈ ℝ⁺ ↦ (𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) ⇝_𝑟 0)
6	3, 4, 5	syl2anc 584	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (𝑚 ∈ ℝ⁺ ↦ (𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) ⇝_𝑟 0)
7		reefcl 15796	. . . . . . . 8 ⊢ (𝑧 ∈ ℝ → (exp‘𝑧) ∈ ℝ)
8	7	adantl 482	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) → (exp‘𝑧) ∈ ℝ)
9		1re 10975	. . . . . . 7 ⊢ 1 ∈ ℝ
10		ifcl 4504	. . . . . . 7 ⊢ (((exp‘𝑧) ∈ ℝ ∧ 1 ∈ ℝ) → if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) ∈ ℝ)
11	8, 9, 10	sylancl 586	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) → if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) ∈ ℝ)
12		rpre 12738	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℝ)
13		maxlt 12927	. . . . . . . . . 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 1466	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ 𝑛 ∈ ℝ⁺) → (if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) < 𝑛 ↔ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛)))
15		simprrr 779	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘𝑧) < 𝑛)
16		reeflog 25736	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℝ⁺ → (exp‘(log‘𝑛)) = 𝑛)
17	16	ad2antrl 725	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘(log‘𝑛)) = 𝑛)
18	15, 17	breqtrrd 5102	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘𝑧) < (exp‘(log‘𝑛)))
19		simplr 766	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝑧 ∈ ℝ)
20	12	ad2antrl 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝑛 ∈ ℝ)
21		simprrl 778	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 1 < 𝑛)
22	20, 21	rplogcld 25784	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (log‘𝑛) ∈ ℝ⁺)
23	22	rpred 12772	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (log‘𝑛) ∈ ℝ)
24		eflt 15826	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℝ ∧ (log‘𝑛) ∈ ℝ) → (𝑧 < (log‘𝑛) ↔ (exp‘𝑧) < (exp‘(log‘𝑛))))
25	19, 23, 24	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (𝑧 < (log‘𝑛) ↔ (exp‘𝑧) < (exp‘(log‘𝑛))))
26	18, 25	mpbird 256	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝑧 < (log‘𝑛))
27		breq2 5078	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (log‘𝑛) → (𝑧 < 𝑚 ↔ 𝑧 < (log‘𝑛)))
28		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = (log‘𝑛) → 𝑚 = (log‘𝑛))
29		oveq2 7283	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = (log‘𝑛) → ((exp‘𝐴)↑_𝑐𝑚) = ((exp‘𝐴)↑_𝑐(log‘𝑛)))
30	28, 29	oveq12d 7293	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = (log‘𝑛) → (𝑚 / ((exp‘𝐴)↑_𝑐𝑚)) = ((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛))))
31	30	fveq2d 6778	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (log‘𝑛) → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) = (abs‘((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛)))))
32	31	breq1d 5084	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (log‘𝑛) → ((abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥 ↔ (abs‘((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛)))) < 𝑥))
33	27, 32	imbi12d 345	. . . . . . . . . . . . . 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 3557	. . . . . . . . . . . . 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 17	. . . . . . . . . . . 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 44	. . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝐴 ∈ ℝ)
38	37	relogefd 25783	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (log‘(exp‘𝐴)) = 𝐴)
39	38	oveq2d 7291	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((log‘𝑛) · (log‘(exp‘𝐴))) = ((log‘𝑛) · 𝐴))
40	22	rpcnd 12774	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (log‘𝑛) ∈ ℂ)
41		rpcn 12740	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
42	41	ad2antrr 723	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝐴 ∈ ℂ)
43	40, 42	mulcomd 10996	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((log‘𝑛) · 𝐴) = (𝐴 · (log‘𝑛)))
44	39, 43	eqtrd 2778	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((log‘𝑛) · (log‘(exp‘𝐴))) = (𝐴 · (log‘𝑛)))
45	44	fveq2d 6778	. . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘𝐴) ∈ ℝ)
47	46	recnd 11003	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘𝐴) ∈ ℂ)
48		efne0 15806	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ≠ 0)
49	42, 48	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘𝐴) ≠ 0)
