Step | Hyp | Ref
| Expression |
1 | | rpssre 12593 |
. . . 4
⊢
ℝ+ ⊆ ℝ |
2 | | ax-1cn 10787 |
. . . 4
⊢ 1 ∈
ℂ |
3 | | o1const 15181 |
. . . 4
⊢
((ℝ+ ⊆ ℝ ∧ 1 ∈ ℂ) →
(𝑥 ∈
ℝ+ ↦ 1) ∈ 𝑂(1)) |
4 | 1, 2, 3 | mp2an 692 |
. . 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 13546 |
. . . . 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 726 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑋 ∈ 𝐷) |
14 | | elfzelz 13112 |
. . . . . . . 8
⊢ (𝑑 ∈
(1...(⌊‘𝑥))
→ 𝑑 ∈
ℤ) |
15 | 14 | adantl 485 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ∈
ℤ) |
16 | 8, 9, 10, 11, 13, 15 | dchrzrhcl 26126 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝑋‘(𝐿‘𝑑)) ∈ ℂ) |
17 | | elfznn 13141 |
. . . . . . . . 9
⊢ (𝑑 ∈
(1...(⌊‘𝑥))
→ 𝑑 ∈
ℕ) |
18 | 17 | adantl 485 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ∈
ℕ) |
19 | | mucl 26023 |
. . . . . . . . . 10
⊢ (𝑑 ∈ ℕ →
(μ‘𝑑) ∈
ℤ) |
20 | 19 | zred 12282 |
. . . . . . . . 9
⊢ (𝑑 ∈ ℕ →
(μ‘𝑑) ∈
ℝ) |
21 | | nndivre 11871 |
. . . . . . . . 9
⊢
(((μ‘𝑑)
∈ ℝ ∧ 𝑑
∈ ℕ) → ((μ‘𝑑) / 𝑑) ∈ ℝ) |
22 | 20, 21 | mpancom 688 |
. . . . . . . 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 10861 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((μ‘𝑑) /
𝑑) ∈
ℂ) |
25 | 16, 24 | mulcld 10853 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ) |
26 | 7, 25 | fsumcl 15297 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
Σ𝑑 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ) |
27 | | dchrisumn0.t |
. . . . . 6
⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑇) |
28 | | climcl 15060 |
. . . . . 6
⊢ (seq1( +
, 𝐹) ⇝ 𝑇 → 𝑇 ∈ ℂ) |
29 | 27, 28 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝑇 ∈ ℂ) |
30 | 29 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑇 ∈
ℂ) |
31 | 26, 30 | mulcld 10853 |
. . 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 11077 |
. . . . 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 590 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1
− (Σ𝑑 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) ∈ ℂ) |
35 | | 1red 10834 |
. . . 4
⊢ (𝜑 → 1 ∈
ℝ) |
36 | | dchrisumn0.c |
. . . . . 6
⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) |
37 | | elrege0 13042 |
. . . . . 6
⊢ (𝐶 ∈ (0[,)+∞) ↔
(𝐶 ∈ ℝ ∧ 0
≤ 𝐶)) |
38 | 36, 37 | sylib 221 |
. . . . 5
⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) |
39 | 38 | simpld 498 |
. . . 4
⊢ (𝜑 → 𝐶 ∈ ℝ) |
40 | | fzfid 13546 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(1...(⌊‘𝑥))
∈ Fin) |
41 | 25 | adantlrr 721 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ) |
42 | | nnuz 12477 |
. . . . . . . . . . . 12
⊢ ℕ =
(ℤ≥‘1) |
43 | | 1zzd 12208 |
. . . . . . . . . . . 12
⊢ (𝜑 → 1 ∈
ℤ) |
44 | 12 | adantr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋 ∈ 𝐷) |
45 | | nnz 12199 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℤ) |
46 | 45 | adantl 485 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ) |
47 | 8, 9, 10, 11, 44, 46 | dchrzrhcl 26126 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ) |
48 | | nncn 11838 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℂ) |
49 | 48 | adantl 485 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ) |
50 | | nnne0 11864 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0) |
51 | 50 | adantl 485 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0) |
52 | 47, 49, 51 | divcld 11608 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ) |
53 | | dchrisumn0.f |
. . . . . . . . . . . . . . 15
⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) |
