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Theorem logexprlim 27284
Description: The sum Σ𝑛𝑥, 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.
StepHypRef Expression
1 fzfid 14011 . . . . . 6 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → (1...(⌊‘𝑥)) ∈ Fin)
2 simpr 484 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
3 elfznn 13590 . . . . . . . . . 10 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
43nnrpd 13073 . . . . . . . . 9 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ+)
5 rpdivcl 13058 . . . . . . . . 9 ((𝑥 ∈ ℝ+𝑛 ∈ ℝ+) → (𝑥 / 𝑛) ∈ ℝ+)
62, 4, 5syl2an 596 . . . . . . . 8 (((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
76relogcld 26680 . . . . . . 7 (((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
8 simpll 767 . . . . . . 7 (((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑁 ∈ ℕ0)
97, 8reexpcld 14200 . . . . . 6 (((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛))↑𝑁) ∈ ℝ)
101, 9fsumrecl 15767 . . . . 5 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) ∈ ℝ)
11 relogcl 26632 . . . . . . 7 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
12 id 22 . . . . . . 7 (𝑁 ∈ ℕ0𝑁 ∈ ℕ0)
13 reexpcl 14116 . . . . . . 7 (((log‘𝑥) ∈ ℝ ∧ 𝑁 ∈ ℕ0) → ((log‘𝑥)↑𝑁) ∈ ℝ)
1411, 12, 13syl2anr 597 . . . . . 6 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → ((log‘𝑥)↑𝑁) ∈ ℝ)
15 faccl 14319 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ)
1615adantr 480 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → (!‘𝑁) ∈ ℕ)
1716nnred 12279 . . . . . . 7 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → (!‘𝑁) ∈ ℝ)
18 fzfid 14011 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → (0...𝑁) ∈ Fin)
1911adantl 481 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℝ)
20 elfznn0 13657 . . . . . . . . . 10 (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0)
21 reexpcl 14116 . . . . . . . . . 10 (((log‘𝑥) ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((log‘𝑥)↑𝑘) ∈ ℝ)
2219, 20, 21syl2an 596 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → ((log‘𝑥)↑𝑘) ∈ ℝ)
2320adantl 481 . . . . . . . . . 10 (((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ0)
2423faccld 14320 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℕ)
2522, 24nndivred 12318 . . . . . . . 8 (((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℝ)
2618, 25fsumrecl 15767 . . . . . . 7 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℝ)
2717, 26remulcld 11289 . . . . . 6 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) ∈ ℝ)
2814, 27resubcld 11689 . . . . 5 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) ∈ ℝ)
2910, 28resubcld 11689 . . . 4 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) ∈ ℝ)
3029, 2rerpdivcld 13106 . . 3 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) ∈ ℝ)
31 rerpdivcl 13063 . . . 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)
3228, 31sylancom 588 . . 3 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → ((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) / 𝑥) ∈ ℝ)
33 1red 11260 . . . 4 (𝑁 ∈ ℕ0 → 1 ∈ ℝ)
3415nncnd 12280 . . . 4 (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℂ)
35 simpl 482 . . . . . . . . 9 ((𝑘 = 𝑁𝑥 ∈ ℝ+) → 𝑘 = 𝑁)
3635oveq2d 7447 . . . . . . . 8 ((𝑘 = 𝑁𝑥 ∈ ℝ+) → ((log‘𝑥)↑𝑘) = ((log‘𝑥)↑𝑁))
3736oveq1d 7446 . . . . . . 7 ((𝑘 = 𝑁𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑘) / 𝑥) = (((log‘𝑥)↑𝑁) / 𝑥))
3837mpteq2dva 5248 . . . . . 6 (𝑘 = 𝑁 → (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑘) / 𝑥)) = (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑁) / 𝑥)))
3938breq1d 5158 . . . . 5 (𝑘 = 𝑁 → ((𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑘) / 𝑥)) ⇝𝑟 0 ↔ (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑁) / 𝑥)) ⇝𝑟 0))
4011recnd 11287 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℂ)
41 id 22 . . . . . . . . 9 (𝑘 ∈ ℕ0𝑘 ∈ ℕ0)
42 cxpexp 26725 . . . . . . . . 9 (((log‘𝑥) ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((log‘𝑥)↑𝑐𝑘) = ((log‘𝑥)↑𝑘))
4340, 41, 42syl2anr 597 . . . . . . . 8 ((𝑘 ∈ ℕ0𝑥 ∈ ℝ+) → ((log‘𝑥)↑𝑐𝑘) = ((log‘𝑥)↑𝑘))
44 rpcn 13043 . . . . . . . . . 10 (𝑥 ∈ ℝ+𝑥 ∈ ℂ)
4544adantl 481 . . . . . . . . 9 ((𝑘 ∈ ℕ0𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ)
4645cxp1d 26763 . . . . . . . 8 ((𝑘 ∈ ℕ0𝑥 ∈ ℝ+) → (𝑥𝑐1) = 𝑥)
4743, 46oveq12d 7449 . . . . . . 7 ((𝑘 ∈ ℕ0𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑐𝑘) / (𝑥𝑐1)) = (((log‘𝑥)↑𝑘) / 𝑥))
4847mpteq2dva 5248 . . . . . 6 (𝑘 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑐𝑘) / (𝑥𝑐1))) = (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑘) / 𝑥)))
49 nn0cn 12534 . . . . . . 7 (𝑘 ∈ ℕ0𝑘 ∈ ℂ)
50 1rp 13036 . . . . . . 7 1 ∈ ℝ+
51 cxploglim2 27037 . . . . . . 7 ((𝑘 ∈ ℂ ∧ 1 ∈ ℝ+) → (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑐𝑘) / (𝑥𝑐1))) ⇝𝑟 0)
5249, 50, 51sylancl 586 . . . . . 6 (𝑘 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑐𝑘) / (𝑥𝑐1))) ⇝𝑟 0)
5348, 52eqbrtrrd 5172 . . . . 5 (𝑘 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑘) / 𝑥)) ⇝𝑟 0)
5439, 53vtoclga 3577 . . . 4 (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑁) / 𝑥)) ⇝𝑟 0)
