Theorem logexprlim 27193

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 13996	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (1...(⌊‘𝑥)) ∈ Fin)
2		simpr 484	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
3		elfznn 13575	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
4	3	nnrpd 13054	. . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ+)
5		rpdivcl 13039	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) → (𝑥 / 𝑛) ∈ ℝ+)
6	2, 4, 5	syl2an 596	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
7	6	relogcld 26589	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
8		simpll 766	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑁 ∈ ℕ0)
9	7, 8	reexpcld 14186	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛))↑𝑁) ∈ ℝ)
10	1, 9	fsumrecl 15755	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) ∈ ℝ)
11		relogcl 26541	. . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
12		id 22	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0)
13		reexpcl 14101	. . . . . . 7 ⊢ (((log‘𝑥) ∈ ℝ ∧ 𝑁 ∈ ℕ0) → ((log‘𝑥)↑𝑁) ∈ ℝ)
14	11, 12, 13	syl2anr 597	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥)↑𝑁) ∈ ℝ)
15		faccl 14306	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ)
16	15	adantr 480	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (!‘𝑁) ∈ ℕ)
17	16	nnred 12260	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (!‘𝑁) ∈ ℝ)
18		fzfid 13996	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (0...𝑁) ∈ Fin)
19	11	adantl 481	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℝ)
20		elfznn0 13642	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0)
21		reexpcl 14101	. . . . . . . . . 10 ⊢ (((log‘𝑥) ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((log‘𝑥)↑𝑘) ∈ ℝ)
22	19, 20, 21	syl2an 596	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → ((log‘𝑥)↑𝑘) ∈ ℝ)
23	20	adantl 481	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ0)
24	23	faccld 14307	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℕ)
25	22, 24	nndivred 12299	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℝ)
26	18, 25	fsumrecl 15755	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℝ)
27	17, 26	remulcld 11270	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) ∈ ℝ)
28	14, 27	resubcld 11670	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) ∈ ℝ)
29	10, 28	resubcld 11670	. . . 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 13087	. . 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 13044	. . . 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 588	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → ((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) / 𝑥) ∈ ℝ)
33		1red 11241	. . . 4 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℝ)
34	15	nncnd 12261	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℂ)
35		simpl 482	. . . . . . . . 9 ⊢ ((𝑘 = 𝑁 ∧ 𝑥 ∈ ℝ+) → 𝑘 = 𝑁)
36	35	oveq2d 7426	. . . . . . . 8 ⊢ ((𝑘 = 𝑁 ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥)↑𝑘) = ((log‘𝑥)↑𝑁))
37	36	oveq1d 7425	. . . . . . 7 ⊢ ((𝑘 = 𝑁 ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑘) / 𝑥) = (((log‘𝑥)↑𝑁) / 𝑥))
38	37	mpteq2dva 5219	. . . . . 6 ⊢ (𝑘 = 𝑁 → (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑘) / 𝑥)) = (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑁) / 𝑥)))
39	38	breq1d 5134	. . . . 5 ⊢ (𝑘 = 𝑁 → ((𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑘) / 𝑥)) ⇝𝑟 0 ↔ (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑁) / 𝑥)) ⇝𝑟 0))
40	11	recnd 11268	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℂ)
41		id 22	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℕ0)
42		cxpexp 26634	. . . . . . . . 9 ⊢ (((log‘𝑥) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((log‘𝑥)↑𝑐𝑘) = ((log‘𝑥)↑𝑘))
43	40, 41, 42	syl2anr 597	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥)↑𝑐𝑘) = ((log‘𝑥)↑𝑘))
44		rpcn 13024	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℂ)
45	44	adantl 481	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ)
46	45	cxp1d 26672	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (𝑥↑𝑐1) = 𝑥)
47	43, 46	oveq12d 7428	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑐𝑘) / (𝑥↑𝑐1)) = (((log‘𝑥)↑𝑘) / 𝑥))
48	47	mpteq2dva 5219	. . . . . 6 ⊢ (𝑘 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑐𝑘) / (𝑥↑𝑐1))) = (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑘) / 𝑥)))
49		nn0cn 12516	. . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → 𝑘 ∈ ℂ)
50		1rp 13017	. . . . . . 7 ⊢ 1 ∈ ℝ+
51		cxploglim2 26946	. . . . . . 7 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℝ+) → (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑐𝑘) / (𝑥↑𝑐1))) ⇝𝑟 0)
