| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | rpssre 13042 | . . . 4
⊢
ℝ+ ⊆ ℝ | 
| 2 |  | ax-1cn 11213 | . . . 4
⊢ 1 ∈
ℂ | 
| 3 |  | o1const 15656 | . . . 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 14014 | . . . . 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 13564 | . . . . . . . 8
⊢ (𝑑 ∈
(1...(⌊‘𝑥))
→ 𝑑 ∈
ℤ) | 
| 15 | 14 | adantl 481 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ∈
ℤ) | 
| 16 | 8, 9, 10, 11, 13, 15 | dchrzrhcl 27289 | . . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝑋‘(𝐿‘𝑑)) ∈ ℂ) | 
| 17 |  | elfznn 13593 | . . . . . . . . 9
⊢ (𝑑 ∈
(1...(⌊‘𝑥))
→ 𝑑 ∈
ℕ) | 
| 18 | 17 | adantl 481 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ∈
ℕ) | 
| 19 |  | mucl 27184 | . . . . . . . . . 10
⊢ (𝑑 ∈ ℕ →
(μ‘𝑑) ∈
ℤ) | 
| 20 | 19 | zred 12722 | . . . . . . . . 9
⊢ (𝑑 ∈ ℕ →
(μ‘𝑑) ∈
ℝ) | 
| 21 |  | nndivre 12307 | . . . . . . . . 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 11289 | . . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((μ‘𝑑) /
𝑑) ∈
ℂ) | 
| 25 | 16, 24 | mulcld 11281 | . . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ) | 
| 26 | 7, 25 | fsumcl 15769 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
Σ𝑑 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) ∈ ℂ) | 
| 27 |  | dchrisumn0.t | . . . . . 6
⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑇) | 
| 28 |  | climcl 15535 | . . . . . 6
⊢ (seq1( +
, 𝐹) ⇝ 𝑇 → 𝑇 ∈ ℂ) | 
| 29 | 27, 28 | syl 17 | . . . . 5
⊢ (𝜑 → 𝑇 ∈ ℂ) | 
| 30 | 29 | adantr 480 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑇 ∈
ℂ) | 
| 31 | 26, 30 | mulcld 11281 | . . 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 11507 | . . . . 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 587 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1
− (Σ𝑑 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) ∈ ℂ) | 
| 35 |  | 1red 11262 | . . . 4
⊢ (𝜑 → 1 ∈
ℝ) | 
| 36 |  | dchrisumn0.c | . . . . . 6
⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) | 
| 37 |  | elrege0 13494 | . . . . . 6
⊢ (𝐶 ∈ (0[,)+∞) ↔
(𝐶 ∈ ℝ ∧ 0
≤ 𝐶)) | 
| 38 | 36, 37 | sylib 218 | . . . . 5
⊢ (𝜑 → (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) | 
| 39 | 38 | simpld 494 | . . . 4
⊢ (𝜑 → 𝐶 ∈ ℝ) | 
| 40 |  | fzfid 14014 | . . . . . . . 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 12921 | . . . . . . . . . . . 12
⊢ ℕ =
(ℤ≥‘1) | 
| 43 |  | 1zzd 12648 | . . . . . . . . . . . 12
⊢ (𝜑 → 1 ∈
ℤ) | 
| 44 | 12 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋 ∈ 𝐷) | 
| 45 |  | nnz 12634 | . . . . . . . . . . . . . . . . 17
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℤ) | 
| 46 | 45 | adantl 481 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ) | 
| 47 | 8, 9, 10, 11, 44, 46 | dchrzrhcl 27289 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ) | 
| 48 |  | nncn 12274 | . . . . . . . . . . . . . . . 16
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℂ) | 
| 49 | 48 | adantl 481 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ) | 
| 50 |  | nnne0 12300 | . . . . . . . . . . . . . . . 16
⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0) | 
| 51 | 50 | adantl 481 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0) | 
| 52 | 47, 49, 51 | divcld 12043 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ) | 
| 53 |  | dchrisumn0.f | . . . . . . . . . . . . . . 15
⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) | 
| 54 |  | 2fveq3 6911 | . . . . . . . . . . . . . . . . 17
⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚))) | 
| 55 |  | id 22 | . . . . . . . . . . . . . . . . 17
⊢ (𝑎 = 𝑚 → 𝑎 = 𝑚) | 
