| Step | Hyp | Ref
| Expression |
| 1 | | breq2 5147 |
. . . . 5
⊢ (𝑗 = 𝑢 → (𝑥 ∥ 𝑗 ↔ 𝑥 ∥ 𝑢)) |
| 2 | 1 | rabbidv 3444 |
. . . 4
⊢ (𝑗 = 𝑢 → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑗} = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑢}) |
| 3 | | csbeq1a 3913 |
. . . . 5
⊢ (𝑗 = 𝑢 → 𝐴 = ⦋𝑢 / 𝑗⦌𝐴) |
| 4 | 3 | adantr 480 |
. . . 4
⊢ ((𝑗 = 𝑢 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑗}) → 𝐴 = ⦋𝑢 / 𝑗⦌𝐴) |
| 5 | 2, 4 | sumeq12dv 15742 |
. . 3
⊢ (𝑗 = 𝑢 → Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑗}𝐴 = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑢}⦋𝑢 / 𝑗⦌𝐴) |
| 6 | | nfcv 2905 |
. . 3
⊢
Ⅎ𝑢Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑗}𝐴 |
| 7 | | nfcv 2905 |
. . . 4
⊢
Ⅎ𝑗{𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑢} |
| 8 | | nfcsb1v 3923 |
. . . 4
⊢
Ⅎ𝑗⦋𝑢 / 𝑗⦌𝐴 |
| 9 | 7, 8 | nfsum 15727 |
. . 3
⊢
Ⅎ𝑗Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑢}⦋𝑢 / 𝑗⦌𝐴 |
| 10 | 5, 6, 9 | cbvsum 15731 |
. 2
⊢
Σ𝑗 ∈
{𝑥 ∈ ℕ ∣
𝑥 ∥ 𝑁}Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑗}𝐴 = Σ𝑢 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑢}⦋𝑢 / 𝑗⦌𝐴 |
| 11 | | breq2 5147 |
. . . . . 6
⊢ (𝑢 = (𝑁 / 𝑣) → (𝑥 ∥ 𝑢 ↔ 𝑥 ∥ (𝑁 / 𝑣))) |
| 12 | 11 | rabbidv 3444 |
. . . . 5
⊢ (𝑢 = (𝑁 / 𝑣) → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑢} = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑣)}) |
| 13 | | csbeq1 3902 |
. . . . . 6
⊢ (𝑢 = (𝑁 / 𝑣) → ⦋𝑢 / 𝑗⦌𝐴 = ⦋(𝑁 / 𝑣) / 𝑗⦌𝐴) |
| 14 | 13 | adantr 480 |
. . . . 5
⊢ ((𝑢 = (𝑁 / 𝑣) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑢}) → ⦋𝑢 / 𝑗⦌𝐴 = ⦋(𝑁 / 𝑣) / 𝑗⦌𝐴) |
| 15 | 12, 14 | sumeq12dv 15742 |
. . . 4
⊢ (𝑢 = (𝑁 / 𝑣) → Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑢}⦋𝑢 / 𝑗⦌𝐴 = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑣)}⦋(𝑁 / 𝑣) / 𝑗⦌𝐴) |
| 16 | | fzfid 14014 |
. . . . 5
⊢ (𝜑 → (1...𝑁) ∈ Fin) |
| 17 | | fsumdvdscom.1 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 18 | | dvdsssfz1 16355 |
. . . . . 6
⊢ (𝑁 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ⊆ (1...𝑁)) |
| 19 | 17, 18 | syl 17 |
. . . . 5
⊢ (𝜑 → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ⊆ (1...𝑁)) |
| 20 | 16, 19 | ssfid 9301 |
. . . 4
⊢ (𝜑 → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ∈ Fin) |
| 21 | | eqid 2737 |
. . . . . 6
⊢ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} = {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} |
| 22 | | eqid 2737 |
. . . . . 6
⊢ (𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ↦ (𝑁 / 𝑧)) = (𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ↦ (𝑁 / 𝑧)) |
| 23 | 21, 22 | dvdsflip 16354 |
. . . . 5
⊢ (𝑁 ∈ ℕ → (𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ↦ (𝑁 / 𝑧)):{𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
| 24 | 17, 23 | syl 17 |
. . . 4
⊢ (𝜑 → (𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ↦ (𝑁 / 𝑧)):{𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
| 25 | | oveq2 7439 |
. . . . . 6
⊢ (𝑧 = 𝑣 → (𝑁 / 𝑧) = (𝑁 / 𝑣)) |
| 26 | | ovex 7464 |
. . . . . 6
⊢ (𝑁 / 𝑧) ∈ V |
| 27 | 25, 22, 26 | fvmpt3i 7021 |
. . . . 5
⊢ (𝑣 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} → ((𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ↦ (𝑁 / 𝑧))‘𝑣) = (𝑁 / 𝑣)) |
| 28 | 27 | adantl 481 |
. . . 4
