Step | Hyp | Ref
| Expression |
1 | | binomlem.4 |
. . . . . 6
⊢ (𝜓 → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)))) |
2 | 1 | adantl 275 |
. . . . 5
⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)))) |
3 | 2 | oveq1d 5841 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + 𝐵)↑𝑁) · 𝐴) = (Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐴)) |
4 | | 0zd 9184 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈
ℤ) |
5 | | binomlem.3 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
6 | 5 | nn0zd 9289 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ ℤ) |
7 | 4, 6 | fzfigd 10339 |
. . . . . . 7
⊢ (𝜑 → (0...𝑁) ∈ Fin) |
8 | | binomlem.1 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ ℂ) |
9 | | fzelp1 9982 |
. . . . . . . . 9
⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ (0...(𝑁 + 1))) |
10 | | elfzelz 9934 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ (0...(𝑁 + 1)) → 𝑘 ∈ ℤ) |
11 | | bccl 10652 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈ ℤ)
→ (𝑁C𝑘) ∈
ℕ0) |
12 | 5, 10, 11 | syl2an 287 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝑁C𝑘) ∈
ℕ0) |
13 | 12 | nn0cnd 9150 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝑁C𝑘) ∈ ℂ) |
14 | 9, 13 | sylan2 284 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈ ℂ) |
15 | | fznn0sub 9965 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (0...𝑁) → (𝑁 − 𝑘) ∈
ℕ0) |
16 | | expcl 10446 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 𝑘) ∈ ℕ0) → (𝐴↑(𝑁 − 𝑘)) ∈ ℂ) |
17 | 8, 15, 16 | syl2an 287 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐴↑(𝑁 − 𝑘)) ∈ ℂ) |
18 | | binomlem.2 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐵 ∈ ℂ) |
19 | | elfznn0 10022 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ (0...(𝑁 + 1)) → 𝑘 ∈ ℕ0) |
20 | | expcl 10446 |
. . . . . . . . . . 11
⊢ ((𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ0)
→ (𝐵↑𝑘) ∈
ℂ) |
21 | 18, 19, 20 | syl2an 287 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝐵↑𝑘) ∈ ℂ) |
22 | 9, 21 | sylan2 284 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐵↑𝑘) ∈ ℂ) |
23 | 17, 22 | mulcld 7900 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)) ∈ ℂ) |
24 | 14, 23 | mulcld 7900 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) ∈ ℂ) |
25 | 7, 8, 24 | fsummulc1 11357 |
. . . . . 6
⊢ (𝜑 → (Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐴) = Σ𝑘 ∈ (0...𝑁)(((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐴)) |
26 | 8 | adantr 274 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ ℂ) |
27 | 14, 23, 26 | mulassd 7903 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐴) = ((𝑁C𝑘) · (((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)) · 𝐴))) |
28 | 5 | nn0cnd 9150 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ ℂ) |
29 | 28 | adantr 274 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ∈ ℂ) |
30 | | 1cnd 7896 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 1 ∈ ℂ) |
31 | | elfzelz 9934 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ) |
32 | 31 | adantl 275 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℤ) |
33 | 32 | zcnd 9292 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℂ) |
34 | 29, 30, 33 | addsubd 8211 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁 + 1) − 𝑘) = ((𝑁 − 𝑘) + 1)) |
35 | 34 | oveq2d 5842 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐴↑((𝑁 + 1) − 𝑘)) = (𝐴↑((𝑁 − 𝑘) + 1))) |
36 | | expp1 10435 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 𝑘) ∈ ℕ0) → (𝐴↑((𝑁 − 𝑘) + 1)) = ((𝐴↑(𝑁 − 𝑘)) · 𝐴)) |
