| Step | Hyp | Ref
| Expression |
| 1 | | fzfid 14014 |
. . 3
⊢ (𝜑 → (1...(⌊‘𝐴)) ∈ Fin) |
| 2 | | fzfid 14014 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (1...𝑛) ∈ Fin) |
| 3 | | elfznn 13593 |
. . . . . 6
⊢ (𝑛 ∈
(1...(⌊‘𝐴))
→ 𝑛 ∈
ℕ) |
| 4 | 3 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ) |
| 5 | | dvdsssfz1 16355 |
. . . . 5
⊢ (𝑛 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ⊆ (1...𝑛)) |
| 6 | 4, 5 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ⊆ (1...𝑛)) |
| 7 | 2, 6 | ssfid 9301 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ∈ Fin) |
| 8 | | nnre 12273 |
. . . . . . . . . . . 12
⊢ (𝑑 ∈ ℕ → 𝑑 ∈
ℝ) |
| 9 | 8 | ad2antrl 728 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) → 𝑑 ∈ ℝ) |
| 10 | 4 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) → 𝑛 ∈ ℕ) |
| 11 | 10 | nnred 12281 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) → 𝑛 ∈ ℝ) |
| 12 | | dvdsflsumcom.2 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 13 | 12 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) → 𝐴 ∈ ℝ) |
| 14 | | nnz 12634 |
. . . . . . . . . . . . 13
⊢ (𝑑 ∈ ℕ → 𝑑 ∈
ℤ) |
| 15 | | dvdsle 16347 |
. . . . . . . . . . . . 13
⊢ ((𝑑 ∈ ℤ ∧ 𝑛 ∈ ℕ) → (𝑑 ∥ 𝑛 → 𝑑 ≤ 𝑛)) |
| 16 | 14, 4, 15 | syl2anr 597 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑑 ∈ ℕ) → (𝑑 ∥ 𝑛 → 𝑑 ≤ 𝑛)) |
| 17 | 16 | impr 454 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) → 𝑑 ≤ 𝑛) |
| 18 | | fznnfl 13902 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ ℝ → (𝑛 ∈
(1...(⌊‘𝐴))
↔ (𝑛 ∈ ℕ
∧ 𝑛 ≤ 𝐴))) |
| 19 | 12, 18 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑛 ∈ (1...(⌊‘𝐴)) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ 𝐴))) |
| 20 | 19 | simplbda 499 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ≤ 𝐴) |
| 21 | 20 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) → 𝑛 ≤ 𝐴) |
| 22 | 9, 11, 13, 17, 21 | letrd 11418 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) → 𝑑 ≤ 𝐴) |
| 23 | 22 | ex 412 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛) → 𝑑 ≤ 𝐴)) |
| 24 | 23 | pm4.71rd 562 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛) ↔ (𝑑 ≤ 𝐴 ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)))) |
| 25 | | ancom 460 |
. . . . . . . . 9
⊢ ((𝑑 ≤ 𝐴 ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) ↔ ((𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛) ∧ 𝑑 ≤ 𝐴)) |
| 26 | | an32 646 |
. . . . . . . . 9
⊢ (((𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛) ∧ 𝑑 ≤ 𝐴) ↔ ((𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝐴) ∧ 𝑑 ∥ 𝑛)) |
| 27 | 25, 26 | bitri 275 |
. . . . . . . 8
⊢ ((𝑑 ≤ 𝐴 ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) ↔ ((𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝐴) ∧ 𝑑 ∥ 𝑛)) |
| 28 | 24, 27 | bitrdi 287 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛) ↔ ((𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝐴) ∧ 𝑑 ∥ 𝑛))) |
| 29 | | fznnfl 13902 |
. . . . . . . . . 10
⊢ (𝐴 ∈ ℝ → (𝑑 ∈
(1...(⌊‘𝐴))
↔ (𝑑 ∈ ℕ
∧ 𝑑 ≤ 𝐴))) |
| 30 | 12, 29 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝐴))) |
| 31 | 30 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑑 ∈ (1...(⌊‘𝐴)) ↔ (𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝐴))) |
| 32 | 31 | anbi1d 631 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛) ↔ ((𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝐴) ∧ 𝑑 ∥ 𝑛))) |
| 33 | 28, 32 | bitr4d 282 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛))) |
| 34 | 33 | pm5.32da 579 |
. . . . 5
⊢ (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛)))) |