| 56 | 54, 55 | oveq12d 7449 | . . . . . . . . . . . . . . . 16
⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) | 
| 57 | 56 | cbvmptv 5255 | . . . . . . . . . . . . . . 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 7134 | . . . . . . . . . . . . 13
⊢ (𝜑 → 𝐹:ℕ⟶ℂ) | 
| 60 | 59 | ffvelcdmda 7104 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∈ ℂ) | 
| 61 | 42, 43, 60 | serf 14071 | . . . . . . . . . . 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 13077 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝑥 ∈
ℝ) | 
| 65 |  | nndivre 12307 | . . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℝ ∧ 𝑑 ∈ ℕ) → (𝑥 / 𝑑) ∈ ℝ) | 
| 66 | 64, 17, 65 | syl2an 596 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝑥 / 𝑑) ∈
ℝ) | 
| 67 | 17 | adantl 481 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ∈
ℕ) | 
| 68 | 67 | nncnd 12282 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ∈
ℂ) | 
| 69 | 68 | mullidd 11279 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (1 · 𝑑) =
𝑑) | 
| 70 |  | fznnfl 13902 | . . . . . . . . . . . . . . 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 499 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ≤ 𝑥) | 
| 73 | 69, 72 | eqbrtrd 5165 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (1 · 𝑑) ≤
𝑥) | 
| 74 |  | 1red 11262 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 1 ∈ ℝ) | 
| 75 | 64 | adantr 480 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑥 ∈
ℝ) | 
| 76 | 67 | nnrpd 13075 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ∈
ℝ+) | 
| 77 | 74, 75, 76 | lemuldivd 13126 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((1 · 𝑑) ≤
𝑥 ↔ 1 ≤ (𝑥 / 𝑑))) | 
| 78 | 73, 77 | mpbid 232 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 1 ≤ (𝑥 / 𝑑)) | 
| 79 |  | flge1nn 13861 | . . . . . . . . . . 11
⊢ (((𝑥 / 𝑑) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑑)) → (⌊‘(𝑥 / 𝑑)) ∈ ℕ) | 
| 80 | 66, 78, 79 | syl2anc 584 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (⌊‘(𝑥 /
𝑑)) ∈
ℕ) | 
| 81 | 62, 80 | ffvelcdmd 7105 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) ∈ ℂ) | 
| 82 | 41, 81 | mulcld 11281 | . . . . . . . 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 11281 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇) ∈ ℂ) | 
| 85 | 40, 82, 84 | fsumsub 15824 | . . . . . . 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 11719 | . . . . . . . 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 15738 | . . . . . . 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 13564 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑))) → 𝑚 ∈
ℤ) | 
| 91 | 90 | adantl 481 | . . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → 𝑚 ∈
ℤ) | 
| 92 | 8, 9, 10, 11, 88, 89, 91 | dchrzrhmul 27290 | . . . . . . . . . . . . . . . . 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 7446 | . . . . . . . . . . . . . . . 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 480 | . . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → (𝑋‘(𝐿‘𝑑)) ∈ ℂ) | 
| 96 | 68 | adantr 480 | . . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → 𝑑 ∈
ℂ) | 
| 97 | 8, 9, 10, 11, 88, 91 | dchrzrhcl 27289 | . . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ) | 
| 98 |  | elfznn 13593 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑))) → 𝑚 ∈
ℕ) | 
| 99 | 98 | adantl 481 | . . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → 𝑚 ∈
ℕ) | 
| 100 | 99 | nncnd 12282 | . . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → 𝑚 ∈
ℂ) | 
| 101 | 67 | nnne0d 12316 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ≠
0) | 
