Proof of Theorem basellem2
| Step | Hyp | Ref
| Expression |
| 1 | | basel.p |
. . 3
⊢ 𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (𝑡↑𝑗))) |
| 2 | | ssidd 4007 |
. . . 4
⊢ (𝑀 ∈ ℕ → ℂ
⊆ ℂ) |
| 3 | | nnnn0 12533 |
. . . 4
⊢ (𝑀 ∈ ℕ → 𝑀 ∈
ℕ0) |
| 4 | | elfznn0 13660 |
. . . . . . 7
⊢ (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℕ0) |
| 5 | | oveq2 7439 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑗 → (2 · 𝑛) = (2 · 𝑗)) |
| 6 | 5 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝑛 = 𝑗 → (𝑁C(2 · 𝑛)) = (𝑁C(2 · 𝑗))) |
| 7 | | oveq2 7439 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑗 → (𝑀 − 𝑛) = (𝑀 − 𝑗)) |
| 8 | 7 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝑛 = 𝑗 → (-1↑(𝑀 − 𝑛)) = (-1↑(𝑀 − 𝑗))) |
| 9 | 6, 8 | oveq12d 7449 |
. . . . . . . 8
⊢ (𝑛 = 𝑗 → ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))) = ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗)))) |
| 10 | | eqid 2737 |
. . . . . . . 8
⊢ (𝑛 ∈ ℕ0
↦ ((𝑁C(2 ·
𝑛)) ·
(-1↑(𝑀 − 𝑛)))) = (𝑛 ∈ ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛)))) |
| 11 | | ovex 7464 |
. . . . . . . 8
⊢ ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) ∈ V |
| 12 | 9, 10, 11 | fvmpt 7016 |
. . . . . . 7
⊢ (𝑗 ∈ ℕ0
→ ((𝑛 ∈
ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) = ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗)))) |
| 13 | 4, 12 | syl 17 |
. . . . . 6
⊢ (𝑗 ∈ (0...𝑀) → ((𝑛 ∈ ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) = ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗)))) |
| 14 | 13 | adantl 481 |
. . . . 5
⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) = ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗)))) |
| 15 | | basel.n |
. . . . . . . . . . . 12
⊢ 𝑁 = ((2 · 𝑀) + 1) |
| 16 | | 2nn 12339 |
. . . . . . . . . . . . . 14
⊢ 2 ∈
ℕ |
| 17 | | nnmulcl 12290 |
. . . . . . . . . . . . . 14
⊢ ((2
∈ ℕ ∧ 𝑀
∈ ℕ) → (2 · 𝑀) ∈ ℕ) |
| 18 | 16, 17 | mpan 690 |
. . . . . . . . . . . . 13
⊢ (𝑀 ∈ ℕ → (2
· 𝑀) ∈
ℕ) |
| 19 | 18 | peano2nnd 12283 |
. . . . . . . . . . . 12
⊢ (𝑀 ∈ ℕ → ((2
· 𝑀) + 1) ∈
ℕ) |
| 20 | 15, 19 | eqeltrid 2845 |
. . . . . . . . . . 11
⊢ (𝑀 ∈ ℕ → 𝑁 ∈
ℕ) |
| 21 | 20 | nnnn0d 12587 |
. . . . . . . . . 10
⊢ (𝑀 ∈ ℕ → 𝑁 ∈
ℕ0) |
| 22 | | 2z 12649 |
. . . . . . . . . . 11
⊢ 2 ∈
ℤ |
| 23 | | nn0z 12638 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ0
→ 𝑛 ∈
ℤ) |
| 24 | | zmulcl 12666 |
. . . . . . . . . . 11
⊢ ((2
∈ ℤ ∧ 𝑛
∈ ℤ) → (2 · 𝑛) ∈ ℤ) |
| 25 | 22, 23, 24 | sylancr 587 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ0
→ (2 · 𝑛)
∈ ℤ) |
| 26 | | bccl 14361 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ0
