Proof of Theorem lcmineqlem1
Step | Hyp | Ref
| Expression |
1 | | lcmineqlem1.1 |
. 2
⊢ 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥 |
2 | | elunitcn 12885 |
. . . . 5
⊢ (𝑥 ∈ (0[,]1) → 𝑥 ∈
ℂ) |
3 | | ax-1cn 10618 |
. . . . . . . . . 10
⊢ 1 ∈
ℂ |
4 | | negsub 10957 |
. . . . . . . . . 10
⊢ ((1
∈ ℂ ∧ 𝑥
∈ ℂ) → (1 + -𝑥) = (1 − 𝑥)) |
5 | 3, 4 | mpan 690 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℂ → (1 +
-𝑥) = (1 − 𝑥)) |
6 | 5 | oveq1d 7158 |
. . . . . . . 8
⊢ (𝑥 ∈ ℂ → ((1 +
-𝑥)↑(𝑁 − 𝑀)) = ((1 − 𝑥)↑(𝑁 − 𝑀))) |
7 | 6 | adantl 486 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((1 + -𝑥)↑(𝑁 − 𝑀)) = ((1 − 𝑥)↑(𝑁 − 𝑀))) |
8 | | negcl 10909 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℂ → -𝑥 ∈
ℂ) |
9 | | 1cnd 10659 |
. . . . . . . . . 10
⊢ (𝜑 → 1 ∈
ℂ) |
10 | | lcmineqlem1.4 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ≤ 𝑁) |
11 | | lcmineqlem1.3 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀 ∈ ℕ) |
12 | 11 | nnnn0d 11979 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 ∈
ℕ0) |
13 | | lcmineqlem1.2 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑁 ∈ ℕ) |
14 | 13 | nnnn0d 11979 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
15 | | nn0sub 11969 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℕ0) → (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈
ℕ0)) |
16 | 12, 14, 15 | syl2anc 588 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈
ℕ0)) |
17 | 10, 16 | mpbid 235 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁 − 𝑀) ∈
ℕ0) |
18 | | binom 15218 |
. . . . . . . . . . . 12
⊢ ((1
∈ ℂ ∧ -𝑥
∈ ℂ ∧ (𝑁
− 𝑀) ∈
ℕ0) → ((1 + -𝑥)↑(𝑁 − 𝑀)) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)))) |
19 | 18 | 3com23 1124 |
. . . . . . . . . . 11
⊢ ((1
∈ ℂ ∧ (𝑁
− 𝑀) ∈
ℕ0 ∧ -𝑥 ∈ ℂ) → ((1 + -𝑥)↑(𝑁 − 𝑀)) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)))) |
20 | 19 | 3expia 1119 |
. . . . . . . . . 10
⊢ ((1
∈ ℂ ∧ (𝑁
− 𝑀) ∈
ℕ0) → (-𝑥 ∈ ℂ → ((1 + -𝑥)↑(𝑁 − 𝑀)) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘))))) |
21 | 9, 17, 20 | syl2anc 588 |
. . . . . . . . 9
⊢ (𝜑 → (-𝑥 ∈ ℂ → ((1 + -𝑥)↑(𝑁 − 𝑀)) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘))))) |
22 | 8, 21 | syl5 34 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ ℂ → ((1 + -𝑥)↑(𝑁 − 𝑀)) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘))))) |
23 | 22 | imp 411 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((1 + -𝑥)↑(𝑁 − 𝑀)) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)))) |
24 | 7, 23 | eqtr3d 2796 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((1 − 𝑥)↑(𝑁 − 𝑀)) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)))) |
25 | | elfzelz 12941 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → 𝑘 ∈ ℤ) |
26 | 13 | nnzd 12110 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑁 ∈ ℤ) |
27 | 11 | nnzd 12110 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑀 ∈ ℤ) |
28 | | zsubcl 12048 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 − 𝑀) ∈ ℤ) |
29 | 26, 27, 28 | syl2anc 588 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℤ) |
30 | | zsubcl 12048 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑁 − 𝑀) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((𝑁 − 𝑀) − 𝑘) ∈ ℤ) |
31 | 29, 30 | sylan 584 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → ((𝑁 − 𝑀) − 𝑘) ∈ ℤ) |
32 | 25, 31 | sylan2 596 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((𝑁 − 𝑀) − 𝑘) ∈ ℤ) |
33 | | 1exp 13493 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑁 − 𝑀) − 𝑘) ∈ ℤ → (1↑((𝑁 − 𝑀) − 𝑘)) = 1) |
34 | 32, 33 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (1↑((𝑁 − 𝑀) − 𝑘)) = 1) |
35 | 34 | 3adant2 1129 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (1↑((𝑁 − 𝑀) − 𝑘)) = 1) |
36 | 35 | oveq1d 7158 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)) = (1 · (-𝑥↑𝑘))) |
37 | 8 | 3ad2ant2 1132 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → -𝑥 ∈ ℂ) |
38 | | elfznn0 13034 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → 𝑘 ∈ ℕ0) |
39 | 38 | 3ad2ant3 1133 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 𝑘 ∈ ℕ0) |
40 | | expcl 13482 |
. . . . . . . . . . . . . . . 16
⊢ ((-𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0)
