Proof of Theorem dchrisum0lem2
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | 2cnd 12344 | . . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 2 ∈
ℂ) | 
| 2 |  | rpcn 13045 | . . . . 5
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈
ℂ) | 
| 3 | 2 | adantl 481 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈
ℂ) | 
| 4 |  | fzfid 14014 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(1...(⌊‘𝑥))
∈ Fin) | 
| 5 |  | rpvmasum2.g | . . . . . . 7
⊢ 𝐺 = (DChr‘𝑁) | 
| 6 |  | rpvmasum.z | . . . . . . 7
⊢ 𝑍 =
(ℤ/nℤ‘𝑁) | 
| 7 |  | rpvmasum2.d | . . . . . . 7
⊢ 𝐷 = (Base‘𝐺) | 
| 8 |  | rpvmasum.l | . . . . . . 7
⊢ 𝐿 = (ℤRHom‘𝑍) | 
| 9 |  | rpvmasum2.w | . . . . . . . . . . 11
⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0} | 
| 10 | 9 | ssrab3 4082 | . . . . . . . . . 10
⊢ 𝑊 ⊆ (𝐷 ∖ { 1 }) | 
| 11 |  | dchrisum0.b | . . . . . . . . . 10
⊢ (𝜑 → 𝑋 ∈ 𝑊) | 
| 12 | 10, 11 | sselid 3981 | . . . . . . . . 9
⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 })) | 
| 13 | 12 | eldifad 3963 | . . . . . . . 8
⊢ (𝜑 → 𝑋 ∈ 𝐷) | 
| 14 | 13 | ad2antrr 726 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ 𝑋 ∈ 𝐷) | 
| 15 |  | elfzelz 13564 | . . . . . . . 8
⊢ (𝑚 ∈
(1...(⌊‘𝑥))
→ 𝑚 ∈
ℤ) | 
| 16 | 15 | adantl 481 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ 𝑚 ∈
ℤ) | 
| 17 | 5, 6, 7, 8, 14, 16 | dchrzrhcl 27289 | . . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (𝑋‘(𝐿‘𝑚)) ∈ ℂ) | 
| 18 |  | elfznn 13593 | . . . . . . . . 9
⊢ (𝑚 ∈
(1...(⌊‘𝑥))
→ 𝑚 ∈
ℕ) | 
| 19 | 18 | nnrpd 13075 | . . . . . . . 8
⊢ (𝑚 ∈
(1...(⌊‘𝑥))
→ 𝑚 ∈
ℝ+) | 
| 20 | 19 | adantl 481 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ 𝑚 ∈
ℝ+) | 
| 21 | 20 | rpcnd 13079 | . . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ 𝑚 ∈
ℂ) | 
| 22 | 20 | rpne0d 13082 | . . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ 𝑚 ≠
0) | 
| 23 | 17, 21, 22 | divcld 12043 | . . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ) | 
| 24 | 4, 23 | fsumcl 15769 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ) | 
| 25 | 3, 24 | mulcld 11281 | . . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝑥 · Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ ℂ) | 
| 26 |  | rpssre 13042 | . . . . 5
⊢
ℝ+ ⊆ ℝ | 
| 27 |  | 2cn 12341 | . . . . 5
⊢ 2 ∈
ℂ | 
| 28 |  | o1const 15656 | . . . . 5
⊢
((ℝ+ ⊆ ℝ ∧ 2 ∈ ℂ) →
(𝑥 ∈
ℝ+ ↦ 2) ∈ 𝑂(1)) | 
| 29 | 26, 27, 28 | mp2an 692 | . . . 4
⊢ (𝑥 ∈ ℝ+
↦ 2) ∈ 𝑂(1) | 
| 30 | 29 | a1i 11 | . . 3
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ 2) ∈
𝑂(1)) | 
| 31 | 26 | a1i 11 | . . . 4
⊢ (𝜑 → ℝ+
⊆ ℝ) | 
| 32 |  | 1red 11262 | . . . 4
⊢ (𝜑 → 1 ∈
ℝ) | 
| 33 |  | dchrisum0lem2.e | . . . . 5
⊢ (𝜑 → 𝐸 ∈ (0[,)+∞)) | 
| 34 |  | elrege0 13494 | . . . . . 6
⊢ (𝐸 ∈ (0[,)+∞) ↔
(𝐸 ∈ ℝ ∧ 0
≤ 𝐸)) | 
| 35 | 34 | simplbi 497 | . . . . 5
⊢ (𝐸 ∈ (0[,)+∞) →
𝐸 ∈
ℝ) | 
| 36 | 33, 35 | syl 17 | . . . 4
⊢ (𝜑 → 𝐸 ∈ ℝ) | 
| 37 | 3, 24 | absmuld 15493 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(abs‘(𝑥 ·
Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) = ((abs‘𝑥) · (abs‘Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)))) | 
| 38 |  | rprege0 13050 | . . . . . . . . . 10
⊢ (𝑥 ∈ ℝ+
→ (𝑥 ∈ ℝ
∧ 0 ≤ 𝑥)) | 
| 39 | 38 | adantl 481 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝑥 ∈ ℝ ∧ 0 ≤
