| Step | Hyp | Ref
| Expression |
| 1 | | binomlem.4 |
. . . . . 6
⊢ (𝜓 → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)))) |
| 2 | 1 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)))) |
| 3 | 2 | oveq1d 7446 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + 𝐵)↑𝑁) · 𝐴) = (Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐴)) |
| 4 | | fzfid 14014 |
. . . . . . 7
⊢ (𝜑 → (0...𝑁) ∈ Fin) |
| 5 | | binomlem.1 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 6 | | fzelp1 13616 |
. . . . . . . . 9
⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ (0...(𝑁 + 1))) |
| 7 | | binomlem.3 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
| 8 | | elfzelz 13564 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ (0...(𝑁 + 1)) → 𝑘 ∈ ℤ) |
| 9 | | bccl 14361 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈ ℤ)
→ (𝑁C𝑘) ∈
ℕ0) |
| 10 | 7, 8, 9 | syl2an 596 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝑁C𝑘) ∈
ℕ0) |
| 11 | 10 | nn0cnd 12589 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝑁C𝑘) ∈ ℂ) |
| 12 | 6, 11 | sylan2 593 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈ ℂ) |
| 13 | | fznn0sub 13596 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (0...𝑁) → (𝑁 − 𝑘) ∈
ℕ0) |
| 14 | | expcl 14120 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 𝑘) ∈ ℕ0) → (𝐴↑(𝑁 − 𝑘)) ∈ ℂ) |
| 15 | 5, 13, 14 | syl2an 596 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐴↑(𝑁 − 𝑘)) ∈ ℂ) |
| 16 | | binomlem.2 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐵 ∈ ℂ) |
| 17 | | elfznn0 13660 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ (0...(𝑁 + 1)) → 𝑘 ∈ ℕ0) |
| 18 | | expcl 14120 |
. . . . . . . . . . 11
⊢ ((𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ0)
→ (𝐵↑𝑘) ∈
ℂ) |
| 19 | 16, 17, 18 | syl2an 596 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝐵↑𝑘) ∈ ℂ) |
| 20 | 6, 19 | sylan2 593 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐵↑𝑘) ∈ ℂ) |
| 21 | 15, 20 | mulcld 11281 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)) ∈ ℂ) |
| 22 | 12, 21 | mulcld 11281 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) ∈ ℂ) |
| 23 | 4, 5, 22 | fsummulc1 15821 |
. . . . . 6
⊢ (𝜑 → (Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐴) = Σ𝑘 ∈ (0...𝑁)(((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐴)) |
| 24 | 5 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ ℂ) |
| 25 | 12, 21, 24 | mulassd 11284 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐴) = ((𝑁C𝑘) · (((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)) · 𝐴))) |
| 26 | 7 | nn0cnd 12589 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 27 | 26 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ∈ ℂ) |
| 28 | | 1cnd 11256 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 1 ∈ ℂ) |
| 29 | | elfzelz 13564 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ) |
| 30 | 29 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℤ) |
| 31 | 30 | zcnd 12723 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℂ) |
| 32 | 27, 28, 31 | addsubd 11641 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁 + 1) − 𝑘) = ((𝑁 − 𝑘) + 1)) |
| 33 | 32 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐴↑((𝑁 + 1) − 𝑘)) = (𝐴↑((𝑁 − 𝑘) + 1))) |