∧ (2 · 𝑛) ∈
ℤ) → (𝑁C(2
· 𝑛)) ∈
ℕ0) |
| 27 | 21, 25, 26 | syl2an 596 |
. . . . . . . . 9
⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
→ (𝑁C(2 · 𝑛)) ∈
ℕ0) |
| 28 | 27 | nn0cnd 12589 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
→ (𝑁C(2 · 𝑛)) ∈
ℂ) |
| 29 | | neg1cn 12380 |
. . . . . . . . 9
⊢ -1 ∈
ℂ |
| 30 | | neg1ne0 12382 |
. . . . . . . . 9
⊢ -1 ≠
0 |
| 31 | | nnz 12634 |
. . . . . . . . . 10
⊢ (𝑀 ∈ ℕ → 𝑀 ∈
ℤ) |
| 32 | | zsubcl 12659 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑀 − 𝑛) ∈ ℤ) |
| 33 | 31, 23, 32 | syl2an 596 |
. . . . . . . . 9
⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
→ (𝑀 − 𝑛) ∈
ℤ) |
| 34 | | expclz 14125 |
. . . . . . . . 9
⊢ ((-1
∈ ℂ ∧ -1 ≠ 0 ∧ (𝑀 − 𝑛) ∈ ℤ) → (-1↑(𝑀 − 𝑛)) ∈ ℂ) |
| 35 | 29, 30, 33, 34 | mp3an12i 1467 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
→ (-1↑(𝑀 −
𝑛)) ∈
ℂ) |
| 36 | 28, 35 | mulcld 11281 |
. . . . . . 7
⊢ ((𝑀 ∈ ℕ ∧ 𝑛 ∈ ℕ0)
→ ((𝑁C(2 ·
𝑛)) ·
(-1↑(𝑀 − 𝑛))) ∈
ℂ) |
| 37 | 36 | fmpttd 7135 |
. . . . . 6
⊢ (𝑀 ∈ ℕ → (𝑛 ∈ ℕ0
↦ ((𝑁C(2 ·
𝑛)) ·
(-1↑(𝑀 − 𝑛)))):ℕ0⟶ℂ) |
| 38 | | ffvelcdm 7101 |
. . . . . 6
⊢ (((𝑛 ∈ ℕ0
↦ ((𝑁C(2 ·
𝑛)) ·
(-1↑(𝑀 − 𝑛)))):ℕ0⟶ℂ ∧
𝑗 ∈
ℕ0) → ((𝑛 ∈ ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) ∈ ℂ) |
| 39 | 37, 4, 38 | syl2an 596 |
. . . . 5
⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ (0...𝑀)) → ((𝑛 ∈ ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) ∈ ℂ) |
| 40 | 14, 39 | eqeltrrd 2842 |
. . . 4
⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ (0...𝑀)) → ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) ∈ ℂ) |
| 41 | 2, 3, 40 | elplyd 26241 |
. . 3
⊢ (𝑀 ∈ ℕ → (𝑡 ∈ ℂ ↦
Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (𝑡↑𝑗))) ∈
(Poly‘ℂ)) |
| 42 | 1, 41 | eqeltrid 2845 |
. 2
⊢ (𝑀 ∈ ℕ → 𝑃 ∈
(Poly‘ℂ)) |
| 43 | | nnre 12273 |
. . . . . . . 8
⊢ (𝑀 ∈ ℕ → 𝑀 ∈
ℝ) |
| 44 | | nn0re 12535 |
. . . . . . . 8
⊢ (𝑗 ∈ ℕ0
→ 𝑗 ∈
ℝ) |
| 45 | | ltnle 11340 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑀 < 𝑗 ↔ ¬ 𝑗 ≤ 𝑀)) |
| 46 | 43, 44, 45 | syl2an 596 |
. . . . . . 7
⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
→ (𝑀 < 𝑗 ↔ ¬ 𝑗 ≤ 𝑀)) |
| 47 | 12 | ad2antlr 727 |
. . . . . . . . 9
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → ((𝑛 ∈ ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) = ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗)))) |
| 48 | 21 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → 𝑁 ∈
ℕ0) |
| 49 | | nn0z 12638 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℕ0
→ 𝑗 ∈
ℤ) |
| 50 | 49 | ad2antlr 727 |
. . . . . . . . . . . 12
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → 𝑗 ∈ ℤ) |
