Step | Hyp | Ref
| Expression |
1 | | binomlem.4 |
. . . . . 6
⊢ (𝜓 → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)))) |
2 | 1 | adantl 482 |
. . . . 5
⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + 𝐵)↑𝑁) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)))) |
3 | 2 | oveq1d 6665 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + 𝐵)↑𝑁) · 𝐴) = (Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐴)) |
4 | | fzfid 12772 |
. . . . . . 7
⊢ (𝜑 → (0...𝑁) ∈ Fin) |
5 | | binomlem.1 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ ℂ) |
6 | | fzelp1 12393 |
. . . . . . . . 9
⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ (0...(𝑁 + 1))) |
7 | | binomlem.3 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
8 | | elfzelz 12342 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ (0...(𝑁 + 1)) → 𝑘 ∈ ℤ) |
9 | | bccl 13109 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈ ℤ)
→ (𝑁C𝑘) ∈
ℕ0) |
10 | 7, 8, 9 | syl2an 494 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝑁C𝑘) ∈
ℕ0) |
11 | 10 | nn0cnd 11353 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝑁C𝑘) ∈ ℂ) |
12 | 6, 11 | sylan2 491 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝑁C𝑘) ∈ ℂ) |
13 | | fznn0sub 12373 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (0...𝑁) → (𝑁 − 𝑘) ∈
ℕ0) |
14 | | expcl 12878 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 𝑘) ∈ ℕ0) → (𝐴↑(𝑁 − 𝑘)) ∈ ℂ) |
15 | 5, 13, 14 | syl2an 494 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐴↑(𝑁 − 𝑘)) ∈ ℂ) |
16 | | binomlem.2 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐵 ∈ ℂ) |
17 | | elfznn0 12433 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ (0...(𝑁 + 1)) → 𝑘 ∈ ℕ0) |
18 | | expcl 12878 |
. . . . . . . . . . 11
⊢ ((𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ0)
→ (𝐵↑𝑘) ∈
ℂ) |
19 | 16, 17, 18 | syl2an 494 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝐵↑𝑘) ∈ ℂ) |
20 | 6, 19 | sylan2 491 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐵↑𝑘) ∈ ℂ) |
21 | 15, 20 | mulcld 10060 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)) ∈ ℂ) |
22 | 12, 21 | mulcld 10060 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) ∈ ℂ) |
23 | 4, 5, 22 | fsummulc1 14517 |
. . . . . 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 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ∈ ℂ) |
25 | 12, 21, 24 | mulassd 10063 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐴) = ((𝑁C𝑘) · (((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)) · 𝐴))) |
26 | 7 | nn0cnd 11353 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈ ℂ) |
27 | 26 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑁 ∈ ℂ) |
28 | | 1cnd 10056 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 1 ∈ ℂ) |
29 | | elfzelz 12342 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℤ) |
30 | 29 | adantl 482 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℤ) |
31 | 30 | zcnd 11483 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℂ) |
32 | 27, 28, 31 | addsubd 10413 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁 + 1) − 𝑘) = ((𝑁 − 𝑘) + 1)) |
33 | 32 | oveq2d 6666 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐴↑((𝑁 + 1) − 𝑘)) = (𝐴↑((𝑁 − 𝑘) + 1))) |
34 | | expp1 12867 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℂ ∧ (𝑁 − 𝑘) ∈ ℕ0) → (𝐴↑((𝑁 − 𝑘) + 1)) = ((𝐴↑(𝑁 − 𝑘)) · 𝐴)) |
35 | 5, 13, 34 | syl2an 494 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐴↑((𝑁 − 𝑘) + 1)) = ((𝐴↑(𝑁 − 𝑘)) · 𝐴)) |
36 | 33, 35 | eqtrd 2656 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝐴↑((𝑁 + 1) − 𝑘)) = ((𝐴↑(𝑁 − 𝑘)) · 𝐴)) |
37 | 36 | oveq1d 6665 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)) = (((𝐴↑(𝑁 − 𝑘)) · 𝐴) · (𝐵↑𝑘))) |
38 | 15, 24, 20 | mul32d 10246 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((𝐴↑(𝑁 − 𝑘)) · 𝐴) · (𝐵↑𝑘)) = (((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)) · 𝐴)) |