54 | | 2fveq3 6722 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚))) |
55 | | id 22 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 = 𝑚 → 𝑎 = 𝑚) |
56 | 54, 55 | oveq12d 7231 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
57 | 56 | cbvmptv 5158 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) = (𝑚 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
58 | 53, 57 | eqtri 2765 |
. . . . . . . . . . . . . 14
⊢ 𝐹 = (𝑚 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
59 | 52, 58 | fmptd 6931 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐹:ℕ⟶ℂ) |
60 | 59 | ffvelrnda 6904 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∈ ℂ) |
61 | 42, 43, 60 | serf 13604 |
. . . . . . . . . . 11
⊢ (𝜑 → seq1( + , 𝐹):ℕ⟶ℂ) |
62 | 61 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ seq1( + , 𝐹):ℕ⟶ℂ) |
63 | | simprl 771 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝑥 ∈
ℝ+) |
64 | 63 | rpred 12628 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝑥 ∈
ℝ) |
65 | | nndivre 11871 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ ∧ 𝑑 ∈ ℕ) → (𝑥 / 𝑑) ∈ ℝ) |
66 | 64, 17, 65 | syl2an 599 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝑥 / 𝑑) ∈
ℝ) |
67 | 17 | adantl 485 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ∈
ℕ) |
68 | 67 | nncnd 11846 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ∈
ℂ) |
69 | 68 | mulid2d 10851 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (1 · 𝑑) =
𝑑) |
70 | | fznnfl 13435 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ℝ → (𝑑 ∈
(1...(⌊‘𝑥))
↔ (𝑑 ∈ ℕ
∧ 𝑑 ≤ 𝑥))) |
71 | 64, 70 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (𝑑 ∈
(1...(⌊‘𝑥))
↔ (𝑑 ∈ ℕ
∧ 𝑑 ≤ 𝑥))) |
72 | 71 | simplbda 503 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ≤ 𝑥) |
73 | 69, 72 | eqbrtrd 5075 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (1 · 𝑑) ≤
𝑥) |
74 | | 1red 10834 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 1 ∈ ℝ) |
75 | 64 | adantr 484 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑥 ∈
ℝ) |
76 | 67 | nnrpd 12626 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ∈
ℝ+) |
77 | 74, 75, 76 | lemuldivd 12677 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((1 · 𝑑) ≤
𝑥 ↔ 1 ≤ (𝑥 / 𝑑))) |
78 | 73, 77 | mpbid 235 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 1 ≤ (𝑥 / 𝑑)) |
79 | | flge1nn 13396 |
. . . . . . . . . . 11
⊢ (((𝑥 / 𝑑) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑑)) → (⌊‘(𝑥 / 𝑑)) ∈ ℕ) |
80 | 66, 78, 79 | syl2anc 587 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (⌊‘(𝑥 /
𝑑)) ∈
ℕ) |
81 | 62, 80 | ffvelrnd 6905 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) ∈ ℂ) |
82 | 41, 81 | mulcld 10853 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) ∈ ℂ) |
83 | 29 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑇 ∈
ℂ) |
84 | 41, 83 | mulcld 10853 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇) ∈ ℂ) |
85 | 40, 82, 84 | fsumsub 15352 |
. . . . . . 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))) |
86 | 41, 81, 83 | subdid 11288 |
. . . . . . . 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))) |
87 | 86 | sumeq2dv 15267 |
. . . . . . 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))) |
88 | 12 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → 𝑋 ∈ 𝐷) |
89 | 14 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → 𝑑 ∈
ℤ) |
90 | | elfzelz 13112 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑))) → 𝑚 ∈
ℤ) |
91 | 90 | adantl 485 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → 𝑚 ∈
ℤ) |