52	49, 50, 51	sylancl 586	. . . . . 6 ⊢ (𝑘 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑐𝑘) / (𝑥↑𝑐1))) ⇝𝑟 0)
53	48, 52	eqbrtrrd 5148	. . . . 5 ⊢ (𝑘 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑘) / 𝑥)) ⇝𝑟 0)
54	39, 53	vtoclga 3561	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑁) / 𝑥)) ⇝𝑟 0)
55		rerpdivcl 13044	. . . . . 6 ⊢ ((((log‘𝑥)↑𝑁) ∈ ℝ ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑁) / 𝑥) ∈ ℝ)
56	14, 55	sylancom 588	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑁) / 𝑥) ∈ ℝ)
57	56	recnd 11268	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑁) / 𝑥) ∈ ℂ)
58	10	recnd 11268	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) ∈ ℂ)
59	14	recnd 11268	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥)↑𝑁) ∈ ℂ)
60	34	adantr 480	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (!‘𝑁) ∈ ℂ)
61	26	recnd 11268	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℂ)
62	60, 61	mulcld 11260	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) ∈ ℂ)
63	59, 62	subcld 11599	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) ∈ ℂ)
64	58, 63	subcld 11599	. . . . 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 13032	. . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
66	65	adantl 481	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
67	66	simpld 494	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ)
68	66	simprd 495	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0)
69	64, 67, 68	divcld 12022	. . . 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 717	. . . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (!‘𝑁) ∈ ℕ)
72	71	nncnd 12261	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (!‘𝑁) ∈ ℂ)
73	70, 72	subcld 11599	. . . . . . 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 15460	. . . . . 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 717	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (((log‘𝑥)↑𝑁) / 𝑥) ∈ ℝ)
76	75	recnd 11268	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (((log‘𝑥)↑𝑁) / 𝑥) ∈ ℂ)
77	76	abscld 15460	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘(((log‘𝑥)↑𝑁) / 𝑥)) ∈ ℝ)
78		ioorp 13447	. . . . . . . . . 10 ⊢ (0(,)+∞) = ℝ+
79	78	eqcomi 2745	. . . . . . . . 9 ⊢ ℝ+ = (0(,)+∞)
80		nnuz 12900	. . . . . . . . 9 ⊢ ℕ = (ℤ≥‘1)
81		1z 12627	. . . . . . . . . 10 ⊢ 1 ∈ ℤ
82	81	a1i 11	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ∈ ℤ)
83		1red 11241	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ∈ ℝ)
84		1re 11240	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
85		1nn0 12522	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ0
86	84, 85	nn0addge1i 12554	. . . . . . . . . 10 ⊢ 1 ≤ (1 + 1)
87	86	a1i 11	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ≤ (1 + 1))
88		0red 11243	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ∈ ℝ)
89	71	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (!‘𝑁) ∈ ℕ)
90	89	nnred 12260	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (!‘𝑁) ∈ ℝ)
91		rpre 13022	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ)
92	91	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ)
93		fzfid 13996	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (0...𝑁) ∈ Fin)
94		simprl 770	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ+)
95		rpdivcl 13039	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+) → (𝑥 / 𝑦) ∈ ℝ+)
96	94, 95	sylan 580	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (𝑥 / 𝑦) ∈ ℝ+)
97	96	relogcld 26589	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (log‘(𝑥 / 𝑦)) ∈ ℝ)
98		reexpcl 14101	. . . . . . . . . . . . . 14 ⊢ (((log‘(𝑥 / 𝑦)) ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((log‘(𝑥 / 𝑦))↑𝑘) ∈ ℝ)
99	97, 20, 98	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → ((log‘(𝑥 / 𝑦))↑𝑘) ∈ ℝ)
100	20	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ0)
101	100	faccld 14307	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℕ)
102	99, 101	nndivred 12299	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)) ∈ ℝ)
103	93, 102	fsumrecl 15755	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)) ∈ ℝ)
104	92, 103	remulcld 11270	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))) ∈ ℝ)
105	90, 104	remulcld 11270	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)))) ∈ ℝ)
106		simpll 766	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → 𝑁 ∈ ℕ0)
107	97, 106	reexpcld 14186	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → ((log‘(𝑥 / 𝑦))↑𝑁) ∈ ℝ)
108		nnrp 13025	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ+)
109	108, 107	sylan2 593	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℕ) → ((log‘(𝑥 / 𝑦))↑𝑁) ∈ ℝ)
110		reelprrecn 11226	. . . . . . . . . . . 12 ⊢ ℝ ∈ {ℝ, ℂ}
111	110	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ℝ ∈ {ℝ, ℂ})