𝑥)) | 
| 40 |  | absid 15335 | . . . . . . . . 9
⊢ ((𝑥 ∈ ℝ ∧ 0 ≤
𝑥) → (abs‘𝑥) = 𝑥) | 
| 41 | 39, 40 | syl 17 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(abs‘𝑥) = 𝑥) | 
| 42 | 41 | oveq1d 7446 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
((abs‘𝑥) ·
(abs‘Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) = (𝑥 · (abs‘Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)))) | 
| 43 | 37, 42 | eqtrd 2777 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(abs‘(𝑥 ·
Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) = (𝑥 · (abs‘Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)))) | 
| 44 | 43 | adantrr 717 | . . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(abs‘(𝑥 ·
Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) = (𝑥 · (abs‘Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)))) | 
| 45 | 24 | adantrr 717 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ) | 
| 46 | 45 | subid1d 11609 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚) − 0) = Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) | 
| 47 | 18 | adantl 481 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ 𝑚 ∈
ℕ) | 
| 48 |  | 2fveq3 6911 | . . . . . . . . . . . . . . 15
⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚))) | 
| 49 |  | id 22 | . . . . . . . . . . . . . . 15
⊢ (𝑎 = 𝑚 → 𝑎 = 𝑚) | 
| 50 | 48, 49 | oveq12d 7449 | . . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) | 
| 51 |  | dchrisum0lem2.k | . . . . . . . . . . . . . 14
⊢ 𝐾 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) | 
| 52 |  | ovex 7464 | . . . . . . . . . . . . . 14
⊢ ((𝑋‘(𝐿‘𝑎)) / 𝑎) ∈ V | 
| 53 | 50, 51, 52 | fvmpt3i 7021 | . . . . . . . . . . . . 13
⊢ (𝑚 ∈ ℕ → (𝐾‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) | 
| 54 | 47, 53 | syl 17 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (𝐾‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) | 
| 55 | 54 | adantlrr 721 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (𝐾‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) | 
| 56 |  | rpregt0 13049 | . . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ℝ+
→ (𝑥 ∈ ℝ
∧ 0 < 𝑥)) | 
| 57 | 56 | ad2antrl 728 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (𝑥 ∈ ℝ ∧ 0 <
𝑥)) | 
| 58 | 57 | simpld 494 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝑥 ∈
ℝ) | 
| 59 |  | simprr 773 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 1 ≤ 𝑥) | 
| 60 |  | flge1nn 13861 | . . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℝ ∧ 1 ≤
𝑥) →
(⌊‘𝑥) ∈
ℕ) | 
| 61 | 58, 59, 60 | syl2anc 584 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(⌊‘𝑥) ∈
ℕ) | 
| 62 |  | nnuz 12921 | . . . . . . . . . . . 12
⊢ ℕ =
(ℤ≥‘1) | 
| 63 | 61, 62 | eleqtrdi 2851 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(⌊‘𝑥) ∈
(ℤ≥‘1)) | 
| 64 | 23 | adantlrr 721 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ) | 
| 65 | 55, 63, 64 | fsumser 15766 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚) = (seq1( + , 𝐾)‘(⌊‘𝑥))) | 
| 66 |  | rpvmasum.a | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑁 ∈ ℕ) | 
| 67 |  | rpvmasum2.1 | . . . . . . . . . . . . . 14
⊢  1 =
(0g‘𝐺) | 
| 68 |  | eldifsni 4790 | . . . . . . . . . . . . . . 15