| 51 | | zmulcl 12666 |
. . . . . . . . . . . 12
⊢ ((2
∈ ℤ ∧ 𝑗
∈ ℤ) → (2 · 𝑗) ∈ ℤ) |
| 52 | 22, 50, 51 | sylancr 587 |
. . . . . . . . . . 11
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → (2 · 𝑗) ∈
ℤ) |
| 53 | | ax-1cn 11213 |
. . . . . . . . . . . . . . . . 17
⊢ 1 ∈
ℂ |
| 54 | 53 | 2timesi 12404 |
. . . . . . . . . . . . . . . 16
⊢ (2
· 1) = (1 + 1) |
| 55 | 54 | oveq2i 7442 |
. . . . . . . . . . . . . . 15
⊢ ((2
· 𝑀) + (2 ·
1)) = ((2 · 𝑀) + (1
+ 1)) |
| 56 | | 2cnd 12344 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → 2 ∈
ℂ) |
| 57 | | nncn 12274 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈ ℕ → 𝑀 ∈
ℂ) |
| 58 | 57 | ad2antrr 726 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → 𝑀 ∈ ℂ) |
| 59 | 53 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → 1 ∈
ℂ) |
| 60 | 56, 58, 59 | adddid 11285 |
. . . . . . . . . . . . . . 15
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → (2 · (𝑀 + 1)) = ((2 · 𝑀) + (2 ·
1))) |
| 61 | 15 | oveq1i 7441 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 + 1) = (((2 · 𝑀) + 1) + 1) |
| 62 | 18 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → (2 · 𝑀) ∈
ℕ) |
| 63 | 62 | nncnd 12282 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → (2 · 𝑀) ∈
ℂ) |
| 64 | 63, 59, 59 | addassd 11283 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → (((2 · 𝑀) + 1) + 1) = ((2 · 𝑀) + (1 + 1))) |
| 65 | 61, 64 | eqtrid 2789 |
. . . . . . . . . . . . . . 15
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → (𝑁 + 1) = ((2 · 𝑀) + (1 + 1))) |
| 66 | 55, 60, 65 | 3eqtr4a 2803 |
. . . . . . . . . . . . . 14
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → (2 · (𝑀 + 1)) = (𝑁 + 1)) |
| 67 | | zltp1le 12667 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑀 < 𝑗 ↔ (𝑀 + 1) ≤ 𝑗)) |
| 68 | 31, 49, 67 | syl2an 596 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
→ (𝑀 < 𝑗 ↔ (𝑀 + 1) ≤ 𝑗)) |
| 69 | 68 | biimpa 476 |
. . . . . . . . . . . . . . 15
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → (𝑀 + 1) ≤ 𝑗) |
| 70 | 43 | ad2antrr 726 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → 𝑀 ∈ ℝ) |
| 71 | | peano2re 11434 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈
ℝ) |
| 72 | 70, 71 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → (𝑀 + 1) ∈ ℝ) |
| 73 | 44 | ad2antlr 727 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → 𝑗 ∈ ℝ) |
| 74 | | 2re 12340 |
. . . . . . . . . . . . . . . . . 18
⊢ 2 ∈
ℝ |
| 75 | | 2pos 12369 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 <
2 |
| 76 | 74, 75 | pm3.2i 470 |
. . . . . . . . . . . . . . . . 17
⊢ (2 ∈
ℝ ∧ 0 < 2) |