92 | 8, 9, 10, 11, 88, 89, 91 | dchrzrhmul 26127 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → (𝑋‘(𝐿‘(𝑑 · 𝑚))) = ((𝑋‘(𝐿‘𝑑)) · (𝑋‘(𝐿‘𝑚)))) |
93 | 92 | oveq1d 7228 |
. . . . . . . . . . . . . . . 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))) |
94 | 16 | adantlrr 721 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝑋‘(𝐿‘𝑑)) ∈ ℂ) |
95 | 94 | adantr 484 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → (𝑋‘(𝐿‘𝑑)) ∈ ℂ) |
96 | 68 | adantr 484 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → 𝑑 ∈
ℂ) |
97 | 8, 9, 10, 11, 88, 91 | dchrzrhcl 26126 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ) |
98 | | elfznn 13141 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑))) → 𝑚 ∈
ℕ) |
99 | 98 | adantl 485 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → 𝑚 ∈
ℕ) |
100 | 99 | nncnd 11846 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → 𝑚 ∈
ℂ) |
101 | 67 | nnne0d 11880 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ≠
0) |
102 | 101 | adantr 484 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → 𝑑 ≠ 0) |
103 | 99 | nnne0d 11880 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → 𝑚 ≠ 0) |
104 | 95, 96, 97, 100, 102, 103 | divmuldivd 11649 |
. . . . . . . . . . . . . . . 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))) |
105 | 93, 104 | eqtr4d 2780 |
. . . . . . . . . . . . . . 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))) |
106 | 105 | oveq2d 7229 |
. . . . . . . . . . . . . 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)))) |
107 | 67, 19 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (μ‘𝑑)
∈ ℤ) |
108 | 107 | zcnd 12283 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (μ‘𝑑)
∈ ℂ) |
109 | 108 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) →
(μ‘𝑑) ∈
ℂ) |
110 | 95, 96, 102 | divcld 11608 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → ((𝑋‘(𝐿‘𝑑)) / 𝑑) ∈ ℂ) |
111 | 97, 100, 103 | divcld 11608 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ) |
112 | 109, 110,
111 | mulassd 10856 |
. . . . . . . . . . . . . 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)))) |
113 | 109, 95, 96, 102 | div12d 11644 |
. . . . . . . . . . . . . . 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))) |
114 | 113 | oveq1d 7228 |
. . . . . . . . . . . . . 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))) |
115 | 106, 112,
114 | 3eqtr2d 2783 |
. . . . . . . . . . . . 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))) |
116 | 115 | sumeq2dv 15267 |
. . . . . . . . . . . 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))) |
117 | | fzfid 13546 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (1...(⌊‘(𝑥 / 𝑑))) ∈ Fin) |
118 | | simpll 767 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝜑) |
119 | 118, 98, 52 | syl2an 599 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ) |
120 | 117, 41, 119 | fsummulc2 15348 |
. . . . . . . . . . . 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))) |
121 | | ovex 7246 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ V |
122 | 56, 53, 121 | fvmpt 6818 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ ℕ → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
123 | 99, 122 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
124 | 80, 42 | eleqtrdi 2848 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (⌊‘(𝑥 /
𝑑)) ∈
(ℤ≥‘1)) |
125 | 123, 124,
119 | fsumser 15294 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))((𝑋‘(𝐿‘𝑚)) / 𝑚) = (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) |