⊢ ((𝜑 ∧ 𝑣 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → ((𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ↦ (𝑁 / 𝑧))‘𝑣) = (𝑁 / 𝑣)) |
| 29 | | fzfid 14014 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑢 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (1...𝑢) ∈ Fin) |
| 30 | | ssrab2 4080 |
. . . . . . . 8
⊢ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ⊆ ℕ |
| 31 | | simpr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑢 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → 𝑢 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
| 32 | 30, 31 | sselid 3981 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑢 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → 𝑢 ∈ ℕ) |
| 33 | | dvdsssfz1 16355 |
. . . . . . 7
⊢ (𝑢 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑢} ⊆ (1...𝑢)) |
| 34 | 32, 33 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑢 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑢} ⊆ (1...𝑢)) |
| 35 | 29, 34 | ssfid 9301 |
. . . . 5
⊢ ((𝜑 ∧ 𝑢 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑢} ∈ Fin) |
| 36 | | fsumdvdscom.3 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑗})) → 𝐴 ∈ ℂ) |
| 37 | 36 | ralrimivva 3202 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}∀𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑗}𝐴 ∈ ℂ) |
| 38 | | nfv 1914 |
. . . . . . . . 9
⊢
Ⅎ𝑢∀𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑗}𝐴 ∈ ℂ |
| 39 | 8 | nfel1 2922 |
. . . . . . . . . 10
⊢
Ⅎ𝑗⦋𝑢 / 𝑗⦌𝐴 ∈ ℂ |
| 40 | 7, 39 | nfralw 3311 |
. . . . . . . . 9
⊢
Ⅎ𝑗∀𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑢}⦋𝑢 / 𝑗⦌𝐴 ∈ ℂ |
| 41 | 3 | eleq1d 2826 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑢 → (𝐴 ∈ ℂ ↔ ⦋𝑢 / 𝑗⦌𝐴 ∈ ℂ)) |
| 42 | 2, 41 | raleqbidv 3346 |
. . . . . . . . 9
⊢ (𝑗 = 𝑢 → (∀𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑗}𝐴 ∈ ℂ ↔ ∀𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑢}⦋𝑢 / 𝑗⦌𝐴 ∈ ℂ)) |
| 43 | 38, 40, 42 | cbvralw 3306 |
. . . . . . . 8
⊢
(∀𝑗 ∈
{𝑥 ∈ ℕ ∣
𝑥 ∥ 𝑁}∀𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑗}𝐴 ∈ ℂ ↔ ∀𝑢 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}∀𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑢}⦋𝑢 / 𝑗⦌𝐴 ∈ ℂ) |
| 44 | 37, 43 | sylib 218 |
. . . . . . 7
⊢ (𝜑 → ∀𝑢 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}∀𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑢}⦋𝑢 / 𝑗⦌𝐴 ∈ ℂ) |
| 45 | 44 | r19.21bi 3251 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑢 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → ∀𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑢}⦋𝑢 / 𝑗⦌𝐴 ∈ ℂ) |
| 46 | 45 | r19.21bi 3251 |
. . . . 5
⊢ (((𝜑 ∧ 𝑢 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑢}) → ⦋𝑢 / 𝑗⦌𝐴 ∈ ℂ) |
| 47 | 35, 46 | fsumcl 15769 |
. . . 4
⊢ ((𝜑 ∧ 𝑢 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑢}⦋𝑢 / 𝑗⦌𝐴 ∈ ℂ) |
| 48 | 15, 20, 24, 28, 47 | fsumf1o 15759 |
. . 3
⊢ (𝜑 → Σ𝑢 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑢}⦋𝑢 / 𝑗⦌𝐴 = Σ𝑣 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑣)}⦋(𝑁 / 𝑣) / 𝑗⦌𝐴) |
| 49 | 13 | eleq1d 2826 |
. . . . . . . 8
⊢ (𝑢 = (𝑁 / 𝑣) → (⦋𝑢 / 𝑗⦌𝐴 ∈ ℂ ↔ ⦋(𝑁 / 𝑣) / 𝑗⦌𝐴 ∈ ℂ)) |
| 50 | 12, 49 | raleqbidv 3346 |
. . . . . . 7
⊢ (𝑢 = (𝑁 / 𝑣) → (∀𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑢}⦋𝑢 / 𝑗⦌𝐴 ∈ ℂ ↔ ∀𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑣)}⦋(𝑁 / 𝑣) / 𝑗⦌𝐴 ∈ ℂ)) |