39 | 37, 38 | eqtrd 2656 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)) = (((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)) · 𝐴)) |
40 | 39 | oveq2d 6666 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = ((𝑁C𝑘) · (((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)) · 𝐴))) |
41 | 25, 40 | eqtr4d 2659 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐴) = ((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
42 | 41 | sumeq2dv 14433 |
. . . . . 6
⊢ (𝜑 → Σ𝑘 ∈ (0...𝑁)(((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐴) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
43 | | fzssp1 12384 |
. . . . . . . 8
⊢
(0...𝑁) ⊆
(0...(𝑁 +
1)) |
44 | 43 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (0...𝑁) ⊆ (0...(𝑁 + 1))) |
45 | | fznn0sub 12373 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ (0...(𝑁 + 1)) → ((𝑁 + 1) − 𝑘) ∈
ℕ0) |
46 | | expcl 12878 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧ ((𝑁 + 1) − 𝑘) ∈ ℕ0) → (𝐴↑((𝑁 + 1) − 𝑘)) ∈ ℂ) |
47 | 5, 45, 46 | syl2an 494 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝐴↑((𝑁 + 1) − 𝑘)) ∈ ℂ) |
48 | 47, 19 | mulcld 10060 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)) ∈ ℂ) |
49 | 11, 48 | mulcld 10060 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → ((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) ∈ ℂ) |
50 | 6, 49 | sylan2 491 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) ∈ ℂ) |
51 | 7 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁))) → 𝑁 ∈
ℕ0) |
52 | | eldifi 3732 |
. . . . . . . . . . . 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 482 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁))) → 𝑘 ∈ ℤ) |
55 | | eldifn 3733 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁)) → ¬ 𝑘 ∈ (0...𝑁)) |
56 | 55 | adantl 482 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁))) → ¬ 𝑘 ∈ (0...𝑁)) |
57 | | bcval3 13093 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈ ℤ
∧ ¬ 𝑘 ∈
(0...𝑁)) → (𝑁C𝑘) = 0) |
58 | 51, 54, 56, 57 | syl3anc 1326 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁))) → (𝑁C𝑘) = 0) |
59 | 58 | oveq1d 6665 |
. . . . . . . 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 10234 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (0 · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = 0) |
61 | 52, 60 | sylan2 491 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁))) → (0 · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = 0) |
62 | 59, 61 | eqtrd 2656 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (0...𝑁))) → ((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = 0) |
63 | | fzssuz 12382 |
. . . . . . . 8
⊢
(0...(𝑁 + 1))
⊆ (ℤ≥‘0) |
64 | 63 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (0...(𝑁 + 1)) ⊆
(ℤ≥‘0)) |
65 | 44, 50, 62, 64 | sumss 14455 |
. . . . . 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 2660 |
. . . . 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 481 |
. . . 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 2656 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + 𝐵)↑𝑁) · 𝐴) = Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
69 | 1 | oveq1d 6665 |
. . . 4
⊢ (𝜓 → (((𝐴 + 𝐵)↑𝑁) · 𝐵) = (Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐵)) |
70 | 4, 16, 22 | fsummulc1 14517 |
. . . . 5
⊢ (𝜑 → (Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐵) = Σ𝑘 ∈ (0...𝑁)(((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐵)) |
71 | | 1zzd 11408 |
. . . . . . . 8
⊢ (𝜑 → 1 ∈
ℤ) |
72 | | 0z 11388 |
. . . . . . . . 9
⊢ 0 ∈
ℤ |
73 | 72 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈
ℤ) |
74 | 7 | nn0zd 11480 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ ℤ) |
75 | 16 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ ℂ) |