126 | 125 | oveq2d 7229 |
. . . . . . . . . . . 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))))) |
127 | 116, 120,
126 | 3eqtr2rd 2784 |
. . . . . . . . . . 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)))) |
128 | 127 | sumeq2dv 15267 |
. . . . . . . . . 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)))) |
129 | | 2fveq3 6722 |
. . . . . . . . . . . . 13
⊢ (𝑛 = (𝑑 · 𝑚) → (𝑋‘(𝐿‘𝑛)) = (𝑋‘(𝐿‘(𝑑 · 𝑚)))) |
130 | | id 22 |
. . . . . . . . . . . . 13
⊢ (𝑛 = (𝑑 · 𝑚) → 𝑛 = (𝑑 · 𝑚)) |
131 | 129, 130 | oveq12d 7231 |
. . . . . . . . . . . 12
⊢ (𝑛 = (𝑑 · 𝑚) → ((𝑋‘(𝐿‘𝑛)) / 𝑛) = ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚))) |
132 | 131 | oveq2d 7229 |
. . . . . . . . . . 11
⊢ (𝑛 = (𝑑 · 𝑚) → ((μ‘𝑑) · ((𝑋‘(𝐿‘𝑛)) / 𝑛)) = ((μ‘𝑑) · ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)))) |
133 | | elrabi 3596 |
. . . . . . . . . . . . . . 15
⊢ (𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} → 𝑑 ∈ ℕ) |
134 | 133 | ad2antll 729 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ (𝑛 ∈
(1...(⌊‘𝑥))
∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ ℕ) |
135 | 134, 19 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ (𝑛 ∈
(1...(⌊‘𝑥))
∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (μ‘𝑑) ∈ ℤ) |
136 | 135 | zcnd 12283 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ (𝑛 ∈
(1...(⌊‘𝑥))
∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (μ‘𝑑) ∈ ℂ) |
137 | 12 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑋 ∈ 𝐷) |
138 | | elfzelz 13112 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈
(1...(⌊‘𝑥))
→ 𝑛 ∈
ℤ) |
139 | 138 | adantl 485 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℤ) |
140 | 8, 9, 10, 11, 137, 139 | dchrzrhcl 26126 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑋‘(𝐿‘𝑛)) ∈ ℂ) |
141 | | fz1ssnn 13143 |
. . . . . . . . . . . . . . . . 17
⊢
(1...(⌊‘𝑥)) ⊆ ℕ |
142 | 141 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(1...(⌊‘𝑥))
⊆ ℕ) |
143 | 142 | sselda 3901 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℕ) |
144 | 143 | nncnd 11846 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℂ) |
145 | 143 | nnne0d 11880 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ≠
0) |
146 | 140, 144,
145 | divcld 11608 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ ((𝑋‘(𝐿‘𝑛)) / 𝑛) ∈ ℂ) |
147 | 146 | adantrr 717 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ (𝑛 ∈
(1...(⌊‘𝑥))
∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((𝑋‘(𝐿‘𝑛)) / 𝑛) ∈ ℂ) |
148 | 136, 147 | mulcld 10853 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ (𝑛 ∈
(1...(⌊‘𝑥))
∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((μ‘𝑑) · ((𝑋‘(𝐿‘𝑛)) / 𝑛)) ∈ ℂ) |
149 | 132, 64, 148 | dvdsflsumcom 26070 |
. . . . . . . . . 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)))) |
150 | | 2fveq3 6722 |
. . . . . . . . . . . 12
⊢ (𝑛 = 1 → (𝑋‘(𝐿‘𝑛)) = (𝑋‘(𝐿‘1))) |
151 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝑛 = 1 → 𝑛 = 1) |
152 | 150, 151 | oveq12d 7231 |
. . . . . . . . . . 11
⊢ (𝑛 = 1 → ((𝑋‘(𝐿‘𝑛)) / 𝑛) = ((𝑋‘(𝐿‘1)) / 1)) |
153 | | simprr 773 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 1 ≤ 𝑥) |
154 | | flge1nn 13396 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ 1 ≤
𝑥) →
(⌊‘𝑥) ∈
ℕ) |
155 | 64, 153, 154 | syl2anc 587 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(⌊‘𝑥) ∈
ℕ) |
156 | 155, 42 | eleqtrdi 2848 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(⌊‘𝑥) ∈
(ℤ≥‘1)) |
157 | | eluzfz1 13119 |
. . . . . . . . . . . 12
⊢
((⌊‘𝑥)
∈ (ℤ≥‘1) → 1 ∈
(1...(⌊‘𝑥))) |