112	104	recnd 11268	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))) ∈ ℂ)
113	107, 89	nndivred 12299	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (((log‘(𝑥 / 𝑦))↑𝑁) / (!‘𝑁)) ∈ ℝ)
114		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑁 ∈ ℕ0)
115		advlogexp 26621	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 25925	. . . . . . . . . 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 11268	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → ((log‘(𝑥 / 𝑦))↑𝑁) ∈ ℂ)
119	72	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (!‘𝑁) ∈ ℂ)
120	71	nnne0d 12295	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (!‘𝑁) ≠ 0)
121	120	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (!‘𝑁) ≠ 0)
122	118, 119, 121	divcan2d 12024	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → ((!‘𝑁) · (((log‘(𝑥 / 𝑦))↑𝑁) / (!‘𝑁))) = ((log‘(𝑥 / 𝑦))↑𝑁))
123	122	mpteq2dva 5219	. . . . . . . . . 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 2771	. . . . . . . . 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 7418	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑛 → (𝑥 / 𝑦) = (𝑥 / 𝑛))
126	125	fveq2d 6885	. . . . . . . . . 10 ⊢ (𝑦 = 𝑛 → (log‘(𝑥 / 𝑦)) = (log‘(𝑥 / 𝑛)))
127	126	oveq1d 7425	. . . . . . . . 9 ⊢ (𝑦 = 𝑛 → ((log‘(𝑥 / 𝑦))↑𝑁) = ((log‘(𝑥 / 𝑛))↑𝑁))
128	94	rpxrd 13057	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ*)
129		simp1rl 1239	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑥 ∈ ℝ+)
130		simp2r 1201	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ∈ ℝ+)
131	129, 130	rpdivcld 13073	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → (𝑥 / 𝑛) ∈ ℝ+)
132	131	relogcld 26589	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
133		simp2l 1200	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑦 ∈ ℝ+)
134	129, 133	rpdivcld 13073	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → (𝑥 / 𝑦) ∈ ℝ+)
135	134	relogcld 26589	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → (log‘(𝑥 / 𝑦)) ∈ ℝ)
136		simp1l 1198	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑁 ∈ ℕ0)
137		log1 26551	. . . . . . . . . . 11 ⊢ (log‘1) = 0
138	130	rpcnd 13058	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ∈ ℂ)
139	138	mullidd 11258	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → (1 · 𝑛) = 𝑛)
140		simp33 1212	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑛 ≤ 𝑥)
141	139, 140	eqbrtrd 5146	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → (1 · 𝑛) ≤ 𝑥)
142		1red 11241	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 1 ∈ ℝ)
143	129	rpred 13056	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑥 ∈ ℝ)
144	142, 143, 130	lemuldivd 13105	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 1 ≤ (𝑥 / 𝑛))
146		logleb 26569	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℝ+ ∧ (𝑥 / 𝑛) ∈ ℝ+) → (1 ≤ (𝑥 / 𝑛) ↔ (log‘1) ≤ (log‘(𝑥 / 𝑛))))
147	50, 131, 146	sylancr 587	. . . . . . . . . . . 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 232	. . . . . . . . . . 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 5160	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 0 ≤ (log‘(𝑥 / 𝑛)))
150		simp32 1211	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝑦 ≤ 𝑛)
151	133, 130, 129	lediv2d 13080	. . . . . . . . . . . 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 232	. . . . . . . . . . 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 26593	. . . . . . . . . . 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 232	. . . . . . . . . 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 14198	. . . . . . . . . 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 1380	. . . . . . . . 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 2736	. . . . . . . . 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 1189	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → (𝑥 / 𝑦) ∈ ℝ+)
159	158	relogcld 26589	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → (log‘(𝑥 / 𝑦)) ∈ ℝ)
160		simpll 766	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → 𝑁 ∈ ℕ0)
161		rpcn 13024	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℂ)
162	161	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℂ)
163	162	3ad2antr1 1189	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → 𝑦 ∈ ℂ)
164	163	mullidd 11258	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → (1 · 𝑦) = 𝑦)
165		simpr3 1197	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → 𝑦 ≤ 𝑥)
166	164, 165	eqbrtrd 5146	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → (1 · 𝑦) ≤ 𝑥)
167		1red 11241	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → 1 ∈ ℝ)
168	94	rpred 13056	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
169	168	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → 𝑥 ∈ ℝ)
