Step | Hyp | Ref
| Expression |
1 | | rpssre 11843 |
. . . 4
⊢
ℝ+ ⊆ ℝ |
2 | | ax-1cn 9994 |
. . . 4
⊢ 1 ∈
ℂ |
3 | | o1const 14350 |
. . . 4
⊢
((ℝ+ ⊆ ℝ ∧ 1 ∈ ℂ) →
(𝑥 ∈
ℝ+ ↦ 1) ∈ 𝑂(1)) |
4 | 1, 2, 3 | mp2an 708 |
. . 3
⊢ (𝑥 ∈ ℝ+
↦ 1) ∈ 𝑂(1) |
5 | 4 | a1i 11 |
. 2
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ 1) ∈
𝑂(1)) |
6 | 2 | a1i 11 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 1 ∈
ℂ) |
7 | | fzfid 12772 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(1...(⌊‘𝑥))
∈ Fin) |
8 | | rpvmasum.g |
. . . . . . 7
⊢ 𝐺 = (DChr‘𝑁) |
9 | | rpvmasum.z |
. . . . . . 7
⊢ 𝑍 =
(ℤ/nℤ‘𝑁) |
10 | | rpvmasum.d |
. . . . . . 7
⊢ 𝐷 = (Base‘𝐺) |
11 | | rpvmasum.l |
. . . . . . 7
⊢ 𝐿 = (ℤRHom‘𝑍) |
12 | | dchrisum.b |
. . . . . . . 8
⊢ (𝜑 → 𝑋 ∈ 𝐷) |
13 | 12 | ad2antrr 762 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑋 ∈ 𝐷) |
14 | | elfzelz 12342 |
. . . . . . . 8
⊢ (𝑑 ∈
(1...(⌊‘𝑥))
→ 𝑑 ∈
ℤ) |
15 | 14 | adantl 482 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ∈
ℤ) |
16 | 8, 9, 10, 11, 13, 15 | dchrzrhcl 24970 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝑋‘(𝐿‘𝑑)) ∈ ℂ) |
17 | | elfznn 12370 |
. . . . . . . . 9
⊢ (𝑑 ∈
(1...(⌊‘𝑥))
→ 𝑑 ∈
ℕ) |
18 | 17 | adantl 482 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ∈
ℕ) |
19 | | mucl 24867 |
. . . . . . . . . 10
⊢ (𝑑 ∈ ℕ →
(μ‘𝑑) ∈
ℤ) |
20 | 19 | zred 11482 |
. . . . . . . . 9
⊢ (𝑑 ∈ ℕ →
(μ‘𝑑) ∈
ℝ) |
21 | | nndivre 11056 |
. . . . . . . . 9
⊢
(((μ‘𝑑)
∈ ℝ ∧ 𝑑
∈ ℕ) → ((μ‘𝑑) / 𝑑) ∈ ℝ) |
22 | 20, 21 | mpancom 703 |
. . . . . . . 8
⊢ (𝑑 ∈ ℕ →
((μ‘𝑑) / 𝑑) ∈
ℝ) |
23 | 18, 22 | syl 17 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((μ‘𝑑) /
𝑑) ∈
ℝ) |
24 | 23 | recnd 10068 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((μ‘𝑑) /
𝑑) ∈
ℂ) |
25 | 16, 24 | mulcld 10060 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ) |
26 | 7, 25 | fsumcl 14464 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
Σ𝑑 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ) |
27 | | dchrisumn0.t |
. . . . . 6
⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑇) |
28 | | climcl 14230 |
. . . . . 6
⊢ (seq1( +
, 𝐹) ⇝ 𝑇 → 𝑇 ∈ ℂ) |
29 | 27, 28 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝑇 ∈ ℂ) |
30 | 29 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑇 ∈
ℂ) |
31 | 26, 30 | mulcld 10060 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(Σ𝑑 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇) ∈ ℂ) |
32 | 1 | a1i 11 |
. . . 4
⊢ (𝜑 → ℝ+
⊆ ℝ) |
33 | | subcl 10280 |
. . . . 5
⊢ ((1
∈ ℂ ∧ (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇) ∈ ℂ) → (1 −
(Σ𝑑 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) ∈ ℂ) |
34 | 2, 31, 33 | sylancr 695 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1
− (Σ𝑑 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) ∈ ℂ) |
35 | | 1red 10055 |
. . . 4
⊢ (𝜑 → 1 ∈
ℝ) |
36 | | dchrisumn0.c |
. . . . . 6
⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) |
37 | | elrege0 12278 |
. . . . . 6
⊢ (𝐶 ∈ (0[,)+∞) ↔
(𝐶 ∈ ℝ ∧ 0
≤ 𝐶)) |
38 | 36, 37 | sylib 208 |
. . . . 5
⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) |
39 | 38 | simpld 475 |
. . . 4
⊢ (𝜑 → 𝐶 ∈ ℝ) |
40 | | fzfid 12772 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(1...(⌊‘𝑥))
∈ Fin) |
41 | 25 | adantlrr 757 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ) |
42 | | nnuz 11723 |
. . . . . . . . . . . 12
⊢ ℕ =
(ℤ≥‘1) |
43 | | 1zzd 11408 |
. . . . . . . . . . . 12
⊢ (𝜑 → 1 ∈
ℤ) |
44 | 12 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋 ∈ 𝐷) |
45 | | nnz 11399 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℤ) |
46 | 45 | adantl 482 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ) |
47 | 8, 9, 10, 11, 44, 46 | dchrzrhcl 24970 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ) |
48 | | nncn 11028 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℂ) |
49 | 48 | adantl 482 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ) |
50 | | nnne0 11053 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0) |
