Theorem logexprlim 27534

Description: The sum Σ𝑛 ≤ 𝑥, log↑𝑁(𝑥 / 𝑛) has the asymptotic expansion (𝑁!)𝑥 + 𝑜(𝑥). (More precisely, the omitted term has order 𝑂(log↑𝑁(𝑥) / 𝑥).) (Contributed by Mario Carneiro, 22-May-2016.)

Assertion

Ref	Expression
logexprlim	⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) / 𝑥)) ⇝𝑟 (!‘𝑁))

Distinct variable group:   𝑥,𝑛,𝑁

Proof of Theorem logexprlim

Dummy variables 𝑘 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzfid 14096	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (1...(⌊‘𝑥)) ∈ Fin)
2		simpr 490	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
3		elfznn 13667	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
4	3	nnrpd 13143	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ+)
5		rpdivcl 13128	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) → (𝑥 / 𝑛) ∈ ℝ+)
6	2, 4, 5	syl2an 608	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
7	6	relogcld 26933	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
8		simpll 779	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑁 ∈ ℕ0)
9	7, 8	reexpcld 14286	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛))↑𝑁) ∈ ℝ)
10	1, 9	fsumrecl 15880	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) ∈ ℝ)
11		relogcl 26885	. . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
12		id 23	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0)
13		reexpcl 14201	. . . . . . 7 ⊢ (((log‘𝑥) ∈ ℝ ∧ 𝑁 ∈ ℕ0) → ((log‘𝑥)↑𝑁) ∈ ℝ)
14	11, 12, 13	syl2anr 609	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥)↑𝑁) ∈ ℝ)
15		faccl 14407	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ)
16	15	adantr 486	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (!‘𝑁) ∈ ℕ)
17	16	nnred 12331	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (!‘𝑁) ∈ ℝ)
18		fzfid 14096	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (0...𝑁) ∈ Fin)
19	11	adantl 487	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℝ)
20		elfznn0 13734	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0)
21		reexpcl 14201	. . . . . . . . . 10 ⊢ (((log‘𝑥) ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((log‘𝑥)↑𝑘) ∈ ℝ)
22	19, 20, 21	syl2an 608	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → ((log‘𝑥)↑𝑘) ∈ ℝ)
23	20	adantl 487	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ0)
24	23	faccld 14408	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℕ)
25	22, 24	nndivred 12373	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℝ)
26	18, 25	fsumrecl 15880	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℝ)
27	17, 26	remulcld 11320	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) ∈ ℝ)
28	14, 27	resubcld 11725	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) ∈ ℝ)
29	10, 28	resubcld 11725	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) ∈ ℝ)
30	29, 2	rerpdivcld 13176	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) ∈ ℝ)
31		rerpdivcl 13133	. . . 4 ⊢ (((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) ∈ ℝ ∧ 𝑥 ∈ ℝ+) → ((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) / 𝑥) ∈ ℝ)
32	28, 31	sylancom 600	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → ((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) / 𝑥) ∈ ℝ)
33		1red 11290	. . . 4 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℝ)
34	15	nncnd 12332	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℂ)
35		simpl 488	. . . . . . . . 9 ⊢ ((𝑘 = 𝑁 ∧ 𝑥 ∈ ℝ+) → 𝑘 = 𝑁)
36	35	oveq2d 7428	. . . . . . . 8 ⊢ ((𝑘 = 𝑁 ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥)↑𝑘) = ((log‘𝑥)↑𝑁))
37	36	oveq1d 7427	. . . . . . 7 ⊢ ((𝑘 = 𝑁 ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑘) / 𝑥) = (((log‘𝑥)↑𝑁) / 𝑥))
38	37	mpteq2dva 5198	. . . . . 6 ⊢ (𝑘 = 𝑁 → (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑘) / 𝑥)) = (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑁) / 𝑥)))
39	38	breq1d 5113	. . . . 5 ⊢ (𝑘 = 𝑁 → ((𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑘) / 𝑥)) ⇝𝑟 0 ↔ (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑁) / 𝑥)) ⇝𝑟 0))
40	11	recnd 11318	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℂ)
41		id 23	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℕ0)
42		cxpexp 26978	. . . . . . . . 9 ⊢ (((log‘𝑥) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((log‘𝑥)↑𝑐𝑘) = ((log‘𝑥)↑𝑘))
43	40, 41, 42	syl2anr 609	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥)↑𝑐𝑘) = ((log‘𝑥)↑𝑘))
44		rpcn 13112	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℂ)
45	44	adantl 487	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ)
46	45	cxp1d 27016	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (𝑥↑𝑐1) = 𝑥)
47	43, 46	oveq12d 7430	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑐𝑘) / (𝑥↑𝑐1)) = (((log‘𝑥)↑𝑘) / 𝑥))
48	47	mpteq2dva 5198	. . . . . 6 ⊢ (𝑘 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑐𝑘) / (𝑥↑𝑐1))) = (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑘) / 𝑥)))
49		nn0cn 12597	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ)
50		1rp 13105	. . . . . . 7 ⊢ 1 ∈ ℝ+
51		cxploglim2 27288	. . . . . . 7 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℝ+) → (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑐𝑘) / (𝑥↑𝑐1))) ⇝𝑟 0)
52	49, 50, 51	sylancl 598	. . . . . 6 ⊢ (𝑘 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑐𝑘) / (𝑥↑𝑐1))) ⇝𝑟 0)
53	48, 52	eqbrtrrd 5129	. . . . 5 ⊢ (𝑘 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑘) / 𝑥)) ⇝𝑟 0)
54	39, 53	vtoclga 3537	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑁) / 𝑥)) ⇝𝑟 0)
55		rerpdivcl 13133	. . . . . 6 ⊢ ((((log‘𝑥)↑𝑁) ∈ ℝ ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑁) / 𝑥) ∈ ℝ)
56	14, 55	sylancom 600	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑁) / 𝑥) ∈ ℝ)
57	56	recnd 11318	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑁) / 𝑥) ∈ ℂ)