| 34 | | expp1 14109 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 𝑘) ∈ ℕ0) → (𝐴↑((𝑁 − 𝑘) + 1)) = ((𝐴↑(𝑁 − 𝑘)) · 𝐴)) |
| 35 | 5, 13, 34 | syl2an 596 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐴↑((𝑁 − 𝑘) + 1)) = ((𝐴↑(𝑁 − 𝑘)) · 𝐴)) |
| 36 | 33, 35 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐴↑((𝑁 + 1) − 𝑘)) = ((𝐴↑(𝑁 − 𝑘)) · 𝐴)) |
| 37 | 36 | oveq1d 7446 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)) = (((𝐴↑(𝑁 − 𝑘)) · 𝐴) · (𝐵↑𝑘))) |
| 38 | 15, 24, 20 | mul32d 11471 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((𝐴↑(𝑁 − 𝑘)) · 𝐴) · (𝐵↑𝑘)) = (((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)) · 𝐴)) |
| 39 | 37, 38 | eqtrd 2777 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)) = (((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)) · 𝐴)) |
| 40 | 39 | oveq2d 7447 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = ((𝑁C𝑘) · (((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)) · 𝐴))) |
| 41 | 25, 40 | eqtr4d 2780 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐴) = ((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
| 42 | 41 | sumeq2dv 15738 |
. . . . . 6
⊢ (𝜑 → Σ𝑘 ∈ (0...𝑁)(((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
| 43 | | fzssp1 13607 |
. . . . . . . 8
⊢
(0...𝑁) ⊆
(0...(𝑁 +
1)) |
| 44 | 43 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (0...𝑁) ⊆ (0...(𝑁 + 1))) |
| 45 | | fznn0sub 13596 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ (0...(𝑁 + 1)) → ((𝑁 + 1) − 𝑘) ∈
ℕ0) |
| 46 | | expcl 14120 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧ ((𝑁 + 1) − 𝑘) ∈ ℕ0) → (𝐴↑((𝑁 + 1) − 𝑘)) ∈ ℂ) |
| 47 | 5, 45, 46 | syl2an 596 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝐴↑((𝑁 + 1) − 𝑘)) ∈ ℂ) |
| 48 | 47, 19 | mulcld 11281 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)) ∈ ℂ) |
| 49 | 11, 48 | mulcld 11281 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → ((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) ∈ ℂ) |
| 50 | 6, 49 | sylan2 593 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) ∈ ℂ) |
| 51 | 7 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁))) → 𝑁 ∈
ℕ0) |
| 52 | | eldifi 4131 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁)) → 𝑘 ∈ (0...(𝑁 + 1))) |
| 53 | 52, 8 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁)) → 𝑘 ∈ ℤ) |
| 54 | 53 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁))) → 𝑘 ∈ ℤ) |
| 55 | | eldifn 4132 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁)) → ¬ 𝑘 ∈ (0...𝑁)) |
| 56 | 55 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁))) → ¬ 𝑘 ∈ (0...𝑁)) |
| 57 | | bcval3 14345 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈ ℤ
∧ ¬ 𝑘 ∈
(0...𝑁)) → (𝑁C𝑘) = 0) |
| 58 | 51, 54, 56, 57 | syl3anc 1373 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁))) → (𝑁C𝑘) = 0) |
| 59 | 58 | oveq1d 7446 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁))) → ((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = (0 · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
| 60 | 48 | mul02d 11459 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (0 · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = 0) |
| 61 | 52, 60 | sylan2 593 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁))) → (0 · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = 0) |