51 | 50 | adantl 482 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0) |
52 | 47, 49, 51 | divcld 10801 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ) |
53 | | dchrisumn0.f |
. . . . . . . . . . . . . . 15
⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) |
54 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑎 = 𝑚 → (𝐿‘𝑎) = (𝐿‘𝑚)) |
55 | 54 | fveq2d 6195 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚))) |
56 | | id 22 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 = 𝑚 → 𝑎 = 𝑚) |
57 | 55, 56 | oveq12d 6668 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
58 | 57 | cbvmptv 4750 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) = (𝑚 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
59 | 53, 58 | eqtri 2644 |
. . . . . . . . . . . . . 14
⊢ 𝐹 = (𝑚 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
60 | 52, 59 | fmptd 6385 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐹:ℕ⟶ℂ) |
61 | 60 | ffvelrnda 6359 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∈ ℂ) |
62 | 42, 43, 61 | serf 12829 |
. . . . . . . . . . 11
⊢ (𝜑 → seq1( + , 𝐹):ℕ⟶ℂ) |
63 | 62 | ad2antrr 762 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ seq1( + , 𝐹):ℕ⟶ℂ) |
64 | | simprl 794 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝑥 ∈
ℝ+) |
65 | 64 | rpred 11872 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝑥 ∈
ℝ) |
66 | | nndivre 11056 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ ∧ 𝑑 ∈ ℕ) → (𝑥 / 𝑑) ∈ ℝ) |
67 | 65, 17, 66 | syl2an 494 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝑥 / 𝑑) ∈
ℝ) |
68 | 17 | adantl 482 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ∈
ℕ) |
69 | 68 | nncnd 11036 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ∈
ℂ) |
70 | 69 | mulid2d 10058 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (1 · 𝑑) =
𝑑) |
71 | | fznnfl 12661 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ℝ → (𝑑 ∈
(1...(⌊‘𝑥))
↔ (𝑑 ∈ ℕ
∧ 𝑑 ≤ 𝑥))) |
72 | 65, 71 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (𝑑 ∈
(1...(⌊‘𝑥))
↔ (𝑑 ∈ ℕ
∧ 𝑑 ≤ 𝑥))) |
73 | 72 | simplbda 654 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ≤ 𝑥) |
74 | 70, 73 | eqbrtrd 4675 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (1 · 𝑑) ≤
𝑥) |
75 | | 1red 10055 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 1 ∈ ℝ) |
76 | 65 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑥 ∈
ℝ) |
77 | 68 | nnrpd 11870 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ∈
ℝ+) |
78 | 75, 76, 77 | lemuldivd 11921 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((1 · 𝑑) ≤
𝑥 ↔ 1 ≤ (𝑥 / 𝑑))) |
79 | 74, 78 | mpbid 222 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 1 ≤ (𝑥 / 𝑑)) |
80 | | flge1nn 12622 |
. . . . . . . . . . 11
⊢ (((𝑥 / 𝑑) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑑)) → (⌊‘(𝑥 / 𝑑)) ∈ ℕ) |
81 | 67, 79, 80 | syl2anc 693 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (⌊‘(𝑥 /
𝑑)) ∈
ℕ) |
82 | 63, 81 | ffvelrnd 6360 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) ∈ ℂ) |
83 | 41, 82 | mulcld 10060 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) ∈ ℂ) |
84 | 29 | ad2antrr 762 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑇 ∈
ℂ) |
85 | 41, 84 | mulcld 10060 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇) ∈ ℂ) |
86 | 40, 83, 85 | fsumsub 14520 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑑 ∈
(1...(⌊‘𝑥))((((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) − (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) = (Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) − Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇))) |
87 | 41, 82, 84 | subdid 10486 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) = ((((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) − (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇))) |
88 | 87 | sumeq2dv 14433 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑑 ∈
(1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) = Σ𝑑 ∈ (1...(⌊‘𝑥))((((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) − (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇))) |