170		simpr1 1195	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → 𝑦 ∈ ℝ+)
171	167, 169, 170	lemuldivd 13105	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → ((1 · 𝑦) ≤ 𝑥 ↔ 1 ≤ (𝑥 / 𝑦)))
172	166, 171	mpbid 232	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → 1 ≤ (𝑥 / 𝑦))
173		logleb 26569	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℝ+ ∧ (𝑥 / 𝑦) ∈ ℝ+) → (1 ≤ (𝑥 / 𝑦) ↔ (log‘1) ≤ (log‘(𝑥 / 𝑦))))
174	50, 158, 173	sylancr 587	. . . . . . . . . . . 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 232	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → (log‘1) ≤ (log‘(𝑥 / 𝑦)))
176	137, 175	eqbrtrrid 5160	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦 ∧ 𝑦 ≤ 𝑥)) → 0 ≤ (log‘(𝑥 / 𝑦)))
177	159, 160, 176	expge0d 14187	. . . . . . . . 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 11870	. . . . . . . . . 10 ⊢ 1 ≤ 1
180	179	a1i 11	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ≤ 1)
181		simprr 772	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
182	168	leidd 11808	. . . . . . . . 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 25993	. . . . . . . 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 13996	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
185	94, 4, 5	syl2an 596	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
186	185	relogcld 26589	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
187		simpll 766	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑁 ∈ ℕ0)
188	186, 187	reexpcld 14186	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛))↑𝑁) ∈ ℝ)
189	184, 188	fsumrecl 15755	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) ∈ ℝ)
190	189	recnd 11268	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) ∈ ℂ)
191	94	rpcnd 13058	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℂ)
192	72, 191	mulcld 11260	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((!‘𝑁) · 𝑥) ∈ ℂ)
193	11	ad2antrl 728	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
194	193	recnd 11268	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℂ)
195	194, 114	expcld 14169	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((log‘𝑥)↑𝑁) ∈ ℂ)
196		fzfid 13996	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (0...𝑁) ∈ Fin)
197	193, 20, 21	syl2an 596	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (0...𝑁)) → ((log‘𝑥)↑𝑘) ∈ ℝ)
198	20	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ0)
199	198	faccld 14307	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℕ)
200	197, 199	nndivred 12299	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℝ)
201	200	recnd 11268	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℂ)
202	196, 201	fsumcl 15754	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℂ)
203	72, 202	mulcld 11260	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) ∈ ℂ)
204	195, 203	subcld 11599	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) ∈ ℂ)
205	190, 192, 204	sub32d 11631	. . . . . . . . . . 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 2737	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → 𝑦 = 𝑥)
208	207	fveq2d 6885	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → (⌊‘𝑦) = (⌊‘𝑥))
209	208	oveq2d 7426	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → (1...(⌊‘𝑦)) = (1...(⌊‘𝑥)))
210	209	sumeq1d 15721	. . . . . . . . . . . . . 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 7418	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = 𝑥 → (𝑥 / 𝑦) = (𝑥 / 𝑥))
212	65	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
213		divid 11932	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (𝑥 / 𝑥) = 1)
214	212, 213	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑥 / 𝑥) = 1)
215	211, 214	sylan9eqr 2793	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → (𝑥 / 𝑦) = 1)
216	215	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 / 𝑦) = 1)
217	216	fveq2d 6885	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ (0...𝑁)) → (log‘(𝑥 / 𝑦)) = (log‘1))
218	217, 137	eqtrdi 2787	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ (0...𝑁)) → (log‘(𝑥 / 𝑦)) = 0)
219	218	oveq1d 7425	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ (0...𝑁)) → ((log‘(𝑥 / 𝑦))↑𝑘) = (0↑𝑘))
220	219	oveq1d 7425	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)) = ((0↑𝑘) / (!‘𝑘)))
221	220	sumeq2dv 15723	. . . . . . . . . . . . . . . . . . 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 12899	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ℕ0 = (ℤ≥‘0)
223	114, 222	eleqtrdi 2845	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑁 ∈ (ℤ≥‘0))
224		eluzfz1 13553	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ (ℤ≥‘0) → 0 ∈ (0...𝑁))
225	223, 224	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ∈ (0...𝑁))