76 | 22, 75 | mulcld 10060 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) · 𝐵) ∈ ℂ) |
77 | | oveq2 6658 |
. . . . . . . . . 10
⊢ (𝑘 = (𝑗 − 1) → (𝑁C𝑘) = (𝑁C(𝑗 − 1))) |
78 | | oveq2 6658 |
. . . . . . . . . . . 12
⊢ (𝑘 = (𝑗 − 1) → (𝑁 − 𝑘) = (𝑁 − (𝑗 − 1))) |
79 | 78 | oveq2d 6666 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑗 − 1) → (𝐴↑(𝑁 − 𝑘)) = (𝐴↑(𝑁 − (𝑗 − 1)))) |
80 | | oveq2 6658 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝑗 − 1) → (𝐵↑𝑘) = (𝐵↑(𝑗 − 1))) |
81 | 79, 80 | oveq12d 6668 |
. . . . . . . . . 10
⊢ (𝑘 = (𝑗 − 1) → ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑(𝑁 − (𝑗 − 1))) · (𝐵↑(𝑗 − 1)))) |
82 | 77, 81 | oveq12d 6668 |
. . . . . . . . 9
⊢ (𝑘 = (𝑗 − 1) → ((𝑁C𝑘) · ((𝐴↑(𝑁 − 𝑘)) · (𝐵↑𝑘))) = ((𝑁C(𝑗 − 1)) · ((𝐴↑(𝑁 − (𝑗 − 1))) · (𝐵↑(𝑗 − 1))))) |
83 | 82 | oveq1d 6665 |
. . . . . . . 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 14512 |
. . . . . . 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 6657 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑘 → (𝑗 − 1) = (𝑘 − 1)) |
86 | 85 | oveq2d 6666 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑘 → (𝑁C(𝑗 − 1)) = (𝑁C(𝑘 − 1))) |
87 | 85 | oveq2d 6666 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑘 → (𝑁 − (𝑗 − 1)) = (𝑁 − (𝑘 − 1))) |
88 | 87 | oveq2d 6666 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑘 → (𝐴↑(𝑁 − (𝑗 − 1))) = (𝐴↑(𝑁 − (𝑘 − 1)))) |
89 | 85 | oveq2d 6666 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑘 → (𝐵↑(𝑗 − 1)) = (𝐵↑(𝑘 − 1))) |
90 | 88, 89 | oveq12d 6668 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑘 → ((𝐴↑(𝑁 − (𝑗 − 1))) · (𝐵↑(𝑗 − 1))) = ((𝐴↑(𝑁 − (𝑘 − 1))) · (𝐵↑(𝑘 − 1)))) |
91 | 86, 90 | oveq12d 6668 |
. . . . . . . . 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 6665 |
. . . . . . . 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 14426 |
. . . . . . 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 | syl6eq 2672 |
. . . . . 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 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → 𝑁 ∈ ℂ) |
96 | | elfzelz 12342 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ((0 + 1)...(𝑁 + 1)) → 𝑘 ∈ ℤ) |
97 | 96 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → 𝑘 ∈ ℤ) |
98 | 97 | zcnd 11483 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → 𝑘 ∈ ℂ) |
99 | | 1cnd 10056 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → 1 ∈
ℂ) |
100 | 95, 98, 99 | subsub3d 10422 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (𝑁 − (𝑘 − 1)) = ((𝑁 + 1) − 𝑘)) |
101 | 100 | oveq2d 6666 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (𝐴↑(𝑁 − (𝑘 − 1))) = (𝐴↑((𝑁 + 1) − 𝑘))) |
102 | 101 | oveq1d 6665 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → ((𝐴↑(𝑁 − (𝑘 − 1))) · (𝐵↑(𝑘 − 1))) = ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑(𝑘 − 1)))) |
103 | 102 | oveq2d 6666 |
. . . . . . . . 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 6665 |
. . . . . . . 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 12392 |
. . . . . . . . . . . 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 3599 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ((0 + 1)...(𝑁 + 1)) → 𝑘 ∈ (0...(𝑁 + 1))) |
108 | 7 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → 𝑁 ∈
ℕ0) |
109 | 8 | adantl 482 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → 𝑘 ∈ ℤ) |
110 | | peano2zm 11420 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℤ → (𝑘 − 1) ∈
ℤ) |
111 | 109, 110 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝑘 − 1) ∈ ℤ) |
112 | | bccl 13109 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ0
∧ (𝑘 − 1) ∈
ℤ) → (𝑁C(𝑘 − 1)) ∈
ℕ0) |
113 | 108, 111,
112 | syl2anc 693 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝑁C(𝑘 − 1)) ∈
ℕ0) |
114 | 113 | nn0cnd 11353 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝑁C(𝑘 − 1)) ∈ ℂ) |