55 rerpdivcl 13063 . . . . . 6 ((((log‘𝑥)↑𝑁) ∈ ℝ ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑁) / 𝑥) ∈ ℝ)
5614, 55sylancom 588 . . . . 5 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑁) / 𝑥) ∈ ℝ)
5756recnd 11287 . . . 4 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑁) / 𝑥) ∈ ℂ)
5810recnd 11287 . . . . . 6 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) ∈ ℂ)
5914recnd 11287 . . . . . . 7 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → ((log‘𝑥)↑𝑁) ∈ ℂ)
6034adantr 480 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → (!‘𝑁) ∈ ℂ)
6126recnd 11287 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℂ)
6260, 61mulcld 11279 . . . . . . 7 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) ∈ ℂ)
6359, 62subcld 11618 . . . . . 6 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) ∈ ℂ)
6458, 63subcld 11618 . . . . 5 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) ∈ ℂ)
65 rpcnne0 13051 . . . . . . 7 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
6665adantl 481 . . . . . 6 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
6766simpld 494 . . . . 5 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ)
6866simprd 495 . . . . 5 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → 𝑥 ≠ 0)
6964, 67, 68divcld 12041 . . . 4 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))))) / 𝑥) ∈ ℂ)
7069adantrr 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)
7115adantr 480 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (!‘𝑁) ∈ ℕ)
7271nncnd 12280 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (!‘𝑁) ∈ ℂ)
7370, 72subcld 11618 . . . . . . 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)
7473abscld 15472 . . . . . 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)
7556adantrr 717 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (((log‘𝑥)↑𝑁) / 𝑥) ∈ ℝ)
7675recnd 11287 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (((log‘𝑥)↑𝑁) / 𝑥) ∈ ℂ)
7776abscld 15472 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘(((log‘𝑥)↑𝑁) / 𝑥)) ∈ ℝ)
78 ioorp 13462 . . . . . . . . . 10 (0(,)+∞) = ℝ+
7978eqcomi 2744 . . . . . . . . 9 + = (0(,)+∞)
80 nnuz 12919 . . . . . . . . 9 ℕ = (ℤ‘1)
81 1z 12645 . . . . . . . . . 10 1 ∈ ℤ
8281a1i 11 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ∈ ℤ)
83 1red 11260 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ∈ ℝ)
84 1re 11259 . . . . . . . . . . 11 1 ∈ ℝ
85 1nn0 12540 . . . . . . . . . . 11 1 ∈ ℕ0
8684, 85nn0addge1i 12572 . . . . . . . . . 10 1 ≤ (1 + 1)
8786a1i 11 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ≤ (1 + 1))
88 0red 11262 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ∈ ℝ)
8971adantr 480 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (!‘𝑁) ∈ ℕ)
9089nnred 12279 . . . . . . . . . 10 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (!‘𝑁) ∈ ℝ)
91 rpre 13041 . . . . . . . . . . . 12 (𝑦 ∈ ℝ+𝑦 ∈ ℝ)
9291adantl 481 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℝ)
93 fzfid 14011 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (0...𝑁) ∈ Fin)
94 simprl 771 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ+)
95 rpdivcl 13058 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ+𝑦 ∈ ℝ+) → (𝑥 / 𝑦) ∈ ℝ+)
9694, 95sylan 580 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (𝑥 / 𝑦) ∈ ℝ+)
9796relogcld 26680 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (log‘(𝑥 / 𝑦)) ∈ ℝ)
98 reexpcl 14116 . . . . . . . . . . . . . 14 (((log‘(𝑥 / 𝑦)) ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((log‘(𝑥 / 𝑦))↑𝑘) ∈ ℝ)
9997, 20, 98syl2an 596 . . . . . . . . . . . . 13 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → ((log‘(𝑥 / 𝑦))↑𝑘) ∈ ℝ)
10020adantl 481 . . . . . . . . . . . . . 14 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ0)
101100faccld 14320 . . . . . . . . . . . . 13 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℕ)
10299, 101nndivred 12318 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)) ∈ ℝ)
10393, 102fsumrecl 15767 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)) ∈ ℝ)
10492, 103remulcld 11289 . . . . . . . . . 10 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))) ∈ ℝ)
10590, 104remulcld 11289 . . . . . . . . 9 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)))) ∈ ℝ)
106 simpll 767 . . . . . . . . . 10 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → 𝑁 ∈ ℕ0)
10797, 106reexpcld 14200 . . . . . . . . 9 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → ((log‘(𝑥 / 𝑦))↑𝑁) ∈ ℝ)
108 nnrp 13044 . . . . . . . . . 10 (𝑦 ∈ ℕ → 𝑦 ∈ ℝ+)
109108, 107sylan2 593 . . . . . . . . 9 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℕ) → ((log‘(𝑥 / 𝑦))↑𝑁) ∈ ℝ)
110 reelprrecn 11245 . . . . . . . . . . . 12 ℝ ∈ {ℝ, ℂ}
111110a1i 11 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ℝ ∈ {ℝ, ℂ})
112104recnd 11287 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))) ∈ ℂ)
113107, 89nndivred 12318 . . . . . . . . . . 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 26712 . . . . . . . . . . . 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))))