89 | 12 | ad3antrrr 766 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → 𝑋 ∈ 𝐷) |
90 | 14 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → 𝑑 ∈
ℤ) |
91 | | elfzelz 12342 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑))) → 𝑚 ∈
ℤ) |
92 | 91 | adantl 482 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → 𝑚 ∈
ℤ) |
93 | 8, 9, 10, 11, 89, 90, 92 | dchrzrhmul 24971 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → (𝑋‘(𝐿‘(𝑑 · 𝑚))) = ((𝑋‘(𝐿‘𝑑)) · (𝑋‘(𝐿‘𝑚)))) |
94 | 93 | oveq1d 6665 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)) = (((𝑋‘(𝐿‘𝑑)) · (𝑋‘(𝐿‘𝑚))) / (𝑑 · 𝑚))) |
95 | 16 | adantlrr 757 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝑋‘(𝐿‘𝑑)) ∈ ℂ) |
96 | 95 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → (𝑋‘(𝐿‘𝑑)) ∈ ℂ) |
97 | 69 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → 𝑑 ∈
ℂ) |
98 | 8, 9, 10, 11, 89, 92 | dchrzrhcl 24970 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ) |
99 | | elfznn 12370 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑))) → 𝑚 ∈
ℕ) |
100 | 99 | adantl 482 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → 𝑚 ∈
ℕ) |
101 | 100 | nncnd 11036 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → 𝑚 ∈
ℂ) |
102 | 68 | nnne0d 11065 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ≠
0) |
103 | 102 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → 𝑑 ≠ 0) |
104 | 100 | nnne0d 11065 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → 𝑚 ≠ 0) |
105 | 96, 97, 98, 101, 103, 104 | divmuldivd 10842 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → (((𝑋‘(𝐿‘𝑑)) / 𝑑) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (((𝑋‘(𝐿‘𝑑)) · (𝑋‘(𝐿‘𝑚))) / (𝑑 · 𝑚))) |
106 | 94, 105 | eqtr4d 2659 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)) = (((𝑋‘(𝐿‘𝑑)) / 𝑑) · ((𝑋‘(𝐿‘𝑚)) / 𝑚))) |
107 | 106 | oveq2d 6666 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) →
((μ‘𝑑) ·
((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚))) = ((μ‘𝑑) · (((𝑋‘(𝐿‘𝑑)) / 𝑑) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)))) |
108 | 68, 19 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (μ‘𝑑)
∈ ℤ) |
109 | 108 | zcnd 11483 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (μ‘𝑑)
∈ ℂ) |
110 | 109 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) →
(μ‘𝑑) ∈
ℂ) |
111 | 96, 97, 103 | divcld 10801 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → ((𝑋‘(𝐿‘𝑑)) / 𝑑) ∈ ℂ) |
112 | 98, 101, 104 | divcld 10801 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ) |
113 | 110, 111,
112 | mulassd 10063 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) →
(((μ‘𝑑) ·
((𝑋‘(𝐿‘𝑑)) / 𝑑)) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)) = ((μ‘𝑑) · (((𝑋‘(𝐿‘𝑑)) / 𝑑) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)))) |
114 | 110, 96, 97, 103 | div12d 10837 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) →
((μ‘𝑑) ·
((𝑋‘(𝐿‘𝑑)) / 𝑑)) = ((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) |
115 | 114 | oveq1d 6665 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) →
(((μ‘𝑑) ·
((𝑋‘(𝐿‘𝑑)) / 𝑑)) · ((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((𝑋‘(𝐿‘𝑚)) / 𝑚))) |
116 | 107, 113,
115 | 3eqtr2d 2662 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) →
((μ‘𝑑) ·
((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚))) = (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((𝑋‘(𝐿‘𝑚)) / 𝑚))) |
117 | 116 | sumeq2dv 14433 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))((μ‘𝑑) · ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚))) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((𝑋‘(𝐿‘𝑚)) / 𝑚))) |
118 | | fzfid 12772 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (1...(⌊‘(𝑥 / 𝑑))) ∈ Fin) |
119 | | simpll 790 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝜑) |
120 | 119, 99, 52 | syl2an 494 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ) |