115 | 107, 114 | sylan2 491 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (𝑁C(𝑘 − 1)) ∈ ℂ) |
116 | 107, 47 | sylan2 491 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (𝐴↑((𝑁 + 1) − 𝑘)) ∈ ℂ) |
117 | 16 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → 𝐵 ∈ ℂ) |
118 | | elfznn 12370 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ (1...(𝑁 + 1)) → 𝑘 ∈ ℕ) |
119 | | 0p1e1 11132 |
. . . . . . . . . . . . . . 15
⊢ (0 + 1) =
1 |
120 | 119 | oveq1i 6660 |
. . . . . . . . . . . . . 14
⊢ ((0 +
1)...(𝑁 + 1)) = (1...(𝑁 + 1)) |
121 | 118, 120 | eleq2s 2719 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ((0 + 1)...(𝑁 + 1)) → 𝑘 ∈ ℕ) |
122 | 121 | adantl 482 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → 𝑘 ∈ ℕ) |
123 | | nnm1nn0 11334 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈
ℕ0) |
124 | 122, 123 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (𝑘 − 1) ∈
ℕ0) |
125 | 117, 124 | expcld 13008 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (𝐵↑(𝑘 − 1)) ∈ ℂ) |
126 | 116, 125 | mulcld 10060 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑(𝑘 − 1))) ∈
ℂ) |
127 | 115, 126,
117 | mulassd 10063 |
. . . . . . . 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))) |
128 | 116, 125,
117 | mulassd 10063 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑(𝑘 − 1))) · 𝐵) = ((𝐴↑((𝑁 + 1) − 𝑘)) · ((𝐵↑(𝑘 − 1)) · 𝐵))) |
129 | | expm1t 12888 |
. . . . . . . . . . . 12
⊢ ((𝐵 ∈ ℂ ∧ 𝑘 ∈ ℕ) → (𝐵↑𝑘) = ((𝐵↑(𝑘 − 1)) · 𝐵)) |
130 | 16, 121, 129 | syl2an 494 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (𝐵↑𝑘) = ((𝐵↑(𝑘 − 1)) · 𝐵)) |
131 | 130 | oveq2d 6666 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)) = ((𝐴↑((𝑁 + 1) − 𝑘)) · ((𝐵↑(𝑘 − 1)) · 𝐵))) |
132 | 128, 131 | eqtr4d 2659 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → (((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑(𝑘 − 1))) · 𝐵) = ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) |
133 | 132 | oveq2d 6666 |
. . . . . . . 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)))) |
134 | 104, 127,
133 | 3eqtrd 2660 |
. . . . . . 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)))) |
135 | 134 | sumeq2dv 14433 |
. . . . . 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)))) |
136 | 106 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → ((0 + 1)...(𝑁 + 1)) ⊆ (0...(𝑁 + 1))) |
137 | 114, 48 | mulcld 10060 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → ((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) ∈ ℂ) |
138 | 107, 137 | sylan2 491 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) → ((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) ∈ ℂ) |
139 | 7 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → 𝑁 ∈
ℕ0) |
140 | | eldifi 3732 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1))) → 𝑘 ∈ (0...(𝑁 + 1))) |
141 | 140 | adantl 482 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → 𝑘 ∈ (0...(𝑁 + 1))) |
142 | 141, 8 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → 𝑘 ∈ ℤ) |
143 | 142, 110 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → (𝑘 − 1) ∈ ℤ) |
144 | | eldifn 3733 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1))) → ¬ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) |
145 | 144 | adantl 482 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → ¬ 𝑘 ∈ ((0 + 1)...(𝑁 + 1))) |
146 | 72 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → 0 ∈
ℤ) |
147 | 139 | nn0zd 11480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → 𝑁 ∈ ℤ) |
148 | | 1zzd 11408 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → 1 ∈
ℤ) |
149 | | fzaddel 12375 |
. . . . . . . . . . . . 13
⊢ (((0
∈ ℤ ∧ 𝑁
∈ ℤ) ∧ ((𝑘
− 1) ∈ ℤ ∧ 1 ∈ ℤ)) → ((𝑘 − 1) ∈ (0...𝑁) ↔ ((𝑘 − 1) + 1) ∈ ((0 + 1)...(𝑁 + 1)))) |
150 | 146, 147,
143, 148, 149 | syl22anc 1327 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → ((𝑘 − 1) ∈ (0...𝑁) ↔ ((𝑘 − 1) + 1) ∈ ((0 + 1)...(𝑁 + 1)))) |