37 | 8, 15, 36 | syl2an 287 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐴↑((𝑁 − 𝑘) + 1)) = ((𝐴↑(𝑁 − 𝑘)) · 𝐴)) |
38 | 35, 37 | eqtrd 2190 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐴↑((𝑁 + 1) − 𝑘)) = ((𝐴↑(𝑁 − 𝑘)) · 𝐴)) |
39 | 38 | oveq1d 5841 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)) = (((𝐴↑(𝑁 − 𝑘)) · 𝐴) · (𝐵↑𝑘))) |
40 | 17, 26, 22 | mul32d 8032 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((𝐴↑(𝑁 − 𝑘)) · 𝐴) · (𝐵↑𝑘)) = (((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)) · 𝐴)) |
41 | 39, 40 | eqtrd 2190 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)) = (((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)) · 𝐴)) |
42 | 41 | oveq2d 5842 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = ((𝑁C𝑘) · (((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)) · 𝐴))) |
43 | 27, 42 | eqtr4d 2193 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐴) = ((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
44 | 43 | sumeq2dv 11276 |
. . . . . 6
⊢ (𝜑 → Σ𝑘 ∈ (0...𝑁)(((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
45 | | fzssp1 9975 |
. . . . . . . 8
⊢
(0...𝑁) ⊆
(0...(𝑁 +
1)) |
46 | 45 | a1i 9 |
. . . . . . 7
⊢ (𝜑 → (0...𝑁) ⊆ (0...(𝑁 + 1))) |
47 | | fznn0sub 9965 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ (0...(𝑁 + 1)) → ((𝑁 + 1) − 𝑘) ∈
ℕ0) |
48 | | expcl 10446 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧ ((𝑁 + 1) − 𝑘) ∈ ℕ0) → (𝐴↑((𝑁 + 1) − 𝑘)) ∈ ℂ) |
49 | 8, 47, 48 | syl2an 287 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝐴↑((𝑁 + 1) − 𝑘)) ∈ ℂ) |
50 | 49, 21 | mulcld 7900 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)) ∈ ℂ) |
51 | 13, 50 | mulcld 7900 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → ((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) ∈ ℂ) |
52 | 9, 51 | sylan2 284 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) ∈ ℂ) |
53 | 5 | adantr 274 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁))) → 𝑁 ∈
ℕ0) |
54 | | eldifi 3230 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁)) → 𝑘 ∈ (0...(𝑁 + 1))) |
55 | 54, 10 | syl 14 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁)) → 𝑘 ∈ ℤ) |
56 | 55 | adantl 275 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁))) → 𝑘 ∈ ℤ) |
57 | | eldifn 3231 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁)) → ¬ 𝑘 ∈ (0...𝑁)) |
58 | 57 | adantl 275 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁))) → ¬ 𝑘 ∈ (0...𝑁)) |
59 | | bcval3 10636 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈ ℤ
∧ ¬ 𝑘 ∈
(0...𝑁)) → (𝑁C𝑘) = 0) |
60 | 53, 56, 58, 59 | syl3anc 1220 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁))) → (𝑁C𝑘) = 0) |
61 | 60 | oveq1d 5841 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁))) → ((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = (0 · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
62 | 50 | mul02d 8271 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (0 · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = 0) |
63 | 54, 62 | sylan2 284 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁))) → (0 · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = 0) |
64 | 61, 63 | eqtrd 2190 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁))) → ((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = 0) |
65 | | eluzelz 9453 |
. . . . . . . . . 10
⊢ (𝑛 ∈
(ℤ≥‘0) → 𝑛 ∈ ℤ) |
66 | 65 | adantl 275 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘0))
→ 𝑛 ∈
ℤ) |
67 | | 0zd 9184 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘0))
→ 0 ∈ ℤ) |
68 | 6 | adantr 274 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘0))
→ 𝑁 ∈
ℤ) |