226	225	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → 0 ∈ (0...𝑁))
227	226	snssd 4790	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → {0} ⊆ (0...𝑁))
228		elsni 4623	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ {0} → 𝑘 = 0)
229	228	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ {0}) → 𝑘 = 0)
230		oveq2 7418	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = 0 → (0↑𝑘) = (0↑0))
231		0exp0e1 14089	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0↑0) = 1
232	230, 231	eqtrdi 2787	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = 0 → (0↑𝑘) = 1)
233		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 = 0 → (!‘𝑘) = (!‘0))
234		fac0 14299	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (!‘0) = 1
235	233, 234	eqtrdi 2787	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = 0 → (!‘𝑘) = 1)
236	232, 235	oveq12d 7428	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 0 → ((0↑𝑘) / (!‘𝑘)) = (1 / 1))
237		1div1e1 11937	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1 / 1) = 1
238	236, 237	eqtrdi 2787	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 0 → ((0↑𝑘) / (!‘𝑘)) = 1)
239	229, 238	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ {0}) → ((0↑𝑘) / (!‘𝑘)) = 1)
240		ax-1cn 11192	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 1 ∈ ℂ
241	239, 240	eqeltrdi 2843	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ {0}) → ((0↑𝑘) / (!‘𝑘)) ∈ ℂ)
242		eldifi 4111	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑘 ∈ ((0...𝑁) ∖ {0}) → 𝑘 ∈ (0...𝑁))
243	242	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → 𝑘 ∈ (0...𝑁))
244	243, 20	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → 𝑘 ∈ ℕ0)
245		eldifsni 4771	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑘 ∈ ((0...𝑁) ∖ {0}) → 𝑘 ≠ 0)
246	245	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → 𝑘 ≠ 0)
247		eldifsn 4767	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (ℕ0 ∖ {0}) ↔ (𝑘 ∈ ℕ0 ∧ 𝑘 ≠ 0))
248	244, 246, 247	sylanbrc 583	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → 𝑘 ∈ (ℕ0 ∖ {0}))
249		dfn2 12519	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ℕ = (ℕ0 ∖ {0})
250	248, 249	eleqtrrdi 2846	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → 𝑘 ∈ ℕ)
251	250	0expd 14162	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → (0↑𝑘) = 0)
252	251	oveq1d 7425	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → ((0↑𝑘) / (!‘𝑘)) = (0 / (!‘𝑘)))
253	244	faccld 14307	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → (!‘𝑘) ∈ ℕ)
254	253	nncnd 12261	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → (!‘𝑘) ∈ ℂ)
255	253	nnne0d 12295	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → (!‘𝑘) ≠ 0)
256	254, 255	div0d 12021	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → (0 / (!‘𝑘)) = 0)
257	252, 256	eqtrd 2771	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → ((0↑𝑘) / (!‘𝑘)) = 0)
258		fzfid 13996	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → (0...𝑁) ∈ Fin)
259	227, 241, 257, 258	fsumss 15746	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → Σ𝑘 ∈ {0} ((0↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ (0...𝑁)((0↑𝑘) / (!‘𝑘)))
260	221, 259	eqtr4d 2774	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ {0} ((0↑𝑘) / (!‘𝑘)))
261		0cn 11232	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℂ
262	238	sumsn 15767	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℂ ∧ 1 ∈ ℂ) → Σ𝑘 ∈ {0} ((0↑𝑘) / (!‘𝑘)) = 1)
263	261, 240, 262	mp2an 692	. . . . . . . . . . . . . . . . . 18 ⊢ Σ𝑘 ∈ {0} ((0↑𝑘) / (!‘𝑘)) = 1
264	260, 263	eqtrdi 2787	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)) = 1)
265	207, 264	oveq12d 7428	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))) = (𝑥 · 1))
266	191	mulridd 11257	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑥 · 1) = 𝑥)
267	266	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → (𝑥 · 1) = 𝑥)
268	265, 267	eqtrd 2771	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))) = 𝑥)
269	268	oveq2d 7426	. . . . . . . . . . . . . 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 7428	. . . . . . . . . . . . 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 7445	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · 𝑥)) ∈ V)
272	206, 270, 94, 271	fvmptd 6998	. . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → 𝑦 = 1)
274	273	fveq2d 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (⌊‘𝑦) = (⌊‘1))
275		flid 13830	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 ∈ ℤ → (⌊‘1) = 1)
276	81, 275	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (⌊‘1) = 1
277	274, 276	eqtrdi 2787	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (⌊‘𝑦) = 1)