11694, 114, 115syl2anc 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))))
117111, 112, 113, 116, 72dvmptcmul 26017 . . . . . . . . . 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)))))
118107recnd 11287 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → ((log‘(𝑥 / 𝑦))↑𝑁) ∈ ℂ)
11972adantr 480 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (!‘𝑁) ∈ ℂ)
12071nnne0d 12314 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (!‘𝑁) ≠ 0)
121120adantr 480 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → (!‘𝑁) ≠ 0)
122118, 119, 121divcan2d 12043 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → ((!‘𝑁) · (((log‘(𝑥 / 𝑦))↑𝑁) / (!‘𝑁))) = ((log‘(𝑥 / 𝑦))↑𝑁))
123122mpteq2dva 5248 . . . . . . . . . 10 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑦 ∈ ℝ+ ↦ ((!‘𝑁) · (((log‘(𝑥 / 𝑦))↑𝑁) / (!‘𝑁)))) = (𝑦 ∈ ℝ+ ↦ ((log‘(𝑥 / 𝑦))↑𝑁)))
124117, 123eqtrd 2775 . . . . . . . . 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 7439 . . . . . . . . . . 11 (𝑦 = 𝑛 → (𝑥 / 𝑦) = (𝑥 / 𝑛))
126125fveq2d 6911 . . . . . . . . . 10 (𝑦 = 𝑛 → (log‘(𝑥 / 𝑦)) = (log‘(𝑥 / 𝑛)))
127126oveq1d 7446 . . . . . . . . 9 (𝑦 = 𝑛 → ((log‘(𝑥 / 𝑦))↑𝑁) = ((log‘(𝑥 / 𝑛))↑𝑁))
12894rpxrd 13076 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ*)
129 simp1rl 1237 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → 𝑥 ∈ ℝ+)
130 simp2r 1199 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → 𝑛 ∈ ℝ+)
131129, 130rpdivcld 13092 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → (𝑥 / 𝑛) ∈ ℝ+)
132131relogcld 26680 . . . . . . . . . 10 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
133 simp2l 1198 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → 𝑦 ∈ ℝ+)
134129, 133rpdivcld 13092 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → (𝑥 / 𝑦) ∈ ℝ+)
135134relogcld 26680 . . . . . . . . . 10 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → (log‘(𝑥 / 𝑦)) ∈ ℝ)
136 simp1l 1196 . . . . . . . . . 10 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → 𝑁 ∈ ℕ0)
137 log1 26642 . . . . . . . . . . 11 (log‘1) = 0
138130rpcnd 13077 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → 𝑛 ∈ ℂ)
139138mullidd 11277 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → (1 · 𝑛) = 𝑛)
140 simp33 1210 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → 𝑛𝑥)
141139, 140eqbrtrd 5170 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → (1 · 𝑛) ≤ 𝑥)
142 1red 11260 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → 1 ∈ ℝ)
143129rpred 13075 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → 𝑥 ∈ ℝ)
144142, 143, 130lemuldivd 13124 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → ((1 · 𝑛) ≤ 𝑥 ↔ 1 ≤ (𝑥 / 𝑛)))
145141, 144mpbid 232 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → 1 ≤ (𝑥 / 𝑛))
146 logleb 26660 . . . . . . . . . . . . 13 ((1 ∈ ℝ+ ∧ (𝑥 / 𝑛) ∈ ℝ+) → (1 ≤ (𝑥 / 𝑛) ↔ (log‘1) ≤ (log‘(𝑥 / 𝑛))))
14750, 131, 146sylancr 587 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → (1 ≤ (𝑥 / 𝑛) ↔ (log‘1) ≤ (log‘(𝑥 / 𝑛))))
148145, 147mpbid 232 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → (log‘1) ≤ (log‘(𝑥 / 𝑛)))
149137, 148eqbrtrrid 5184 . . . . . . . . . 10 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → 0 ≤ (log‘(𝑥 / 𝑛)))
150 simp32 1209 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → 𝑦𝑛)
151133, 130, 129lediv2d 13099 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → (𝑦𝑛 ↔ (𝑥 / 𝑛) ≤ (𝑥 / 𝑦)))
152150, 151mpbid 232 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → (𝑥 / 𝑛) ≤ (𝑥 / 𝑦))
153131, 134logled 26684 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → ((𝑥 / 𝑛) ≤ (𝑥 / 𝑦) ↔ (log‘(𝑥 / 𝑛)) ≤ (log‘(𝑥 / 𝑦))))
154152, 153mpbid 232 . . . . . . . . . 10 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → (log‘(𝑥 / 𝑛)) ≤ (log‘(𝑥 / 𝑦)))
155 leexp1a 14212 . . . . . . . . . 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))
156132, 135, 136, 149, 154, 155syl32anc 1377 . . . . . . . . 9 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+𝑛 ∈ ℝ+) ∧ (1 ≤ 𝑦𝑦𝑛𝑛𝑥)) → ((log‘(𝑥 / 𝑛))↑𝑁) ≤ ((log‘(𝑥 / 𝑦))↑𝑁))
157 eqid 2735 . . . . . . . . 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))))))
158963ad2antr1 1187 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦𝑦𝑥)) → (𝑥 / 𝑦) ∈ ℝ+)
159158relogcld 26680 . . . . . . . . . 10 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦𝑦𝑥)) → (log‘(𝑥 / 𝑦)) ∈ ℝ)
160 simpll 767 . . . . . . . . . 10 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦𝑦𝑥)) → 𝑁 ∈ ℕ0)
161 rpcn 13043 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ ℝ+𝑦 ∈ ℂ)
162161adantl 481 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 ∈ ℝ+) → 𝑦 ∈ ℂ)
1631623ad2antr1 1187 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦𝑦𝑥)) → 𝑦 ∈ ℂ)
164163mullidd 11277 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦𝑦𝑥)) → (1 · 𝑦) = 𝑦)
165 simpr3 1195 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦𝑦𝑥)) → 𝑦𝑥)
166164, 165eqbrtrd 5170 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦𝑦𝑥)) → (1 · 𝑦) ≤ 𝑥)
167 1red 11260 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦𝑦𝑥)) → 1 ∈ ℝ)
16894rpred 13075 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℝ)