| 62 | 59, 61 | eqtrd 2777 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁))) → ((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = 0) |
| 63 | | fzssuz 13605 |
. . . . . . . 8
⊢
(0...(𝑁 + 1))
⊆ (ℤ≥‘0) |
| 64 | 63 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (0...(𝑁 + 1)) ⊆
(ℤ≥‘0)) |
| 65 | 44, 50, 62, 64 | sumss 15760 |
. . . . . 6
⊢ (𝜑 → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
| 66 | 23, 42, 65 | 3eqtrd 2781 |
. . . . 5
⊢ (𝜑 → (Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐴) = Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
| 67 | 66 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → (Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐴) = Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
| 68 | 3, 67 | eqtrd 2777 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + 𝐵)↑𝑁) · 𝐴) = Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
| 69 | 1 | oveq1d 7446 |
. . . 4
⊢ (𝜓 → (((𝐴 + 𝐵)↑𝑁) · 𝐵) = (Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐵)) |
| 70 | 4, 16, 22 | fsummulc1 15821 |
. . . . 5
⊢ (𝜑 → (Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐵) = Σ𝑘 ∈ (0...𝑁)(((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐵)) |
| 71 | | 1zzd 12648 |
. . . . . . . 8
⊢ (𝜑 → 1 ∈
ℤ) |
| 72 | | 0z 12624 |
. . . . . . . . 9
⊢ 0 ∈
ℤ |
| 73 | 72 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈
ℤ) |
| 74 | 7 | nn0zd 12639 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 75 | 16 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ ℂ) |
| 76 | 22, 75 | mulcld 11281 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐵) ∈ ℂ) |
| 77 | | oveq2 7439 |
. . . . . . . . . 10
⊢ (𝑘 = (𝑗 − 1) → (𝑁C𝑘) = (𝑁C(𝑗 − 1))) |
| 78 | | oveq2 7439 |
. . . . . . . . . . . 12
⊢ (𝑘 = (𝑗 − 1) → (𝑁 − 𝑘) = (𝑁 − (𝑗 − 1))) |
| 79 | 78 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑗 − 1) → (𝐴↑(𝑁 − 𝑘)) = (𝐴↑(𝑁 − (𝑗 − 1)))) |
| 80 | | oveq2 7439 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑗 − 1) → (𝐵↑𝑘) = (𝐵↑(𝑗 − 1))) |
| 81 | 79, 80 | oveq12d 7449 |
. . . . . . . . . 10
⊢ (𝑘 = (𝑗 − 1) → ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑(𝑁 − (𝑗 − 1))) · (𝐵↑(𝑗 − 1)))) |
| 82 | 77, 81 | oveq12d 7449 |
. . . . . . . . 9
⊢ (𝑘 = (𝑗 − 1) → ((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) = ((𝑁C(𝑗 − 1)) · ((𝐴↑(𝑁 − (𝑗 − 1))) · (𝐵↑(𝑗 − 1))))) |
| 83 | 82 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝑘 = (𝑗 − 1) → (((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐵) = (((𝑁C(𝑗 − 1)) · ((𝐴↑(𝑁 − (𝑗 − 1))) · (𝐵↑(𝑗 − 1)))) · 𝐵)) |
| 84 | 71, 73, 74, 76, 83 | fsumshft 15816 |
. . . . . . 7
⊢ (𝜑 → Σ𝑘 ∈ (0...𝑁)(((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐵) = Σ𝑗 ∈ ((0 + 1)...(𝑁 + 1))(((𝑁C(𝑗 − 1)) · ((𝐴↑(𝑁 − (𝑗 − 1))) · (𝐵↑(𝑗 − 1)))) · 𝐵)) |
| 85 | | oveq1 7438 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑘 → (𝑗 − 1) = (𝑘 − 1)) |
| 86 | 85 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑘 → (𝑁C(𝑗 − 1)) = (𝑁C(𝑘 − 1))) |
| 87 | 85 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑘 → (𝑁 − (𝑗 − 1)) = (𝑁 − (𝑘 − 1))) |
| 88 | 87 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑘 → (𝐴↑(𝑁 − (𝑗 − 1))) = (𝐴↑(𝑁 − (𝑘 − 1)))) |
