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Theorem cxploglim 27153

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 13031	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		reefcl 16147	. . . 4 ⊢ (𝐴 ∈ ℝ → (exp‘𝐴) ∈ ℝ)
3	1, 2	syl 18	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘𝐴) ∈ ℝ)
4		efgt1 16178	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → 1 < (exp‘𝐴))
5		cxp2limlem 27151	. . 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 16147	. . . . . . . 8 ⊢ (𝑧 ∈ ℝ → (exp‘𝑧) ∈ ℝ)
8	7	adantl 486	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) → (exp‘𝑧) ∈ ℝ)
9		1re 11214	. . . . . . 7 ⊢ 1 ∈ ℝ
10		ifcl 4532	. . . . . . 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 13031	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℝ)
13		maxlt 13225	. . . . . . . . . 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 1495	. . . . . . . . 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 26756	. . . . . . . . . . . . . . 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 5138	. . . . . . . . . . . . 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 26805	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (log‘𝑛) ∈ ℝ⁺)
23	22	rpred 13066	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (log‘𝑛) ∈ ℝ)
24		eflt 16179	. . . . . . . . . . . . . 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 5112	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (log‘𝑛) → (𝑧 < 𝑚 ↔ 𝑧 < (log‘𝑛)))
28		id 23	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = (log‘𝑛) → 𝑚 = (log‘𝑛))
29		oveq2 7420	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = (log‘𝑛) → ((exp‘𝐴)↑_𝑐𝑚) = ((exp‘𝐴)↑_𝑐(log‘𝑛)))
30	28, 29	oveq12d 7430	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = (log‘𝑛) → (𝑚 / ((exp‘𝐴)↑_𝑐𝑚)) = ((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛))))
31	30	fveq2d 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (log‘𝑛) → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) = (abs‘((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛)))))
32	31	breq1d 5118	. . . . . . . . . . . . . . 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 3576	. . . . . . . . . . . . 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 26804	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (log‘(exp‘𝐴)) = 𝐴)
39	38	oveq2d 7428	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((log‘𝑛) · (log‘(exp‘𝐴))) = ((log‘𝑛) · 𝐴))
40	22	rpcnd 13068	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (log‘𝑛) ∈ ℂ)
41		rpcn 13033	. . . . . . . . . . . . . . . . . . 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 11236	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((log‘𝑛) · 𝐴) = (𝐴 · (log‘𝑛)))
44	39, 43	eqtrd 2797	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((log‘𝑛) · (log‘(exp‘𝐴))) = (𝐴 · (log‘𝑛)))
45	44	fveq2d 6885	. . . . . . . . . . . . . . 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 11243	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘𝐴) ∈ ℂ)
48		efne0 16158	. . . . . . . . . . . . . . . . 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 26888	. . . . . . . . . . . . . . 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 13033	. . . . . . . . . . . . . . . . 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 13039	. . . . . . . . . . . . . . . . 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 26888	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (𝑛↑_𝑐𝐴) = (exp‘(𝐴 · (log‘𝑛))))
56	45, 50, 55	3eqtr4d 2807	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((exp‘𝐴)↑_𝑐(log‘𝑛)) = (𝑛↑_𝑐𝐴))
57	56	oveq2d 7428	. . . . . . . . . . . . 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 6885	. . . . . . . . . . . 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 5118	. . . . . . . . . . 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 3164	. . . . . 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 5111	. . . . . . 7 ⊢ (𝑦 = if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) → (𝑦 < 𝑛 ↔ if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) < 𝑛))
66	65	rspceaimv 3586	. . . . . 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 3165	. . . 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 3178	. . 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 16167	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → (exp‘𝐴) ∈ ℝ⁺)
73		rpre 13031	. . . . . . . . 9 ⊢ (𝑚 ∈ ℝ⁺ → 𝑚 ∈ ℝ)
74	73	adantl 486	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → 𝑚 ∈ ℝ)
75	72, 74	rpcxpcld 26909	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → ((exp‘𝐴)↑_𝑐𝑚) ∈ ℝ⁺)
76	70, 75	rpdivcld 13083	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → (𝑚 / ((exp‘𝐴)↑_𝑐𝑚)) ∈ ℝ⁺)
77	76	rpcnd 13068	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → (𝑚 / ((exp‘𝐴)↑_𝑐𝑚)) ∈ ℂ)
78	77	ralrimiva 3156	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ∀𝑚 ∈ ℝ⁺ (𝑚 / ((exp‘𝐴)↑_𝑐𝑚)) ∈ ℂ)
79		rpssre 13030	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
80	79	a1i 11	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ℝ⁺ ⊆ ℝ)
81	78, 80	rlim0lt 15567	. . 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 26751	. . . . . . . 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 26909	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑛↑_𝑐𝐴) ∈ ℝ⁺)
87	83, 86	rerpdivcld 13097	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → ((log‘𝑛) / (𝑛↑_𝑐𝐴)) ∈ ℝ)
88	87	recnd 11243	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → ((log‘𝑛) / (𝑛↑_𝑐𝐴)) ∈ ℂ)
89	88	ralrimiva 3156	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ∀𝑛 ∈ ℝ⁺ ((log‘𝑛) / (𝑛↑_𝑐𝐴)) ∈ ℂ)
90	89, 80	rlim0lt 15567	. . 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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1569 ∈ wcel 2142 ≠ wne 2957 ∀wral 3078 ∃wrex 3088 ⊆ wss 3904 ifcif 4486 class class class wbr 5108 ↦ cmpt 5191 ‘cfv 6536 (class class class)co 7412 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 · cmul 11111 < clt 11249 ≤ cle 11250 / cdiv 11877 ℝ⁺crp 13022 abscabs 15292 ⇝_𝑟 crli 15543 expce 16121 logclog 26730 ↑_𝑐ccxp 26731