| 51 | 44 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑣 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → ∀𝑢 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}∀𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑢}⦋𝑢 / 𝑗⦌𝐴 ∈ ℂ) |
| 52 | | dvdsdivcl 16353 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ ∧ 𝑣 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑁 / 𝑣) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
| 53 | 17, 52 | sylan 580 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑣 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑁 / 𝑣) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
| 54 | 50, 51, 53 | rspcdva 3623 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑣 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → ∀𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑣)}⦋(𝑁 / 𝑣) / 𝑗⦌𝐴 ∈ ℂ) |
| 55 | 54 | r19.21bi 3251 |
. . . . 5
⊢ (((𝜑 ∧ 𝑣 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑣)}) → ⦋(𝑁 / 𝑣) / 𝑗⦌𝐴 ∈ ℂ) |
| 56 | 55 | anasss 466 |
. . . 4
⊢ ((𝜑 ∧ (𝑣 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑣)})) → ⦋(𝑁 / 𝑣) / 𝑗⦌𝐴 ∈ ℂ) |
| 57 | 17, 56 | fsumdvdsdiag 27227 |
. . 3
⊢ (𝜑 → Σ𝑣 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑣)}⦋(𝑁 / 𝑣) / 𝑗⦌𝐴 = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}Σ𝑣 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}⦋(𝑁 / 𝑣) / 𝑗⦌𝐴) |
| 58 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑣 = ((𝑁 / 𝑘) / 𝑚) → (𝑁 / 𝑣) = (𝑁 / ((𝑁 / 𝑘) / 𝑚))) |
| 59 | 58 | csbeq1d 3903 |
. . . . . 6
⊢ (𝑣 = ((𝑁 / 𝑘) / 𝑚) → ⦋(𝑁 / 𝑣) / 𝑗⦌𝐴 = ⦋(𝑁 / ((𝑁 / 𝑘) / 𝑚)) / 𝑗⦌𝐴) |
| 60 | | fzfid 14014 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (1...(𝑁 / 𝑘)) ∈ Fin) |
| 61 | | dvdsdivcl 16353 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑁 / 𝑘) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) |
| 62 | 30, 61 | sselid 3981 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℕ ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑁 / 𝑘) ∈ ℕ) |
| 63 | 17, 62 | sylan 580 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑁 / 𝑘) ∈ ℕ) |
| 64 | | dvdsssfz1 16355 |
. . . . . . . 8
⊢ ((𝑁 / 𝑘) ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)} ⊆ (1...(𝑁 / 𝑘))) |
| 65 | 63, 64 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)} ⊆ (1...(𝑁 / 𝑘))) |
| 66 | 60, 65 | ssfid 9301 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)} ∈ Fin) |
| 67 | | eqid 2737 |
. . . . . . . 8
⊢ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)} = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)} |
| 68 | | eqid 2737 |
. . . . . . . 8
⊢ (𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)} ↦ ((𝑁 / 𝑘) / 𝑧)) = (𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)} ↦ ((𝑁 / 𝑘) / 𝑧)) |
| 69 | 67, 68 | dvdsflip 16354 |
. . . . . . 7
⊢ ((𝑁 / 𝑘) ∈ ℕ → (𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)} ↦ ((𝑁 / 𝑘) / 𝑧)):{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) |
| 70 | 63, 69 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → (𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)} ↦ ((𝑁 / 𝑘) / 𝑧)):{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}–1-1-onto→{𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) |
| 71 | | oveq2 7439 |
. . . . . . . 8