158 | 156, 157 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 1 ∈
(1...(⌊‘𝑥))) |
159 | 152, 40, 142, 158, 146 | musumsum 26074 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑛 ∈
(1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((𝑋‘(𝐿‘𝑛)) / 𝑛)) = ((𝑋‘(𝐿‘1)) / 1)) |
160 | 128, 149,
159 | 3eqtr2d 2783 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑑 ∈
(1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) = ((𝑋‘(𝐿‘1)) / 1)) |
161 | 8, 9, 10, 11, 12 | dchrzrh1 26125 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑋‘(𝐿‘1)) = 1) |
162 | 161 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (𝑋‘(𝐿‘1)) = 1) |
163 | 162 | oveq1d 7228 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → ((𝑋‘(𝐿‘1)) / 1) = (1 / 1)) |
164 | | 1div1e1 11522 |
. . . . . . . . . 10
⊢ (1 / 1) =
1 |
165 | 163, 164 | eqtrdi 2794 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → ((𝑋‘(𝐿‘1)) / 1) = 1) |
166 | 160, 165 | eqtr2d 2778 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 1 = Σ𝑑 ∈
(1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))))) |
167 | 29 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝑇 ∈
ℂ) |
168 | 40, 167, 41 | fsummulc1 15349 |
. . . . . . . 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)) |
169 | 166, 168 | oveq12d 7231 |
. . . . . . 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))) |
170 | 85, 87, 169 | 3eqtr4rd 2788 |
. . . . . 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))) |
171 | 170 | fveq2d 6721 |
. . . . 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)))) |
172 | 81, 83 | subcld 11189 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇) ∈ ℂ) |
173 | 41, 172 | mulcld 10853 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ∈ ℂ) |
174 | 40, 173 | fsumcl 15297 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑑 ∈
(1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ∈ ℂ) |
175 | 174 | abscld 15000 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(abs‘Σ𝑑 ∈
(1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ∈ ℝ) |
176 | 173 | abscld 15000 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ∈ ℝ) |
177 | 40, 176 | fsumrecl 15298 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑑 ∈
(1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ∈ ℝ) |
178 | 39 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝐶 ∈
ℝ) |
179 | 40, 173 | fsumabs 15365 |
. . . . . 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)))) |
180 | | reflcl 13371 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ →
(⌊‘𝑥) ∈
ℝ) |
181 | 64, 180 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(⌊‘𝑥) ∈
ℝ) |
182 | 181, 178 | remulcld 10863 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
((⌊‘𝑥) ·
𝐶) ∈
ℝ) |
183 | 182, 63 | rerpdivcld 12659 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(((⌊‘𝑥)
· 𝐶) / 𝑥) ∈
ℝ) |
184 | 178, 63 | rerpdivcld 12659 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (𝐶 / 𝑥) ∈ ℝ) |
185 | 184 | adantr 484 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝐶 / 𝑥) ∈
ℝ) |
186 | 41 | abscld 15000 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) ∈ ℝ) |
187 | 67 | nnrecred 11881 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (1 / 𝑑) ∈
ℝ) |
188 | 172 | abscld 15000 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ∈ ℝ) |
189 | 76 | rpred 12628 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ∈
ℝ) |
190 | 185, 189 | remulcld 10863 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((𝐶 / 𝑥) · 𝑑) ∈ ℝ) |