58	10	recnd 11318	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) ∈ ℂ)
59	14	recnd 11318	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥)↑𝑁) ∈ ℂ)
60	34	adantr 486	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (!‘𝑁) ∈ ℂ)
61	26	recnd 11318	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℂ)
62	60, 61	mulcld 11310	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) ∈ ℂ)
63	59, 62	subcld 11650	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) ∈ ℂ)
64	58, 63	subcld 11650	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) ∈ ℂ)
65		rpcnne0 13120	. . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
66	65	adantl 487	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
67	66	simpld 500	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ)
68	66	simprd 501	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0)
69	64, 67, 68	divcld 12074	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) ∈ ℂ)
70	69	adantrr 730	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) ∈ ℂ)
71	15	adantr 486	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (!‘𝑁) ∈ ℕ)
72	71	nncnd 12332	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (!‘𝑁) ∈ ℂ)
73	70, 72	subcld 11650	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) − (!‘𝑁)) ∈ ℂ)
74	73	abscld 15586	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘(((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) − (!‘𝑁))) ∈ ℝ)
75	56	adantrr 730	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (((log‘𝑥)↑𝑁) / 𝑥) ∈ ℝ)
76	75	recnd 11318	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (((log‘𝑥)↑𝑁) / 𝑥) ∈ ℂ)
77	76	abscld 15586	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘(((log‘𝑥)↑𝑁) / 𝑥)) ∈ ℝ)
78		ioorp 13537	. . . . . . . . . 10 ⊢ (0(,)+∞) = ℝ+
79	78	eqcomi 2770	. . . . . . . . 9 ⊢ ℝ+ = (0(,)+∞)
80		nnuz 12985	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
81		1z 12707	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
82	81	a1i 11	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ∈ ℤ)
83		1red 11290	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ∈ ℝ)
84		1re 11289	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
85		1nn0 12603	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ0
86	84, 85	nn0addge1i 12635	. . . . . . . . . 10 ⊢ 1 ≤ (1 + 1)
87	86	a1i 11	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ≤ (1 + 1))
88		0red 11292	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ∈ ℝ)
89	71	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (!‘𝑁) ∈ ℕ)
90	89	nnred 12331	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (!‘𝑁) ∈ ℝ)
91		rpre 13110	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ)
92	91	adantl 487	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ)
93		fzfid 14096	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (0...𝑁) ∈ Fin)
94		simprl 783	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ+)
95		rpdivcl 13128	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+) → (𝑥 / 𝑦) ∈ ℝ+)
96	94, 95	sylan 592	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (𝑥 / 𝑦) ∈ ℝ+)
97	96	relogcld 26933	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (log‘(𝑥 / 𝑦)) ∈ ℝ)
98		reexpcl 14201	. . . . . . . . . . . . . 14 ⊢ (((log‘(𝑥 / 𝑦)) ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((log‘(𝑥 / 𝑦))↑𝑘) ∈ ℝ)
99	97, 20, 98	syl2an 608	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → ((log‘(𝑥 / 𝑦))↑𝑘) ∈ ℝ)
100	20	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ0)
101	100	faccld 14408	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℕ)
102	99, 101	nndivred 12373	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)) ∈ ℝ)
103	93, 102	fsumrecl 15880	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)) ∈ ℝ)
104	92, 103	remulcld 11320	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))) ∈ ℝ)
105	90, 104	remulcld 11320	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)))) ∈ ℝ)
106		simpll 779	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → 𝑁 ∈ ℕ0)
107	97, 106	reexpcld 14286	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → ((log‘(𝑥 / 𝑦))↑𝑁) ∈ ℝ)
108		nnrp 13113	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ+)
109	108, 107	sylan2 605	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℕ) → ((log‘(𝑥 / 𝑦))↑𝑁) ∈ ℝ)
110		reelprrecn 11273	. . . . . . . . . . . 12 ⊢ ℝ ∈ {ℝ, ℂ}
111	110	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ℝ ∈ {ℝ, ℂ})
112	104	recnd 11318	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))) ∈ ℂ)
113	107, 89	nndivred 12373	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (((log‘(𝑥 / 𝑦))↑𝑁) / (!‘𝑁)) ∈ ℝ)
114		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑁 ∈ ℕ0)
115		advlogexp 26965	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑁 ∈ ℕ0) → (ℝ D (𝑦 ∈ ℝ+ ↦ (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))))) = (𝑦 ∈ ℝ+ ↦ (((log‘(𝑥 / 𝑦))↑𝑁) / (!‘𝑁))))
116	94, 114, 115	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (ℝ D (𝑦 ∈ ℝ+ ↦ (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))))) = (𝑦 ∈ ℝ+ ↦ (((log‘(𝑥 / 𝑦))↑𝑁) / (!‘𝑁))))
117	111, 112, 113, 116, 72	dvmptcmul 26264	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (ℝ D (𝑦 ∈ ℝ+ ↦ ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)))))) = (𝑦 ∈ ℝ+ ↦ ((!‘𝑁) · (((log‘(𝑥 / 𝑦))↑𝑁) / (!‘𝑁)))))
118	107	recnd 11318	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → ((log‘(𝑥 / 𝑦))↑𝑁) ∈ ℂ)
119	72	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (!‘𝑁) ∈ ℂ)
120	71	nnne0d 12369	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (!‘𝑁) ≠ 0)
121	120	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (!‘𝑁) ≠ 0)
122	118, 119, 121	divcan2d 12076	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → ((!‘𝑁) · (((log‘(𝑥 / 𝑦))↑𝑁) / (!‘𝑁))) = ((log‘(𝑥 / 𝑦))↑𝑁))