121 | 118, 41, 120 | fsummulc2 14516 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑚)) / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((𝑋‘(𝐿‘𝑚)) / 𝑚))) |
122 | | ovex 6678 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ V |
123 | 57, 53, 122 | fvmpt 6282 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ ℕ → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
124 | 100, 123 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
125 | 81, 42 | syl6eleq 2711 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (⌊‘(𝑥 /
𝑑)) ∈
(ℤ≥‘1)) |
126 | 124, 125,
120 | fsumser 14461 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))((𝑋‘(𝐿‘𝑚)) / 𝑚) = (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) |
127 | 126 | oveq2d 6666 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))))) |
128 | 117, 121,
127 | 3eqtr2rd 2663 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)))) |
129 | 128 | sumeq2dv 14433 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑑 ∈
(1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) = Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)))) |
130 | | fveq2 6191 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = (𝑑 · 𝑚) → (𝐿‘𝑛) = (𝐿‘(𝑑 · 𝑚))) |
131 | 130 | fveq2d 6195 |
. . . . . . . . . . . . 13
⊢ (𝑛 = (𝑑 · 𝑚) → (𝑋‘(𝐿‘𝑛)) = (𝑋‘(𝐿‘(𝑑 · 𝑚)))) |
132 | | id 22 |
. . . . . . . . . . . . 13
⊢ (𝑛 = (𝑑 · 𝑚) → 𝑛 = (𝑑 · 𝑚)) |
133 | 131, 132 | oveq12d 6668 |
. . . . . . . . . . . 12
⊢ (𝑛 = (𝑑 · 𝑚) → ((𝑋‘(𝐿‘𝑛)) / 𝑛) = ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚))) |
134 | 133 | oveq2d 6666 |
. . . . . . . . . . 11
⊢ (𝑛 = (𝑑 · 𝑚) → ((μ‘𝑑) · ((𝑋‘(𝐿‘𝑛)) / 𝑛)) = ((μ‘𝑑) · ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)))) |
135 | | elrabi 3359 |
. . . . . . . . . . . . . . 15
⊢ (𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} → 𝑑 ∈ ℕ) |
136 | 135 | ad2antll 765 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ (𝑛 ∈
(1...(⌊‘𝑥))
∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ ℕ) |
137 | 136, 19 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ (𝑛 ∈
(1...(⌊‘𝑥))
∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (μ‘𝑑) ∈ ℤ) |
138 | 137 | zcnd 11483 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ (𝑛 ∈
(1...(⌊‘𝑥))
∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (μ‘𝑑) ∈ ℂ) |
139 | 12 | ad2antrr 762 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑋 ∈ 𝐷) |
140 | | elfzelz 12342 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈
(1...(⌊‘𝑥))
→ 𝑛 ∈
ℤ) |
141 | 140 | adantl 482 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℤ) |
142 | 8, 9, 10, 11, 139, 141 | dchrzrhcl 24970 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑋‘(𝐿‘𝑛)) ∈ ℂ) |
143 | 17 | ssriv 3607 |
. . . . . . . . . . . . . . . . 17
⊢
(1...(⌊‘𝑥)) ⊆ ℕ |
144 | 143 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(1...(⌊‘𝑥))
⊆ ℕ) |
145 | 144 | sselda 3603 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℕ) |
146 | 145 | nncnd 11036 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℂ) |
147 | 145 | nnne0d 11065 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ≠
0) |
148 | 142, 146,
147 | divcld 10801 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((𝑋‘(𝐿‘𝑛)) / 𝑛) ∈ ℂ) |
149 | 148 | adantrr 753 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ (𝑛 ∈
(1...(⌊‘𝑥))
∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((𝑋‘(𝐿‘𝑛)) / 𝑛) ∈ ℂ) |
150 | 138, 149 | mulcld 10060 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ (𝑛 ∈
(1...(⌊‘𝑥))
∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((μ‘𝑑) · ((𝑋‘(𝐿‘𝑛)) / 𝑛)) ∈ ℂ) |
151 | 134, 65, 150 | dvdsflsumcom 24914 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑛 ∈
(1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((𝑋‘(𝐿‘𝑛)) / 𝑛)) = Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)))) |
152 | | fveq2 6191 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 1 → (𝐿‘𝑛) = (𝐿‘1)) |
153 | 152 | fveq2d 6195 |
. . . . . . . . . . . 12
⊢ (𝑛 = 1 → (𝑋‘(𝐿‘𝑛)) = (𝑋‘(𝐿‘1))) |
154 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝑛 = 1 → 𝑛 = 1) |