191 | 41 | absge0d 15008 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)))) |
192 | 172 | absge0d 15008 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) |
193 | 94 | abscld 15000 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘(𝑋‘(𝐿‘𝑑))) ∈ ℝ) |
194 | 24 | adantlrr 721 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((μ‘𝑑) /
𝑑) ∈
ℂ) |
195 | 194 | abscld 15000 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((μ‘𝑑) / 𝑑)) ∈ ℝ) |
196 | 94 | absge0d 15008 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ (abs‘(𝑋‘(𝐿‘𝑑)))) |
197 | 194 | absge0d 15008 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ (abs‘((μ‘𝑑) / 𝑑))) |
198 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢
(Base‘𝑍) =
(Base‘𝑍) |
199 | 12 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑋 ∈ 𝐷) |
200 | | rpvmasum.a |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑁 ∈ ℕ) |
201 | 200 | nnnn0d 12150 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
202 | 9, 198, 11 | znzrhfo 20512 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℕ0
→ 𝐿:ℤ–onto→(Base‘𝑍)) |
203 | | fof 6633 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐿:ℤ–onto→(Base‘𝑍) → 𝐿:ℤ⟶(Base‘𝑍)) |
204 | 201, 202,
203 | 3syl 18 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍)) |
205 | 204 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝐿:ℤ⟶(Base‘𝑍)) |
206 | | ffvelrn 6902 |
. . . . . . . . . . . . . . 15
⊢ ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑑 ∈ ℤ) → (𝐿‘𝑑) ∈ (Base‘𝑍)) |
207 | 205, 14, 206 | syl2an 599 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝐿‘𝑑) ∈ (Base‘𝑍)) |
208 | 8, 10, 9, 198, 199, 207 | dchrabs2 26143 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘(𝑋‘(𝐿‘𝑑))) ≤ 1) |
209 | 108, 68, 101 | absdivd 15019 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((μ‘𝑑) / 𝑑)) = ((abs‘(μ‘𝑑)) / (abs‘𝑑))) |
210 | 76 | rprege0d 12635 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝑑 ∈ ℝ
∧ 0 ≤ 𝑑)) |
211 | | absid 14860 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑑 ∈ ℝ ∧ 0 ≤
𝑑) → (abs‘𝑑) = 𝑑) |
212 | 210, 211 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘𝑑) =
𝑑) |
213 | 212 | oveq2d 7229 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((abs‘(μ‘𝑑)) / (abs‘𝑑)) = ((abs‘(μ‘𝑑)) / 𝑑)) |
214 | 209, 213 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((μ‘𝑑) / 𝑑)) = ((abs‘(μ‘𝑑)) / 𝑑)) |
215 | 108 | abscld 15000 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘(μ‘𝑑)) ∈ ℝ) |
216 | | mule1 26030 |
. . . . . . . . . . . . . . . 16
⊢ (𝑑 ∈ ℕ →
(abs‘(μ‘𝑑))
≤ 1) |
217 | 67, 216 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘(μ‘𝑑)) ≤ 1) |
218 | 215, 74, 76, 217 | lediv1dd 12686 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((abs‘(μ‘𝑑)) / 𝑑) ≤ (1 / 𝑑)) |
219 | 214, 218 | eqbrtrd 5075 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((μ‘𝑑) / 𝑑)) ≤ (1 / 𝑑)) |
220 | 193, 74, 195, 187, 196, 197, 208, 219 | lemul12ad 11774 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((abs‘(𝑋‘(𝐿‘𝑑))) · (abs‘((μ‘𝑑) / 𝑑))) ≤ (1 · (1 / 𝑑))) |
221 | 94, 194 | absmuld 15018 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) = ((abs‘(𝑋‘(𝐿‘𝑑))) · (abs‘((μ‘𝑑) / 𝑑)))) |
222 | 187 | recnd 10861 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (1 / 𝑑) ∈
ℂ) |
223 | 222 | mulid2d 10851 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (1 · (1 / 𝑑))
= (1 / 𝑑)) |
224 | 223 | eqcomd 2743 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (1 / 𝑑) = (1
· (1 / 𝑑))) |
225 | 220, 221,
224 | 3brtr4d 5085 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) ≤ (1 / 𝑑)) |