123	122	mpteq2dva 5198	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑦 ∈ ℝ+ ↦ ((!‘𝑁) · (((log‘(𝑥 / 𝑦))↑𝑁) / (!‘𝑁)))) = (𝑦 ∈ ℝ+ ↦ ((log‘(𝑥 / 𝑦))↑𝑁)))
124	117, 123	eqtrd 2796	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (ℝ D (𝑦 ∈ ℝ+ ↦ ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)))))) = (𝑦 ∈ ℝ+ ↦ ((log‘(𝑥 / 𝑦))↑𝑁)))
125		oveq2 7420	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑛 → (𝑥 / 𝑦) = (𝑥 / 𝑛))
126	125	fveq2d 6881	. . . . . . . . . 10 ⊢ (𝑦 = 𝑛 → (log‘(𝑥 / 𝑦)) = (log‘(𝑥 / 𝑛)))
127	126	oveq1d 7427	. . . . . . . . 9 ⊢ (𝑦 = 𝑛 → ((log‘(𝑥 / 𝑦))↑𝑁) = ((log‘(𝑥 / 𝑛))↑𝑁))
128	94	rpxrd 13146	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ*)
129		simp1rl 1257	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑥 ∈ ℝ+)
130		simp2r 1219	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ∈ ℝ+)
131	129, 130	rpdivcld 13162	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → (𝑥 / 𝑛) ∈ ℝ+)
132	131	relogcld 26933	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
133		simp2l 1218	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑦 ∈ ℝ+)
134	129, 133	rpdivcld 13162	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → (𝑥 / 𝑦) ∈ ℝ+)
135	134	relogcld 26933	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → (log‘(𝑥 / 𝑦)) ∈ ℝ)
136		simp1l 1216	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑁 ∈ ℕ0)
137		log1 26895	. . . . . . . . . . 11 ⊢ (log‘1) = 0
138	130	rpcnd 13147	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ∈ ℂ)
139	138	mullidd 11308	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → (1 · 𝑛) = 𝑛)
140		simp33 1230	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ≤ 𝑥)
141	139, 140	eqbrtrd 5127	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → (1 · 𝑛) ≤ 𝑥)
142		1red 11290	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 1 ∈ ℝ)
143	129	rpred 13145	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑥 ∈ ℝ)
144	142, 143, 130	lemuldivd 13194	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → ((1 · 𝑛) ≤ 𝑥 ↔ 1 ≤ (𝑥 / 𝑛)))
145	141, 144	mpbid 235	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 1 ≤ (𝑥 / 𝑛))
146		logleb 26913	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℝ+ ∧ (𝑥 / 𝑛) ∈ ℝ+) → (1 ≤ (𝑥 / 𝑛) ↔ (log‘1) ≤ (log‘(𝑥 / 𝑛))))
147	50, 131, 146	sylancr 599	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → (1 ≤ (𝑥 / 𝑛) ↔ (log‘1) ≤ (log‘(𝑥 / 𝑛))))
148	145, 147	mpbid 235	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → (log‘1) ≤ (log‘(𝑥 / 𝑛)))
149	137, 148	eqbrtrrid 5141	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 0 ≤ (log‘(𝑥 / 𝑛)))
150		simp32 1229	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑦 ≤ 𝑛)
151	133, 130, 129	lediv2d 13169	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → (𝑦 ≤ 𝑛 ↔ (𝑥 / 𝑛) ≤ (𝑥 / 𝑦)))
152	150, 151	mpbid 235	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → (𝑥 / 𝑛) ≤ (𝑥 / 𝑦))
153	131, 134	logled 26937	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → ((𝑥 / 𝑛) ≤ (𝑥 / 𝑦) ↔ (log‘(𝑥 / 𝑛)) ≤ (log‘(𝑥 / 𝑦))))
154	152, 153	mpbid 235	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → (log‘(𝑥 / 𝑛)) ≤ (log‘(𝑥 / 𝑦)))
155		leexp1a 14298	. . . . . . . . . 10 ⊢ ((((log‘(𝑥 / 𝑛)) ∈ ℝ ∧ (log‘(𝑥 / 𝑦)) ∈ ℝ ∧ 𝑁 ∈ ℕ0) ∧ (0 ≤ (log‘(𝑥 / 𝑛)) ∧ (log‘(𝑥 / 𝑛)) ≤ (log‘(𝑥 / 𝑦)))) → ((log‘(𝑥 / 𝑛))↑𝑁) ≤ ((log‘(𝑥 / 𝑦))↑𝑁))
156	132, 135, 136, 149, 154, 155	syl32anc 1405	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → ((log‘(𝑥 / 𝑛))↑𝑁) ≤ ((log‘(𝑥 / 𝑦))↑𝑁))
157		eqid 2761	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)))))) = (𝑦 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))))))
158	96	3ad2antr1 1207	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → (𝑥 / 𝑦) ∈ ℝ+)
159	158	relogcld 26933	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → (log‘(𝑥 / 𝑦)) ∈ ℝ)
160		simpll 779	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → 𝑁 ∈ ℕ0)
161		rpcn 13112	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℂ)
162	161	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℂ)
163	162	3ad2antr1 1207	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → 𝑦 ∈ ℂ)
164	163	mullidd 11308	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → (1 · 𝑦) = 𝑦)
165		simpr3 1215	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → 𝑦 ≤ 𝑥)
166	164, 165	eqbrtrd 5127	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → (1 · 𝑦) ≤ 𝑥)
167		1red 11290	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → 1 ∈ ℝ)
168	94	rpred 13145	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
169	168	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → 𝑥 ∈ ℝ)
170		simpr1 1213	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → 𝑦 ∈ ℝ+)
171	167, 169, 170	lemuldivd 13194	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → ((1 · 𝑦) ≤ 𝑥 ↔ 1 ≤ (𝑥 / 𝑦)))
172	166, 171	mpbid 235	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → 1 ≤ (𝑥 / 𝑦))
173		logleb 26913	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℝ+ ∧ (𝑥 / 𝑦) ∈ ℝ+) → (1 ≤ (𝑥 / 𝑦) ↔ (log‘1) ≤ (log‘(𝑥 / 𝑦))))
174	50, 158, 173	sylancr 599	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → (1 ≤ (𝑥 / 𝑦) ↔ (log‘1) ≤ (log‘(𝑥 / 𝑦))))
175	172, 174	mpbid 235	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → (log‘1) ≤ (log‘(𝑥 / 𝑦)))
176	137, 175	eqbrtrrid 5141	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → 0 ≤ (log‘(𝑥 / 𝑦)))
177	159, 160, 176	expge0d 14287	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → 0 ≤ ((log‘(𝑥 / 𝑦))↑𝑁))
178	50	a1i 11	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ∈ ℝ+)
179		1le1 11925	. . . . . . . . . 10 ⊢ 1 ≤ 1
180	179	a1i 11	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ≤ 1)
181		simprr 785	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
182	168	leidd 11863	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ≤ 𝑥)