155 | 153, 154 | oveq12d 6668 |
. . . . . . . . . . 11
⊢ (𝑛 = 1 → ((𝑋‘(𝐿‘𝑛)) / 𝑛) = ((𝑋‘(𝐿‘1)) / 1)) |
156 | | simprr 796 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 1 ≤ 𝑥) |
157 | | flge1nn 12622 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ 1 ≤
𝑥) →
(⌊‘𝑥) ∈
ℕ) |
158 | 65, 156, 157 | syl2anc 693 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(⌊‘𝑥) ∈
ℕ) |
159 | 158, 42 | syl6eleq 2711 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(⌊‘𝑥) ∈
(ℤ≥‘1)) |
160 | | eluzfz1 12348 |
. . . . . . . . . . . 12
⊢
((⌊‘𝑥)
∈ (ℤ≥‘1) → 1 ∈
(1...(⌊‘𝑥))) |
161 | 159, 160 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 1 ∈
(1...(⌊‘𝑥))) |
162 | 155, 40, 144, 161, 148 | musumsum 24918 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑛 ∈
(1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((𝑋‘(𝐿‘𝑛)) / 𝑛)) = ((𝑋‘(𝐿‘1)) / 1)) |
163 | 129, 151,
162 | 3eqtr2d 2662 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑑 ∈
(1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) = ((𝑋‘(𝐿‘1)) / 1)) |
164 | 8, 9, 10, 11, 12 | dchrzrh1 24969 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑋‘(𝐿‘1)) = 1) |
165 | 164 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (𝑋‘(𝐿‘1)) = 1) |
166 | 165 | oveq1d 6665 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → ((𝑋‘(𝐿‘1)) / 1) = (1 / 1)) |
167 | | 1div1e1 10717 |
. . . . . . . . . 10
⊢ (1 / 1) =
1 |
168 | 166, 167 | syl6eq 2672 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → ((𝑋‘(𝐿‘1)) / 1) = 1) |
169 | 163, 168 | eqtr2d 2657 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 1 = Σ𝑑 ∈
(1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))))) |
170 | 29 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝑇 ∈
ℂ) |
171 | 40, 170, 41 | fsummulc1 14517 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (Σ𝑑 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇) = Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) |
172 | 169, 171 | oveq12d 6668 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (1 −
(Σ𝑑 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) = (Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) − Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇))) |
173 | 86, 88, 172 | 3eqtr4rd 2667 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (1 −
(Σ𝑑 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) = Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) |
174 | 173 | fveq2d 6195 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (abs‘(1
− (Σ𝑑 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇))) = (abs‘Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)))) |
175 | 82, 84 | subcld 10392 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇) ∈ ℂ) |
176 | 41, 175 | mulcld 10060 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ∈ ℂ) |
177 | 40, 176 | fsumcl 14464 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑑 ∈
(1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ∈ ℂ) |
178 | 177 | abscld 14175 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(abs‘Σ𝑑 ∈
(1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ∈ ℝ) |
179 | 176 | abscld 14175 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ∈ ℝ) |
180 | 40, 179 | fsumrecl 14465 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑑 ∈
(1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ∈ ℝ) |
181 | 39 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝐶 ∈
ℝ) |
182 | 40, 176 | fsumabs 14533 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(abs‘Σ𝑑 ∈
(1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ≤ Σ𝑑 ∈ (1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)))) |
183 | | reflcl 12597 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ →
(⌊‘𝑥) ∈
ℝ) |
184 | 65, 183 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(⌊‘𝑥) ∈
ℝ) |
185 | 184, 181 | remulcld 10070 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
((⌊‘𝑥) ·
𝐶) ∈
ℝ) |
186 | 185, 64 | rerpdivcld 11903 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(((⌊‘𝑥)