69 | | fzdcel 9948 |
. . . . . . . . 9
⊢ ((𝑛 ∈ ℤ ∧ 0 ∈
ℤ ∧ 𝑁 ∈
ℤ) → DECID 𝑛 ∈ (0...𝑁)) |
70 | 66, 67, 68, 69 | syl3anc 1220 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘0))
→ DECID 𝑛 ∈ (0...𝑁)) |
71 | 70 | ralrimiva 2530 |
. . . . . . 7
⊢ (𝜑 → ∀𝑛 ∈
(ℤ≥‘0)DECID 𝑛 ∈ (0...𝑁)) |
72 | | fzssuz 9973 |
. . . . . . . 8
⊢
(0...(𝑁 + 1))
⊆ (ℤ≥‘0) |
73 | 72 | a1i 9 |
. . . . . . 7
⊢ (𝜑 → (0...(𝑁 + 1)) ⊆
(ℤ≥‘0)) |
74 | 68 | peano2zd 9294 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘0))
→ (𝑁 + 1) ∈
ℤ) |
75 | | fzdcel 9948 |
. . . . . . . . 9
⊢ ((𝑛 ∈ ℤ ∧ 0 ∈
ℤ ∧ (𝑁 + 1)
∈ ℤ) → DECID 𝑛 ∈ (0...(𝑁 + 1))) |
76 | 66, 67, 74, 75 | syl3anc 1220 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘0))
→ DECID 𝑛 ∈ (0...(𝑁 + 1))) |
77 | 76 | ralrimiva 2530 |
. . . . . . 7
⊢ (𝜑 → ∀𝑛 ∈
(ℤ≥‘0)DECID 𝑛 ∈ (0...(𝑁 + 1))) |
78 | 46, 52, 64, 71, 4, 73, 77 | isumss 11299 |
. . . . . 6
⊢ (𝜑 → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
79 | 25, 44, 78 | 3eqtrd 2194 |
. . . . 5
⊢ (𝜑 → (Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐴) = Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
80 | 79 | adantr 274 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → (Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐴) = Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
81 | 3, 80 | eqtrd 2190 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + 𝐵)↑𝑁) · 𝐴) = Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
82 | 1 | oveq1d 5841 |
. . . 4
⊢ (𝜓 → (((𝐴 + 𝐵)↑𝑁) · 𝐵) = (Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐵)) |
83 | 7, 18, 24 | fsummulc1 11357 |
. . . . 5
⊢ (𝜑 → (Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐵) = Σ𝑘 ∈ (0...𝑁)(((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐵)) |
84 | | 1zzd 9199 |
. . . . . . . 8
⊢ (𝜑 → 1 ∈
ℤ) |
85 | 18 | adantr 274 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ ℂ) |
86 | 24, 85 | mulcld 7900 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐵) ∈ ℂ) |
87 | | oveq2 5834 |
. . . . . . . . . 10
⊢ (𝑘 = (𝑗 − 1) → (𝑁C𝑘) = (𝑁C(𝑗 − 1))) |
88 | | oveq2 5834 |
. . . . . . . . . . . 12
⊢ (𝑘 = (𝑗 − 1) → (𝑁 − 𝑘) = (𝑁 − (𝑗 − 1))) |
89 | 88 | oveq2d 5842 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑗 − 1) → (𝐴↑(𝑁 − 𝑘)) = (𝐴↑(𝑁 − (𝑗 − 1)))) |
90 | | oveq2 5834 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑗 − 1) → (𝐵↑𝑘) = (𝐵↑(𝑗 − 1))) |
91 | 89, 90 | oveq12d 5844 |
. . . . . . . . . 10
⊢ (𝑘 = (𝑗 − 1) → ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑(𝑁 − (𝑗 − 1))) · (𝐵↑(𝑗 − 1)))) |
92 | 87, 91 | oveq12d 5844 |
. . . . . . . . 9
⊢ (𝑘 = (𝑗 − 1) → ((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) = ((𝑁C(𝑗 − 1)) · ((𝐴↑(𝑁 − (𝑗 − 1))) · (𝐵↑(𝑗 − 1))))) |
93 | 92 | oveq1d 5841 |
. . . . . . . 8
⊢ (𝑘 = (𝑗 − 1) → (((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐵) = (((𝑁C(𝑗 − 1)) · ((𝐴↑(𝑁 − (𝑗 − 1))) · (𝐵↑(𝑗 − 1)))) · 𝐵)) |
94 | 84, 4, 6, 86, 93 | fsumshft 11352 |
. . . . . . 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)) |
95 | | oveq1 5833 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑘 → (𝑗 − 1) = (𝑘 − 1)) |
96 | 95 | oveq2d 5842 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑘 → (𝑁C(𝑗 − 1)) = (𝑁C(𝑘 − 1))) |
97 | 95 | oveq2d 5842 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑘 → (𝑁 − (𝑗 − 1)) = (𝑁 − (𝑘 − 1))) |
98 | 97 | oveq2d 5842 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑘 → (𝐴↑(𝑁 − (𝑗 − 1))) = (𝐴↑(𝑁 − (𝑘 − 1)))) |
99 | 95 | oveq2d 5842 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑘 → (𝐵↑(𝑗 − 1)) = (𝐵↑(𝑘 − 1))) |