| 35 | | an12 645 |
. . . . 5
⊢ ((𝑛 ∈
(1...(⌊‘𝐴))
∧ (𝑑 ∈
(1...(⌊‘𝐴))
∧ 𝑑 ∥ 𝑛)) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛))) |
| 36 | 34, 35 | bitrdi 287 |
. . . 4
⊢ (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛)))) |
| 37 | | breq1 5146 |
. . . . . 6
⊢ (𝑥 = 𝑑 → (𝑥 ∥ 𝑛 ↔ 𝑑 ∥ 𝑛)) |
| 38 | 37 | elrab 3692 |
. . . . 5
⊢ (𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} ↔ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛)) |
| 39 | 38 | anbi2i 623 |
. . . 4
⊢ ((𝑛 ∈
(1...(⌊‘𝐴))
∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}) ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ (𝑑 ∈ ℕ ∧ 𝑑 ∥ 𝑛))) |
| 40 | | breq2 5147 |
. . . . . 6
⊢ (𝑥 = 𝑛 → (𝑑 ∥ 𝑥 ↔ 𝑑 ∥ 𝑛)) |
| 41 | 40 | elrab 3692 |
. . . . 5
⊢ (𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥} ↔ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛)) |
| 42 | 41 | anbi2i 623 |
. . . 4
⊢ ((𝑑 ∈
(1...(⌊‘𝐴))
∧ 𝑛 ∈ {𝑥 ∈
(1...(⌊‘𝐴))
∣ 𝑑 ∥ 𝑥}) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∥ 𝑛))) |
| 43 | 36, 39, 42 | 3bitr4g 314 |
. . 3
⊢ (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}) ↔ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥}))) |
| 44 | | dvdsflsumcom.3 |
. . 3
⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) → 𝐵 ∈ ℂ) |
| 45 | 1, 1, 7, 43, 44 | fsumcom2 15810 |
. 2
⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}𝐵 = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥}𝐵) |
| 46 | | dvdsflsumcom.1 |
. . . 4
⊢ (𝑛 = (𝑑 · 𝑚) → 𝐵 = 𝐶) |
| 47 | | fzfid 14014 |
. . . 4
⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (1...(⌊‘(𝐴 / 𝑑))) ∈ Fin) |
| 48 | 12 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ) |
| 49 | 30 | simprbda 498 |
. . . . 5
⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → 𝑑 ∈ ℕ) |
| 50 | | eqid 2737 |
. . . . 5
⊢ (𝑦 ∈
(1...(⌊‘(𝐴 /
𝑑))) ↦ (𝑑 · 𝑦)) = (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦)) |
| 51 | 48, 49, 50 | dvdsflf1o 27230 |
. . . 4
⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → (𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦)):(1...(⌊‘(𝐴 / 𝑑)))–1-1-onto→{𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥}) |
| 52 | | oveq2 7439 |
. . . . . 6
⊢ (𝑦 = 𝑚 → (𝑑 · 𝑦) = (𝑑 · 𝑚)) |
| 53 | | ovex 7464 |
. . . . . 6
⊢ (𝑑 · 𝑚) ∈ V |
| 54 | 52, 50, 53 | fvmpt 7016 |
. . . . 5
⊢ (𝑚 ∈
(1...(⌊‘(𝐴 /
𝑑))) → ((𝑦 ∈
(1...(⌊‘(𝐴 /
𝑑))) ↦ (𝑑 · 𝑦))‘𝑚) = (𝑑 · 𝑚)) |
| 55 | 54 | adantl 481 |
. . . 4
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))) → ((𝑦 ∈ (1...(⌊‘(𝐴 / 𝑑))) ↦ (𝑑 · 𝑦))‘𝑚) = (𝑑 · 𝑚)) |
| 56 | 43 | biimpar 477 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥})) → (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛})) |
| 57 | 56, 44 | syldan 591 |
. . . . 5
⊢ ((𝜑 ∧ (𝑑 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥})) → 𝐵 ∈ ℂ) |
| 58 | 57 | anassrs 467 |
. . . 4
⊢ (((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) ∧ 𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥}) → 𝐵 ∈ ℂ) |
| 59 | 46, 47, 51, 55, 58 | fsumf1o 15759 |
. . 3
⊢ ((𝜑 ∧ 𝑑 ∈ (1...(⌊‘𝐴))) → Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥}𝐵 = Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶) |
| 60 | 59 | sumeq2dv 15738 |
. 2
⊢ (𝜑 → Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑛 ∈ {𝑥 ∈ (1...(⌊‘𝐴)) ∣ 𝑑 ∥ 𝑥}𝐵 = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶) |
| 61 | 45, 60 | eqtrd 2777 |
1
⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))Σ𝑑 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛}𝐵 = Σ𝑑 ∈ (1...(⌊‘𝐴))Σ𝑚 ∈ (1...(⌊‘(𝐴 / 𝑑)))𝐶) |