· 𝐶) / 𝑥) ∈
ℝ) |
187 | 181, 64 | rerpdivcld 11903 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (𝐶 / 𝑥) ∈ ℝ) |
188 | 187 | adantr 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝐶 / 𝑥) ∈
ℝ) |
189 | 41 | abscld 14175 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) ∈ ℝ) |
190 | 68 | nnrecred 11066 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (1 / 𝑑) ∈
ℝ) |
191 | 175 | abscld 14175 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ∈ ℝ) |
192 | 77 | rpred 11872 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ∈
ℝ) |
193 | 188, 192 | remulcld 10070 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((𝐶 / 𝑥) · 𝑑) ∈ ℝ) |
194 | 41 | absge0d 14183 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)))) |
195 | 175 | absge0d 14183 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) |
196 | 95 | abscld 14175 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘(𝑋‘(𝐿‘𝑑))) ∈ ℝ) |
197 | 24 | adantlrr 757 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((μ‘𝑑) /
𝑑) ∈
ℂ) |
198 | 197 | abscld 14175 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((μ‘𝑑) / 𝑑)) ∈ ℝ) |
199 | 95 | absge0d 14183 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ (abs‘(𝑋‘(𝐿‘𝑑)))) |
200 | 197 | absge0d 14183 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ (abs‘((μ‘𝑑) / 𝑑))) |
201 | | eqid 2622 |
. . . . . . . . . . . . . 14
⊢
(Base‘𝑍) =
(Base‘𝑍) |
202 | 12 | ad2antrr 762 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑋 ∈ 𝐷) |
203 | | rpvmasum.a |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑁 ∈ ℕ) |
204 | 203 | nnnn0d 11351 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
205 | 9, 201, 11 | znzrhfo 19896 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℕ0
→ 𝐿:ℤ–onto→(Base‘𝑍)) |
206 | | fof 6115 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐿:ℤ–onto→(Base‘𝑍) → 𝐿:ℤ⟶(Base‘𝑍)) |
207 | 204, 205,
206 | 3syl 18 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍)) |
208 | 207 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝐿:ℤ⟶(Base‘𝑍)) |
209 | | ffvelrn 6357 |
. . . . . . . . . . . . . . 15
⊢ ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑑 ∈ ℤ) → (𝐿‘𝑑) ∈ (Base‘𝑍)) |
210 | 208, 14, 209 | syl2an 494 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝐿‘𝑑) ∈ (Base‘𝑍)) |
211 | 8, 10, 9, 201, 202, 210 | dchrabs2 24987 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘(𝑋‘(𝐿‘𝑑))) ≤ 1) |
212 | 109, 69, 102 | absdivd 14194 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((μ‘𝑑) / 𝑑)) = ((abs‘(μ‘𝑑)) / (abs‘𝑑))) |
213 | 77 | rprege0d 11879 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝑑 ∈ ℝ
∧ 0 ≤ 𝑑)) |
214 | | absid 14036 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑑 ∈ ℝ ∧ 0 ≤
𝑑) → (abs‘𝑑) = 𝑑) |
215 | 213, 214 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘𝑑) =
𝑑) |
216 | 215 | oveq2d 6666 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((abs‘(μ‘𝑑)) / (abs‘𝑑)) = ((abs‘(μ‘𝑑)) / 𝑑)) |
217 | 212, 216 | eqtrd 2656 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((μ‘𝑑) / 𝑑)) = ((abs‘(μ‘𝑑)) / 𝑑)) |
218 | 109 | abscld 14175 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘(μ‘𝑑)) ∈ ℝ) |
219 | | mule1 24874 |
. . . . . . . . . . . . . . . 16
⊢ (𝑑 ∈ ℕ →
(abs‘(μ‘𝑑))
≤ 1) |
220 | 68, 219 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘(μ‘𝑑)) ≤ 1) |
221 | 218, 75, 77, 220 | lediv1dd 11930 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((abs‘(μ‘𝑑)) / 𝑑) ≤ (1 / 𝑑)) |
222 | 217, 221 | eqbrtrd 4675 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((μ‘𝑑) / 𝑑)) ≤ (1 / 𝑑)) |
223 | 196, 75, 198, 190, 199, 200, 211, 222 | lemul12ad 10966 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((abs‘(𝑋‘(𝐿‘𝑑))) · (abs‘((μ‘𝑑) / 𝑑))) ≤ (1 · (1 / 𝑑))) |
224 | 95, 197 | absmuld 14193 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) = ((abs‘(𝑋‘(𝐿‘𝑑))) · (abs‘((μ‘𝑑) / 𝑑)))) |
225 | 190 | recnd 10068 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (1 / 𝑑) ∈
ℂ) |
226 | 225 | mulid2d 10058 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (1 · (1 / 𝑑))