100 | 98, 99 | oveq12d 5844 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑘 → ((𝐴↑(𝑁 − (𝑗 − 1))) · (𝐵↑(𝑗 − 1))) = ((𝐴↑(𝑁 − (𝑘 − 1))) · (𝐵↑(𝑘 − 1)))) |
101 | 96, 100 | oveq12d 5844 |
. . . . . . . . 9
⊢ (𝑗 = 𝑘 → ((𝑁C(𝑗 − 1)) · ((𝐴↑(𝑁 − (𝑗 − 1))) · (𝐵↑(𝑗 − 1)))) = ((𝑁C(𝑘 − 1)) · ((𝐴↑(𝑁 − (𝑘 − 1))) · (𝐵↑(𝑘 − 1))))) |
102 | 101 | oveq1d 5841 |
. . . . . . . 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)) |
103 | 102 | cbvsumv 11269 |
. . . . . . 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) |
104 | 94, 103 | eqtrdi 2206 |
. . . . . 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)) |
105 | 28 | adantr 274 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → 𝑁 ∈ ℂ) |
106 | | elfzelz 9934 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ((0 + 1)...(𝑁 + 1)) → 𝑘 ∈ ℤ) |
107 | 106 | adantl 275 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → 𝑘 ∈ ℤ) |
108 | 107 | zcnd 9292 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → 𝑘 ∈ ℂ) |
109 | | 1cnd 7896 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → 1 ∈
ℂ) |
110 | 105, 108,
109 | subsub3d 8220 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (𝑁 − (𝑘 − 1)) = ((𝑁 + 1) − 𝑘)) |
111 | 110 | oveq2d 5842 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (𝐴↑(𝑁 − (𝑘 − 1))) = (𝐴↑((𝑁 + 1) − 𝑘))) |
112 | 111 | oveq1d 5841 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → ((𝐴↑(𝑁 − (𝑘 − 1))) · (𝐵↑(𝑘 − 1))) = ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑(𝑘 − 1)))) |
113 | 112 | oveq2d 5842 |
. . . . . . . . 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))))) |
114 | 113 | oveq1d 5841 |
. . . . . . . 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)) |
115 | | 0z 9183 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℤ |
116 | | fzp1ss 9981 |
. . . . . . . . . . . 12
⊢ (0 ∈
ℤ → ((0 + 1)...(𝑁 + 1)) ⊆ (0...(𝑁 + 1))) |
117 | 115, 116 | ax-mp 5 |
. . . . . . . . . . 11
⊢ ((0 +
1)...(𝑁 + 1)) ⊆
(0...(𝑁 +
1)) |
118 | 117 | sseli 3124 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ((0 + 1)...(𝑁 + 1)) → 𝑘 ∈ (0...(𝑁 + 1))) |
119 | 10 | adantl 275 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → 𝑘 ∈ ℤ) |
120 | | peano2zm 9210 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℤ → (𝑘 − 1) ∈
ℤ) |
121 | 119, 120 | syl 14 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝑘 − 1) ∈ ℤ) |
122 | | bccl 10652 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ0
∧ (𝑘 − 1) ∈
ℤ) → (𝑁C(𝑘 − 1)) ∈
ℕ0) |
123 | 5, 121, 122 | syl2an2r 585 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝑁C(𝑘 − 1)) ∈
ℕ0) |
124 | 123 | nn0cnd 9150 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝑁C(𝑘 − 1)) ∈ ℂ) |
125 | 118, 124 | sylan2 284 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (𝑁C(𝑘 − 1)) ∈ ℂ) |
126 | 118, 49 | sylan2 284 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (𝐴↑((𝑁 + 1) − 𝑘)) ∈ ℂ) |
127 | 18 | adantr 274 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → 𝐵 ∈ ℂ) |
128 | | elfznn 9962 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ (1...(𝑁 + 1)) → 𝑘 ∈ ℕ) |
129 | | 0p1e1 8952 |
. . . . . . . . . . . . . . 15
⊢ (0 + 1) =
1 |
130 | 129 | oveq1i 5836 |
. . . . . . . . . . . . . 14
⊢ ((0 +
1)...(𝑁 + 1)) = (1...(𝑁 + 1)) |
131 | 128, 130 | eleq2s 2252 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ((0 + 1)...(𝑁 + 1)) → 𝑘 ∈ ℕ) |
132 | 131 | adantl 275 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → 𝑘 ∈ ℕ) |
133 | | nnm1nn0 9136 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈
ℕ0) |
134 | 132, 133 | syl 14 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (𝑘 − 1) ∈
ℕ0) |
135 | 127, 134 | expcld 10560 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (𝐵↑(𝑘 − 1)) ∈ ℂ) |