169168adantr 480 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦𝑦𝑥)) → 𝑥 ∈ ℝ)
170 simpr1 1193 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦𝑦𝑥)) → 𝑦 ∈ ℝ+)
171167, 169, 170lemuldivd 13124 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦𝑦𝑥)) → ((1 · 𝑦) ≤ 𝑥 ↔ 1 ≤ (𝑥 / 𝑦)))
172166, 171mpbid 232 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦𝑦𝑥)) → 1 ≤ (𝑥 / 𝑦))
173 logleb 26660 . . . . . . . . . . . . 13 ((1 ∈ ℝ+ ∧ (𝑥 / 𝑦) ∈ ℝ+) → (1 ≤ (𝑥 / 𝑦) ↔ (log‘1) ≤ (log‘(𝑥 / 𝑦))))
17450, 158, 173sylancr 587 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦𝑦𝑥)) → (1 ≤ (𝑥 / 𝑦) ↔ (log‘1) ≤ (log‘(𝑥 / 𝑦))))
175172, 174mpbid 232 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦𝑦𝑥)) → (log‘1) ≤ (log‘(𝑥 / 𝑦)))
176137, 175eqbrtrrid 5184 . . . . . . . . . 10 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦𝑦𝑥)) → 0 ≤ (log‘(𝑥 / 𝑦)))
177159, 160, 176expge0d 14201 . . . . . . . . 9 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ (𝑦 ∈ ℝ+ ∧ 1 ≤ 𝑦𝑦𝑥)) → 0 ≤ ((log‘(𝑥 / 𝑦))↑𝑁))
17850a1i 11 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ∈ ℝ+)
179 1le1 11889 . . . . . . . . . 10 1 ≤ 1
180179a1i 11 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ≤ 1)
181 simprr 773 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ≤ 𝑥)
182168leidd 11827 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥𝑥)
18379, 80, 82, 83, 87, 88, 105, 107, 109, 124, 127, 128, 156, 157, 177, 178, 94, 180, 181, 182dvfsumlem4 26085 . . . . . . . 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 14011 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (1...(⌊‘𝑥)) ∈ Fin)
18594, 4, 5syl2an 596 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
186185relogcld 26680 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
187 simpll 767 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑁 ∈ ℕ0)
188186, 187reexpcld 14200 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛))↑𝑁) ∈ ℝ)
189184, 188fsumrecl 15767 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) ∈ ℝ)
190189recnd 11287 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) ∈ ℂ)
19194rpcnd 13077 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ∈ ℂ)
19272, 191mulcld 11279 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((!‘𝑁) · 𝑥) ∈ ℂ)
19311ad2antrl 728 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℝ)
194193recnd 11287 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (log‘𝑥) ∈ ℂ)
195194, 114expcld 14183 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((log‘𝑥)↑𝑁) ∈ ℂ)
196 fzfid 14011 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (0...𝑁) ∈ Fin)
197193, 20, 21syl2an 596 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (0...𝑁)) → ((log‘𝑥)↑𝑘) ∈ ℝ)
19820adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ0)
199198faccld 14320 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℕ)
200197, 199nndivred 12318 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℝ)
201200recnd 11287 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℂ)
202196, 201fsumcl 15766 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℂ)
20372, 202mulcld 11279 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) ∈ ℂ)
204195, 203subcld 11618 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) ∈ ℂ)
205190, 192, 204sub32d 11650 . . . . . . . . . . 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 2736 . . . . . . . . . . . . 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)
208207fveq2d 6911 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → (⌊‘𝑦) = (⌊‘𝑥))
209208oveq2d 7447 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → (1...(⌊‘𝑦)) = (1...(⌊‘𝑥)))
210209sumeq1d 15733 . . . . . . . . . . . . . 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 7439 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑥 → (𝑥 / 𝑦) = (𝑥 / 𝑥))
21265ad2antrl 728 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
213 divid 11951 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (𝑥 / 𝑥) = 1)
214212, 213syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑥 / 𝑥) = 1)
215211, 214sylan9eqr 2797 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → (𝑥 / 𝑦) = 1)
216215adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 / 𝑦) = 1)
217216fveq2d 6911 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ (0...𝑁)) → (log‘(𝑥 / 𝑦)) = (log‘1))
218217, 137eqtrdi 2791 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ (0...𝑁)) → (log‘(𝑥 / 𝑦)) = 0)
219218oveq1d 7446 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ (0...𝑁)) → ((log‘(𝑥 / 𝑦))↑𝑘) = (0↑𝑘))
220219oveq1d 7446 . . . . . . . . . . . . . . . . . . . 20 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)) = ((0↑𝑘) / (!‘𝑘)))
221220sumeq2dv 15735 . . . . . . . . . . . . . . . . . . 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 12918 . . . . . . . . . . . . . . . . . . . . . . . 24 0 = (ℤ‘0)
223114, 222eleqtrdi 2849 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑁 ∈ (ℤ‘0))
224 eluzfz1 13568 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ (ℤ‘0) → 0 ∈ (0...𝑁))
225223, 224syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ∈ (0...𝑁))