183	79, 80, 82, 83, 87, 88, 105, 107, 109, 124, 127, 128, 156, 157, 177, 178, 94, 180, 181, 182	dvfsumlem4 26329	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘(((𝑦 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))))))‘𝑥) − ((𝑦 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))))))‘1))) ≤ ⦋1 / 𝑦⦌((log‘(𝑥 / 𝑦))↑𝑁))
184		fzfid 14096	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
185	94, 4, 5	syl2an 608	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
186	185	relogcld 26933	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
187		simpll 779	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑁 ∈ ℕ0)
188	186, 187	reexpcld 14286	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛))↑𝑁) ∈ ℝ)
189	184, 188	fsumrecl 15880	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) ∈ ℝ)
190	189	recnd 11318	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) ∈ ℂ)
191	94	rpcnd 13147	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℂ)
192	72, 191	mulcld 11310	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((!‘𝑁) · 𝑥) ∈ ℂ)
193	11	ad2antrl 741	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
194	193	recnd 11318	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℂ)
195	194, 114	expcld 14269	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((log‘𝑥)↑𝑁) ∈ ℂ)
196		fzfid 14096	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (0...𝑁) ∈ Fin)
197	193, 20, 21	syl2an 608	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (0...𝑁)) → ((log‘𝑥)↑𝑘) ∈ ℝ)
198	20	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ0)
199	198	faccld 14408	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℕ)
200	197, 199	nndivred 12373	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℝ)
201	200	recnd 11318	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℂ)
202	196, 201	fsumcl 15879	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℂ)
203	72, 202	mulcld 11310	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) ∈ ℂ)
204	195, 203	subcld 11650	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) ∈ ℂ)
205	190, 192, 204	sub32d 11682	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · 𝑥)) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) − ((!‘𝑁) · 𝑥)))
206		eqidd 2762	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑦 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)))))) = (𝑦 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)))))))
207		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
208	207	fveq2d 6881	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → (⌊‘𝑦) = (⌊‘𝑥))
209	208	oveq2d 7428	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → (1...(⌊‘𝑦)) = (1...(⌊‘𝑥)))
210	209	sumeq1d 15847	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) = Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁))
211		oveq2 7420	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = 𝑥 → (𝑥 / 𝑦) = (𝑥 / 𝑥))
212	65	ad2antrl 741	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
213		divid 11985	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (𝑥 / 𝑥) = 1)
214	212, 213	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑥 / 𝑥) = 1)
215	211, 214	sylan9eqr 2818	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → (𝑥 / 𝑦) = 1)
216	215	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 / 𝑦) = 1)
217	216	fveq2d 6881	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ (0...𝑁)) → (log‘(𝑥 / 𝑦)) = (log‘1))
218	217, 137	eqtrdi 2812	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ (0...𝑁)) → (log‘(𝑥 / 𝑦)) = 0)
219	218	oveq1d 7427	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ (0...𝑁)) → ((log‘(𝑥 / 𝑦))↑𝑘) = (0↑𝑘))
220	219	oveq1d 7427	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)) = ((0↑𝑘) / (!‘𝑘)))
221	220	sumeq2dv 15849	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ (0...𝑁)((0↑𝑘) / (!‘𝑘)))
222		nn0uz 12984	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ℕ0 = (ℤ≥‘0)
223	114, 222	eleqtrdi 2871	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑁 ∈ (ℤ≥‘0))
224		eluzfz1 13644	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ (ℤ≥‘0) → 0 ∈ (0...𝑁))
225	223, 224	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ∈ (0...𝑁))
226	225	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → 0 ∈ (0...𝑁))
227	226	snssd 4747	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → {0} ⊆ (0...𝑁))
228		elsni 4601	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ {0} → 𝑘 = 0)
229	228	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ {0}) → 𝑘 = 0)
230		oveq2 7420	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = 0 → (0↑𝑘) = (0↑0))
231		0exp0e1 14189	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0↑0) = 1
232	230, 231	eqtrdi 2812	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = 0 → (0↑𝑘) = 1)
233		fveq2 6877	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = 0 → (!‘𝑘) = (!‘0))
234		fac0 14400	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (!‘0) = 1
235	233, 234	eqtrdi 2812	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = 0 → (!‘𝑘) = 1)
236	232, 235	oveq12d 7430	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 0 → ((0↑𝑘) / (!‘𝑘)) = (1 / 1))
237		1div1e1 11988	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1 / 1) = 1
238	236, 237	eqtrdi 2812	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 0 → ((0↑𝑘) / (!‘𝑘)) = 1)
239	229, 238	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ {0}) → ((0↑𝑘) / (!‘𝑘)) = 1)
240		ax-1cn 11239	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℂ
241	239, 240	eqeltrdi 2869	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ {0}) → ((0↑𝑘) / (!‘𝑘)) ∈ ℂ)
242		eldifi 4078	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 ∈ ((0...𝑁) ∖ {0}) → 𝑘 ∈ (0...𝑁))
243	242	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → 𝑘 ∈ (0...𝑁))