226 | | 2fveq3 6722 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = (𝑥 / 𝑑) → (seq1( + , 𝐹)‘(⌊‘𝑦)) = (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) |
227 | 226 | fvoveq1d 7235 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑥 / 𝑑) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) = (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) |
228 | | oveq2 7221 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑥 / 𝑑) → (𝐶 / 𝑦) = (𝐶 / (𝑥 / 𝑑))) |
229 | 227, 228 | breq12d 5066 |
. . . . . . . . . . . . 13
⊢ (𝑦 = (𝑥 / 𝑑) → ((abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐶 / 𝑦) ↔ (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ≤ (𝐶 / (𝑥 / 𝑑)))) |
230 | | dchrisumn0.1 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + ,
𝐹)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐶 / 𝑦)) |
231 | 230 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ∀𝑦 ∈
(1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐶 / 𝑦)) |
232 | | 1re 10833 |
. . . . . . . . . . . . . . 15
⊢ 1 ∈
ℝ |
233 | | elicopnf 13033 |
. . . . . . . . . . . . . . 15
⊢ (1 ∈
ℝ → ((𝑥 / 𝑑) ∈ (1[,)+∞) ↔
((𝑥 / 𝑑) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑑)))) |
234 | 232, 233 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 / 𝑑) ∈ (1[,)+∞) ↔ ((𝑥 / 𝑑) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑑))) |
235 | 66, 78, 234 | sylanbrc 586 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝑥 / 𝑑) ∈
(1[,)+∞)) |
236 | 229, 231,
235 | rspcdva 3539 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ≤ (𝐶 / (𝑥 / 𝑑))) |
237 | 178 | recnd 10861 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝐶 ∈
ℂ) |
238 | 237 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝐶 ∈
ℂ) |
239 | | rpcnne0 12604 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ℝ+
→ (𝑥 ∈ ℂ
∧ 𝑥 ≠
0)) |
240 | 239 | ad2antrl 728 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) |
241 | 240 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝑥 ∈ ℂ
∧ 𝑥 ≠
0)) |
242 | | divdiv2 11544 |
. . . . . . . . . . . . . 14
⊢ ((𝐶 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0)) → (𝐶 / (𝑥 / 𝑑)) = ((𝐶 · 𝑑) / 𝑥)) |
243 | 238, 241,
68, 101, 242 | syl112anc 1376 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝐶 / (𝑥 / 𝑑)) = ((𝐶 · 𝑑) / 𝑥)) |
244 | | div23 11509 |
. . . . . . . . . . . . . 14
⊢ ((𝐶 ∈ ℂ ∧ 𝑑 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → ((𝐶 · 𝑑) / 𝑥) = ((𝐶 / 𝑥) · 𝑑)) |
245 | 238, 68, 241, 244 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((𝐶 · 𝑑) / 𝑥) = ((𝐶 / 𝑥) · 𝑑)) |
246 | 243, 245 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝐶 / (𝑥 / 𝑑)) = ((𝐶 / 𝑥) · 𝑑)) |
247 | 236, 246 | breqtrd 5079 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ≤ ((𝐶 / 𝑥) · 𝑑)) |
248 | 186, 187,
188, 190, 191, 192, 225, 247 | lemul12ad 11774 |
. . . . . . . . . 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))) |
249 | 41, 172 | absmuld 15018 |
. . . . . . . . . 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)))) |
250 | 184 | recnd 10861 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (𝐶 / 𝑥) ∈ ℂ) |
251 | 250 | adantr 484 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝐶 / 𝑥) ∈
ℂ) |
252 | 251, 68, 101 | divcan4d 11614 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (((𝐶 / 𝑥) · 𝑑) / 𝑑) = (𝐶 / 𝑥)) |
253 | 251, 68 | mulcld 10853 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((𝐶 / 𝑥) · 𝑑) ∈ ℂ) |
254 | 253, 68, 101 | divrec2d 11612 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (((𝐶 / 𝑥) · 𝑑) / 𝑑) = ((1 / 𝑑) · ((𝐶 / 𝑥) · 𝑑))) |
255 | 252, 254 | eqtr3d 2779 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝐶 / 𝑥) = ((1 / 𝑑) · ((𝐶 / 𝑥) · 𝑑))) |
256 | 248, 249,
255 | 3brtr4d 5085 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ≤ (𝐶 / 𝑥)) |
257 | 40, 176, 185, 256 | fsumle 15363 |
. . . . . . . 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)) |
258 | 155 | nnnn0d 12150 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(⌊‘𝑥) ∈
ℕ0) |
259 | | hashfz1 13912 |
. . . . . . . . . . 11
⊢
((⌊‘𝑥)
∈ ℕ0 → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥)) |
260 | 258, 259 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥)) |
261 | 260 | oveq1d 7228 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
((♯‘(1...(⌊‘𝑥))) · (𝐶 / 𝑥)) = ((⌊‘𝑥) · (𝐶 / 𝑥))) |
262 | | fsumconst 15354 |
. . . . . . . . . 10
⊢
(((1...(⌊‘𝑥)) ∈ Fin ∧ (𝐶 / 𝑥) ∈ ℂ) → Σ𝑑 ∈
(1...(⌊‘𝑥))(𝐶 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (𝐶 / 𝑥))) |
263 | 40, 250, 262 | syl2anc 587 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑑 ∈
(1...(⌊‘𝑥))(𝐶 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (𝐶 / 𝑥))) |
264 | 155 | nncnd 11846 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(⌊‘𝑥) ∈
ℂ) |
265 | | divass 11508 |
. . . . . . . . . 10
⊢
(((⌊‘𝑥)
∈ ℂ ∧ 𝐶
∈ ℂ ∧ (𝑥
∈ ℂ ∧ 𝑥 ≠
0)) → (((⌊‘𝑥) · 𝐶) / 𝑥) = ((⌊‘𝑥) · (𝐶 / 𝑥))) |
266 | 264, 237,
240, 265 | syl3anc 1373 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(((⌊‘𝑥)
· 𝐶) / 𝑥) = ((⌊‘𝑥) · (𝐶 / 𝑥))) |
267 | 261, 263,
266 | 3eqtr4d 2787 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑑 ∈
(1...(⌊‘𝑥))(𝐶 / 𝑥) = (((⌊‘𝑥) · 𝐶) / 𝑥)) |
268 | 257, 267 | breqtrd 5079 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑑 ∈
(1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ≤ (((⌊‘𝑥) · 𝐶) / 𝑥)) |
269 | 38 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (𝐶 ∈ ℝ ∧ 0 ≤
𝐶)) |
270 | | flle 13374 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℝ →
(⌊‘𝑥) ≤
𝑥) |
271 | 64, 270 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(⌊‘𝑥) ≤
𝑥) |
272 | | lemul1a 11686 |
. . . . . . . . 9
⊢
((((⌊‘𝑥)
∈ ℝ ∧ 𝑥
∈ ℝ ∧ (𝐶
∈ ℝ ∧ 0 ≤ 𝐶)) ∧ (⌊‘𝑥) ≤ 𝑥) → ((⌊‘𝑥) · 𝐶) ≤ (𝑥 · 𝐶)) |
273 | 181, 64, 269, 271, 272 | syl31anc 1375 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
((⌊‘𝑥) ·
𝐶) ≤ (𝑥 · 𝐶)) |
274 | 182, 178,
63 | ledivmuld 12681 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
((((⌊‘𝑥)
· 𝐶) / 𝑥) ≤ 𝐶 ↔ ((⌊‘𝑥) · 𝐶) ≤ (𝑥 · 𝐶))) |
275 | 273, 274 | mpbird 260 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(((⌊‘𝑥)
· 𝐶) / 𝑥) ≤ 𝐶) |
276 | 177, 183,
178, 268, 275 | letrd 10989 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑑 ∈
(1...(⌊‘𝑥))(abs‘(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ≤ 𝐶) |
277 | 175, 177,
178, 179, 276 | letrd 10989 |
. . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(abs‘Σ𝑑 ∈
(1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) ≤ 𝐶) |
278 | 171, 277 | eqbrtrd 5075 |
. . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (abs‘(1
− (Σ𝑑 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇))) ≤ 𝐶) |
279 | 32, 34, 35, 39, 278 | elo1d 15097 |
. . 3
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (1
− (Σ𝑑 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇))) ∈ 𝑂(1)) |
280 | 6, 31, 279 | o1dif 15191 |
. 2
⊢ (𝜑 → ((𝑥 ∈ ℝ+ ↦ 1) ∈
𝑂(1) ↔ (𝑥
∈ ℝ+ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) ∈ 𝑂(1))) |
281 | 5, 280 | mpbid 235 |
1
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦
(Σ𝑑 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) ∈ 𝑂(1)) |