| 89 | 85 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑘 → (𝐵↑(𝑗 − 1)) = (𝐵↑(𝑘 − 1))) |
| 90 | 88, 89 | oveq12d 7449 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑘 → ((𝐴↑(𝑁 − (𝑗 − 1))) · (𝐵↑(𝑗 − 1))) = ((𝐴↑(𝑁 − (𝑘 − 1))) · (𝐵↑(𝑘 − 1)))) |
| 91 | 86, 90 | oveq12d 7449 |
. . . . . . . . 9
⊢ (𝑗 = 𝑘 → ((𝑁C(𝑗 − 1)) · ((𝐴↑(𝑁 − (𝑗 − 1))) · (𝐵↑(𝑗 − 1)))) = ((𝑁C(𝑘 − 1)) · ((𝐴↑(𝑁 − (𝑘 − 1))) · (𝐵↑(𝑘 − 1))))) |
| 92 | 91 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝑗 = 𝑘 → (((𝑁C(𝑗 − 1)) · ((𝐴↑(𝑁 − (𝑗 − 1))) · (𝐵↑(𝑗 − 1)))) · 𝐵) = (((𝑁C(𝑘 − 1)) · ((𝐴↑(𝑁 − (𝑘 − 1))) · (𝐵↑(𝑘 − 1)))) · 𝐵)) |
| 93 | 92 | cbvsumv 15732 |
. . . . . . 7
⊢
Σ𝑗 ∈ ((0
+ 1)...(𝑁 + 1))(((𝑁C(𝑗 − 1)) · ((𝐴↑(𝑁 − (𝑗 − 1))) · (𝐵↑(𝑗 − 1)))) · 𝐵) = Σ𝑘 ∈ ((0 + 1)...(𝑁 + 1))(((𝑁C(𝑘 − 1)) · ((𝐴↑(𝑁 − (𝑘 − 1))) · (𝐵↑(𝑘 − 1)))) · 𝐵) |
| 94 | 84, 93 | eqtrdi 2793 |
. . . . . 6
⊢ (𝜑 → Σ𝑘 ∈ (0...𝑁)(((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐵) = Σ𝑘 ∈ ((0 + 1)...(𝑁 + 1))(((𝑁C(𝑘 − 1)) · ((𝐴↑(𝑁 − (𝑘 − 1))) · (𝐵↑(𝑘 − 1)))) · 𝐵)) |
| 95 | 26 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → 𝑁 ∈ ℂ) |
| 96 | | elfzelz 13564 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ((0 + 1)...(𝑁 + 1)) → 𝑘 ∈ ℤ) |
| 97 | 96 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → 𝑘 ∈ ℤ) |
| 98 | 97 | zcnd 12723 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → 𝑘 ∈ ℂ) |
| 99 | | 1cnd 11256 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → 1 ∈
ℂ) |
| 100 | 95, 98, 99 | subsub3d 11650 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (𝑁 − (𝑘 − 1)) = ((𝑁 + 1) − 𝑘)) |
| 101 | 100 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (𝐴↑(𝑁 − (𝑘 − 1))) = (𝐴↑((𝑁 + 1) − 𝑘))) |
| 102 | 101 | oveq1d 7446 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → ((𝐴↑(𝑁 − (𝑘 − 1))) · (𝐵↑(𝑘 − 1))) = ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑(𝑘 − 1)))) |
| 103 | 102 | oveq2d 7447 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → ((𝑁C(𝑘 − 1)) · ((𝐴↑(𝑁 − (𝑘 − 1))) · (𝐵↑(𝑘 − 1)))) = ((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑(𝑘 − 1))))) |
| 104 | 103 | oveq1d 7446 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (((𝑁C(𝑘 − 1)) · ((𝐴↑(𝑁 − (𝑘 − 1))) · (𝐵↑(𝑘 − 1)))) · 𝐵) = (((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑(𝑘 − 1)))) · 𝐵)) |
| 105 | | fzp1ss 13615 |
. . . . . . . . . . . 12
⊢ (0 ∈
ℤ → ((0 + 1)...(𝑁 + 1)) ⊆ (0...(𝑁 + 1))) |
| 106 | 72, 105 | ax-mp 5 |
. . . . . . . . . . 11
⊢ ((0 +
1)...(𝑁 + 1)) ⊆
(0...(𝑁 +
1)) |
| 107 | 106 | sseli 3979 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ((0 + 1)...(𝑁 + 1)) → 𝑘 ∈ (0...(𝑁 + 1))) |
| 108 | 8 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → 𝑘 ∈ ℤ) |
| 109 | | peano2zm 12660 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℤ → (𝑘 − 1) ∈
ℤ) |
| 110 | 108, 109 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝑘 − 1) ∈ ℤ) |
| 111 | | bccl 14361 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ0
∧ (𝑘 − 1) ∈
ℤ) → (𝑁C(𝑘 − 1)) ∈
ℕ0) |
| 112 | 7, 110, 111 | syl2an2r 685 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝑁C(𝑘 − 1)) ∈
ℕ0) |
| 113 | 112 | nn0cnd 12589 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝑁C(𝑘 − 1)) ∈ ℂ) |