226225adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → 0 ∈ (0...𝑁))
227226snssd 4814 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → {0} ⊆ (0...𝑁))
228 elsni 4648 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ {0} → 𝑘 = 0)
229228adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ {0}) → 𝑘 = 0)
230 oveq2 7439 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = 0 → (0↑𝑘) = (0↑0))
231 0exp0e1 14104 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0↑0) = 1
232230, 231eqtrdi 2791 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 0 → (0↑𝑘) = 1)
233 fveq2 6907 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 = 0 → (!‘𝑘) = (!‘0))
234 fac0 14312 . . . . . . . . . . . . . . . . . . . . . . . . 25 (!‘0) = 1
235233, 234eqtrdi 2791 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 0 → (!‘𝑘) = 1)
236232, 235oveq12d 7449 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 0 → ((0↑𝑘) / (!‘𝑘)) = (1 / 1))
237 1div1e1 11956 . . . . . . . . . . . . . . . . . . . . . . 23 (1 / 1) = 1
238236, 237eqtrdi 2791 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = 0 → ((0↑𝑘) / (!‘𝑘)) = 1)
239229, 238syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ {0}) → ((0↑𝑘) / (!‘𝑘)) = 1)
240 ax-1cn 11211 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℂ
241239, 240eqeltrdi 2847 . . . . . . . . . . . . . . . . . . . 20 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ {0}) → ((0↑𝑘) / (!‘𝑘)) ∈ ℂ)
242 eldifi 4141 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 ∈ ((0...𝑁) ∖ {0}) → 𝑘 ∈ (0...𝑁))
243242adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → 𝑘 ∈ (0...𝑁))
244243, 20syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → 𝑘 ∈ ℕ0)
245 eldifsni 4795 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑘 ∈ ((0...𝑁) ∖ {0}) → 𝑘 ≠ 0)
246245adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → 𝑘 ≠ 0)
247 eldifsn 4791 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (ℕ0 ∖ {0}) ↔ (𝑘 ∈ ℕ0𝑘 ≠ 0))
248244, 246, 247sylanbrc 583 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → 𝑘 ∈ (ℕ0 ∖ {0}))
249 dfn2 12537 . . . . . . . . . . . . . . . . . . . . . . . 24 ℕ = (ℕ0 ∖ {0})
250248, 249eleqtrrdi 2850 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → 𝑘 ∈ ℕ)
2512500expd 14176 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → (0↑𝑘) = 0)
252251oveq1d 7446 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → ((0↑𝑘) / (!‘𝑘)) = (0 / (!‘𝑘)))
253244faccld 14320 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → (!‘𝑘) ∈ ℕ)
254253nncnd 12280 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → (!‘𝑘) ∈ ℂ)
255253nnne0d 12314 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → (!‘𝑘) ≠ 0)
256254, 255div0d 12040 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → (0 / (!‘𝑘)) = 0)
257252, 256eqtrd 2775 . . . . . . . . . . . . . . . . . . . 20 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) ∧ 𝑘 ∈ ((0...𝑁) ∖ {0})) → ((0↑𝑘) / (!‘𝑘)) = 0)
258 fzfid 14011 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → (0...𝑁) ∈ Fin)
259227, 241, 257, 258fsumss 15758 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → Σ𝑘 ∈ {0} ((0↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ (0...𝑁)((0↑𝑘) / (!‘𝑘)))
260221, 259eqtr4d 2778 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ {0} ((0↑𝑘) / (!‘𝑘)))
261 0cn 11251 . . . . . . . . . . . . . . . . . . 19 0 ∈ ℂ
262238sumsn 15779 . . . . . . . . . . . . . . . . . . 19 ((0 ∈ ℂ ∧ 1 ∈ ℂ) → Σ𝑘 ∈ {0} ((0↑𝑘) / (!‘𝑘)) = 1)
263261, 240, 262mp2an 692 . . . . . . . . . . . . . . . . . 18 Σ𝑘 ∈ {0} ((0↑𝑘) / (!‘𝑘)) = 1
264260, 263eqtrdi 2791 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)) = 1)
265207, 264oveq12d 7449 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))) = (𝑥 · 1))
266191mulridd 11276 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑥 · 1) = 𝑥)
267266adantr 480 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → (𝑥 · 1) = 𝑥)
268265, 267eqtrd 2775 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘))) = 𝑥)
269268oveq2d 7447 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 𝑥) → ((!‘𝑁) · (𝑦 · Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)))) = ((!‘𝑁) · 𝑥))
270210, 269oveq12d 7449 . . . . . . . . . . . . 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 7466 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) − ((!‘𝑁) · 𝑥)) ∈ V)
272206, 270, 94, 271fvmptd 7023 . . . . . . . . . . . 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)
274273fveq2d 6911 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (⌊‘𝑦) = (⌊‘1))
275 flid 13845 . . . . . . . . . . . . . . . . . . 19 (1 ∈ ℤ → (⌊‘1) = 1)
27681, 275ax-mp 5 . . . . . . . . . . . . . . . . . 18 (⌊‘1) = 1
277274, 276eqtrdi 2791 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (⌊‘𝑦) = 1)
278277oveq2d 7447 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (1...(⌊‘𝑦)) = (1...1))
279278sumeq1d 15733 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) = Σ𝑛 ∈ (1...1)((log‘(𝑥 / 𝑛))↑𝑁))
280191div1d 12033 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑥 / 1) = 𝑥)
281280adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (𝑥 / 1) = 𝑥)