244	243, 20	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → 𝑘 ∈ ℕ0)
245		eldifsni 4753	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ ((0...𝑁) ∖ {0}) → 𝑘 ≠ 0)
246	245	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → 𝑘 ≠ 0)
247		eldifsn 4748	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (ℕ0 ∖ {0}) ↔ (𝑘 ∈ ℕ0 ∧ 𝑘 ≠ 0))
248	244, 246, 247	sylanbrc 595	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → 𝑘 ∈ (ℕ0 ∖ {0}))
249		dfn2 12600	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ℕ = (ℕ0 ∖ {0})
250	248, 249	eleqtrrdi 2872	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → 𝑘 ∈ ℕ)
251	250	0expd 14262	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → (0↑𝑘) = 0)
252	251	oveq1d 7427	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → ((0↑𝑘) / (!‘𝑘)) = (0 / (!‘𝑘)))
253	244	faccld 14408	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → (!‘𝑘) ∈ ℕ)
254	253	nncnd 12332	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → (!‘𝑘) ∈ ℂ)
255	253	nnne0d 12369	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → (!‘𝑘) ≠ 0)
256	254, 255	div0d 12073	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → (0 / (!‘𝑘)) = 0)
257	252, 256	eqtrd 2796	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → ((0↑𝑘) / (!‘𝑘)) = 0)
258		fzfid 14096	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → (0...𝑁) ∈ Fin)
259	227, 241, 257, 258	fsumss 15871	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → Σ𝑘 ∈ {0} ((0↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ (0...𝑁)((0↑𝑘) / (!‘𝑘)))
260	221, 259	eqtr4d 2799	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ {0} ((0↑𝑘) / (!‘𝑘)))
261		0cn 11279	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℂ
262	238	sumsn 15892	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℂ ∧ 1 ∈ ℂ) → Σ𝑘 ∈ {0} ((0↑𝑘) / (!‘𝑘)) = 1)
263	261, 240, 262	mp2an 705	. . . . . . . . . . . . . . . . . 18 ⊢ Σ𝑘 ∈ {0} ((0↑𝑘) / (!‘𝑘)) = 1
264	260, 263	eqtrdi 2812	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)) = 1)
265	207, 264	oveq12d 7430	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))) = (𝑥 · 1))
266	191	mulridd 11307	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑥 · 1) = 𝑥)
267	266	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → (𝑥 · 1) = 𝑥)
268	265, 267	eqtrd 2796	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))) = 𝑥)
269	268	oveq2d 7428	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)))) = ((!‘𝑁) · 𝑥))
270	210, 269	oveq12d 7430	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → (Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · 𝑥)))
271		ovexd 7447	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · 𝑥)) ∈ V)
272	206, 270, 94, 271	fvmptd 6993	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((𝑦 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))))))‘𝑥) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · 𝑥)))
273		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → 𝑦 = 1)
274	273	fveq2d 6881	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (⌊‘𝑦) = (⌊‘1))
275		flid 13928	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 ∈ ℤ → (⌊‘1) = 1)
276	81, 275	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (⌊‘1) = 1
277	274, 276	eqtrdi 2812	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (⌊‘𝑦) = 1)
278	277	oveq2d 7428	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (1...(⌊‘𝑦)) = (1...1))
279	278	sumeq1d 15847	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) = Σ𝑛 ∈ (1...1)((log‘(𝑥 / 𝑛))↑𝑁))
280	191	div1d 12066	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑥 / 1) = 𝑥)
281	280	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (𝑥 / 1) = 𝑥)
282	281	fveq2d 6881	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (log‘(𝑥 / 1)) = (log‘𝑥))
283	282	oveq1d 7427	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → ((log‘(𝑥 / 1))↑𝑁) = ((log‘𝑥)↑𝑁))
284	195	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → ((log‘𝑥)↑𝑁) ∈ ℂ)
285	283, 284	eqeltrd 2861	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → ((log‘(𝑥 / 1))↑𝑁) ∈ ℂ)
286		oveq2 7420	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 1 → (𝑥 / 𝑛) = (𝑥 / 1))
287	286	fveq2d 6881	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 1 → (log‘(𝑥 / 𝑛)) = (log‘(𝑥 / 1)))
288	287	oveq1d 7427	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 1 → ((log‘(𝑥 / 𝑛))↑𝑁) = ((log‘(𝑥 / 1))↑𝑁))
289	288	fsum1 15893	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℤ ∧ ((log‘(𝑥 / 1))↑𝑁) ∈ ℂ) → Σ𝑛 ∈ (1...1)((log‘(𝑥 / 𝑛))↑𝑁) = ((log‘(𝑥 / 1))↑𝑁))
290	81, 285, 289	sylancr 599	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → Σ𝑛 ∈ (1...1)((log‘(𝑥 / 𝑛))↑𝑁) = ((log‘(𝑥 / 1))↑𝑁))
291	279, 290, 283	3eqtrd 2800	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) = ((log‘𝑥)↑𝑁))
292	273	oveq2d 7428	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (𝑥 / 𝑦) = (𝑥 / 1))
293	292, 281	eqtrd 2796	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (𝑥 / 𝑦) = 𝑥)
294	293	fveq2d 6881	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (log‘(𝑥 / 𝑦)) = (log‘𝑥))
295	294	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) ∧ 𝑘 ∈ (0...𝑁)) → (log‘(𝑥 / 𝑦)) = (log‘𝑥))
296	295	oveq1d 7427	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) ∧ 𝑘 ∈ (0...𝑁)) → ((log‘(𝑥 / 𝑦))↑𝑘) = ((log‘𝑥)↑𝑘))
297	296	oveq1d 7427	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)) = (((log‘𝑥)↑𝑘) / (!‘𝑘)))
298	297	sumeq2dv 15849	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))
299	273, 298	oveq12d 7430	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))) = (1 · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))
300	202	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℂ)
301	300	mullidd 11308	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (1 · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) = Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))