| 114 | 107, 113 | sylan2 593 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (𝑁C(𝑘 − 1)) ∈ ℂ) |
| 115 | 107, 47 | sylan2 593 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (𝐴↑((𝑁 + 1) − 𝑘)) ∈ ℂ) |
| 116 | 16 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → 𝐵 ∈ ℂ) |
| 117 | | elfznn 13593 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ (1...(𝑁 + 1)) → 𝑘 ∈ ℕ) |
| 118 | | 0p1e1 12388 |
. . . . . . . . . . . . . . 15
⊢ (0 + 1) =
1 |
| 119 | 118 | oveq1i 7441 |
. . . . . . . . . . . . . 14
⊢ ((0 +
1)...(𝑁 + 1)) = (1...(𝑁 + 1)) |
| 120 | 117, 119 | eleq2s 2859 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ((0 + 1)...(𝑁 + 1)) → 𝑘 ∈ ℕ) |
| 121 | 120 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → 𝑘 ∈ ℕ) |
| 122 | | nnm1nn0 12567 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈
ℕ0) |
| 123 | 121, 122 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (𝑘 − 1) ∈
ℕ0) |
| 124 | 116, 123 | expcld 14186 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (𝐵↑(𝑘 − 1)) ∈ ℂ) |
| 125 | 115, 124 | mulcld 11281 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑(𝑘 − 1))) ∈
ℂ) |
| 126 | 114, 125,
116 | mulassd 11284 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑(𝑘 − 1)))) · 𝐵) = ((𝑁C(𝑘 − 1)) · (((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑(𝑘 − 1))) · 𝐵))) |
| 127 | 115, 124,
116 | mulassd 11284 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑(𝑘 − 1))) · 𝐵) = ((𝐴↑((𝑁 + 1) − 𝑘)) · ((𝐵↑(𝑘 − 1)) · 𝐵))) |
| 128 | | expm1t 14131 |
. . . . . . . . . . . 12
⊢ ((𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (𝐵↑𝑘) = ((𝐵↑(𝑘 − 1)) · 𝐵)) |
| 129 | 16, 120, 128 | syl2an 596 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (𝐵↑𝑘) = ((𝐵↑(𝑘 − 1)) · 𝐵)) |
| 130 | 129 | oveq2d 7447 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑((𝑁 + 1) − 𝑘)) · ((𝐵↑(𝑘 − 1)) · 𝐵))) |
| 131 | 127, 130 | eqtr4d 2780 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑(𝑘 − 1))) · 𝐵) = ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) |
| 132 | 131 | oveq2d 7447 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → ((𝑁C(𝑘 − 1)) · (((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑(𝑘 − 1))) · 𝐵)) = ((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
| 133 | 104, 126,
132 | 3eqtrd 2781 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (((𝑁C(𝑘 − 1)) · ((𝐴↑(𝑁 − (𝑘 − 1))) · (𝐵↑(𝑘 − 1)))) · 𝐵) = ((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
| 134 | 133 | sumeq2dv 15738 |
. . . . . 6
⊢ (𝜑 → Σ𝑘 ∈ ((0 + 1)...(𝑁 + 1))(((𝑁C(𝑘 − 1)) · ((𝐴↑(𝑁 − (𝑘 − 1))) · (𝐵↑(𝑘 − 1)))) · 𝐵) = Σ𝑘 ∈ ((0 + 1)...(𝑁 + 1))((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
| 135 | 106 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → ((0 + 1)...(𝑁 + 1)) ⊆ (0...(𝑁 + 1))) |
| 136 | 113, 48 | mulcld 11281 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → ((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) ∈ ℂ) |
| 137 | 107, 136 | sylan2 593 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → ((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) ∈ ℂ) |
| 138 | 7 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → 𝑁 ∈
ℕ0) |
| 139 | | eldifi 4131 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1))) → 𝑘 ∈ (0...(𝑁 + 1))) |
| 140 | 139 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → 𝑘 ∈ (0...(𝑁 + 1))) |