| 77 | 76 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → (2 ∈ ℝ ∧
0 < 2)) |
| 78 | | lemul2 12120 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑀 + 1) ∈ ℝ ∧ 𝑗 ∈ ℝ ∧ (2 ∈
ℝ ∧ 0 < 2)) → ((𝑀 + 1) ≤ 𝑗 ↔ (2 · (𝑀 + 1)) ≤ (2 · 𝑗))) |
| 79 | 72, 73, 77, 78 | syl3anc 1373 |
. . . . . . . . . . . . . . 15
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → ((𝑀 + 1) ≤ 𝑗 ↔ (2 · (𝑀 + 1)) ≤ (2 · 𝑗))) |
| 80 | 69, 79 | mpbid 232 |
. . . . . . . . . . . . . 14
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → (2 · (𝑀 + 1)) ≤ (2 · 𝑗)) |
| 81 | 66, 80 | eqbrtrrd 5167 |
. . . . . . . . . . . . 13
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → (𝑁 + 1) ≤ (2 · 𝑗)) |
| 82 | 20 | nnzd 12640 |
. . . . . . . . . . . . . . 15
⊢ (𝑀 ∈ ℕ → 𝑁 ∈
ℤ) |
| 83 | 82 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → 𝑁 ∈ ℤ) |
| 84 | | zltp1le 12667 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℤ ∧ (2
· 𝑗) ∈ ℤ)
→ (𝑁 < (2 ·
𝑗) ↔ (𝑁 + 1) ≤ (2 · 𝑗))) |
| 85 | 83, 52, 84 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → (𝑁 < (2 · 𝑗) ↔ (𝑁 + 1) ≤ (2 · 𝑗))) |
| 86 | 81, 85 | mpbird 257 |
. . . . . . . . . . . 12
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → 𝑁 < (2 · 𝑗)) |
| 87 | 86 | olcd 875 |
. . . . . . . . . . 11
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → ((2 · 𝑗) < 0 ∨ 𝑁 < (2 · 𝑗))) |
| 88 | | bcval4 14346 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ0
∧ (2 · 𝑗) ∈
ℤ ∧ ((2 · 𝑗) < 0 ∨ 𝑁 < (2 · 𝑗))) → (𝑁C(2 · 𝑗)) = 0) |
| 89 | 48, 52, 87, 88 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → (𝑁C(2 · 𝑗)) = 0) |
| 90 | 89 | oveq1d 7446 |
. . . . . . . . 9
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → ((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) = (0 · (-1↑(𝑀 − 𝑗)))) |
| 91 | | zsubcl 12659 |
. . . . . . . . . . . . 13
⊢ ((𝑀 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑀 − 𝑗) ∈ ℤ) |
| 92 | 31, 49, 91 | syl2an 596 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
→ (𝑀 − 𝑗) ∈
ℤ) |
| 93 | | expclz 14125 |
. . . . . . . . . . . 12
⊢ ((-1
∈ ℂ ∧ -1 ≠ 0 ∧ (𝑀 − 𝑗) ∈ ℤ) → (-1↑(𝑀 − 𝑗)) ∈ ℂ) |
| 94 | 29, 30, 92, 93 | mp3an12i 1467 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
→ (-1↑(𝑀 −
𝑗)) ∈
ℂ) |
| 95 | 94 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → (-1↑(𝑀 − 𝑗)) ∈ ℂ) |
| 96 | 95 | mul02d 11459 |
. . . . . . . . 9
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → (0 ·
(-1↑(𝑀 − 𝑗))) = 0) |
| 97 | 47, 90, 96 | 3eqtrd 2781 |
. . . . . . . 8
⊢ (((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
∧ 𝑀 < 𝑗) → ((𝑛 ∈ ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) = 0) |
| 98 | 97 | ex 412 |
. . . . . . 7
⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
→ (𝑀 < 𝑗 → ((𝑛 ∈ ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) = 0)) |
| 99 | 46, 98 | sylbird 260 |
. . . . . 6
⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
→ (¬ 𝑗 ≤ 𝑀 → ((𝑛 ∈ ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) = 0)) |
| 100 | 99 | necon1ad 2957 |
. . . . 5
⊢ ((𝑀 ∈ ℕ ∧ 𝑗 ∈ ℕ0)
→ (((𝑛 ∈
ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) ≠ 0 → 𝑗 ≤ 𝑀)) |
| 101 | 100 | ralrimiva 3146 |
. . . 4
⊢ (𝑀 ∈ ℕ →
∀𝑗 ∈
ℕ0 (((𝑛
∈ ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) ≠ 0 → 𝑗 ≤ 𝑀)) |
| 102 | | plyco0 26231 |
. . . . 5
⊢ ((𝑀 ∈ ℕ0
∧ (𝑛 ∈
ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛)))):ℕ0⟶ℂ)
→ (((𝑛 ∈
ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛)))) “
(ℤ≥‘(𝑀 + 1))) = {0} ↔ ∀𝑗 ∈ ℕ0
(((𝑛 ∈
ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) ≠ 0 → 𝑗 ≤ 𝑀))) |
| 103 | 3, 37, 102 | syl2anc 584 |
. . . 4
⊢ (𝑀 ∈ ℕ → (((𝑛 ∈ ℕ0
↦ ((𝑁C(2 ·
𝑛)) ·
(-1↑(𝑀 − 𝑛)))) “
(ℤ≥‘(𝑀 + 1))) = {0} ↔ ∀𝑗 ∈ ℕ0
(((𝑛 ∈
ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) ≠ 0 → 𝑗 ≤ 𝑀))) |
| 104 | 101, 103 | mpbird 257 |
. . 3
⊢ (𝑀 ∈ ℕ → ((𝑛 ∈ ℕ0
↦ ((𝑁C(2 ·
𝑛)) ·
(-1↑(𝑀 − 𝑛)))) “
(ℤ≥‘(𝑀 + 1))) = {0}) |
| 105 | 13 | oveq1d 7446 |
. . . . . . 7
⊢ (𝑗 ∈ (0...𝑀) → (((𝑛 ∈ ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) · (𝑡↑𝑗)) = (((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (𝑡↑𝑗))) |
| 106 | 105 | sumeq2i 15734 |
. . . . . 6
⊢
Σ𝑗 ∈
(0...𝑀)(((𝑛 ∈ ℕ0
↦ ((𝑁C(2 ·
𝑛)) ·
(-1↑(𝑀 − 𝑛))))‘𝑗) · (𝑡↑𝑗)) = Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (𝑡↑𝑗)) |
| 107 | 106 | mpteq2i 5247 |
. . . . 5
⊢ (𝑡 ∈ ℂ ↦
Σ𝑗 ∈ (0...𝑀)(((𝑛 ∈ ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) · (𝑡↑𝑗))) = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑁C(2 · 𝑗)) · (-1↑(𝑀 − 𝑗))) · (𝑡↑𝑗))) |
| 108 | 1, 107 | eqtr4i 2768 |
. . . 4
⊢ 𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑛 ∈ ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) · (𝑡↑𝑗))) |
| 109 | 108 | a1i 11 |
. . 3
⊢ (𝑀 ∈ ℕ → 𝑃 = (𝑡 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑀)(((𝑛 ∈ ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑗) · (𝑡↑𝑗)))) |
| 110 | | oveq2 7439 |
. . . . . . . . 9
⊢ (𝑛 = 𝑀 → (2 · 𝑛) = (2 · 𝑀)) |
| 111 | 110 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑛 = 𝑀 → (𝑁C(2 · 𝑛)) = (𝑁C(2 · 𝑀))) |
| 112 | | oveq2 7439 |
. . . . . . . . 9
⊢ (𝑛 = 𝑀 → (𝑀 − 𝑛) = (𝑀 − 𝑀)) |
| 113 | 112 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑛 = 𝑀 → (-1↑(𝑀 − 𝑛)) = (-1↑(𝑀 − 𝑀))) |
| 114 | 111, 113 | oveq12d 7449 |
. . . . . . 7
⊢ (𝑛 = 𝑀 → ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))) = ((𝑁C(2 · 𝑀)) · (-1↑(𝑀 − 𝑀)))) |