136 | 126, 135 | mulcld 7900 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑(𝑘 − 1))) ∈
ℂ) |
137 | 125, 136,
127 | mulassd 7903 |
. . . . . . . 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))) |
138 | 126, 135,
127 | mulassd 7903 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑(𝑘 − 1))) · 𝐵) = ((𝐴↑((𝑁 + 1) − 𝑘)) · ((𝐵↑(𝑘 − 1)) · 𝐵))) |
139 | | expm1t 10456 |
. . . . . . . . . . . 12
⊢ ((𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (𝐵↑𝑘) = ((𝐵↑(𝑘 − 1)) · 𝐵)) |
140 | 18, 131, 139 | syl2an 287 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (𝐵↑𝑘) = ((𝐵↑(𝑘 − 1)) · 𝐵)) |
141 | 140 | oveq2d 5842 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑((𝑁 + 1) − 𝑘)) · ((𝐵↑(𝑘 − 1)) · 𝐵))) |
142 | 138, 141 | eqtr4d 2193 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑(𝑘 − 1))) · 𝐵) = ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) |
143 | 142 | oveq2d 5842 |
. . . . . . . 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)))) |
144 | 114, 137,
143 | 3eqtrd 2194 |
. . . . . . 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)))) |
145 | 144 | sumeq2dv 11276 |
. . . . . 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)))) |
146 | 117 | a1i 9 |
. . . . . . 7
⊢ (𝜑 → ((0 + 1)...(𝑁 + 1)) ⊆ (0...(𝑁 + 1))) |
147 | 124, 50 | mulcld 7900 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → ((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) ∈ ℂ) |
148 | 118, 147 | sylan2 284 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → ((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) ∈ ℂ) |
149 | 5 | adantr 274 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → 𝑁 ∈
ℕ0) |
150 | | eldifi 3230 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1))) → 𝑘 ∈ (0...(𝑁 + 1))) |
151 | 150 | adantl 275 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → 𝑘 ∈ (0...(𝑁 + 1))) |
152 | 151, 10 | syl 14 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → 𝑘 ∈ ℤ) |
153 | 152, 120 | syl 14 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → (𝑘 − 1) ∈ ℤ) |
154 | | eldifn 3231 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1))) → ¬ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) |
155 | 154 | adantl 275 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → ¬ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) |
156 | | 0zd 9184 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → 0 ∈
ℤ) |
157 | 149 | nn0zd 9289 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → 𝑁 ∈ ℤ) |
158 | | 1zzd 9199 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → 1 ∈
ℤ) |
159 | | fzaddel 9967 |
. . . . . . . . . . . . 13
⊢ (((0
∈ ℤ ∧ 𝑁
∈ ℤ) ∧ ((𝑘
− 1) ∈ ℤ ∧ 1 ∈ ℤ)) → ((𝑘 − 1) ∈ (0...𝑁) ↔ ((𝑘 − 1) + 1) ∈ ((0 + 1)...(𝑁 + 1)))) |
160 | 156, 157,
153, 158, 159 | syl22anc 1221 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → ((𝑘 − 1) ∈ (0...𝑁) ↔ ((𝑘 − 1) + 1) ∈ ((0 + 1)...(𝑁 + 1)))) |
161 | 152 | zcnd 9292 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → 𝑘 ∈ ℂ) |
162 | | ax-1cn 7827 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
ℂ |
163 | | npcan 8088 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑘 −
1) + 1) = 𝑘) |
164 | 161, 162,
163 | sylancl 410 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → ((𝑘 − 1) + 1) = 𝑘) |
165 | 164 | eleq1d 2226 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → (((𝑘 − 1) + 1) ∈ ((0 + 1)...(𝑁 + 1)) ↔ 𝑘 ∈ ((0 + 1)...(𝑁 + 1)))) |
166 | 160, 165 | bitrd 187 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → ((𝑘 − 1) ∈ (0...𝑁) ↔ 𝑘 ∈ ((0 + 1)...(𝑁 + 1)))) |