302	299, 301	eqtrd 2796	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))) = Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))
303	302	oveq2d 7428	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)))) = ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))
304	291, 303	oveq12d 7430	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))))) = (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))))
305		ovexd 7447	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) ∈ V)
306	206, 304, 178, 305	fvmptd 6993	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((𝑦 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))))))‘1) = (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))))
307	272, 306	oveq12d 7430	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (((𝑦 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))))))‘𝑥) − ((𝑦 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))))))‘1)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · 𝑥)) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))))
308	70, 72, 191	subdird 11754	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) − (!‘𝑁)) · 𝑥) = ((((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) · 𝑥) − ((!‘𝑁) · 𝑥)))
309	64	adantrr 730	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) ∈ ℂ)
310	212	simprd 501	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ≠ 0)
311	309, 191, 310	divcan1d 12075	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) · 𝑥) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))))
312	311	oveq1d 7427	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) · 𝑥) − ((!‘𝑁) · 𝑥)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) − ((!‘𝑁) · 𝑥)))
313	308, 312	eqtrd 2796	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) − (!‘𝑁)) · 𝑥) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) − ((!‘𝑁) · 𝑥)))
314	205, 307, 313	3eqtr4d 2806	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (((𝑦 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))))))‘𝑥) − ((𝑦 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))))))‘1)) = ((((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) − (!‘𝑁)) · 𝑥))
315	314	fveq2d 6881	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘(((𝑦 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))))))‘𝑥) − ((𝑦 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))))))‘1))) = (abs‘((((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) − (!‘𝑁)) · 𝑥)))
316	73, 191	absmuld 15604	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘((((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) − (!‘𝑁)) · 𝑥)) = ((abs‘(((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) − (!‘𝑁))) · (abs‘𝑥)))
317		rprege0 13117	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
318	317	ad2antrl 741	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
319		absid 15443	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (abs‘𝑥) = 𝑥)
320	318, 319	syl 18	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘𝑥) = 𝑥)
321	320	oveq2d 7428	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((abs‘(((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) − (!‘𝑁))) · (abs‘𝑥)) = ((abs‘(((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) − (!‘𝑁))) · 𝑥))
322	315, 316, 321	3eqtrd 2800	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘(((𝑦 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))))))‘𝑥) − ((𝑦 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))))))‘1))) = ((abs‘(((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) − (!‘𝑁))) · 𝑥))
323		1cnd 11283	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ∈ ℂ)
324	294	oveq1d 7427	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → ((log‘(𝑥 / 𝑦))↑𝑁) = ((log‘𝑥)↑𝑁))
325	323, 324	csbied 3883	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ⦋1 / 𝑦⦌((log‘(𝑥 / 𝑦))↑𝑁) = ((log‘𝑥)↑𝑁))
326	183, 322, 325	3brtr3d 5136	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((abs‘(((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) − (!‘𝑁))) · 𝑥) ≤ ((log‘𝑥)↑𝑁))
327	14	adantrr 730	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((log‘𝑥)↑𝑁) ∈ ℝ)
328	74, 327, 94	lemuldivd 13194	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (((abs‘(((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) − (!‘𝑁))) · 𝑥) ≤ ((log‘𝑥)↑𝑁) ↔ (abs‘(((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) − (!‘𝑁))) ≤ (((log‘𝑥)↑𝑁) / 𝑥)))
329	326, 328	mpbid 235	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘(((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) − (!‘𝑁))) ≤ (((log‘𝑥)↑𝑁) / 𝑥))
330	75	leabsd 15562	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (((log‘𝑥)↑𝑁) / 𝑥) ≤ (abs‘(((log‘𝑥)↑𝑁) / 𝑥)))
331	74, 75, 77, 329, 330	letrd 11448	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘(((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) − (!‘𝑁))) ≤ (abs‘(((log‘𝑥)↑𝑁) / 𝑥)))
332	57	adantrr 730	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (((log‘𝑥)↑𝑁) / 𝑥) ∈ ℂ)
333	332	subid1d 11639	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((((log‘𝑥)↑𝑁) / 𝑥) − 0) = (((log‘𝑥)↑𝑁) / 𝑥))
334	333	fveq2d 6881	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘((((log‘𝑥)↑𝑁) / 𝑥) − 0)) = (abs‘(((log‘𝑥)↑𝑁) / 𝑥)))
335	331, 334	breqtrrd 5133	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘(((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) − (!‘𝑁))) ≤ (abs‘((((log‘𝑥)↑𝑁) / 𝑥) − 0)))
336	33, 34, 54, 57, 69, 335	rlimsqzlem 15796	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥)) ⇝𝑟 (!‘𝑁))
337		divsubdir 11991	. . . . . 6 ⊢ ((((log‘𝑥)↑𝑁) ∈ ℂ ∧ ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → ((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) / 𝑥) = ((((log‘𝑥)↑𝑁) / 𝑥) − (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥)))