⊢ (𝑧 = 𝑚 → ((𝑁 / 𝑘) / 𝑧) = ((𝑁 / 𝑘) / 𝑚)) |
| 72 | | ovex 7464 |
. . . . . . . 8
⊢ ((𝑁 / 𝑘) / 𝑧) ∈ V |
| 73 | 71, 68, 72 | fvmpt3i 7021 |
. . . . . . 7
⊢ (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)} → ((𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)} ↦ ((𝑁 / 𝑘) / 𝑧))‘𝑚) = ((𝑁 / 𝑘) / 𝑚)) |
| 74 | 73 | adantl 481 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) → ((𝑧 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)} ↦ ((𝑁 / 𝑘) / 𝑧))‘𝑚) = ((𝑁 / 𝑘) / 𝑚)) |
| 75 | 17 | fsumdvdsdiaglem 27226 |
. . . . . . . 8
⊢ (𝜑 → ((𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ∧ 𝑣 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) → (𝑣 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑣)}))) |
| 76 | 56 | ex 412 |
. . . . . . . 8
⊢ (𝜑 → ((𝑣 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑣)}) → ⦋(𝑁 / 𝑣) / 𝑗⦌𝐴 ∈ ℂ)) |
| 77 | 75, 76 | syld 47 |
. . . . . . 7
⊢ (𝜑 → ((𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} ∧ 𝑣 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) → ⦋(𝑁 / 𝑣) / 𝑗⦌𝐴 ∈ ℂ)) |
| 78 | 77 | impl 455 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑣 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) → ⦋(𝑁 / 𝑣) / 𝑗⦌𝐴 ∈ ℂ) |
| 79 | 59, 66, 70, 74, 78 | fsumf1o 15759 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → Σ𝑣 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}⦋(𝑁 / 𝑣) / 𝑗⦌𝐴 = Σ𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}⦋(𝑁 / ((𝑁 / 𝑘) / 𝑚)) / 𝑗⦌𝐴) |
| 80 | | ovexd 7466 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) → (𝑁 / ((𝑁 / 𝑘) / 𝑚)) ∈ V) |
| 81 | | nncn 12274 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℂ) |
| 82 | | nnne0 12300 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈ ℕ → 𝑁 ≠ 0) |
| 83 | 81, 82 | jca 511 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 ∈ ℕ → (𝑁 ∈ ℂ ∧ 𝑁 ≠ 0)) |
| 84 | 17, 83 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑁 ∈ ℂ ∧ 𝑁 ≠ 0)) |
| 85 | 84 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) → (𝑁 ∈ ℂ ∧ 𝑁 ≠ 0)) |
| 86 | 85 | simpld 494 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) → 𝑁 ∈ ℂ) |
| 87 | | elrabi 3687 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁} → 𝑘 ∈ ℕ) |
| 88 | 87 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → 𝑘 ∈ ℕ) |
| 89 | 88 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) → 𝑘 ∈ ℕ) |
| 90 | | nncn 12274 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℂ) |
| 91 | | nnne0 12300 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0) |
| 92 | 90, 91 | jca 511 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ → (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0)) |
| 93 | 89, 92 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) → (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0)) |
| 94 | | elrabi 3687 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)} → 𝑚 ∈ ℕ) |
| 95 | 94 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) → 𝑚 ∈ ℕ) |
| 96 | | nncn 12274 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℂ) |
| 97 | | nnne0 12300 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0) |
| 98 | 96, 97 | jca 511 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ ℕ → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0)) |
| 99 | 95, 98 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0)) |