282281fveq2d 6911 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (log‘(𝑥 / 1)) = (log‘𝑥))
283282oveq1d 7446 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → ((log‘(𝑥 / 1))↑𝑁) = ((log‘𝑥)↑𝑁))
284195adantr 480 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → ((log‘𝑥)↑𝑁) ∈ ℂ)
285283, 284eqeltrd 2839 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → ((log‘(𝑥 / 1))↑𝑁) ∈ ℂ)
286 oveq2 7439 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 1 → (𝑥 / 𝑛) = (𝑥 / 1))
287286fveq2d 6911 . . . . . . . . . . . . . . . . . 18 (𝑛 = 1 → (log‘(𝑥 / 𝑛)) = (log‘(𝑥 / 1)))
288287oveq1d 7446 . . . . . . . . . . . . . . . . 17 (𝑛 = 1 → ((log‘(𝑥 / 𝑛))↑𝑁) = ((log‘(𝑥 / 1))↑𝑁))
289288fsum1 15780 . . . . . . . . . . . . . . . 16 ((1 ∈ ℤ ∧ ((log‘(𝑥 / 1))↑𝑁) ∈ ℂ) → Σ𝑛 ∈ (1...1)((log‘(𝑥 / 𝑛))↑𝑁) = ((log‘(𝑥 / 1))↑𝑁))
29081, 285, 289sylancr 587 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → Σ𝑛 ∈ (1...1)((log‘(𝑥 / 𝑛))↑𝑁) = ((log‘(𝑥 / 1))↑𝑁))
291279, 290, 2833eqtrd 2779 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → Σ𝑛 ∈ (1...(⌊‘𝑦))((log‘(𝑥 / 𝑛))↑𝑁) = ((log‘𝑥)↑𝑁))
292273oveq2d 7447 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (𝑥 / 𝑦) = (𝑥 / 1))
293292, 281eqtrd 2775 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (𝑥 / 𝑦) = 𝑥)
294293fveq2d 6911 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (log‘(𝑥 / 𝑦)) = (log‘𝑥))
295294adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) ∧ 𝑘 ∈ (0...𝑁)) → (log‘(𝑥 / 𝑦)) = (log‘𝑥))
296295oveq1d 7446 . . . . . . . . . . . . . . . . . . 19 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) ∧ 𝑘 ∈ (0...𝑁)) → ((log‘(𝑥 / 𝑦))↑𝑘) = ((log‘𝑥)↑𝑘))
297296oveq1d 7446 . . . . . . . . . . . . . . . . . 18 ((((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)) = (((log‘𝑥)↑𝑘) / (!‘𝑘)))
298297sumeq2dv 15735 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → Σ𝑘 ∈ (0...𝑁)(((log‘(𝑥 / 𝑦))↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))
299273, 298oveq12d 7449 . . . . . . . . . . . . . . . 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))))
300202adantr 480 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℂ)
301300mullidd 11277 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → (1 · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) = Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))
302299, 301eqtrd 2775 . . . . . . . . . . . . . . 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)))
303302oveq2d 7447 . . . . . . . . . . . . . 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))))
304291, 303oveq12d 7449 . . . . . . . . . . . . 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 7466 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) ∈ V)
306206, 304, 178, 305fvmptd 7023 . . . . . . . . . . . 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)))))
307272, 306oveq12d 7449 . . . . . . . . . . 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))))))
30870, 72, 191subdird 11718 . . . . . . . . . . . 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)))
30964adantrr 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)
310212simprd 495 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 𝑥 ≠ 0)
311309, 191, 310divcan1d 12042 . . . . . . . . . . . . 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))))))
312311oveq1d 7446 . . . . . . . . . . . 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)))
313308, 312eqtrd 2775 . . . . . . . . . . 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)))
314205, 307, 3133eqtr4d 2785 . . . . . . . . . 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))
315314fveq2d 6911 . . . . . . . . 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)))
31673, 191absmuld 15490 . . . . . . . . 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 13048 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
318317ad2antrl 728 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
319 absid 15332 . . . . . . . . . . 11 ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (abs‘𝑥) = 𝑥)
320318, 319syl 17 . . . . . . . . . 10 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘𝑥) = 𝑥)
321320oveq2d 7447 . . . . . . . . 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))
322315, 316, 3213eqtrd 2779 . . . . . . . 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 11254 . . . . . . . . 9 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 ∈ ℂ)
324294oveq1d 7446 . . . . . . . . 9 (((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) ∧ 𝑦 = 1) → ((log‘(𝑥 / 𝑦))↑𝑁) = ((log‘𝑥)↑𝑁))
325323, 324csbied 3946 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 1 / 𝑦((log‘(𝑥 / 𝑦))↑𝑁) = ((log‘𝑥)↑𝑁))
326183, 322, 3253brtr3d 5179 . . . . . . 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))
32714adantrr 717 . . . . . . . 8 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((log‘𝑥)↑𝑁) ∈ ℝ)
32874, 327, 94lemuldivd 13124 . . . . . . 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)))
329326, 328mpbid 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))
33075leabsd 15450 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (((log‘𝑥)↑𝑁) / 𝑥) ≤ (abs‘(((log‘𝑥)↑𝑁) / 𝑥)))