| 115 | | ovex 7464 |
. . . . . . 7
⊢ ((𝑁C(2 · 𝑀)) · (-1↑(𝑀 − 𝑀))) ∈ V |
| 116 | 114, 10, 115 | fvmpt 7016 |
. . . . . 6
⊢ (𝑀 ∈ ℕ0
→ ((𝑛 ∈
ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛))))‘𝑀) = ((𝑁C(2 · 𝑀)) · (-1↑(𝑀 − 𝑀)))) |
| 117 | 3, 116 | syl 17 |
. . . . 5
⊢ (𝑀 ∈ ℕ → ((𝑛 ∈ ℕ0
↦ ((𝑁C(2 ·
𝑛)) ·
(-1↑(𝑀 − 𝑛))))‘𝑀) = ((𝑁C(2 · 𝑀)) · (-1↑(𝑀 − 𝑀)))) |
| 118 | 57 | subidd 11608 |
. . . . . . . 8
⊢ (𝑀 ∈ ℕ → (𝑀 − 𝑀) = 0) |
| 119 | 118 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑀 ∈ ℕ →
(-1↑(𝑀 − 𝑀)) =
(-1↑0)) |
| 120 | | exp0 14106 |
. . . . . . . 8
⊢ (-1
∈ ℂ → (-1↑0) = 1) |
| 121 | 29, 120 | ax-mp 5 |
. . . . . . 7
⊢
(-1↑0) = 1 |
| 122 | 119, 121 | eqtrdi 2793 |
. . . . . 6
⊢ (𝑀 ∈ ℕ →
(-1↑(𝑀 − 𝑀)) = 1) |
| 123 | 122 | oveq2d 7447 |
. . . . 5
⊢ (𝑀 ∈ ℕ → ((𝑁C(2 · 𝑀)) · (-1↑(𝑀 − 𝑀))) = ((𝑁C(2 · 𝑀)) · 1)) |
| 124 | 18 | nnred 12281 |
. . . . . . . . . . 11
⊢ (𝑀 ∈ ℕ → (2
· 𝑀) ∈
ℝ) |
| 125 | 124 | lep1d 12199 |
. . . . . . . . . 10
⊢ (𝑀 ∈ ℕ → (2
· 𝑀) ≤ ((2
· 𝑀) +
1)) |
| 126 | 125, 15 | breqtrrdi 5185 |
. . . . . . . . 9
⊢ (𝑀 ∈ ℕ → (2
· 𝑀) ≤ 𝑁) |
| 127 | 18 | nnnn0d 12587 |
. . . . . . . . . . 11
⊢ (𝑀 ∈ ℕ → (2
· 𝑀) ∈
ℕ0) |
| 128 | | nn0uz 12920 |
. . . . . . . . . . 11
⊢
ℕ0 = (ℤ≥‘0) |
| 129 | 127, 128 | eleqtrdi 2851 |
. . . . . . . . . 10
⊢ (𝑀 ∈ ℕ → (2
· 𝑀) ∈
(ℤ≥‘0)) |
| 130 | | elfz5 13556 |
. . . . . . . . . 10
⊢ (((2
· 𝑀) ∈
(ℤ≥‘0) ∧ 𝑁 ∈ ℤ) → ((2 · 𝑀) ∈ (0...𝑁) ↔ (2 · 𝑀) ≤ 𝑁)) |
| 131 | 129, 82, 130 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝑀 ∈ ℕ → ((2
· 𝑀) ∈
(0...𝑁) ↔ (2 ·
𝑀) ≤ 𝑁)) |
| 132 | 126, 131 | mpbird 257 |
. . . . . . . 8
⊢ (𝑀 ∈ ℕ → (2
· 𝑀) ∈
(0...𝑁)) |
| 133 | | bccl2 14362 |
. . . . . . . 8
⊢ ((2
· 𝑀) ∈
(0...𝑁) → (𝑁C(2 · 𝑀)) ∈ ℕ) |
| 134 | 132, 133 | syl 17 |
. . . . . . 7
⊢ (𝑀 ∈ ℕ → (𝑁C(2 · 𝑀)) ∈ ℕ) |
| 135 | 134 | nncnd 12282 |
. . . . . 6
⊢ (𝑀 ∈ ℕ → (𝑁C(2 · 𝑀)) ∈ ℂ) |
| 136 | 135 | mulridd 11278 |
. . . . 5
⊢ (𝑀 ∈ ℕ → ((𝑁C(2 · 𝑀)) · 1) = (𝑁C(2 · 𝑀))) |
| 137 | 117, 123,
136 | 3eqtrd 2781 |
. . . 4
⊢ (𝑀 ∈ ℕ → ((𝑛 ∈ ℕ0
↦ ((𝑁C(2 ·
𝑛)) ·
(-1↑(𝑀 − 𝑛))))‘𝑀) = (𝑁C(2 · 𝑀))) |
| 138 | 134 | nnne0d 12316 |
. . . 4
⊢ (𝑀 ∈ ℕ → (𝑁C(2 · 𝑀)) ≠ 0) |
| 139 | 137, 138 | eqnetrd 3008 |
. . 3
⊢ (𝑀 ∈ ℕ → ((𝑛 ∈ ℕ0
↦ ((𝑁C(2 ·
𝑛)) ·
(-1↑(𝑀 − 𝑛))))‘𝑀) ≠ 0) |
| 140 | 42, 3, 37, 104, 109, 139 | dgreq 26283 |
. 2
⊢ (𝑀 ∈ ℕ →
(deg‘𝑃) = 𝑀) |
| 141 | 42, 3, 37, 104, 109 | coeeq 26266 |
. 2
⊢ (𝑀 ∈ ℕ →
(coeff‘𝑃) = (𝑛 ∈ ℕ0
↦ ((𝑁C(2 ·
𝑛)) ·
(-1↑(𝑀 − 𝑛))))) |
| 142 | 42, 140, 141 | 3jca 1129 |
1
⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (Poly‘ℂ)
∧ (deg‘𝑃) = 𝑀 ∧ (coeff‘𝑃) = (𝑛 ∈ ℕ0 ↦ ((𝑁C(2 · 𝑛)) · (-1↑(𝑀 − 𝑛)))))) |