⊢ (𝑋 ∈ (𝐷 ∖ { 1 }) → 𝑋 ≠ 1 ) | 
| 69 | 12, 68 | syl 17 | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑋 ≠ 1 ) | 
| 70 |  | dchrisum0lem2.t | . . . . . . . . . . . . . 14
⊢ (𝜑 → seq1( + , 𝐾) ⇝ 𝑇) | 
| 71 |  | dchrisum0lem2.3 | . . . . . . . . . . . . . 14
⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + ,
𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 / 𝑦)) | 
| 72 | 6, 8, 66, 5, 7, 67, 13, 69, 51, 33, 70, 71, 9 | dchrvmaeq0 27548 | . . . . . . . . . . . . 13
⊢ (𝜑 → (𝑋 ∈ 𝑊 ↔ 𝑇 = 0)) | 
| 73 | 11, 72 | mpbid 232 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝑇 = 0) | 
| 74 | 73 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝑇 = 0) | 
| 75 | 74 | eqcomd 2743 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 0 = 𝑇) | 
| 76 | 65, 75 | oveq12d 7449 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚) − 0) = ((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇)) | 
| 77 | 46, 76 | eqtr3d 2779 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚) = ((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇)) | 
| 78 | 77 | fveq2d 6910 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(abs‘Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (abs‘((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇))) | 
| 79 |  | 2fveq3 6911 | . . . . . . . . . 10
⊢ (𝑦 = 𝑥 → (seq1( + , 𝐾)‘(⌊‘𝑦)) = (seq1( + , 𝐾)‘(⌊‘𝑥))) | 
| 80 | 79 | fvoveq1d 7453 | . . . . . . . . 9
⊢ (𝑦 = 𝑥 → (abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) = (abs‘((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇))) | 
| 81 |  | oveq2 7439 | . . . . . . . . 9
⊢ (𝑦 = 𝑥 → (𝐸 / 𝑦) = (𝐸 / 𝑥)) | 
| 82 | 80, 81 | breq12d 5156 | . . . . . . . 8
⊢ (𝑦 = 𝑥 → ((abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 / 𝑦) ↔ (abs‘((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇)) ≤ (𝐸 / 𝑥))) | 
| 83 | 71 | adantr 480 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → ∀𝑦 ∈
(1[,)+∞)(abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 / 𝑦)) | 
| 84 |  | 1re 11261 | . . . . . . . . . 10
⊢ 1 ∈
ℝ | 
| 85 |  | elicopnf 13485 | . . . . . . . . . 10
⊢ (1 ∈
ℝ → (𝑥 ∈
(1[,)+∞) ↔ (𝑥
∈ ℝ ∧ 1 ≤ 𝑥))) | 
| 86 | 84, 85 | ax-mp 5 | . . . . . . . . 9
⊢ (𝑥 ∈ (1[,)+∞) ↔
(𝑥 ∈ ℝ ∧ 1
≤ 𝑥)) | 
| 87 | 58, 59, 86 | sylanbrc 583 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝑥 ∈
(1[,)+∞)) | 
| 88 | 82, 83, 87 | rspcdva 3623 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(abs‘((seq1( + , 𝐾)‘(⌊‘𝑥)) − 𝑇)) ≤ (𝐸 / 𝑥)) | 
| 89 | 78, 88 | eqbrtrd 5165 | . . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(abs‘Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) ≤ (𝐸 / 𝑥)) | 
| 90 | 45 | abscld 15475 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(abs‘Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ ℝ) | 
| 91 | 36 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → 𝐸 ∈
ℝ) | 
| 92 |  | lemuldiv2 12149 | . . . . . . 7
⊢
(((abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ ℝ ∧ 𝐸 ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → ((𝑥 · (abs‘Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) ≤ 𝐸 ↔ (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) ≤ (𝐸 / 𝑥))) | 
| 93 | 90, 91, 57, 92 | syl3anc 1373 | . . . . . 6