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-inf2 9608 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-se 5614 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8155 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-er 8692 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-div 11878 df-nn 12240 df-2 12309 df-3 12310 df-4 12311 df-5 12312 df-6 12313 df-7 12314 df-8 12315 df-9 12316 df-n0 12511 df-z 12598 df-dec 12718 df-uz 12869 df-q 12979 df-rp 13023 df-xneg 13143 df-xadd 13144 df-xmul 13145 df-ioo 13382 df-ioc 13383 df-ico 13384 df-icc 13385 df-fz 13542 df-fzo 13690 df-fl 13832 df-mod 13910 df-seq 14045 df-exp 14105 df-fac 14317 df-bc 14346 df-hash 14374 df-shft 15111 df-cj 15157 df-re 15158 df-im 15159 df-sqrt 15293 df-abs 15294 df-limsup 15529 df-clim 15546 df-rlim 15547 df-sum 15745 df-ef 16127 df-sin 16129 df-cos 16130 df-pi 16132 df-struct 17213 df-sets 17230 df-slot 17248 df-ndx 17260 df-base 17276 df-ress 17297 df-plusg 17329 df-mulr 17330 df-starv 17331 df-sca 17332 df-vsca 17333 df-ip 17334 df-tset 17335 df-ple 17336 df-ds 17338 df-unif 17339 df-hom 17340 df-cco 17341 df-rest 17481 df-topn 17482 df-0g 17500 df-gsum 17501 df-topgen 17502 df-pt 17503 df-prds 17506 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18704 df-sgrp 18783 df-mnd 18799 df-submnd 18848 df-mulg 19140 df-cntz 19393 df-cmn 19858 df-psmet 21525 df-xmet 21526 df-met 21527 df-bl 21528 df-mopn 21529 df-fbas 21530 df-fg 21531 df-cnfld 21534 df-top 23062 df-topon 23079 df-topsp 23101 df-bases 23114 df-cld 23187 df-ntr 23188 df-cls 23189 df-nei 23266 df-lp 23304 df-perf 23305 df-cn 23395 df-cnp 23396 df-haus 23483 df-tx 23730 df-hmeo 23923 df-fil 24014 df-fm 24106 df-flim 24107 df-flf 24108 df-xms 24488 df-ms 24489 df-tms 24490 df-cncf 25048 df-limc 26036 df-dv 26037 df-log 26732 df-cxp 26733

This theorem is used by: cxploglim2 27154 logfacrlim 27399 chtppilimlem2 27649 chpchtlim 27654 dchrvmasumlema 27675 logdivsum 27708

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