338	59, 62, 66, 337	syl3anc 1398	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → ((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) / 𝑥) = ((((log‘𝑥)↑𝑁) / 𝑥) − (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥)))
339	338	mpteq2dva 5198	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ ((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) / 𝑥)) = (𝑥 ∈ ℝ+ ↦ ((((log‘𝑥)↑𝑁) / 𝑥) − (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥))))
340		rerpdivcl 13133	. . . . . . 7 ⊢ ((((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) ∈ ℝ ∧ 𝑥 ∈ ℝ+) → (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥) ∈ ℝ)
341	27, 340	sylancom 600	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥) ∈ ℝ)
342		divass 11973	. . . . . . . . . 10 ⊢ (((!‘𝑁) ∈ ℂ ∧ Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥) = ((!‘𝑁) · (Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) / 𝑥)))
343	60, 61, 66, 342	syl3anc 1398	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥) = ((!‘𝑁) · (Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) / 𝑥)))
344	25	recnd 11318	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℂ)
345	18, 67, 344, 68	fsumdivc 15932	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) / 𝑥) = Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / (!‘𝑘)) / 𝑥))
346	22	recnd 11318	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → ((log‘𝑥)↑𝑘) ∈ ℂ)
347	24	nnrpd 13143	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℝ+)
348	347	rpcnne0d 13154	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → ((!‘𝑘) ∈ ℂ ∧ (!‘𝑘) ≠ 0))
349	66	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
350		divdiv32 12006	. . . . . . . . . . . . 13 ⊢ ((((log‘𝑥)↑𝑘) ∈ ℂ ∧ ((!‘𝑘) ∈ ℂ ∧ (!‘𝑘) ≠ 0) ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → ((((log‘𝑥)↑𝑘) / (!‘𝑘)) / 𝑥) = ((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)))
351	346, 348, 349, 350	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → ((((log‘𝑥)↑𝑘) / (!‘𝑘)) / 𝑥) = ((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)))
352	351	sumeq2dv 15849	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / (!‘𝑘)) / 𝑥) = Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)))
353	345, 352	eqtrd 2796	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) / 𝑥) = Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)))
354	353	oveq2d 7428	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → ((!‘𝑁) · (Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) / 𝑥)) = ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘))))
355	343, 354	eqtrd 2796	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥) = ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘))))
356	355	mpteq2dva 5198	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥)) = (𝑥 ∈ ℝ+ ↦ ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)))))
357	2	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → 𝑥 ∈ ℝ+)
358	22, 357	rerpdivcld 13176	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘𝑥)↑𝑘) / 𝑥) ∈ ℝ)
359	358, 24	nndivred 12373	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → ((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)) ∈ ℝ)
360	18, 359	fsumrecl 15880	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)) ∈ ℝ)
361		rpssre 13109	. . . . . . . . . 10 ⊢ ℝ+ ⊆ ℝ
362		rlimconst 15691	. . . . . . . . . 10 ⊢ ((ℝ+ ⊆ ℝ ∧ (!‘𝑁) ∈ ℂ) → (𝑥 ∈ ℝ+ ↦ (!‘𝑁)) ⇝𝑟 (!‘𝑁))
363	361, 34, 362	sylancr 599	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (!‘𝑁)) ⇝𝑟 (!‘𝑁))
364	361	a1i 11	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → ℝ+ ⊆ ℝ)
365		fzfid 14096	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → (0...𝑁) ∈ Fin)
366	359	anasss 472	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 𝑘 ∈ (0...𝑁))) → ((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)) ∈ ℝ)
367	358	an32s 665	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑘) / 𝑥) ∈ ℝ)
368	20	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ0)
369	368	faccld 14408	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℕ)
370	369	nnred 12331	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℝ)
371	370	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑥 ∈ ℝ+) → (!‘𝑘) ∈ ℝ)
372	368, 53	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑘) / 𝑥)) ⇝𝑟 0)
373	369	nncnd 12332	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℂ)
374		rlimconst 15691	. . . . . . . . . . . . . 14 ⊢ ((ℝ+ ⊆ ℝ ∧ (!‘𝑘) ∈ ℂ) → (𝑥 ∈ ℝ+ ↦ (!‘𝑘)) ⇝𝑟 (!‘𝑘))
375	361, 373, 374	sylancr 599	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 ∈ ℝ+ ↦ (!‘𝑘)) ⇝𝑟 (!‘𝑘))
376	369	nnne0d 12369	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ≠ 0)
377	376	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑥 ∈ ℝ+) → (!‘𝑘) ≠ 0)
378	367, 371, 372, 375, 376, 377	rlimdiv 15793	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 ∈ ℝ+ ↦ ((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘))) ⇝𝑟 (0 / (!‘𝑘)))
379	373, 376	div0d 12073	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) → (0 / (!‘𝑘)) = 0)
380	378, 379	breqtrd 5131	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 ∈ ℝ+ ↦ ((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘))) ⇝𝑟 0)
381	364, 365, 366, 380	fsumrlim 15958	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘))) ⇝𝑟 Σ𝑘 ∈ (0...𝑁)0)
382		fzfi 14095	. . . . . . . . . . . 12 ⊢ (0...𝑁) ∈ Fin
383	382	olci 880	. . . . . . . . . . 11 ⊢ ((0...𝑁) ⊆ (ℤ≥‘0) ∨ (0...𝑁) ∈ Fin)
384		sumz 15868	. . . . . . . . . . 11 ⊢ (((0...𝑁) ⊆ (ℤ≥‘0) ∨ (0...𝑁) ∈ Fin) → Σ𝑘 ∈ (0...𝑁)0 = 0)
385	383, 384	ax-mp 5	. . . . . . . . . 10 ⊢ Σ𝑘 ∈ (0...𝑁)0 = 0
386	381, 385	breqtrdi 5146	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘))) ⇝𝑟 0)
387	17, 360, 363, 386	rlimmul 15792	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)))) ⇝𝑟 ((!‘𝑁) · 0))