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → ((𝑥 ·
(abs‘Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) ≤ 𝐸 ↔ (abs‘Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) ≤ (𝐸 / 𝑥))) | 
| 94 | 89, 93 | mpbird 257 | . . . . 5
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) → (𝑥 ·
(abs‘Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) ≤ 𝐸) | 
| 95 | 44, 94 | eqbrtrd 5165 | . . . 4
⊢ ((𝜑 ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤
𝑥)) →
(abs‘(𝑥 ·
Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) ≤ 𝐸) | 
| 96 | 31, 25, 32, 36, 95 | elo1d 15572 | . . 3
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (𝑥 · Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) ∈ 𝑂(1)) | 
| 97 | 1, 25, 30, 96 | o1mul2 15661 | . 2
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (2
· (𝑥 ·
Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)))) ∈ 𝑂(1)) | 
| 98 |  | fzfid 14014 | . . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (1...(⌊‘((𝑥↑2) / 𝑚))) ∈ Fin) | 
| 99 | 20 | rpsqrtcld 15450 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (√‘𝑚)
∈ ℝ+) | 
| 100 | 99 | rpcnd 13079 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (√‘𝑚)
∈ ℂ) | 
| 101 | 99 | rpne0d 13082 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (√‘𝑚)
≠ 0) | 
| 102 | 17, 100, 101 | divcld 12043 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ) | 
| 103 | 102 | adantr 480 | . . . . . 6
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
∧ 𝑑 ∈
(1...(⌊‘((𝑥↑2) / 𝑚)))) → ((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) ∈ ℂ) | 
| 104 |  | elfznn 13593 | . . . . . . . . . 10
⊢ (𝑑 ∈
(1...(⌊‘((𝑥↑2) / 𝑚))) → 𝑑 ∈ ℕ) | 
| 105 | 104 | adantl 481 | . . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
∧ 𝑑 ∈
(1...(⌊‘((𝑥↑2) / 𝑚)))) → 𝑑 ∈ ℕ) | 
| 106 | 105 | nnrpd 13075 | . . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
∧ 𝑑 ∈
(1...(⌊‘((𝑥↑2) / 𝑚)))) → 𝑑 ∈ ℝ+) | 
| 107 | 106 | rpsqrtcld 15450 | . . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
∧ 𝑑 ∈
(1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘𝑑) ∈
ℝ+) | 
| 108 | 107 | rpcnd 13079 | . . . . . 6
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
∧ 𝑑 ∈
(1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘𝑑) ∈ ℂ) | 
| 109 | 107 | rpne0d 13082 | . . . . . 6
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
∧ 𝑑 ∈
(1...(⌊‘((𝑥↑2) / 𝑚)))) → (√‘𝑑) ≠ 0) | 
| 110 | 103, 108,
109 | divcld 12043 | . . . . 5
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
∧ 𝑑 ∈
(1...(⌊‘((𝑥↑2) / 𝑚)))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ) | 
| 111 | 98, 110 | fsumcl 15769 | . . . 4
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ Σ𝑑 ∈
(1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ) | 
| 112 | 4, 111 | fsumcl 15769 | . . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
Σ𝑚 ∈
(1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) ∈ ℂ) | 
| 113 |  | mulcl 11239 | . . . 4
⊢ ((2
∈ ℂ ∧ (𝑥
· Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) ∈ ℂ) → (2 · (𝑥 · Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) ∈ ℂ) | 
| 114 | 27, 25, 113 | sylancr 587 | . . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (2
· (𝑥 ·
Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) ∈ ℂ) | 