278	277	oveq2d 7426	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (1...(⌊‘𝑦)) = (1...1))
279	278	sumeq1d 15721	. . . . . . . . . . . . . . 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 12014	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑥 / 1) = 𝑥)
281	280	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (𝑥 / 1) = 𝑥)
282	281	fveq2d 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (log‘(𝑥 / 1)) = (log‘𝑥))
283	282	oveq1d 7425	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → ((log‘(𝑥 / 1))↑𝑁) = ((log‘𝑥)↑𝑁))
284	195	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → ((log‘𝑥)↑𝑁) ∈ ℂ)
285	283, 284	eqeltrd 2835	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → ((log‘(𝑥 / 1))↑𝑁) ∈ ℂ)
286		oveq2 7418	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 1 → (𝑥 / 𝑛) = (𝑥 / 1))
287	286	fveq2d 6885	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 1 → (log‘(𝑥 / 𝑛)) = (log‘(𝑥 / 1)))
288	287	oveq1d 7425	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 1 → ((log‘(𝑥 / 𝑛))↑𝑁) = ((log‘(𝑥 / 1))↑𝑁))
289	288	fsum1 15768	. . . . . . . . . . . . . . . 16 ⊢ ((1 ∈ ℤ ∧ ((log‘(𝑥 / 1))↑𝑁) ∈ ℂ) → Σ𝑛 ∈ (1...1)((log‘(𝑥 / 𝑛))↑𝑁) = ((log‘(𝑥 / 1))↑𝑁))
290	81, 285, 289	sylancr 587	. . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) = ((log‘𝑥)↑𝑁))
292	273	oveq2d 7426	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (𝑥 / 𝑦) = (𝑥 / 1))
293	292, 281	eqtrd 2771	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (𝑥 / 𝑦) = 𝑥)
294	293	fveq2d 6885	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (log‘(𝑥 / 𝑦)) = (log‘𝑥))
295	294	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) ∧ 𝑘 ∈ (0...𝑁)) → (log‘(𝑥 / 𝑦)) = (log‘𝑥))
296	295	oveq1d 7425	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) ∧ 𝑘 ∈ (0...𝑁)) → ((log‘(𝑥 / 𝑦))↑𝑘) = ((log‘𝑥)↑𝑘))
297	296	oveq1d 7425	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)) = (((log‘𝑥)↑𝑘) / (!‘𝑘)))
298	297	sumeq2dv 15723	. . . . . . . . . . . . . . . . 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 7428	. . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℂ)
301	300	mullidd 11258	. . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . 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 7426	. . . . . . . . . . . . . 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 7428	. . . . . . . . . . . . 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 7445	. . . . . . . . . . . . 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 6998	. . . . . . . . . . . 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 7428	. . . . . . . . . . 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 11699	. . . . . . . . . . . 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 717	. . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ≠ 0)
311	309, 191, 310	divcan1d 12023	. . . . . . . . . . . . 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 7425	. . . . . . . . . . . 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 2771	. . . . . . . . . . 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 2781	. . . . . . . . . 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 6885	. . . . . . . . 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 15478	. . . . . . . . 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 13029	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
318	317	ad2antrl 728	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
319		absid 15320	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (abs‘𝑥) = 𝑥)
320	318, 319	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘𝑥) = 𝑥)
321	320	oveq2d 7426	. . . . . . . . 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 2775	. . . . . . . 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 11235	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ∈ ℂ)
324	294	oveq1d 7425	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → ((log‘(𝑥 / 𝑦))↑𝑁) = ((log‘𝑥)↑𝑁))
325	323, 324	csbied 3915	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ⦋1 / 𝑦⦌((log‘(𝑥 / 𝑦))↑𝑁) = ((log‘𝑥)↑𝑁))
326	183, 322, 325	3brtr3d 5155	. . . . . . 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 717	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((log‘𝑥)↑𝑁) ∈ ℝ)
328	74, 327, 94	lemuldivd 13105	. . . . . . 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 232	. . . . . 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 15438	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (((log‘𝑥)↑𝑁) / 𝑥) ≤ (abs‘(((log‘𝑥)↑𝑁) / 𝑥)))
331	74, 75, 77, 329, 330	letrd 11397	. . . . 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 717	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (((log‘𝑥)↑𝑁) / 𝑥) ∈ ℂ)
333	332	subid1d 11588	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((((log‘𝑥)↑𝑁) / 𝑥) − 0) = (((log‘𝑥)↑𝑁) / 𝑥))