388	34	mul01d 11490	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → ((!‘𝑁) · 0) = 0)
389	387, 388	breqtrd 5131	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)))) ⇝𝑟 0)
390	356, 389	eqbrtrd 5127	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥)) ⇝𝑟 0)
391	56, 341, 54, 390	rlimsub 15791	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ ((((log‘𝑥)↑𝑁) / 𝑥) − (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥))) ⇝𝑟 (0 − 0))
392		0m0e0 12442	. . . . 5 ⊢ (0 − 0) = 0
393	391, 392	breqtrdi 5146	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ ((((log‘𝑥)↑𝑁) / 𝑥) − (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥))) ⇝𝑟 0)
394	339, 393	eqbrtrd 5127	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ ((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) / 𝑥)) ⇝𝑟 0)
395	30, 32, 336, 394	rlimadd 15790	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) + ((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) / 𝑥))) ⇝𝑟 ((!‘𝑁) + 0))
396		divsubdir 11991	. . . . . 6 ⊢ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) ∈ ℂ ∧ (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) / 𝑥) − ((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) / 𝑥)))
397	58, 63, 66, 396	syl3anc 1398	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) / 𝑥) − ((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) / 𝑥)))
398	397	oveq1d 7427	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) + ((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) / 𝑥)) = (((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) / 𝑥) − ((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) / 𝑥)) + ((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) / 𝑥)))
399	10, 2	rerpdivcld 13176	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) / 𝑥) ∈ ℝ)
400	399	recnd 11318	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) / 𝑥) ∈ ℂ)
401	32	recnd 11318	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → ((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) / 𝑥) ∈ ℂ)
402	400, 401	npcand 11654	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) / 𝑥) − ((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) / 𝑥)) + ((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) / 𝑥)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) / 𝑥))
403	398, 402	eqtrd 2796	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) + ((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) / 𝑥)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) / 𝑥))
404	403	mpteq2dva 5198	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) + ((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) / 𝑥))) = (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) / 𝑥)))
405	34	addridd 11491	. 2 ⊢ (𝑁 ∈ ℕ0 → ((!‘𝑁) + 0) = (!‘𝑁))
406	395, 404, 405	3brtr3d 5136	1 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) / 𝑥)) ⇝𝑟 (!‘𝑁))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  Vcvv 3451  ⦋csb 3847   ∖ cdif 3896   ⊆ wss 3899  {csn 4584  {cpr 4586   class class class wbr 5103   ↦ cmpt 5186  ‘cfv 6531  (class class class)co 7412  Fincfn 8957  ℂcc 11179  ℝcr 11180  0cc0 11181  1c1 11182   + caddc 11184   · cmul 11186  +∞cpnf 11321   ≤ cle 11325   − cmin 11522   / cdiv 11954  ℕcn 12316  ℕ₀cn0 12587  ℤcz 12674  ℤ_≥cuz 12946  ℝ⁺crp 13101  (,)cioo 13457  ...cfz 13620  ⌊cfl 13910  ↑cexp 14184  !cfa 14397  abscabs 15381   ⇝_𝑟 crli 15632  Σcsu 15833   D cdv 26163  logclog 26864  ↑_𝑐ccxp 26865

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-inf2 9626  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258  ax-pre-sup 11259  ax-addf 11260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7682  df-om 7867  df-1st 7990  df-2nd 7991  df-supp 8162  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-2o 8461  df-er 8701  df-map 8833  df-pm 8834  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fsupp 9338  df-fi 9387  df-sup 9418  df-inf 9419  df-oi 9488  df-card 10001  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-div 11955  df-nn 12317  df-2 12386  df-3 12387  df-4 12388  df-5 12389  df-6 12390  df-7 12391  df-8 12392  df-9 12393  df-n0 12588  df-z 12675  df-dec 12796  df-uz 12947  df-q 13057  df-rp 13102  df-xneg 13222  df-xadd 13223  df-xmul 13224  df-ioo 13461  df-ioc 13462  df-ico 13463  df-icc 13464  df-fz 13621  df-fzo 13769  df-fl 13912  df-mod 13990  df-seq 14125  df-exp 14185  df-fac 14398  df-bc 14427  df-hash 14455  df-shft 15200  df-cj 15246  df-re 15247  df-im 15248  df-sqrt 15382  df-abs 15383  df-limsup 15618  df-clim 15635  df-rlim 15636  df-sum 15834  df-ef 16213  df-e 16214  df-sin 16215  df-cos 16216  df-pi 16218  df-struct 17305  df-sets 17322  df-slot 17340  df-ndx 17352  df-base 17368  df-ress 17389  df-plusg 17421  df-mulr 17422  df-starv 17423  df-sca 17424  df-vsca 17425  df-ip 17426  df-tset 17427  df-ple 17428  df-ds 17430  df-unif 17431  df-hom 17432  df-cco 17433  df-rest 17573  df-topn 17574  df-0g 17592  df-gsum 17593  df-topgen 17594  df-pt 17595  df-prds 17598  df-xrs 17654  df-qtop 17659  df-imas 17660  df-xps 17662  df-mre 17736  df-mrc 17737  df-acs 17739  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-submnd 18959  df-mulg 19258  df-cntz 19511  df-cmn 19976  df-psmet 21650  df-xmet 21651  df-met 21652  df-bl 21653  df-mopn 21654  df-fbas 21655  df-fg 21656  df-cnfld 21659  df-top 23192  df-topon 23209  df-topsp 23231  df-bases 23244  df-cld 23317  df-ntr 23318  df-cls 23319  df-nei 23396  df-lp 23434  df-perf 23435  df-cn 23525  df-cnp 23526  df-haus 23613  df-cmp 23685  df-tx 23861  df-hmeo 24054  df-fil 24145  df-fm 24237  df-flim 24238  df-flf 24239  df-xms 24619  df-ms 24620  df-tms 24621  df-cncf 25179  df-limc 26166  df-dv 26167  df-log 26866  df-cxp 26867

This theorem is used by:  logfacrlim2  27535  selberglem2  27855