→ (-𝑥↑𝑘) ∈
ℂ) |
41 | 37, 39, 40 | syl2anc 588 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (-𝑥↑𝑘) ∈ ℂ) |
42 | 41 | mulid2d 10682 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (1 · (-𝑥↑𝑘)) = (-𝑥↑𝑘)) |
43 | 36, 42 | eqtrd 2794 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)) = (-𝑥↑𝑘)) |
44 | | mulm1 11104 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ℂ → (-1
· 𝑥) = -𝑥) |
45 | 44 | oveq1d 7158 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℂ → ((-1
· 𝑥)↑𝑘) = (-𝑥↑𝑘)) |
46 | 45 | 3ad2ant2 1132 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((-1 · 𝑥)↑𝑘) = (-𝑥↑𝑘)) |
47 | 43, 46 | eqtr4d 2797 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)) = ((-1 · 𝑥)↑𝑘)) |
48 | | neg1cn 11773 |
. . . . . . . . . . . . . . 15
⊢ -1 ∈
ℂ |
49 | | mulexp 13503 |
. . . . . . . . . . . . . . 15
⊢ ((-1
∈ ℂ ∧ 𝑥
∈ ℂ ∧ 𝑘
∈ ℕ0) → ((-1 · 𝑥)↑𝑘) = ((-1↑𝑘) · (𝑥↑𝑘))) |
50 | 48, 49 | mp3an1 1446 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0)
→ ((-1 · 𝑥)↑𝑘) = ((-1↑𝑘) · (𝑥↑𝑘))) |
51 | 38, 50 | sylan2 596 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((-1 · 𝑥)↑𝑘) = ((-1↑𝑘) · (𝑥↑𝑘))) |
52 | 51 | 3adant1 1128 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((-1 · 𝑥)↑𝑘) = ((-1↑𝑘) · (𝑥↑𝑘))) |
53 | 47, 52 | eqtrd 2794 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)) = ((-1↑𝑘) · (𝑥↑𝑘))) |
54 | 53 | oveq2d 7159 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘))) = (((𝑁 − 𝑀)C𝑘) · ((-1↑𝑘) · (𝑥↑𝑘)))) |
55 | | bccl 13717 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 − 𝑀) ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → ((𝑁 − 𝑀)C𝑘) ∈
ℕ0) |
56 | 17, 25, 55 | syl2an 599 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((𝑁 − 𝑀)C𝑘) ∈
ℕ0) |
57 | 56 | 3adant2 1129 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((𝑁 − 𝑀)C𝑘) ∈
ℕ0) |
58 | 57 | nn0cnd 11981 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((𝑁 − 𝑀)C𝑘) ∈ ℂ) |
59 | | expcl 13482 |
. . . . . . . . . . . 12
⊢ ((-1
∈ ℂ ∧ 𝑘
∈ ℕ0) → (-1↑𝑘) ∈ ℂ) |
60 | 48, 39, 59 | sylancr 591 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (-1↑𝑘) ∈ ℂ) |
61 | | expcl 13482 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0)
→ (𝑥↑𝑘) ∈
ℂ) |
62 | 38, 61 | sylan2 596 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (𝑥↑𝑘) ∈ ℂ) |
63 | 62 | 3adant1 1128 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (𝑥↑𝑘) ∈ ℂ) |
64 | 58, 60, 63 | mulassd 10687 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((((𝑁 − 𝑀)C𝑘) · (-1↑𝑘)) · (𝑥↑𝑘)) = (((𝑁 − 𝑀)C𝑘) · ((-1↑𝑘) · (𝑥↑𝑘)))) |
65 | 54, 64 | eqtr4d 2797 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘))) = ((((𝑁 − 𝑀)C𝑘) · (-1↑𝑘)) · (𝑥↑𝑘))) |
66 | 58, 60 | mulcomd 10685 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (((𝑁 − 𝑀)C𝑘) · (-1↑𝑘)) = ((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘))) |
67 | 66 | oveq1d 7158 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((((𝑁 − 𝑀)C𝑘) · (-1↑𝑘)) · (𝑥↑𝑘)) = (((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘))) |
68 | 65, 67 | eqtrd 2794 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘))) = (((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘))) |
69 | 68 | 3expa 1116 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘))) = (((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘))) |
70 | 69 | sumeq2dv 15093 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘))) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘))) |
71 | 24, 70 | eqtrd 2794 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((1 − 𝑥)↑(𝑁 − 𝑀)) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘))) |
72 | 2, 71 | sylan2 596 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → ((1 − 𝑥)↑(𝑁 − 𝑀)) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘))) |
73 | 72 | oveq2d 7159 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → ((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) = ((𝑥↑(𝑀 − 1)) · Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘)))) |
74 | 73 | itgeq2dv 24466 |
. 2
⊢ (𝜑 → ∫(0[,]1)((𝑥↑(𝑀 − 1)) · ((1 − 𝑥)↑(𝑁 − 𝑀))) d𝑥 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘))) d𝑥) |
75 | 1, 74 | syl5eq 2806 |
1
⊢ (𝜑 → 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘))) d𝑥) |