334	333	fveq2d 6885	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘((((log‘𝑥)↑𝑁) / 𝑥) − 0)) = (abs‘(((log‘𝑥)↑𝑁) / 𝑥)))
335	331, 334	breqtrrd 5152	. . . 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 15670	. . 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 11940	. . . . . 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 1373	. . . . 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 5219	. . . 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 13044	. . . . . . 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 588	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥) ∈ ℝ)
342		divass 11919	. . . . . . . . . 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 1373	. . . . . . . . 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 11268	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℂ)
345	18, 67, 344, 68	fsumdivc 15807	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) / 𝑥) = Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / (!‘𝑘)) / 𝑥))
346	22	recnd 11268	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → ((log‘𝑥)↑𝑘) ∈ ℂ)
347	24	nnrpd 13054	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℝ+)
348	347	rpcnne0d 13065	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → ((!‘𝑘) ∈ ℂ ∧ (!‘𝑘) ≠ 0))
349	66	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
350		divdiv32 11954	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → ((((log‘𝑥)↑𝑘) / (!‘𝑘)) / 𝑥) = ((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)))
352	351	sumeq2dv 15723	. . . . . . . . . . 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 2771	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) / 𝑥) = Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)))
354	353	oveq2d 7426	. . . . . . . . 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 2771	. . . . . . . 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 5219	. . . . . . 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 480	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → 𝑥 ∈ ℝ+)
358	22, 357	rerpdivcld 13087	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘𝑥)↑𝑘) / 𝑥) ∈ ℝ)
359	358, 24	nndivred 12299	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → ((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)) ∈ ℝ)
360	18, 359	fsumrecl 15755	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)) ∈ ℝ)
361		rpssre 13021	. . . . . . . . . 10 ⊢ ℝ+ ⊆ ℝ
362		rlimconst 15565	. . . . . . . . . 10 ⊢ ((ℝ+ ⊆ ℝ ∧ (!‘𝑁) ∈ ℂ) → (𝑥 ∈ ℝ+ ↦ (!‘𝑁)) ⇝𝑟 (!‘𝑁))
363	361, 34, 362	sylancr 587	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (!‘𝑁)) ⇝𝑟 (!‘𝑁))
364	361	a1i 11	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → ℝ+ ⊆ ℝ)
365		fzfid 13996	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → (0...𝑁) ∈ Fin)
366	359	anasss 466	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 𝑘 ∈ (0...𝑁))) → ((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)) ∈ ℝ)
367	358	an32s 652	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑘) / 𝑥) ∈ ℝ)
368	20	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ0)
369	368	faccld 14307	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℕ)
370	369	nnred 12260	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℝ)
371	370	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑥 ∈ ℝ+) → (!‘𝑘) ∈ ℝ)
372	368, 53	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑘) / 𝑥)) ⇝𝑟 0)
373	369	nncnd 12261	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℂ)
374		rlimconst 15565	. . . . . . . . . . . . . 14 ⊢ ((ℝ+ ⊆ ℝ ∧ (!‘𝑘) ∈ ℂ) → (𝑥 ∈ ℝ+ ↦ (!‘𝑘)) ⇝𝑟 (!‘𝑘))
375	361, 373, 374	sylancr 587	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 ∈ ℝ+ ↦ (!‘𝑘)) ⇝𝑟 (!‘𝑘))
376	369	nnne0d 12295	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ≠ 0)
377	376	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) ∧ 𝑥 ∈ ℝ+) → (!‘𝑘) ≠ 0)
378	367, 371, 372, 375, 376, 377	rlimdiv 15667	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 ∈ ℝ+ ↦ ((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘))) ⇝𝑟 (0 / (!‘𝑘)))
379	373, 376	div0d 12021	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) → (0 / (!‘𝑘)) = 0)
380	378, 379	breqtrd 5150	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 ∈ ℝ+ ↦ ((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘))) ⇝𝑟 0)
381	364, 365, 366, 380	fsumrlim 15832	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘))) ⇝𝑟 Σ𝑘 ∈ (0...𝑁)0)
382		fzfi 13995	. . . . . . . . . . . 12 ⊢ (0...𝑁) ∈ Fin
383	382	olci 866	. . . . . . . . . . 11 ⊢ ((0...𝑁) ⊆ (ℤ≥‘0) ∨ (0...𝑁) ∈ Fin)
384		sumz 15743	. . . . . . . . . . 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 5165	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘))) ⇝𝑟 0)