33174, 75, 77, 329, 330letrd 11416 . . . . 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)))
33257adantrr 717 . . . . . . 7 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (((log‘𝑥)↑𝑁) / 𝑥) ∈ ℂ)
333332subid1d 11607 . . . . . 6 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((((log‘𝑥)↑𝑁) / 𝑥) − 0) = (((log‘𝑥)↑𝑁) / 𝑥))
334333fveq2d 6911 . . . . 5 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘((((log‘𝑥)↑𝑁) / 𝑥) − 0)) = (abs‘(((log‘𝑥)↑𝑁) / 𝑥)))
335331, 334breqtrrd 5176 . . . 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)))
33633, 34, 54, 57, 69, 335rlimsqzlem 15682 . . 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 11959 . . . . . 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)))
33859, 62, 66, 337syl3anc 1370 . . . . 5 ((𝑁 ∈ ℕ0𝑥 ∈ 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)))
339338mpteq2dva 5248 . . . 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 13063 . . . . . . 7 ((((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) ∈ ℝ ∧ 𝑥 ∈ ℝ+) → (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥) ∈ ℝ)
34127, 340sylancom 588 . . . . . 6 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥) ∈ ℝ)
342 divass 11938 . . . . . . . . . 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)))
34360, 61, 66, 342syl3anc 1370 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥) = ((!‘𝑁) · (Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) / 𝑥)))
34425recnd 11287 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘𝑥)↑𝑘) / (!‘𝑘)) ∈ ℂ)
34518, 67, 344, 68fsumdivc 15819 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → (Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) / 𝑥) = Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / (!‘𝑘)) / 𝑥))
34622recnd 11287 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → ((log‘𝑥)↑𝑘) ∈ ℂ)
34724nnrpd 13073 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℝ+)
348347rpcnne0d 13084 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → ((!‘𝑘) ∈ ℂ ∧ (!‘𝑘) ≠ 0))
34966adantr 480 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
350 divdiv32 11973 . . . . . . . . . . . . 13 ((((log‘𝑥)↑𝑘) ∈ ℂ ∧ ((!‘𝑘) ∈ ℂ ∧ (!‘𝑘) ≠ 0) ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → ((((log‘𝑥)↑𝑘) / (!‘𝑘)) / 𝑥) = ((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)))
351346, 348, 349, 350syl3anc 1370 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → ((((log‘𝑥)↑𝑘) / (!‘𝑘)) / 𝑥) = ((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)))
352351sumeq2dv 15735 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / (!‘𝑘)) / 𝑥) = Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)))
353345, 352eqtrd 2775 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → (Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) / 𝑥) = Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)))
354353oveq2d 7447 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → ((!‘𝑁) · (Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)) / 𝑥)) = ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘))))
355343, 354eqtrd 2775 . . . . . . . 8 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥) = ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘))))
356355mpteq2dva 5248 . . . . . . 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)))))
3572adantr 480 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → 𝑥 ∈ ℝ+)
35822, 357rerpdivcld 13106 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → (((log‘𝑥)↑𝑘) / 𝑥) ∈ ℝ)
359358, 24nndivred 12318 . . . . . . . . . 10 (((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) ∧ 𝑘 ∈ (0...𝑁)) → ((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)) ∈ ℝ)
36018, 359fsumrecl 15767 . . . . . . . . 9 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)) ∈ ℝ)
361 rpssre 13040 . . . . . . . . . 10 + ⊆ ℝ
362 rlimconst 15577 . . . . . . . . . 10 ((ℝ+ ⊆ ℝ ∧ (!‘𝑁) ∈ ℂ) → (𝑥 ∈ ℝ+ ↦ (!‘𝑁)) ⇝𝑟 (!‘𝑁))
363361, 34, 362sylancr 587 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (!‘𝑁)) ⇝𝑟 (!‘𝑁))
364361a1i 11 . . . . . . . . . . 11 (𝑁 ∈ ℕ0 → ℝ+ ⊆ ℝ)
365 fzfid 14011 . . . . . . . . . . 11 (𝑁 ∈ ℕ0 → (0...𝑁) ∈ Fin)
366359anasss 466 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0 ∧ (𝑥 ∈ ℝ+𝑘 ∈ (0...𝑁))) → ((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)) ∈ ℝ)
367358an32s 652 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑘 ∈ (0...𝑁)) ∧ 𝑥 ∈ ℝ+) → (((log‘𝑥)↑𝑘) / 𝑥) ∈ ℝ)
36820adantl 481 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ0)
369368faccld 14320 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℕ)
370369nnred 12279 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℝ)
371370adantr 480 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑘 ∈ (0...𝑁)) ∧ 𝑥 ∈ ℝ+) → (!‘𝑘) ∈ ℝ)
372368, 53syl 17 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑘 ∈ (0...𝑁)) → (𝑥 ∈ ℝ+ ↦ (((log‘𝑥)↑𝑘) / 𝑥)) ⇝𝑟 0)
373369nncnd 12280 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℂ)
374 rlimconst 15577 . . . . . . . . . . . . . 14 ((ℝ+ ⊆ ℝ ∧ (!‘𝑘) ∈ ℂ) → (𝑥 ∈ ℝ+ ↦ (!‘𝑘)) ⇝𝑟 (!‘𝑘))