| 141 | 140, 8 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → 𝑘 ∈ ℤ) |
| 142 | 141, 109 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → (𝑘 − 1) ∈ ℤ) |
| 143 | | eldifn 4132 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1))) → ¬ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) |
| 144 | 143 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → ¬ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) |
| 145 | 72 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → 0 ∈
ℤ) |
| 146 | 138 | nn0zd 12639 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → 𝑁 ∈ ℤ) |
| 147 | | 1zzd 12648 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → 1 ∈
ℤ) |
| 148 | | fzaddel 13598 |
. . . . . . . . . . . . 13
⊢ (((0
∈ ℤ ∧ 𝑁
∈ ℤ) ∧ ((𝑘
− 1) ∈ ℤ ∧ 1 ∈ ℤ)) → ((𝑘 − 1) ∈ (0...𝑁) ↔ ((𝑘 − 1) + 1) ∈ ((0 + 1)...(𝑁 + 1)))) |
| 149 | 145, 146,
142, 147, 148 | syl22anc 839 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → ((𝑘 − 1) ∈ (0...𝑁) ↔ ((𝑘 − 1) + 1) ∈ ((0 + 1)...(𝑁 + 1)))) |
| 150 | 141 | zcnd 12723 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → 𝑘 ∈ ℂ) |
| 151 | | ax-1cn 11213 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
ℂ |
| 152 | | npcan 11517 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑘 −
1) + 1) = 𝑘) |
| 153 | 150, 151,
152 | sylancl 586 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → ((𝑘 − 1) + 1) = 𝑘) |
| 154 | 153 | eleq1d 2826 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → (((𝑘 − 1) + 1) ∈ ((0 + 1)...(𝑁 + 1)) ↔ 𝑘 ∈ ((0 + 1)...(𝑁 + 1)))) |
| 155 | 149, 154 | bitrd 279 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → ((𝑘 − 1) ∈ (0...𝑁) ↔ 𝑘 ∈ ((0 + 1)...(𝑁 + 1)))) |
| 156 | 144, 155 | mtbird 325 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → ¬ (𝑘 − 1) ∈ (0...𝑁)) |
| 157 | | bcval3 14345 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ0
∧ (𝑘 − 1) ∈
ℤ ∧ ¬ (𝑘
− 1) ∈ (0...𝑁))
→ (𝑁C(𝑘 − 1)) =
0) |
| 158 | 138, 142,
156, 157 | syl3anc 1373 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → (𝑁C(𝑘 − 1)) = 0) |
| 159 | 158 | oveq1d 7446 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → ((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = (0 · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
| 160 | 139, 60 | sylan2 593 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → (0 · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = 0) |
| 161 | 159, 160 | eqtrd 2777 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → ((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = 0) |
| 162 | 135, 137,
161, 64 | sumss 15760 |
. . . . . 6
⊢ (𝜑 → Σ𝑘 ∈ ((0 + 1)...(𝑁 + 1))((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
| 163 | 94, 134, 162 | 3eqtrd 2781 |
. . . . 5
⊢ (𝜑 → Σ𝑘 ∈ (0...𝑁)(((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐵) = Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
| 164 | 70, 163 | eqtrd 2777 |
. . . 4
⊢ (𝜑 → (Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐵) = Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
| 165 | 69, 164 | sylan9eqr 2799 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + 𝐵)↑𝑁) · 𝐵) = Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
| 166 | 68, 165 | oveq12d 7449 |
. 2