| 102 | 101 | adantr 480 | . . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → 𝑑 ≠ 0) | 
| 103 | 99 | nnne0d 12316 | . . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → 𝑚 ≠ 0) | 
| 104 | 95, 96, 97, 100, 102, 103 | divmuldivd 12084 | . . . . . . . . . . . . . . . 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 7447 | . . . . . . . . . . . . . 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 12723 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (μ‘𝑑)
∈ ℂ) | 
| 109 | 108 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) →
(μ‘𝑑) ∈
ℂ) | 
| 110 | 95, 96, 102 | divcld 12043 | . . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → ((𝑋‘(𝐿‘𝑑)) / 𝑑) ∈ ℂ) | 
| 111 | 97, 100, 103 | divcld 12043 | . . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ) | 
| 112 | 109, 110,
111 | mulassd 11284 | . . . . . . . . . . . . . 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 12079 | . . . . . . . . . . . . . . 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 7446 | . . . . . . . . . . . . . 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 15738 | . . . . . . . . . . . 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 14014 | . . . . . . . . . . . . 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 596 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
∧ 𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ) | 
| 120 | 117, 41, 119 | fsummulc2 15820 | . . . . . . . . . . . 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 7464 | . . . . . . . . . . . . . . . 16
⊢ ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ V | 
| 122 | 56, 53, 121 | fvmpt 7016 | . . . . . . . . . . . . . . 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 2851 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (⌊‘(𝑥 /
𝑑)) ∈
(ℤ≥‘1)) | 
| 125 | 123, 124,
119 | fsumser 15766 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ Σ𝑚 ∈
(1...(⌊‘(𝑥 /
𝑑)))((𝑋‘(𝐿‘𝑚)) / 𝑚) = (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) | 
| 126 | 125 | oveq2d 7447 | . . . . . . . . . . . 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 15738 | . . . . . . . . . 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 6911 | . . . . . . . . . . . . 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 7449 | . . . . . . . . . . . 12
⊢ (𝑛 = (𝑑 · 𝑚) → ((𝑋‘(𝐿‘𝑛)) / 𝑛) = ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚))) | 
| 132 | 131 | oveq2d 7447 | . . . . . . . . . . 11
⊢ (𝑛 = (𝑑 · 𝑚) → ((μ‘𝑑) · ((𝑋‘(𝐿‘𝑛)) / 𝑛)) = ((μ‘𝑑) · ((𝑋‘(𝐿‘(𝑑 · 𝑚))) / (𝑑 · 𝑚)))) | 
| 133 |  | elrabi 3687 | . . . . . . . . . . . . . . 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 12723 | . . . . . . . . . . . 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 13564 | . . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈
(1...(⌊‘𝑥))
→ 𝑛 ∈
ℤ) | 
| 139 | 138 | adantl 481 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℤ) | 
| 140 | 8, 9, 10, 11, 137, 139 | dchrzrhcl 27289 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ (𝑋‘(𝐿‘𝑛)) ∈ ℂ) | 
| 141 |  | fz1ssnn 13595 | . . . . . . . . . . . . . . . . 17
⊢
(1...(⌊‘𝑥)) ⊆ ℕ | 
| 142 | 141 | a1i 11 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(1...(⌊‘𝑥))
⊆ ℕ) | 
| 143 | 142 | sselda 3983 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℕ) | 
| 144 | 143 | nncnd 12282 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ∈
ℂ) | 
| 145 | 143 | nnne0d 12316 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑛 ∈
(1...(⌊‘𝑥)))
→ 𝑛 ≠
0) | 
| 146 | 140, 144,
145 | divcld 12043 | . . . . . . . . . . . . 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 11281 | . . . . . . . . . . 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 27231 | . . . . . . . . . 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 6911 | . . . . . . . . . . . 12
⊢ (𝑛 = 1 → (𝑋‘(𝐿‘𝑛)) = (𝑋‘(𝐿‘1))) | 
| 151 |  | id 22 | . . . . . . . . . . . 12
⊢ (𝑛 = 1 → 𝑛 = 1) | 
| 152 | 150, 151 | oveq12d 7449 | . . . . . . . . . . 11
⊢ (𝑛 = 1 → ((𝑋‘(𝐿‘𝑛)) / 𝑛) = ((𝑋‘(𝐿‘1)) / 1)) | 
| 153 |  | simprr 773 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 1 ≤ 𝑥) | 
| 154 |  | flge1nn 13861 | . . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ ∧ 1 ≤
𝑥) →
(⌊‘𝑥) ∈
ℕ) | 
| 155 | 64, 153, 154 | syl2anc 584 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(⌊‘𝑥) ∈
ℕ) | 
| 156 | 155, 42 | eleqtrdi 2851 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(⌊‘𝑥) ∈
(ℤ≥‘1)) | 
| 157 |  | eluzfz1 13571 | . . . . . . . . . . . 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 27235 | . . . . . . . . . 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 27288 | . . . . . . . . . . . 12
⊢ (𝜑 → (𝑋‘(𝐿‘1)) = 1) | 
| 162 | 161 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (𝑋‘(𝐿‘1)) = 1) | 
| 163 | 162 | oveq1d 7446 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → ((𝑋‘(𝐿‘1)) / 1) = (1 / 1)) | 
| 164 |  | 1div1e1 11958 | . . . . . . . . . 10
⊢ (1 / 1) =
1 | 
| 165 | 163, 164 | eqtrdi 2793 | . . . . . . . . 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 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝑇 ∈
ℂ) | 
| 168 | 40, 167, 41 | fsummulc1 15821 | . . . . . . . 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 7449 | . . . . . . 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 6910 | . . . . 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 11620 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇) ∈ ℂ) | 
| 173 | 41, 172 | mulcld 11281 | . . . . . . . 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 15769 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑑 ∈
(1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · ((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ∈ ℂ) | 
| 175 | 174 | abscld 15475 | . . . . . 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 15475 | . . . . . . 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 15770 | . . . . . 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 480 | . . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝐶 ∈
ℝ) | 
| 179 | 40, 173 | fsumabs 15837 | . . . . . 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 13836 | . . . . . . . . . 10
⊢ (𝑥 ∈ ℝ →
(⌊‘𝑥) ∈
ℝ) | 
| 181 | 64, 180 | syl 17 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(⌊‘𝑥) ∈
ℝ) | 
| 182 | 181, 178 | remulcld 11291 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
((⌊‘𝑥) ·
𝐶) ∈
ℝ) | 
| 183 | 182, 63 | rerpdivcld 13108 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(((⌊‘𝑥)
· 𝐶) / 𝑥) ∈
ℝ) | 
| 184 | 178, 63 | rerpdivcld 13108 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (𝐶 / 𝑥) ∈ ℝ) | 
| 185 | 184 | adantr 480 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝐶 / 𝑥) ∈
ℝ) | 
| 186 | 41 | abscld 15475 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) ∈ ℝ) | 
| 187 | 67 | nnrecred 12317 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (1 / 𝑑) ∈
ℝ) | 
| 188 | 172 | abscld 15475 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ∈ ℝ) | 
| 189 | 76 | rpred 13077 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝑑 ∈
ℝ) | 
| 190 | 185, 189 | remulcld 11291 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((𝐶 / 𝑥) · 𝑑) ∈ ℝ) | 
| 191 | 41 | absge0d 15483 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)))) | 
| 192 | 172 | absge0d 15483 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) | 
| 193 | 94 | abscld 15475 | . . . . . . . . . . . . 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 15475 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((μ‘𝑑) / 𝑑)) ∈ ℝ) | 
| 196 | 94 | absge0d 15483 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 0 ≤ (abs‘(𝑋‘(𝐿‘𝑑)))) | 
| 197 | 194 | absge0d 15483 | . . . . . . . . . . . . 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 12587 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑁 ∈
ℕ0) | 
| 202 | 9, 198, 11 | znzrhfo 21566 | . . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℕ0
→ 𝐿:ℤ–onto→(Base‘𝑍)) | 
| 203 |  | fof 6820 | . . . . . . . . . . . . . . . . 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 480 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝐿:ℤ⟶(Base‘𝑍)) | 
| 206 |  | ffvelcdm 7101 | . . . . . . . . . . . . . . 15
⊢ ((𝐿:ℤ⟶(Base‘𝑍) ∧ 𝑑 ∈ ℤ) → (𝐿‘𝑑) ∈ (Base‘𝑍)) | 
| 207 | 205, 14, 206 | syl2an 596 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝐿‘𝑑) ∈ (Base‘𝑍)) | 
| 208 | 8, 10, 9, 198, 199, 207 | dchrabs2 27306 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘(𝑋‘(𝐿‘𝑑))) ≤ 1) | 
| 209 | 108, 68, 101 | absdivd 15494 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((μ‘𝑑) / 𝑑)) = ((abs‘(μ‘𝑑)) / (abs‘𝑑))) | 
| 210 | 76 | rprege0d 13084 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝑑 ∈ ℝ
∧ 0 ≤ 𝑑)) | 
| 211 |  | absid 15335 | . . . . . . . . . . . . . . . . 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 7447 | . . . . . . . . . . . . . . 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 15475 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘(μ‘𝑑)) ∈ ℝ) | 
| 216 |  | mule1 27191 | . . . . . . . . . . . . . . . 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 13135 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((abs‘(μ‘𝑑)) / 𝑑) ≤ (1 / 𝑑)) | 
| 219 | 214, 218 | eqbrtrd 5165 | . . . . . . . . . . . . 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 12210 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((abs‘(𝑋‘(𝐿‘𝑑))) · (abs‘((μ‘𝑑) / 𝑑))) ≤ (1 · (1 / 𝑑))) | 
| 221 | 94, 194 | absmuld 15493 | . . . . . . . . . . . 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 11289 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (1 / 𝑑) ∈
ℂ) | 
| 223 | 222 | mullidd 11279 | . . . . . . . . . . . . 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 5175 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑))) ≤ (1 / 𝑑)) | 
| 226 |  | 2fveq3 6911 | . . . . . . . . . . . . . . 15
⊢ (𝑦 = (𝑥 / 𝑑) → (seq1( + , 𝐹)‘(⌊‘𝑦)) = (seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑)))) | 
| 227 | 226 | fvoveq1d 7453 | . . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑥 / 𝑑) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) = (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇))) | 
| 228 |  | oveq2 7439 | . . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑥 / 𝑑) → (𝐶 / 𝑦) = (𝐶 / (𝑥 / 𝑑))) | 
| 229 | 227, 228 | breq12d 5156 | . . . . . . . . . . . . 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 11261 | . . . . . . . . . . . . . . 15
⊢ 1 ∈
ℝ | 
| 233 |  | elicopnf 13485 | . . . . . . . . . . . . . . 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 583 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝑥 / 𝑑) ∈
(1[,)+∞)) | 
| 236 | 229, 231,
235 | rspcdva 3623 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (abs‘((seq1( + , 𝐹)‘(⌊‘(𝑥 / 𝑑))) − 𝑇)) ≤ (𝐶 / (𝑥 / 𝑑))) | 
| 237 | 178 | recnd 11289 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝐶 ∈
ℂ) | 
| 238 | 237 | adantr 480 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ 𝐶 ∈
ℂ) | 
| 239 |  | rpcnne0 13053 | . . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ℝ+
→ (𝑥 ∈ ℂ
∧ 𝑥 ≠
0)) | 
| 240 | 239 | ad2antrl 728 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) | 
| 241 | 240 | adantr 480 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝑥 ∈ ℂ
∧ 𝑥 ≠
0)) | 
| 242 |  | divdiv2 11979 | . . . . . . . . . . . . . 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 11941 | . . . . . . . . . . . . . 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 5169 | . . . . . . . . . . 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 12210 | . . . . . . . . . 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 15493 | . . . . . . . . . 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 11289 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (𝐶 / 𝑥) ∈ ℂ) | 
| 251 | 250 | adantr 480 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (𝐶 / 𝑥) ∈
ℂ) | 
| 252 | 251, 68, 101 | divcan4d 12049 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ (((𝐶 / 𝑥) · 𝑑) / 𝑑) = (𝐶 / 𝑥)) | 
| 253 | 251, 68 | mulcld 11281 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑑 ∈
(1...(⌊‘𝑥)))
→ ((𝐶 / 𝑥) · 𝑑) ∈ ℂ) | 
| 254 | 253, 68, 101 | divrec2d 12047 | . . . . . . . . . . 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 5175 | . . . . . . . . 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 15835 | . . . . . . . 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 12587 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(⌊‘𝑥) ∈
ℕ0) | 
| 259 |  | hashfz1 14385 | . . . . . . . . . . 11
⊢
((⌊‘𝑥)
∈ ℕ0 → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥)) | 
| 260 | 258, 259 | syl 17 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥)) | 
| 261 | 260 | oveq1d 7446 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
((♯‘(1...(⌊‘𝑥))) · (𝐶 / 𝑥)) = ((⌊‘𝑥) · (𝐶 / 𝑥))) | 
| 262 |  | fsumconst 15826 | . . . . . . . . . 10
⊢
(((1...(⌊‘𝑥)) ∈ Fin ∧ (𝐶 / 𝑥) ∈ ℂ) → Σ𝑑 ∈
(1...(⌊‘𝑥))(𝐶 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (𝐶 / 𝑥))) | 
| 263 | 40, 250, 262 | syl2anc 584 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑑 ∈
(1...(⌊‘𝑥))(𝐶 / 𝑥) = ((♯‘(1...(⌊‘𝑥))) · (𝐶 / 𝑥))) | 
| 264 | 155 | nncnd 12282 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(⌊‘𝑥) ∈
ℂ) | 
| 265 |  | divass 11940 | . . . . . . . . . 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 5169 | . . . . . . 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 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (𝐶 ∈ ℝ ∧ 0 ≤
𝐶)) | 
| 270 |  | flle 13839 | . . . . . . . . . 10
⊢ (𝑥 ∈ ℝ →
(⌊‘𝑥) ≤
𝑥) | 
| 271 | 64, 270 | syl 17 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(⌊‘𝑥) ≤
𝑥) | 
| 272 |  | lemul1a 12121 | . . . . . . . . 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 13130 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
((((⌊‘𝑥)
· 𝐶) / 𝑥) ≤ 𝐶 ↔ ((⌊‘𝑥) · 𝐶) ≤ (𝑥 · 𝐶))) | 
| 275 | 273, 274 | mpbird 257 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(((⌊‘𝑥)
· 𝐶) / 𝑥) ≤ 𝐶) | 
| 276 | 177, 183,
178, 268, 275 | letrd 11418 | . . . . . 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 11418 | . . . . 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 5165 | . . . 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 15572 | . . 3
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (1
− (Σ𝑑 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇))) ∈ 𝑂(1)) | 
| 280 | 6, 31, 279 | o1dif 15666 | . 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 232 | 1
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦
(Σ𝑑 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · 𝑇)) ∈ 𝑂(1)) |