50	47, 49, 40	cxpefd 25867	. . . . . . . . . . . . . . 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 12740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℂ)
52	51	ad2antrl 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝑛 ∈ ℂ)
53		rpne0 12746	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ≠ 0)
54	53	ad2antrl 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝑛 ≠ 0)
55	52, 54, 42	cxpefd 25867	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (𝑛↑_𝑐𝐴) = (exp‘(𝐴 · (log‘𝑛))))
56	45, 50, 55	3eqtr4d 2788	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((exp‘𝐴)↑_𝑐(log‘𝑛)) = (𝑛↑_𝑐𝐴))
57	56	oveq2d 7291	. . . . . . . . . . . . 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 6778	. . . . . . . . . . . 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 5084	. . . . . . . . . . 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 238	. . . . . . . . . 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 457	. . . . . . . . 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 239	. . . . . . . 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 86	. . . . . . 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 3106	. . . . . 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 5077	. . . . . . 7 ⊢ (𝑦 = if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) → (𝑦 < 𝑛 ↔ if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) < 𝑛))
66	65	rspceaimv 3565	. . . . . 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 681	. . . . 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 3213	. . . 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 3109	. . 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 485	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → 𝑚 ∈ ℝ⁺)
71	1	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
72	71	rpefcld 15814	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → (exp‘𝐴) ∈ ℝ⁺)
73		rpre 12738	. . . . . . . . 9 ⊢ (𝑚 ∈ ℝ⁺ → 𝑚 ∈ ℝ)
74	73	adantl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → 𝑚 ∈ ℝ)
75	72, 74	rpcxpcld 25887	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → ((exp‘𝐴)↑_𝑐𝑚) ∈ ℝ⁺)
76	70, 75	rpdivcld 12789	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → (𝑚 / ((exp‘𝐴)↑_𝑐𝑚)) ∈ ℝ⁺)
77	76	rpcnd 12774	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → (𝑚 / ((exp‘𝐴)↑_𝑐𝑚)) ∈ ℂ)
78	77	ralrimiva 3103	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ∀𝑚 ∈ ℝ⁺ (𝑚 / ((exp‘𝐴)↑_𝑐𝑚)) ∈ ℂ)
79		rpssre 12737	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
80	79	a1i 11	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ℝ⁺ ⊆ ℝ)
81	78, 80	rlim0lt 15218	. . 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 25731	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ⁺ → (log‘𝑛) ∈ ℝ)
83	82	adantl 482	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (log‘𝑛) ∈ ℝ)
84		simpr 485	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → 𝑛 ∈ ℝ⁺)
85	1	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
86	84, 85	rpcxpcld 25887	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑛↑_𝑐𝐴) ∈ ℝ⁺)
87	83, 86	rerpdivcld 12803	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → ((log‘𝑛) / (𝑛↑_𝑐𝐴)) ∈ ℝ)
88	87	recnd 11003	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → ((log‘𝑛) / (𝑛↑_𝑐𝐴)) ∈ ℂ)
89	88	ralrimiva 3103	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ∀𝑛 ∈ ℝ⁺ ((log‘𝑛) / (𝑛↑_𝑐𝐴)) ∈ ℂ)
90	89, 80	rlim0lt 15218	. . 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 294	. 2 ⊢ (𝐴 ∈ ℝ⁺ → ((𝑚 ∈ ℝ⁺ ↦ (𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) ⇝_𝑟 0 → (𝑛 ∈ ℝ⁺ ↦ ((log‘𝑛) / (𝑛↑_𝑐𝐴))) ⇝_𝑟 0))
92	6, 91	mpd 15	1 ⊢ (𝐴 ∈ ℝ⁺ → (𝑛 ∈ ℝ⁺ ↦ ((log‘𝑛) / (𝑛↑_𝑐𝐴))) ⇝_𝑟 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 ⊆ wss 3887 ifcif 4459 class class class wbr 5074 ↦ cmpt 5157 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 · cmul 10876 < clt 11009 ≤ cle 11010 / cdiv 11632 ℝ⁺crp 12730 abscabs 14945 ⇝_𝑟 crli 15194 expce 15771 logclog 25710 ↑_𝑐ccxp 25711

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-ef 15777 df-sin 15779 df-cos 15780 df-pi 15782 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 df-log 25712 df-cxp 25713

This theorem is referenced by: cxploglim2 26128 logfacrlim 26372 chtppilimlem2 26622 chpchtlim 26627 dchrvmasumlema 26648 logdivsum 26681

W3C validator