387	17, 360, 363, 386	rlimmul 15666	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)))) ⇝𝑟 ((!‘𝑁) · 0))
388	34	mul01d 11439	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → ((!‘𝑁) · 0) = 0)
389	387, 388	breqtrd 5150	. . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)))) ⇝𝑟 0)
390	356, 389	eqbrtrd 5146	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥)) ⇝𝑟 0)
391	56, 341, 54, 390	rlimsub 15665	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ ((((log‘𝑥)↑𝑁) / 𝑥) − (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥))) ⇝𝑟 (0 − 0))
392		0m0e0 12365	. . . . 5 ⊢ (0 − 0) = 0
393	391, 392	breqtrdi 5165	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ ((((log‘𝑥)↑𝑁) / 𝑥) − (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥))) ⇝𝑟 0)
394	339, 393	eqbrtrd 5146	. . 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 15664	. 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 11940	. . . . . 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 1373	. . . . 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 7425	. . . 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 13087	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) / 𝑥) ∈ ℝ)
400	399	recnd 11268	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) / 𝑥) ∈ ℂ)
401	32	recnd 11268	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ℝ+) → ((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) / 𝑥) ∈ ℂ)
402	400, 401	npcand 11603	. . . 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 2771	. . 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 5219	. 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 11440	. 2 ⊢ (𝑁 ∈ ℕ0 → ((!‘𝑁) + 0) = (!‘𝑁))
406	395, 404, 405	3brtr3d 5155	1 ⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) / 𝑥)) ⇝𝑟 (!‘𝑁))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  Vcvv 3464  ⦋csb 3879   ∖ cdif 3928   ⊆ wss 3931  {csn 4606  {cpr 4608   class class class wbr 5124   ↦ cmpt 5206  ‘cfv 6536  (class class class)co 7410  Fincfn 8964  ℂcc 11132  ℝcr 11133  0cc0 11134  1c1 11135   + caddc 11137   · cmul 11139  +∞cpnf 11271   ≤ cle 11275   − cmin 11471   / cdiv 11899  ℕcn 12245  ℕ₀cn0 12506  ℤcz 12593  ℤ_≥cuz 12857  ℝ⁺crp 13013  (,)cioo 13367  ...cfz 13529  ⌊cfl 13812  ↑cexp 14084  !cfa 14296  abscabs 15258   ⇝_𝑟 crli 15506  Σcsu 15707   D cdv 25821  logclog 26520  ↑_𝑐ccxp 26521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211  ax-pre-sup 11212  ax-addf 11213

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-iin 4975  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  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 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7676  df-om 7867  df-1st 7993  df-2nd 7994  df-supp 8165  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-er 8724  df-map 8847  df-pm 8848  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9379  df-fi 9428  df-sup 9459  df-inf 9460  df-oi 9529  df-card 9958  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-div 11900  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-z 12594  df-dec 12714  df-uz 12858  df-q 12970  df-rp 13014  df-xneg 13133  df-xadd 13134  df-xmul 13135  df-ioo 13371  df-ioc 13372  df-ico 13373  df-icc 13374  df-fz 13530  df-fzo 13677  df-fl 13814  df-mod 13892  df-seq 14025  df-exp 14085  df-fac 14297  df-bc 14326  df-hash 14354  df-shft 15091  df-cj 15123  df-re 15124  df-im 15125  df-sqrt 15259  df-abs 15260  df-limsup 15492  df-clim 15509  df-rlim 15510  df-sum 15708  df-ef 16088  df-e 16089  df-sin 16090  df-cos 16091  df-pi 16093  df-struct 17171  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-ress 17257  df-plusg 17289  df-mulr 17290  df-starv 17291  df-sca 17292  df-vsca 17293  df-ip 17294  df-tset 17295  df-ple 17296  df-ds 17298  df-unif 17299  df-hom 17300  df-cco 17301  df-rest 17441  df-topn 17442  df-0g 17460  df-gsum 17461  df-topgen 17462  df-pt 17463  df-prds 17466  df-xrs 17521  df-qtop 17526  df-imas 17527  df-xps 17529  df-mre 17603  df-mrc 17604  df-acs 17606  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-submnd 18767  df-mulg 19056  df-cntz 19305  df-cmn 19768  df-psmet 21312  df-xmet 21313  df-met 21314  df-bl 21315  df-mopn 21316  df-fbas 21317  df-fg 21318  df-cnfld 21321  df-top 22837  df-topon 22854  df-topsp 22876  df-bases 22889  df-cld 22962  df-ntr 22963  df-cls 22964  df-nei 23041  df-lp 23079  df-perf 23080  df-cn 23170  df-cnp 23171  df-haus 23258  df-cmp 23330  df-tx 23505  df-hmeo 23698  df-fil 23789  df-fm 23881  df-flim 23882  df-flf 23883  df-xms 24264  df-ms 24265  df-tms 24266  df-cncf 24827  df-limc 25824  df-dv 25825  df-log 26522  df-cxp 26523

This theorem is referenced by:  logfacrlim2  27194  selberglem2  27514