= (1 / 𝑑)) |
227 | 226 | eqcomd 2628 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (1 / 𝑑) = (1
· (1 / 𝑑))) |
228 | 223, 224,
227 | 3brtr4d 4685 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) ≤ (1 / 𝑑)) |
229 | | 1re 10039 |
. . . . . . . . . . . . . . 15
⊢ 1 ∈
ℝ |
230 | | elicopnf 12269 |
. . . . . . . . . . . . . . 15
⊢ (1 ∈
ℝ → ((𝑥 / 𝑑) ∈ (1[,)+∞) ↔
((𝑥 / 𝑑) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑑)))) |
231 | 229, 230 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 / 𝑑) ∈ (1[,)+∞) ↔ ((𝑥 / 𝑑) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑑))) |
232 | 67, 79, 231 | sylanbrc 698 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝑥 / 𝑑) ∈
(1[,)+∞)) |
233 | | dchrisumn0.1 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + ,
𝐹)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐶 / 𝑦)) |
234 | 233 | ad2antrr 762 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ∀𝑦 ∈
(1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐶 / 𝑦)) |
235 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 = (𝑥 / 𝑑) → (⌊‘𝑦) = (⌊‘(𝑥 / 𝑑))) |
236 | 235 | fveq2d 6195 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 = (𝑥 / 𝑑) → (seq1( + , 𝐹)‘(⌊‘𝑦)) = (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) |
237 | 236 | oveq1d 6665 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = (𝑥 / 𝑑) → ((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇) = ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) |
238 | 237 | fveq2d 6195 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = (𝑥 / 𝑑) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) = (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) |
239 | | oveq2 6658 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = (𝑥 / 𝑑) → (𝐶 / 𝑦) = (𝐶 / (𝑥 / 𝑑))) |
240 | 238, 239 | breq12d 4666 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑥 / 𝑑) → ((abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐶 / 𝑦) ↔ (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ≤ (𝐶 / (𝑥 / 𝑑)))) |
241 | 240 | rspcv 3305 |
. . . . . . . . . . . . 13
⊢ ((𝑥 / 𝑑) ∈ (1[,)+∞) → (∀𝑦 ∈
(1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐶 / 𝑦) → (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ≤ (𝐶 / (𝑥 / 𝑑)))) |
242 | 232, 234,
241 | sylc 65 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ≤ (𝐶 / (𝑥 / 𝑑))) |
243 | 181 | recnd 10068 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝐶 ∈
ℂ) |
244 | 243 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝐶 ∈
ℂ) |
245 | | rpcnne0 11850 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ℝ+
→ (𝑥 ∈ ℂ
∧ 𝑥 ≠
0)) |
246 | 245 | ad2antrl 764 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) |
247 | 246 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝑥 ∈ ℂ
∧ 𝑥 ≠
0)) |
248 | | divdiv2 10737 |
. . . . . . . . . . . . . 14
⊢ ((𝐶 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0)) → (𝐶 / (𝑥 / 𝑑)) = ((𝐶 · 𝑑) / 𝑥)) |
249 | 244, 247,
69, 102, 248 | syl112anc 1330 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝐶 / (𝑥 / 𝑑)) = ((𝐶 · 𝑑) / 𝑥)) |
250 | | div23 10704 |
. . . . . . . . . . . . . 14
⊢ ((𝐶 ∈ ℂ ∧ 𝑑 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → ((𝐶 · 𝑑) / 𝑥) = ((𝐶 / 𝑥) · 𝑑)) |
251 | 244, 69, 247, 250 | syl3anc 1326 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((𝐶 · 𝑑) / 𝑥) = ((𝐶 / 𝑥) · 𝑑)) |
252 | 249, 251 | eqtrd 2656 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝐶 / (𝑥 / 𝑑)) = ((𝐶 / 𝑥) · 𝑑)) |
253 | 242, 252 | breqtrd 4679 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ≤ ((𝐶 / 𝑥) · 𝑑)) |
254 | 189, 190,
191, 193, 194, 195, 228, 253 | lemul12ad 10966 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) · (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ≤ ((1 / 𝑑) · ((𝐶 / 𝑥) · 𝑑))) |
255 | 41, 175 | absmuld 14193 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) = ((abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) · (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)))) |