| 100 | | divdiv1 11978 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℂ ∧ (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0) ∧ (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0)) → ((𝑁 / 𝑘) / 𝑚) = (𝑁 / (𝑘 · 𝑚))) |
| 101 | 86, 93, 99, 100 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) → ((𝑁 / 𝑘) / 𝑚) = (𝑁 / (𝑘 · 𝑚))) |
| 102 | 101 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) → (𝑁 / ((𝑁 / 𝑘) / 𝑚)) = (𝑁 / (𝑁 / (𝑘 · 𝑚)))) |
| 103 | | nnmulcl 12290 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ ℕ ∧ 𝑚 ∈ ℕ) → (𝑘 · 𝑚) ∈ ℕ) |
| 104 | 88, 94, 103 | syl2an 596 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) → (𝑘 · 𝑚) ∈ ℕ) |
| 105 | | nncn 12274 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 · 𝑚) ∈ ℕ → (𝑘 · 𝑚) ∈ ℂ) |
| 106 | | nnne0 12300 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 · 𝑚) ∈ ℕ → (𝑘 · 𝑚) ≠ 0) |
| 107 | 105, 106 | jca 511 |
. . . . . . . . . . . . 13
⊢ ((𝑘 · 𝑚) ∈ ℕ → ((𝑘 · 𝑚) ∈ ℂ ∧ (𝑘 · 𝑚) ≠ 0)) |
| 108 | 104, 107 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) → ((𝑘 · 𝑚) ∈ ℂ ∧ (𝑘 · 𝑚) ≠ 0)) |
| 109 | | ddcan 11981 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℂ ∧ 𝑁 ≠ 0) ∧ ((𝑘 · 𝑚) ∈ ℂ ∧ (𝑘 · 𝑚) ≠ 0)) → (𝑁 / (𝑁 / (𝑘 · 𝑚))) = (𝑘 · 𝑚)) |
| 110 | 85, 108, 109 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) → (𝑁 / (𝑁 / (𝑘 · 𝑚))) = (𝑘 · 𝑚)) |
| 111 | 102, 110 | eqtrd 2777 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) → (𝑁 / ((𝑁 / 𝑘) / 𝑚)) = (𝑘 · 𝑚)) |
| 112 | 111 | eqeq2d 2748 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) → (𝑗 = (𝑁 / ((𝑁 / 𝑘) / 𝑚)) ↔ 𝑗 = (𝑘 · 𝑚))) |
| 113 | 112 | biimpa 476 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) ∧ 𝑗 = (𝑁 / ((𝑁 / 𝑘) / 𝑚))) → 𝑗 = (𝑘 · 𝑚)) |
| 114 | | fsumdvdscom.2 |
. . . . . . . 8
⊢ (𝑗 = (𝑘 · 𝑚) → 𝐴 = 𝐵) |
| 115 | 113, 114 | syl 17 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) ∧ 𝑗 = (𝑁 / ((𝑁 / 𝑘) / 𝑚))) → 𝐴 = 𝐵) |
| 116 | 80, 115 | csbied 3935 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) ∧ 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}) → ⦋(𝑁 / ((𝑁 / 𝑘) / 𝑚)) / 𝑗⦌𝐴 = 𝐵) |
| 117 | 116 | sumeq2dv 15738 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → Σ𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}⦋(𝑁 / ((𝑁 / 𝑘) / 𝑚)) / 𝑗⦌𝐴 = Σ𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}𝐵) |
| 118 | 79, 117 | eqtrd 2777 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}) → Σ𝑣 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}⦋(𝑁 / 𝑣) / 𝑗⦌𝐴 = Σ𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}𝐵) |
| 119 | 118 | sumeq2dv 15738 |
. . 3
⊢ (𝜑 → Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}Σ𝑣 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}⦋(𝑁 / 𝑣) / 𝑗⦌𝐴 = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}Σ𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}𝐵) |
| 120 | 48, 57, 119 | 3eqtrd 2781 |
. 2
⊢ (𝜑 → Σ𝑢 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑢}⦋𝑢 / 𝑗⦌𝐴 = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}Σ𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}𝐵) |
| 121 | 10, 120 | eqtrid 2789 |
1
⊢ (𝜑 → Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑗}𝐴 = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑁}Σ𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑁 / 𝑘)}𝐵) |