| 115 |  | 2re 12340 | . . . . . . . . . 10
⊢ 2 ∈
ℝ | 
| 116 |  | simpr 484 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈
ℝ+) | 
| 117 |  | 2z 12649 | . . . . . . . . . . . . . 14
⊢ 2 ∈
ℤ | 
| 118 |  | rpexpcl 14121 | . . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ+
∧ 2 ∈ ℤ) → (𝑥↑2) ∈
ℝ+) | 
| 119 | 116, 117,
118 | sylancl 586 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝑥↑2) ∈
ℝ+) | 
| 120 |  | rpdivcl 13060 | . . . . . . . . . . . . 13
⊢ (((𝑥↑2) ∈
ℝ+ ∧ 𝑚
∈ ℝ+) → ((𝑥↑2) / 𝑚) ∈
ℝ+) | 
| 121 | 119, 19, 120 | syl2an 596 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ ((𝑥↑2) / 𝑚) ∈
ℝ+) | 
| 122 | 121 | rpsqrtcld 15450 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (√‘((𝑥↑2) / 𝑚)) ∈
ℝ+) | 
| 123 | 122 | rpred 13077 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (√‘((𝑥↑2) / 𝑚)) ∈ ℝ) | 
| 124 |  | remulcl 11240 | . . . . . . . . . 10
⊢ ((2
∈ ℝ ∧ (√‘((𝑥↑2) / 𝑚)) ∈ ℝ) → (2 ·
(√‘((𝑥↑2)
/ 𝑚))) ∈
ℝ) | 
| 125 | 115, 123,
124 | sylancr 587 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (2 · (√‘((𝑥↑2) / 𝑚))) ∈ ℝ) | 
| 126 | 125 | recnd 11289 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (2 · (√‘((𝑥↑2) / 𝑚))) ∈ ℂ) | 
| 127 | 102, 126 | mulcld 11281 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 ·
(√‘((𝑥↑2)
/ 𝑚)))) ∈
ℂ) | 
| 128 | 4, 111, 127 | fsumsub 15824 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
Σ𝑚 ∈
(1...(⌊‘𝑥))(Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) − (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 ·
(√‘((𝑥↑2)
/ 𝑚))))) = (Σ𝑚 ∈
(1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) − Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 ·
(√‘((𝑥↑2)
/ 𝑚)))))) | 
| 129 | 107 | rpcnne0d 13086 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
∧ 𝑑 ∈
(1...(⌊‘((𝑥↑2) / 𝑚)))) → ((√‘𝑑) ∈ ℂ ∧
(√‘𝑑) ≠
0)) | 
| 130 |  | reccl 11929 | . . . . . . . . . . 11
⊢
(((√‘𝑑)
∈ ℂ ∧ (√‘𝑑) ≠ 0) → (1 / (√‘𝑑)) ∈
ℂ) | 
| 131 | 129, 130 | syl 17 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
∧ 𝑑 ∈
(1...(⌊‘((𝑥↑2) / 𝑚)))) → (1 / (√‘𝑑)) ∈
ℂ) | 
| 132 | 98, 131 | fsumcl 15769 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ Σ𝑑 ∈
(1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑)) ∈ ℂ) | 
| 133 | 102, 132,
126 | subdid 11719 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑)) − (2 · (√‘((𝑥↑2) / 𝑚))))) = ((((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑))) − (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 ·
(√‘((𝑥↑2)
/ 𝑚)))))) | 
| 134 |  | fveq2 6906 | . . . . . . . . . . . . . 14
⊢ (𝑦 = ((𝑥↑2) / 𝑚) → (⌊‘𝑦) = (⌊‘((𝑥↑2) / 𝑚))) | 
| 135 | 134 | oveq2d 7447 | . . . . . . . . . . . . 13
⊢ (𝑦 = ((𝑥↑2) / 𝑚) → (1...(⌊‘𝑦)) = (1...(⌊‘((𝑥↑2) / 𝑚)))) | 
| 136 | 135 | sumeq1d 15736 | . . . . . . . . . . . 12
⊢ (𝑦 = ((𝑥↑2) / 𝑚) → Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) = Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑))) | 
| 137 |  | fveq2 6906 | . . . . . . . . . . . . 13
⊢ (𝑦 = ((𝑥↑2) / 𝑚) → (√‘𝑦) = (√‘((𝑥↑2) / 𝑚))) | 
| 138 | 137 | oveq2d 7447 | . . . . . . . . . . . 12
⊢ (𝑦 = ((𝑥↑2) / 𝑚) → (2 · (√‘𝑦)) = (2 ·
(√‘((𝑥↑2)
/ 𝑚)))) | 
| 139 | 136, 138 | oveq12d 7449 | . . . . . . . . . . 11
⊢ (𝑦 = ((𝑥↑2) / 𝑚) → (Σ𝑑 ∈ (1...(⌊‘𝑦))(1 / (√‘𝑑)) − (2 ·
(√‘𝑦))) =