256 | 187 | recnd 10068 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (𝐶 / 𝑥) ∈ ℂ) |
257 | 256 | adantr 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝐶 / 𝑥) ∈
ℂ) |
258 | 257, 69, 102 | divcan4d 10807 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (((𝐶 / 𝑥) · 𝑑) / 𝑑) = (𝐶 / 𝑥)) |
259 | 257, 69 | mulcld 10060 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((𝐶 / 𝑥) · 𝑑) ∈ ℂ) |
260 | 259, 69, 102 | divrec2d 10805 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (((𝐶 / 𝑥) · 𝑑) / 𝑑) = ((1 / 𝑑) · ((𝐶 / 𝑥) · 𝑑))) |
261 | 258, 260 | eqtr3d 2658 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝐶 / 𝑥) = ((1 / 𝑑) · ((𝐶 / 𝑥) · 𝑑))) |
262 | 254, 255,
261 | 3brtr4d 4685 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ≤ (𝐶 / 𝑥)) |
263 | 40, 179, 188, 262 | fsumle 14531 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑑 ∈
(1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ≤ Σ𝑑 ∈ (1...(⌊‘𝑥))(𝐶 / 𝑥)) |
264 | 158 | nnnn0d 11351 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(⌊‘𝑥) ∈
ℕ0) |
265 | | hashfz1 13134 |
. . . . . . . . . . 11
⊢
((⌊‘𝑥)
∈ ℕ0 → (#‘(1...(⌊‘𝑥))) = (⌊‘𝑥)) |
266 | 264, 265 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(#‘(1...(⌊‘𝑥))) = (⌊‘𝑥)) |
267 | 266 | oveq1d 6665 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
((#‘(1...(⌊‘𝑥))) · (𝐶 / 𝑥)) = ((⌊‘𝑥) · (𝐶 / 𝑥))) |
268 | | fsumconst 14522 |
. . . . . . . . . 10
⊢
(((1...(⌊‘𝑥)) ∈ Fin ∧ (𝐶 / 𝑥) ∈ ℂ) → Σ𝑑 ∈
(1...(⌊‘𝑥))(𝐶 / 𝑥) = ((#‘(1...(⌊‘𝑥))) · (𝐶 / 𝑥))) |
269 | 40, 256, 268 | syl2anc 693 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑑 ∈
(1...(⌊‘𝑥))(𝐶 / 𝑥) = ((#‘(1...(⌊‘𝑥))) · (𝐶 / 𝑥))) |
270 | 158 | nncnd 11036 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(⌊‘𝑥) ∈
ℂ) |
271 | | divass 10703 |
. . . . . . . . . 10
⊢
(((⌊‘𝑥)
∈ ℂ ∧ 𝐶
∈ ℂ ∧ (𝑥
∈ ℂ ∧ 𝑥 ≠
0)) → (((⌊‘𝑥) · 𝐶) / 𝑥) = ((⌊‘𝑥) · (𝐶 / 𝑥))) |
272 | 270, 243,
246, 271 | syl3anc 1326 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(((⌊‘𝑥)
· 𝐶) / 𝑥) = ((⌊‘𝑥) · (𝐶 / 𝑥))) |
273 | 267, 269,
272 | 3eqtr4d 2666 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑑 ∈
(1...(⌊‘𝑥))(𝐶 / 𝑥) = (((⌊‘𝑥) · 𝐶) / 𝑥)) |
274 | 263, 273 | breqtrd 4679 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑑 ∈
(1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ≤ (((⌊‘𝑥) · 𝐶) / 𝑥)) |
275 | 38 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (𝐶 ∈ ℝ ∧ 0 ≤
𝐶)) |
276 | | flle 12600 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ →
(⌊‘𝑥) ≤
𝑥) |
277 | 65, 276 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(⌊‘𝑥) ≤
𝑥) |
278 | | lemul1a 10877 |
. . . . . . . . 9
⊢
((((⌊‘𝑥)
∈ ℝ ∧ 𝑥
∈ ℝ ∧ (𝐶
∈ ℝ ∧ 0 ≤ 𝐶)) ∧ (⌊‘𝑥) ≤ 𝑥) → ((⌊‘𝑥) · 𝐶) ≤ (𝑥 · 𝐶)) |
279 | 184, 65, 275, 277, 278 | syl31anc 1329 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
((⌊‘𝑥) ·
𝐶) ≤ (𝑥 · 𝐶)) |
280 | 185, 181,
64 | ledivmuld 11925 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
((((⌊‘𝑥)
· 𝐶) / 𝑥) ≤ 𝐶 ↔ ((⌊‘𝑥) · 𝐶) ≤ (𝑥 · 𝐶))) |
281 | 279, 280 | mpbird 247 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(((⌊‘𝑥)
· 𝐶) / 𝑥) ≤ 𝐶) |
282 | 180, 186,
181, 274, 281 | letrd 10194 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑑 ∈
(1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ≤ 𝐶) |
283 | 178, 180,
181, 182, 282 | letrd 10194 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(abs‘Σ𝑑 ∈
(1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ≤ 𝐶) |
284 | 174, 283 | eqbrtrd 4675 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (abs‘(1
− (Σ𝑑 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇))) ≤ 𝐶) |
285 | 32, 34, 35, 39, 284 | elo1d 14267 |
. . 3
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (1
− (Σ𝑑 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇))) ∈ 𝑂(1)) |
286 | 6, 31, 285 | o1dif 14360 |
. 2
⊢ (𝜑 → ((𝑥 ∈ ℝ+ ↦ 1) ∈
𝑂(1) ↔ (𝑥
∈ ℝ+ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) ∈ 𝑂(1))) |
287 | 5, 286 | mpbid 222 |
1
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦
(Σ𝑑 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) ∈ 𝑂(1)) |