151 | 142 | zcnd 11483 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → 𝑘 ∈ ℂ) |
152 | | ax-1cn 9994 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
ℂ |
153 | | npcan 10290 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑘 −
1) + 1) = 𝑘) |
154 | 151, 152,
153 | sylancl 694 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → ((𝑘 − 1) + 1) = 𝑘) |
155 | 154 | eleq1d 2686 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → (((𝑘 − 1) + 1) ∈ ((0 + 1)...(𝑁 + 1)) ↔ 𝑘 ∈ ((0 + 1)...(𝑁 + 1)))) |
156 | 150, 155 | bitrd 268 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → ((𝑘 − 1) ∈ (0...𝑁) ↔ 𝑘 ∈ ((0 + 1)...(𝑁 + 1)))) |
157 | 145, 156 | mtbird 315 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → ¬ (𝑘 − 1) ∈ (0...𝑁)) |
158 | | bcval3 13093 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℕ0
∧ (𝑘 − 1) ∈
ℤ ∧ ¬ (𝑘
− 1) ∈ (0...𝑁))
→ (𝑁C(𝑘 − 1)) =
0) |
159 | 139, 143,
157, 158 | syl3anc 1326 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → (𝑁C(𝑘 − 1)) = 0) |
160 | 159 | oveq1d 6665 |
. . . . . . . 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)))) |
161 | 140, 60 | sylan2 491 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → (0 · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = 0) |
162 | 160, 161 | eqtrd 2656 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ ((0 + 1)...(𝑁 + 1)))) → ((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘))) = 0) |
163 | 136, 138,
162, 64 | sumss 14455 |
. . . . . 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)))) |
164 | 94, 135, 163 | 3eqtrd 2660 |
. . . . 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)))) |
165 | 70, 164 | eqtrd 2656 |
. . . 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)))) |
166 | 69, 165 | sylan9eqr 2678 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → (((𝐴 + 𝐵)↑𝑁) · 𝐵) = Σ𝑘 ∈ (0...(𝑁 + 1))((𝑁C(𝑘 − 1)) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |
167 | 68, 166 | oveq12d 6668 |
. 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))))) |
168 | 5, 16 | addcld 10059 |
. . . . 5
⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ) |
169 | 168, 7 | expp1d 13009 |
. . . 4
⊢ (𝜑 → ((𝐴 + 𝐵)↑(𝑁 + 1)) = (((𝐴 + 𝐵)↑𝑁) · (𝐴 + 𝐵))) |
170 | 168, 7 | expcld 13008 |
. . . . 5
⊢ (𝜑 → ((𝐴 + 𝐵)↑𝑁) ∈ ℂ) |
171 | 170, 5, 16 | adddid 10064 |
. . . 4
⊢ (𝜑 → (((𝐴 + 𝐵)↑𝑁) · (𝐴 + 𝐵)) = ((((𝐴 + 𝐵)↑𝑁) · 𝐴) + (((𝐴 + 𝐵)↑𝑁) · 𝐵))) |
172 | 169, 171 | eqtrd 2656 |
. . 3
⊢ (𝜑 → ((𝐴 + 𝐵)↑(𝑁 + 1)) = ((((𝐴 + 𝐵)↑𝑁) · 𝐴) + (((𝐴 + 𝐵)↑𝑁) · 𝐵))) |
173 | 172 | adantr 481 |
. 2
⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + 𝐵)↑(𝑁 + 1)) = ((((𝐴 + 𝐵)↑𝑁) · 𝐴) + (((𝐴 + 𝐵)↑𝑁) · 𝐵))) |
174 | | bcpasc 13108 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ0
∧ 𝑘 ∈ ℤ)
→ ((𝑁C𝑘) + (𝑁C(𝑘 − 1))) = ((𝑁 + 1)C𝑘)) |
175 | 7, 8, 174 | syl2an 494 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → ((𝑁C𝑘) + (𝑁C(𝑘 − 1))) = ((𝑁 + 1)C𝑘)) |
176 | 175 | oveq1d 6665 |
. . . . . 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)))) |
177 | 11, 114, 48 | adddird 10065 |
. . . . . 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))))) |
178 | 176, 177 | eqtr3d 2658 |
. . . . 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))))) |
179 | 178 | sumeq2dv 14433 |
. . . 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))))) |
180 | | fzfid 12772 |
. . . . 5
⊢ (𝜑 → (0...(𝑁 + 1)) ∈ Fin) |
181 | 180, 49, 137 | fsumadd 14470 |
. . . 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))))) |
182 | 179, 181 | eqtrd 2656 |
. . 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))))) |
183 | 182 | adantr 481 |
. 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))))) |
184 | 167, 173,
183 | 3eqtr4d 2666 |
1
⊢ ((𝜑 ∧ 𝜓) → ((𝐴 + 𝐵)↑(𝑁 + 1)) = Σ𝑘 ∈ (0...(𝑁 + 1))(((𝑁 + 1)C𝑘) · ((𝐴↑((𝑁 + 1) − 𝑘)) · (𝐵↑𝑘)))) |