(Σ𝑑 ∈
(1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑)) − (2 · (√‘((𝑥↑2) / 𝑚))))) | 
| 140 |  | dchrisum0lem2.h | . . . . . . . . . . 11
⊢ 𝐻 = (𝑦 ∈ ℝ+ ↦
(Σ𝑑 ∈
(1...(⌊‘𝑦))(1 /
(√‘𝑑)) −
(2 · (√‘𝑦)))) | 
| 141 |  | ovex 7464 | . . . . . . . . . . 11
⊢
(Σ𝑑 ∈
(1...(⌊‘𝑦))(1 /
(√‘𝑑)) −
(2 · (√‘𝑦))) ∈ V | 
| 142 | 139, 140,
141 | fvmpt3i 7021 | . . . . . . . . . 10
⊢ (((𝑥↑2) / 𝑚) ∈ ℝ+ → (𝐻‘((𝑥↑2) / 𝑚)) = (Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑)) − (2 · (√‘((𝑥↑2) / 𝑚))))) | 
| 143 | 121, 142 | syl 17 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (𝐻‘((𝑥↑2) / 𝑚)) = (Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑)) − (2 · (√‘((𝑥↑2) / 𝑚))))) | 
| 144 | 143 | oveq2d 7447 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) = (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑)) − (2 · (√‘((𝑥↑2) / 𝑚)))))) | 
| 145 | 103, 108,
109 | divrecd 12046 | . . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
∧ 𝑑 ∈
(1...(⌊‘((𝑥↑2) / 𝑚)))) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) = (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (1 / (√‘𝑑)))) | 
| 146 | 145 | sumeq2dv 15738 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ Σ𝑑 ∈
(1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) = Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (1 / (√‘𝑑)))) | 
| 147 | 98, 102, 131 | fsummulc2 15820 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑))) = Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (1 / (√‘𝑑)))) | 
| 148 | 146, 147 | eqtr4d 2780 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ Σ𝑑 ∈
(1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) = (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑)))) | 
| 149 | 148 | oveq1d 7446 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (Σ𝑑 ∈
(1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) − (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 ·
(√‘((𝑥↑2)
/ 𝑚))))) = ((((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(1 / (√‘𝑑))) − (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 ·
(√‘((𝑥↑2)
/ 𝑚)))))) | 
| 150 | 133, 144,
149 | 3eqtr4d 2787 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) = (Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) − (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 ·
(√‘((𝑥↑2)
/ 𝑚)))))) | 
| 151 | 150 | sumeq2dv 15738 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
Σ𝑚 ∈
(1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) = Σ𝑚 ∈ (1...(⌊‘𝑥))(Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) − (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 ·
(√‘((𝑥↑2)
/ 𝑚)))))) | 
| 152 |  | mulcl 11239 | . . . . . . . . . 10
⊢ ((2
∈ ℂ ∧ 𝑥
∈ ℂ) → (2 · 𝑥) ∈ ℂ) | 
| 153 | 27, 3, 152 | sylancr 587 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (2
· 𝑥) ∈
ℂ) | 
| 154 | 4, 153, 23 | fsummulc2 15820 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ((2
· 𝑥) ·
Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘𝑥))((2 · 𝑥) · ((𝑋‘(𝐿‘𝑚)) / 𝑚))) | 
| 155 | 1, 3, 24 | mulassd 11284 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ((2
· 𝑥) ·
Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (2 · (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)))) | 