⊢ ((𝜑 ∧ 𝜓) → ((((𝐴 + 𝐵)↑𝑁) · 𝐴) + (((𝐴 + 𝐵)↑𝑁) · 𝐵)) = (Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) + Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))))) |
| 167 | 5, 16 | addcld 11280 |
. . . . 5
⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ) |
| 168 | 167, 7 | expp1d 14187 |
. . . 4
⊢ (𝜑 → ((𝐴 + 𝐵)↑(𝑁 + 1)) = (((𝐴 + 𝐵)↑𝑁) · (𝐴 + 𝐵))) |
| 169 | 167, 7 | expcld 14186 |
. . . . 5
⊢ (𝜑 → ((𝐴 + 𝐵)↑𝑁) ∈ ℂ) |
| 170 | 169, 5, 16 | adddid 11285 |
. . . 4
⊢ (𝜑 → (((𝐴 + 𝐵)↑𝑁) · (𝐴 + 𝐵)) = ((((𝐴 + 𝐵)↑𝑁) · 𝐴) + (((𝐴 + 𝐵)↑𝑁) · 𝐵))) |
| 171 | 168, 170 | eqtrd 2777 |
. . 3
⊢ (𝜑 → ((𝐴 + 𝐵)↑(𝑁 + 1)) = ((((𝐴 + 𝐵)↑𝑁) · 𝐴) + (((𝐴 + 𝐵)↑𝑁) · 𝐵))) |
| 172 | 171 | adantr 480 |
. 2
⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + 𝐵)↑(𝑁 + 1)) = ((((𝐴 + 𝐵)↑𝑁) · 𝐴) + (((𝐴 + 𝐵)↑𝑁) · 𝐵))) |
| 173 | | bcpasc 14360 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈ ℤ)
→ ((𝑁C𝑘) + (𝑁C(𝑘 − 1))) = ((𝑁 + 1)C𝑘)) |
| 174 | 7, 8, 173 | syl2an 596 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → ((𝑁C𝑘) + (𝑁C(𝑘 − 1))) = ((𝑁 + 1)C𝑘)) |
| 175 | 174 | oveq1d 7446 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (((𝑁C𝑘) + (𝑁C(𝑘 − 1))) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = (((𝑁 + 1)C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
| 176 | 11, 113, 48 | adddird 11286 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (((𝑁C𝑘) + (𝑁C(𝑘 − 1))) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = (((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) + ((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))))) |
| 177 | 175, 176 | eqtr3d 2779 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (((𝑁 + 1)C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = (((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) + ((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))))) |
| 178 | 177 | sumeq2dv 15738 |
. . . 4
⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 + 1))(((𝑁 + 1)C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = Σ𝑘 ∈ (0...(𝑁 + 1))(((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) + ((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))))) |
| 179 | | fzfid 14014 |
. . . . 5
⊢ (𝜑 → (0...(𝑁 + 1)) ∈ Fin) |
| 180 | 179, 49, 136 | fsumadd 15776 |
. . . 4
⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 + 1))(((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) + ((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) = (Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) + Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))))) |
| 181 | 178, 180 | eqtrd 2777 |
. . 3
⊢ (𝜑 → Σ𝑘 ∈ (0...(𝑁 + 1))(((𝑁 + 1)C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = (Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) + Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))))) |
| 182 | 181 | adantr 480 |
. 2
⊢ ((𝜑 ∧ 𝜓) → Σ𝑘 ∈ (0...(𝑁 + 1))(((𝑁 + 1)C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = (Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) + Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))))) |
| 183 | 166, 172,
182 | 3eqtr4d 2787 |
1
⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + 𝐵)↑(𝑁 + 1)) = Σ𝑘 ∈ (0...(𝑁 + 1))(((𝑁 + 1)C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |