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 13508 |
. . . . 5
⊢ (𝑥 ∈ (0[,]1) → 𝑥 ∈
ℂ) |
| 3 | | ax-1cn 11213 |
. . . . . . . . . 10
⊢ 1 ∈
ℂ |
| 4 | | negsub 11557 |
. . . . . . . . . 10
⊢ ((1
∈ ℂ ∧ 𝑥
∈ ℂ) → (1 + -𝑥) = (1 − 𝑥)) |
| 5 | 3, 4 | mpan 690 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℂ → (1 +
-𝑥) = (1 − 𝑥)) |
| 6 | 5 | oveq1d 7446 |
. . . . . . . 8
⊢ (𝑥 ∈ ℂ → ((1 +
-𝑥)↑(𝑁 − 𝑀)) = ((1 − 𝑥)↑(𝑁 − 𝑀))) |
| 7 | 6 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((1 + -𝑥)↑(𝑁 − 𝑀)) = ((1 − 𝑥)↑(𝑁 − 𝑀))) |
| 8 | | negcl 11508 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℂ → -𝑥 ∈
ℂ) |
| 9 | | 1cnd 11256 |
. . . . . . . . . 10
⊢ (𝜑 → 1 ∈
ℂ) |
| 10 | | lcmineqlem1.4 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ≤ 𝑁) |
| 11 | | lcmineqlem1.3 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀 ∈ ℕ) |
| 12 | 11 | nnnn0d 12587 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑀 ∈
ℕ0) |
| 13 | | lcmineqlem1.2 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑁 ∈ ℕ) |
| 14 | 13 | nnnn0d 12587 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
| 15 | | nn0sub 12576 |
. . . . . . . . . . . 12
⊢ ((𝑀 ∈ ℕ0
∧ 𝑁 ∈
ℕ0) → (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈
ℕ0)) |
| 16 | 12, 14, 15 | syl2anc 584 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈
ℕ0)) |
| 17 | 10, 16 | mpbid 232 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁 − 𝑀) ∈
ℕ0) |
| 18 | | binom 15866 |
. . . . . . . . . . . 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 1127 |
. . . . . . . . . . 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 1122 |
. . . . . . . . . 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 584 |
. . . . . . . . 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 406 |
. . . . . . 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 2779 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((1 − 𝑥)↑(𝑁 − 𝑀)) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((𝑁 − 𝑀)C𝑘) · ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)))) |
| 25 | | elfzelz 13564 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → 𝑘 ∈ ℤ) |
| 26 | 13 | nnzd 12640 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 27 | 11 | nnzd 12640 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 28 | | zsubcl 12659 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 − 𝑀) ∈ ℤ) |
| 29 | 26, 27, 28 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℤ) |
| 30 | | zsubcl 12659 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑁 − 𝑀) ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((𝑁 − 𝑀) − 𝑘) ∈ ℤ) |
| 31 | 29, 30 | sylan 580 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → ((𝑁 − 𝑀) − 𝑘) ∈ ℤ) |
| 32 | 25, 31 | sylan2 593 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((𝑁 − 𝑀) − 𝑘) ∈ ℤ) |
| 33 | | 1exp 14132 |
. . . . . . . . . . . . . . . . 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 1132 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (1↑((𝑁 − 𝑀) − 𝑘)) = 1) |
| 36 | 35 | oveq1d 7446 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)) = (1 · (-𝑥↑𝑘))) |
| 37 | 8 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → -𝑥 ∈ ℂ) |
| 38 | | elfznn0 13660 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ (0...(𝑁 − 𝑀)) → 𝑘 ∈ ℕ0) |
| 39 | 38 | 3ad2ant3 1136 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → 𝑘 ∈ ℕ0) |
| 40 | | expcl 14120 |
. . . . . . . . . . . . . . . 16
⊢ ((-𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0)
→ (-𝑥↑𝑘) ∈
ℂ) |
| 41 | 37, 39, 40 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (-𝑥↑𝑘) ∈ ℂ) |
| 42 | 41 | mullidd 11279 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (1 · (-𝑥↑𝑘)) = (-𝑥↑𝑘)) |
| 43 | 36, 42 | eqtrd 2777 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)) = (-𝑥↑𝑘)) |
| 44 | | mulm1 11704 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ℂ → (-1
· 𝑥) = -𝑥) |
| 45 | 44 | oveq1d 7446 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ℂ → ((-1
· 𝑥)↑𝑘) = (-𝑥↑𝑘)) |
| 46 | 45 | 3ad2ant2 1135 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((-1 · 𝑥)↑𝑘) = (-𝑥↑𝑘)) |
| 47 | 43, 46 | eqtr4d 2780 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)) = ((-1 · 𝑥)↑𝑘)) |
| 48 | | neg1cn 12380 |
. . . . . . . . . . . . . . 15
⊢ -1 ∈
ℂ |
| 49 | | mulexp 14142 |
. . . . . . . . . . . . . . 15
⊢ ((-1
∈ ℂ ∧ 𝑥
∈ ℂ ∧ 𝑘
∈ ℕ0) → ((-1 · 𝑥)↑𝑘) = ((-1↑𝑘) · (𝑥↑𝑘))) |
| 50 | 48, 49 | mp3an1 1450 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0)
→ ((-1 · 𝑥)↑𝑘) = ((-1↑𝑘) · (𝑥↑𝑘))) |
| 51 | 38, 50 | sylan2 593 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((-1 · 𝑥)↑𝑘) = ((-1↑𝑘) · (𝑥↑𝑘))) |
| 52 | 51 | 3adant1 1131 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((-1 · 𝑥)↑𝑘) = ((-1↑𝑘) · (𝑥↑𝑘))) |
| 53 | 47, 52 | eqtrd 2777 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((1↑((𝑁 − 𝑀) − 𝑘)) · (-𝑥↑𝑘)) = ((-1↑𝑘) · (𝑥↑𝑘))) |
| 54 | 53 | oveq2d 7447 |
. . . . . . . . . 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 14361 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 − 𝑀) ∈ ℕ0 ∧ 𝑘 ∈ ℤ) → ((𝑁 − 𝑀)C𝑘) ∈
ℕ0) |
| 56 | 17, 25, 55 | syl2an 596 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((𝑁 − 𝑀)C𝑘) ∈
ℕ0) |
| 57 | 56 | 3adant2 1132 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((𝑁 − 𝑀)C𝑘) ∈
ℕ0) |
| 58 | 57 | nn0cnd 12589 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → ((𝑁 − 𝑀)C𝑘) ∈ ℂ) |
| 59 | | expcl 14120 |
. . . . . . . . . . . 12
⊢ ((-1
∈ ℂ ∧ 𝑘
∈ ℕ0) → (-1↑𝑘) ∈ ℂ) |
| 60 | 48, 39, 59 | sylancr 587 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (-1↑𝑘) ∈ ℂ) |
| 61 | | expcl 14120 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ ℕ0)
→ (𝑥↑𝑘) ∈
ℂ) |
| 62 | 38, 61 | sylan2 593 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (𝑥↑𝑘) ∈ ℂ) |
| 63 | 62 | 3adant1 1131 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (𝑥↑𝑘) ∈ ℂ) |
| 64 | 58, 60, 63 | mulassd 11284 |
. . . . . . . . . 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 2780 |
. . . . . . . . 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 11282 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ ∧ 𝑘 ∈ (0...(𝑁 − 𝑀))) → (((𝑁 − 𝑀)C𝑘) · (-1↑𝑘)) = ((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘))) |
| 67 | 66 | oveq1d 7446 |
. . . . . . . . 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 2777 |
. . . . . . . 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 1119 |
. . . . . . 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 15738 |
. . . . . 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 2777 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((1 − 𝑥)↑(𝑁 − 𝑀)) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘))) |
| 72 | 2, 71 | sylan2 593 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → ((1 − 𝑥)↑(𝑁 − 𝑀)) = Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘))) |
| 73 | 72 | oveq2d 7447 |
. . 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 25817 |
. 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 | eqtrid 2789 |
1
⊢ (𝜑 → 𝐹 = ∫(0[,]1)((𝑥↑(𝑀 − 1)) · Σ𝑘 ∈ (0...(𝑁 − 𝑀))(((-1↑𝑘) · ((𝑁 − 𝑀)C𝑘)) · (𝑥↑𝑘))) d𝑥) |