| 156 | 153 | adantr 480 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (2 · 𝑥)
∈ ℂ) | 
| 157 | 156, 102,
100, 101 | div12d 12079 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ ((2 · 𝑥)
· (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑚))) = (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((2 · 𝑥) / (√‘𝑚)))) | 
| 158 | 99 | rpcnne0d 13086 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ ((√‘𝑚)
∈ ℂ ∧ (√‘𝑚) ≠ 0)) | 
| 159 |  | divdiv1 11978 | . . . . . . . . . . . . 13
⊢ (((𝑋‘(𝐿‘𝑚)) ∈ ℂ ∧ ((√‘𝑚) ∈ ℂ ∧
(√‘𝑚) ≠ 0)
∧ ((√‘𝑚)
∈ ℂ ∧ (√‘𝑚) ≠ 0)) → (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑚)) = ((𝑋‘(𝐿‘𝑚)) / ((√‘𝑚) · (√‘𝑚)))) | 
| 160 | 17, 158, 158, 159 | syl3anc 1373 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑚)) = ((𝑋‘(𝐿‘𝑚)) / ((√‘𝑚) · (√‘𝑚)))) | 
| 161 | 20 | rprege0d 13084 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (𝑚 ∈ ℝ
∧ 0 ≤ 𝑚)) | 
| 162 |  | remsqsqrt 15295 | . . . . . . . . . . . . . 14
⊢ ((𝑚 ∈ ℝ ∧ 0 ≤
𝑚) →
((√‘𝑚) ·
(√‘𝑚)) = 𝑚) | 
| 163 | 161, 162 | syl 17 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ ((√‘𝑚)
· (√‘𝑚))
= 𝑚) | 
| 164 | 163 | oveq2d 7447 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ ((𝑋‘(𝐿‘𝑚)) / ((√‘𝑚) · (√‘𝑚))) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) | 
| 165 | 160, 164 | eqtr2d 2778 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑚))) | 
| 166 | 165 | oveq2d 7447 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ ((2 · 𝑥)
· ((𝑋‘(𝐿‘𝑚)) / 𝑚)) = ((2 · 𝑥) · (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑚)))) | 
| 167 | 119 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (𝑥↑2) ∈
ℝ+) | 
| 168 | 167 | rprege0d 13084 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ ((𝑥↑2) ∈
ℝ ∧ 0 ≤ (𝑥↑2))) | 
| 169 |  | sqrtdiv 15304 | . . . . . . . . . . . . . . 15
⊢ ((((𝑥↑2) ∈ ℝ ∧ 0
≤ (𝑥↑2)) ∧
𝑚 ∈
ℝ+) → (√‘((𝑥↑2) / 𝑚)) = ((√‘(𝑥↑2)) / (√‘𝑚))) | 
| 170 | 168, 20, 169 | syl2anc 584 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (√‘((𝑥↑2) / 𝑚)) = ((√‘(𝑥↑2)) / (√‘𝑚))) | 
| 171 | 38 | ad2antlr 727 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (𝑥 ∈ ℝ
∧ 0 ≤ 𝑥)) | 
| 172 |  | sqrtsq 15308 | . . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ℝ ∧ 0 ≤
𝑥) →
(√‘(𝑥↑2))
= 𝑥) | 
| 173 | 171, 172 | syl 17 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (√‘(𝑥↑2)) = 𝑥) | 
| 174 | 173 | oveq1d 7446 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ ((√‘(𝑥↑2)) / (√‘𝑚)) = (𝑥 / (√‘𝑚))) | 
| 175 | 170, 174 | eqtrd 2777 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (√‘((𝑥↑2) / 𝑚)) = (𝑥 / (√‘𝑚))) | 
| 176 | 175 | oveq2d 7447 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (2 · (√‘((𝑥↑2) / 𝑚))) = (2 · (𝑥 / (√‘𝑚)))) | 
| 177 |  | 2cnd 12344 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ 2 ∈ ℂ) | 
| 178 | 3 | adantr 480 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ 𝑥 ∈
ℂ) | 
| 179 |  | divass 11940 | . . . . . . . . . . . . 13
⊢ ((2
∈ ℂ ∧ 𝑥
∈ ℂ ∧ ((√‘𝑚) ∈ ℂ ∧ (√‘𝑚) ≠ 0)) → ((2 ·
𝑥) / (√‘𝑚)) = (2 · (𝑥 / (√‘𝑚)))) | 
| 180 | 177, 178,
158, 179 | syl3anc 1373 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ ((2 · 𝑥) /
(√‘𝑚)) = (2
· (𝑥 /
(√‘𝑚)))) | 
| 181 | 176, 180 | eqtr4d 2780 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (2 · (√‘((𝑥↑2) / 𝑚))) = ((2 · 𝑥) / (√‘𝑚))) | 