375361, 373, 374sylancr 587 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑘 ∈ (0...𝑁)) → (𝑥 ∈ ℝ+ ↦ (!‘𝑘)) ⇝𝑟 (!‘𝑘))
376369nnne0d 12314 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑘 ∈ (0...𝑁)) → (!‘𝑘) ≠ 0)
377376adantr 480 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑘 ∈ (0...𝑁)) ∧ 𝑥 ∈ ℝ+) → (!‘𝑘) ≠ 0)
378367, 371, 372, 375, 376, 377rlimdiv 15679 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ (0...𝑁)) → (𝑥 ∈ ℝ+ ↦ ((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘))) ⇝𝑟 (0 / (!‘𝑘)))
379373, 376div0d 12040 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑘 ∈ (0...𝑁)) → (0 / (!‘𝑘)) = 0)
380378, 379breqtrd 5174 . . . . . . . . . . 11 ((𝑁 ∈ ℕ0𝑘 ∈ (0...𝑁)) → (𝑥 ∈ ℝ+ ↦ ((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘))) ⇝𝑟 0)
381364, 365, 366, 380fsumrlim 15844 . . . . . . . . . 10 (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘))) ⇝𝑟 Σ𝑘 ∈ (0...𝑁)0)
382 fzfi 14010 . . . . . . . . . . . 12 (0...𝑁) ∈ Fin
383382olci 866 . . . . . . . . . . 11 ((0...𝑁) ⊆ (ℤ‘0) ∨ (0...𝑁) ∈ Fin)
384 sumz 15755 . . . . . . . . . . 11 (((0...𝑁) ⊆ (ℤ‘0) ∨ (0...𝑁) ∈ Fin) → Σ𝑘 ∈ (0...𝑁)0 = 0)
385383, 384ax-mp 5 . . . . . . . . . 10 Σ𝑘 ∈ (0...𝑁)0 = 0
386381, 385breqtrdi 5189 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘))) ⇝𝑟 0)
38717, 360, 363, 386rlimmul 15678 . . . . . . . 8 (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)))) ⇝𝑟 ((!‘𝑁) · 0))
38834mul01d 11458 . . . . . . . 8 (𝑁 ∈ ℕ0 → ((!‘𝑁) · 0) = 0)
389387, 388breqtrd 5174 . . . . . . 7 (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)((((log‘𝑥)↑𝑘) / 𝑥) / (!‘𝑘)))) ⇝𝑟 0)
390356, 389eqbrtrd 5170 . . . . . 6 (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥)) ⇝𝑟 0)
39156, 341, 54, 390rlimsub 15677 . . . . 5 (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ ((((log‘𝑥)↑𝑁) / 𝑥) − (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥))) ⇝𝑟 (0 − 0))
392 0m0e0 12384 . . . . 5 (0 − 0) = 0
393391, 392breqtrdi 5189 . . . 4 (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ ((((log‘𝑥)↑𝑁) / 𝑥) − (((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘))) / 𝑥))) ⇝𝑟 0)
394339, 393eqbrtrd 5170 . . 3 (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ ((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) / 𝑥)) ⇝𝑟 0)
39530, 32, 336, 394rlimadd 15676 . 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 11959 . . . . . 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)))
39758, 63, 66, 396syl3anc 1370 . . . . 5 ((𝑁 ∈ ℕ0𝑥 ∈ 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)))
398397oveq1d 7446 . . . 4 ((𝑁 ∈ ℕ0𝑥 ∈ 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)))
39910, 2rerpdivcld 13106 . . . . . 6 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) / 𝑥) ∈ ℝ)
400399recnd 11287 . . . . 5 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) / 𝑥) ∈ ℂ)
40132recnd 11287 . . . . 5 ((𝑁 ∈ ℕ0𝑥 ∈ ℝ+) → ((((log‘𝑥)↑𝑁) − ((!‘𝑁) · Σ𝑘 ∈ (0...𝑁)(((log‘𝑥)↑𝑘) / (!‘𝑘)))) / 𝑥) ∈ ℂ)
402400, 401npcand 11622 . . . 4 ((𝑁 ∈ ℕ0𝑥 ∈ 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))
403398, 402eqtrd 2775 . . 3 ((𝑁 ∈ ℕ0𝑥 ∈ 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))
404403mpteq2dva 5248 . 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)))
40534addridd 11459 . 2 (𝑁 ∈ ℕ0 → ((!‘𝑁) + 0) = (!‘𝑁))
406395, 404, 4053brtr3d 5179 1 (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) / 𝑥)) ⇝𝑟 (!‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  Vcvv 3478  csb 3908  cdif 3960  wss 3963  {csn 4631  {cpr 4633   class class class wbr 5148  cmpt 5231  cfv 6563  (class class class)co 7431  Fincfn 8984  cc 11151  cr 11152  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158  +∞cpnf 11290  cle 11294  cmin 11490   / cdiv 11918  cn 12264  0cn0 12524  cz 12611  cuz 12876  +crp 13032  (,)cioo 13384  ...cfz 13544  cfl 13827  cexp 14099  !cfa 14309  abscabs 15270  𝑟 crli 15518  Σcsu 15719   D cdv 25913  logclog 26611  𝑐ccxp 26612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ioo 13388  df-ioc 13389  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-fl 13829  df-mod 13907  df-seq 14040  df-exp 14100  df-fac 14310  df-bc 14339  df-hash 14367  df-shft 15103  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-limsup 15504  df-clim 15521  df-rlim 15522  df-sum 15720  df-ef 16100  df-e 16101  df-sin 16102  df-cos 16103  df-pi 16105  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-0g 17488  df-gsum 17489  df-topgen 17490  df-pt 17491  df-prds 17494  df-xrs 17549  df-qtop 17554  df-imas 17555  df-xps 17557  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-mulg 19099  df-cntz 19348  df-cmn 19815  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-fbas 21379  df-fg 21380  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-lp 23160  df-perf 23161  df-cn 23251  df-cnp 23252  df-haus 23339  df-cmp 23411  df-tx 23586  df-hmeo 23779  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964  df-xms 24346  df-ms 24347  df-tms 24348  df-cncf 24918  df-limc 25916  df-dv 25917  df-log 26613  df-cxp 26614
This theorem is referenced by:  logfacrlim2  27285  selberglem2  27605
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