167 | 155, 166 | mtbird 663 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → ¬ (𝑘 − 1) ∈ (0...𝑁)) |
168 | | bcval3 10636 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ0
∧ (𝑘 − 1) ∈
ℤ ∧ ¬ (𝑘
− 1) ∈ (0...𝑁))
→ (𝑁C(𝑘 − 1)) =
0) |
169 | 149, 153,
167, 168 | syl3anc 1220 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → (𝑁C(𝑘 − 1)) = 0) |
170 | 169 | oveq1d 5841 |
. . . . . . . 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)))) |
171 | 150, 62 | sylan2 284 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → (0 · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = 0) |
172 | 170, 171 | eqtrd 2190 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → ((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = 0) |
173 | 67 | peano2zd 9294 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘0))
→ (0 + 1) ∈ ℤ) |
174 | | fzdcel 9948 |
. . . . . . . . 9
⊢ ((𝑛 ∈ ℤ ∧ (0 + 1)
∈ ℤ ∧ (𝑁 +
1) ∈ ℤ) → DECID 𝑛 ∈ ((0 + 1)...(𝑁 + 1))) |
175 | 66, 173, 74, 174 | syl3anc 1220 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘0))
→ DECID 𝑛 ∈ ((0 + 1)...(𝑁 + 1))) |
176 | 175 | ralrimiva 2530 |
. . . . . . 7
⊢ (𝜑 → ∀𝑛 ∈
(ℤ≥‘0)DECID 𝑛 ∈ ((0 + 1)...(𝑁 + 1))) |
177 | 146, 148,
172, 176, 4, 73, 77 | isumss 11299 |
. . . . . 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)))) |
178 | 104, 145,
177 | 3eqtrd 2194 |
. . . . 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)))) |
179 | 83, 178 | eqtrd 2190 |
. . . 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)))) |
180 | 82, 179 | sylan9eqr 2212 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + 𝐵)↑𝑁) · 𝐵) = Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
181 | 81, 180 | oveq12d 5844 |
. 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))))) |
182 | 8, 18 | addcld 7899 |
. . . . 5
⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ) |
183 | 182, 5 | expp1d 10561 |
. . . 4
⊢ (𝜑 → ((𝐴 + 𝐵)↑(𝑁 + 1)) = (((𝐴 + 𝐵)↑𝑁) · (𝐴 + 𝐵))) |
184 | 182, 5 | expcld 10560 |
. . . . 5
⊢ (𝜑 → ((𝐴 + 𝐵)↑𝑁) ∈ ℂ) |
185 | 184, 8, 18 | adddid 7904 |
. . . 4
⊢ (𝜑 → (((𝐴 + 𝐵)↑𝑁) · (𝐴 + 𝐵)) = ((((𝐴 + 𝐵)↑𝑁) · 𝐴) + (((𝐴 + 𝐵)↑𝑁) · 𝐵))) |
186 | 183, 185 | eqtrd 2190 |
. . 3
⊢ (𝜑 → ((𝐴 + 𝐵)↑(𝑁 + 1)) = ((((𝐴 + 𝐵)↑𝑁) · 𝐴) + (((𝐴 + 𝐵)↑𝑁) · 𝐵))) |
187 | 186 | adantr 274 |
. 2
⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + 𝐵)↑(𝑁 + 1)) = ((((𝐴 + 𝐵)↑𝑁) · 𝐴) + (((𝐴 + 𝐵)↑𝑁) · 𝐵))) |
188 | | bcpasc 10651 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈ ℤ)
→ ((𝑁C𝑘) + (𝑁C(𝑘 − 1))) = ((𝑁 + 1)C𝑘)) |
189 | 5, 10, 188 | syl2an 287 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → ((𝑁C𝑘) + (𝑁C(𝑘 − 1))) = ((𝑁 + 1)C𝑘)) |
190 | 189 | oveq1d 5841 |
. . . . . 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)))) |
191 | 13, 124, 50 | adddird 7905 |
. . . . . 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))))) |
192 | 190, 191 | eqtr3d 2192 |
. . . . 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))))) |
193 | 192 | sumeq2dv 11276 |
. . . 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))))) |
194 | 6 | peano2zd 9294 |
. . . . . 6
⊢ (𝜑 → (𝑁 + 1) ∈ ℤ) |
195 | 4, 194 | fzfigd 10339 |
. . . . 5
⊢ (𝜑 → (0...(𝑁 + 1)) ∈ Fin) |
196 | 195, 51, 147 | fsumadd 11314 |
. . . 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))))) |
197 | 193, 196 | eqtrd 2190 |
. . 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))))) |
198 | 197 | adantr 274 |
. 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))))) |
199 | 181, 187,
198 | 3eqtr4d 2200 |
1
⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + 𝐵)↑(𝑁 + 1)) = Σ𝑘 ∈ (0...(𝑁 + 1))(((𝑁 + 1)C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |