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Theorem cxploglim 26915

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 12899	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		reefcl 15994	. . . 4 ⊢ (𝐴 ∈ ℝ → (exp‘𝐴) ∈ ℝ)
3	1, 2	syl 17	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘𝐴) ∈ ℝ)
4		efgt1 16025	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → 1 < (exp‘𝐴))
5		cxp2limlem 26913	. . 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 15994	. . . . . . . 8 ⊢ (𝑧 ∈ ℝ → (exp‘𝑧) ∈ ℝ)
8	7	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) → (exp‘𝑧) ∈ ℝ)
9		1re 11112	. . . . . . 7 ⊢ 1 ∈ ℝ
10		ifcl 4518	. . . . . . 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 12899	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℝ)
13		maxlt 13092	. . . . . . . . . 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 1469	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ 𝑛 ∈ ℝ⁺) → (if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) < 𝑛 ↔ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛)))
15		simprrr 781	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘𝑧) < 𝑛)
16		reeflog 26516	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℝ⁺ → (exp‘(log‘𝑛)) = 𝑛)
17	16	ad2antrl 728	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘(log‘𝑛)) = 𝑛)
18	15, 17	breqtrrd 5117	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘𝑧) < (exp‘(log‘𝑛)))
19		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝑧 ∈ ℝ)
20	12	ad2antrl 728	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝑛 ∈ ℝ)
21		simprrl 780	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 1 < 𝑛)
22	20, 21	rplogcld 26565	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (log‘𝑛) ∈ ℝ⁺)
23	22	rpred 12934	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (log‘𝑛) ∈ ℝ)
24		eflt 16026	. . . . . . . . . . . . . 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 257	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝑧 < (log‘𝑛))
27		breq2 5093	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (log‘𝑛) → (𝑧 < 𝑚 ↔ 𝑧 < (log‘𝑛)))
28		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = (log‘𝑛) → 𝑚 = (log‘𝑛))
29		oveq2 7354	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = (log‘𝑛) → ((exp‘𝐴)↑_𝑐𝑚) = ((exp‘𝐴)↑_𝑐(log‘𝑛)))
30	28, 29	oveq12d 7364	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = (log‘𝑛) → (𝑚 / ((exp‘𝐴)↑_𝑐𝑚)) = ((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛))))
31	30	fveq2d 6826	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = (log‘𝑛) → (abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) = (abs‘((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛)))))
32	31	breq1d 5099	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = (log‘𝑛) → ((abs‘(𝑚 / ((exp‘𝐴)↑_𝑐𝑚))) < 𝑥 ↔ (abs‘((log‘𝑛) / ((exp‘𝐴)↑_𝑐(log‘𝑛)))) < 𝑥))
33	27, 32	imbi12d 344	. . . . . . . . . . . . . 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 3568	. . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝐴 ∈ ℝ)
38	37	relogefd 26564	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (log‘(exp‘𝐴)) = 𝐴)
39	38	oveq2d 7362	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((log‘𝑛) · (log‘(exp‘𝐴))) = ((log‘𝑛) · 𝐴))
40	22	rpcnd 12936	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (log‘𝑛) ∈ ℂ)
41		rpcn 12901	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
42	41	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝐴 ∈ ℂ)
43	40, 42	mulcomd 11133	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((log‘𝑛) · 𝐴) = (𝐴 · (log‘𝑛)))
44	39, 43	eqtrd 2766	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((log‘𝑛) · (log‘(exp‘𝐴))) = (𝐴 · (log‘𝑛)))
45	44	fveq2d 6826	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘𝐴) ∈ ℝ)
47	46	recnd 11140	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (exp‘𝐴) ∈ ℂ)
48		efne0 16005	. . . . . . . . . . . . . . . . 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 26648	. . . . . . . . . . . . . . 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 12901	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ∈ ℂ)
52	51	ad2antrl 728	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝑛 ∈ ℂ)
53		rpne0 12907	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℝ⁺ → 𝑛 ≠ 0)
54	53	ad2antrl 728	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → 𝑛 ≠ 0)
55	52, 54, 42	cxpefd 26648	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → (𝑛↑_𝑐𝐴) = (exp‘(𝐴 · (log‘𝑛))))
56	45, 50, 55	3eqtr4d 2776	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ) ∧ (𝑛 ∈ ℝ⁺ ∧ (1 < 𝑛 ∧ (exp‘𝑧) < 𝑛))) → ((exp‘𝐴)↑_𝑐(log‘𝑛)) = (𝑛↑_𝑐𝐴))
57	56	oveq2d 7362	. . . . . . . . . . . . 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 6826	. . . . . . . . . . . 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 5099	. . . . . . . . . . 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 239	. . . . . . . . . 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 456	. . . . . . . . 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 240	. . . . . . . 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 3132	. . . . . 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 5092	. . . . . . 7 ⊢ (𝑦 = if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) → (𝑦 < 𝑛 ↔ if(1 ≤ (exp‘𝑧), (exp‘𝑧), 1) < 𝑛))
66	65	rspceaimv 3578	. . . . . 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 684	. . . . 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 3133	. . . 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 3146	. . 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 484	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → 𝑚 ∈ ℝ⁺)
71	1	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
72	71	rpefcld 16014	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → (exp‘𝐴) ∈ ℝ⁺)
73		rpre 12899	. . . . . . . . 9 ⊢ (𝑚 ∈ ℝ⁺ → 𝑚 ∈ ℝ)
74	73	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → 𝑚 ∈ ℝ)
75	72, 74	rpcxpcld 26669	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → ((exp‘𝐴)↑_𝑐𝑚) ∈ ℝ⁺)
76	70, 75	rpdivcld 12951	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → (𝑚 / ((exp‘𝐴)↑_𝑐𝑚)) ∈ ℝ⁺)
77	76	rpcnd 12936	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ ℝ⁺) → (𝑚 / ((exp‘𝐴)↑_𝑐𝑚)) ∈ ℂ)
78	77	ralrimiva 3124	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ∀𝑚 ∈ ℝ⁺ (𝑚 / ((exp‘𝐴)↑_𝑐𝑚)) ∈ ℂ)
79		rpssre 12898	. . . . 5 ⊢ ℝ⁺ ⊆ ℝ
80	79	a1i 11	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ℝ⁺ ⊆ ℝ)
81	78, 80	rlim0lt 15416	. . 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 26511	. . . . . . . 8 ⊢ (𝑛 ∈ ℝ⁺ → (log‘𝑛) ∈ ℝ)
83	82	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (log‘𝑛) ∈ ℝ)
84		simpr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → 𝑛 ∈ ℝ⁺)
85	1	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
86	84, 85	rpcxpcld 26669	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑛↑_𝑐𝐴) ∈ ℝ⁺)
87	83, 86	rerpdivcld 12965	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → ((log‘𝑛) / (𝑛↑_𝑐𝐴)) ∈ ℝ)
88	87	recnd 11140	. . . . 5 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → ((log‘𝑛) / (𝑛↑_𝑐𝐴)) ∈ ℂ)
89	88	ralrimiva 3124	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ∀𝑛 ∈ ℝ⁺ ((log‘𝑛) / (𝑛↑_𝑐𝐴)) ∈ ℂ)
90	89, 80	rlim0lt 15416	. . 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 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 ⊆ wss 3897 ifcif 4472 class class class wbr 5089 ↦ cmpt 5170 ‘cfv 6481 (class class class)co 7346 ℂcc 11004 ℝcr 11005 0cc0 11006 1c1 11007 · cmul 11011 < clt 11146 ≤ cle 11147 / cdiv 11774 ℝ⁺crp 12890 abscabs 15141 ⇝_𝑟 crli 15392 expce 15968 logclog 26490 ↑_𝑐ccxp 26491

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 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085

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 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 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-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-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-ioc 13250 df-ico 13251 df-icc 13252 df-fz 13408 df-fzo 13555 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-sin 15976 df-cos 15977 df-pi 15979 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19229 df-cmn 19694 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-fbas 21288 df-fg 21289 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cld 22934 df-ntr 22935 df-cls 22936 df-nei 23013 df-lp 23051 df-perf 23052 df-cn 23142 df-cnp 23143 df-haus 23230 df-tx 23477 df-hmeo 23670 df-fil 23761 df-fm 23853 df-flim 23854 df-flf 23855 df-xms 24235 df-ms 24236 df-tms 24237 df-cncf 24798 df-limc 25794 df-dv 25795 df-log 26492 df-cxp 26493

This theorem is referenced by: cxploglim2 26916 logfacrlim 27162 chtppilimlem2 27412 chpchtlim 27417 dchrvmasumlema 27438 logdivsum 27471

W3C validator