| 182 | 181 | oveq2d 7447 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 ·
(√‘((𝑥↑2)
/ 𝑚)))) = (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · ((2 · 𝑥) / (√‘𝑚)))) | 
| 183 | 157, 166,
182 | 3eqtr4d 2787 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈
(1...(⌊‘𝑥)))
→ ((2 · 𝑥)
· ((𝑋‘(𝐿‘𝑚)) / 𝑚)) = (((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 ·
(√‘((𝑥↑2)
/ 𝑚))))) | 
| 184 | 183 | sumeq2dv 15738 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
Σ𝑚 ∈
(1...(⌊‘𝑥))((2
· 𝑥) ·
((𝑋‘(𝐿‘𝑚)) / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 ·
(√‘((𝑥↑2)
/ 𝑚))))) | 
| 185 | 154, 155,
184 | 3eqtr3d 2785 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (2
· (𝑥 ·
Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))) = Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 ·
(√‘((𝑥↑2)
/ 𝑚))))) | 
| 186 | 185 | oveq2d 7447 | . . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(Σ𝑚 ∈
(1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) − (2 · (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)))) = (Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) − Σ𝑚 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (2 ·
(√‘((𝑥↑2)
/ 𝑚)))))) | 
| 187 | 128, 151,
186 | 3eqtr4d 2787 | . . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
Σ𝑚 ∈
(1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚))) = (Σ𝑚 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) − (2 · (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))))) | 
| 188 | 187 | mpteq2dva 5242 | . . . 4
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦
Σ𝑚 ∈
(1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚)))) = (𝑥 ∈ ℝ+ ↦
(Σ𝑚 ∈
(1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) − (2 · (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)))))) | 
| 189 |  | dchrisum0lem1.f | . . . . 5
⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / (√‘𝑎))) | 
| 190 |  | dchrisum0.c | . . . . 5
⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) | 
| 191 |  | dchrisum0.s | . . . . 5
⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆) | 
| 192 |  | dchrisum0.1 | . . . . 5
⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + ,
𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / (√‘𝑦))) | 
| 193 |  | dchrisum0lem2.u | . . . . 5
⊢ (𝜑 → 𝐻 ⇝𝑟 𝑈) | 
| 194 | 6, 8, 66, 5, 7, 67, 9, 11, 189, 190, 191, 192, 140, 193 | dchrisum0lem2a 27561 | . . . 4
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦
Σ𝑚 ∈
(1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) · (𝐻‘((𝑥↑2) / 𝑚)))) ∈ 𝑂(1)) | 
| 195 | 188, 194 | eqeltrrd 2842 | . . 3
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦
(Σ𝑚 ∈
(1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑)) − (2 · (𝑥 · Σ𝑚 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚))))) ∈ 𝑂(1)) | 
| 196 | 112, 114,
195 | o1dif 15666 | . 2
⊢ (𝜑 → ((𝑥 ∈ ℝ+ ↦
Σ𝑚 ∈
(1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1) ↔ (𝑥 ∈ ℝ+
↦ (2 · (𝑥
· Σ𝑚 ∈
(1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑚)) / 𝑚)))) ∈ 𝑂(1))) | 
| 197 | 97, 196 | mpbird 257 | 1
⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦
Σ𝑚 ∈
(1...(⌊‘𝑥))Σ𝑑 ∈ (1...(⌊‘((𝑥↑2) / 𝑚)))(((𝑋‘(𝐿‘𝑚)) / (√‘𝑚)) / (√‘𝑑))) ∈ 𝑂(1)) |