Step |
Hyp |
Ref |
Expression |
1 |
|
lcmineqlem2.1 |
⊢ 𝐹 = ∫ ( 0 [,] 1 ) ( ( 𝑥 ↑ ( 𝑀 − 1 ) ) · ( ( 1 − 𝑥 ) ↑ ( 𝑁 − 𝑀 ) ) ) d 𝑥 |
2 |
|
lcmineqlem2.2 |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
3 |
|
lcmineqlem2.3 |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
4 |
|
lcmineqlem2.4 |
⊢ ( 𝜑 → 𝑀 ≤ 𝑁 ) |
5 |
1 2 3 4
|
lcmineqlem1 |
⊢ ( 𝜑 → 𝐹 = ∫ ( 0 [,] 1 ) ( ( 𝑥 ↑ ( 𝑀 − 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑁 − 𝑀 ) ) ( ( ( - 1 ↑ 𝑘 ) · ( ( 𝑁 − 𝑀 ) C 𝑘 ) ) · ( 𝑥 ↑ 𝑘 ) ) ) d 𝑥 ) |
6 |
|
eqid |
⊢ ( 0 [,] 1 ) = ( 0 [,] 1 ) |
7 |
|
fzfid |
⊢ ( 𝜑 → ( 0 ... ( 𝑁 − 𝑀 ) ) ∈ Fin ) |
8 |
|
0red |
⊢ ( 𝜑 → 0 ∈ ℝ ) |
9 |
|
1red |
⊢ ( 𝜑 → 1 ∈ ℝ ) |
10 |
|
unitsscn |
⊢ ( 0 [,] 1 ) ⊆ ℂ |
11 |
|
resmpt |
⊢ ( ( 0 [,] 1 ) ⊆ ℂ → ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 ↑ ( 𝑀 − 1 ) ) ) ↾ ( 0 [,] 1 ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 ↑ ( 𝑀 − 1 ) ) ) ) |
12 |
10 11
|
ax-mp |
⊢ ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 ↑ ( 𝑀 − 1 ) ) ) ↾ ( 0 [,] 1 ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 ↑ ( 𝑀 − 1 ) ) ) |
13 |
|
nnm1nn0 |
⊢ ( 𝑀 ∈ ℕ → ( 𝑀 − 1 ) ∈ ℕ0 ) |
14 |
|
expcncf |
⊢ ( ( 𝑀 − 1 ) ∈ ℕ0 → ( 𝑥 ∈ ℂ ↦ ( 𝑥 ↑ ( 𝑀 − 1 ) ) ) ∈ ( ℂ –cn→ ℂ ) ) |
15 |
3 13 14
|
3syl |
⊢ ( 𝜑 → ( 𝑥 ∈ ℂ ↦ ( 𝑥 ↑ ( 𝑀 − 1 ) ) ) ∈ ( ℂ –cn→ ℂ ) ) |
16 |
|
rescncf |
⊢ ( ( 0 [,] 1 ) ⊆ ℂ → ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 ↑ ( 𝑀 − 1 ) ) ) ∈ ( ℂ –cn→ ℂ ) → ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 ↑ ( 𝑀 − 1 ) ) ) ↾ ( 0 [,] 1 ) ) ∈ ( ( 0 [,] 1 ) –cn→ ℂ ) ) ) |
17 |
10 16
|
ax-mp |
⊢ ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 ↑ ( 𝑀 − 1 ) ) ) ∈ ( ℂ –cn→ ℂ ) → ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 ↑ ( 𝑀 − 1 ) ) ) ↾ ( 0 [,] 1 ) ) ∈ ( ( 0 [,] 1 ) –cn→ ℂ ) ) |
18 |
15 17
|
syl |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 ↑ ( 𝑀 − 1 ) ) ) ↾ ( 0 [,] 1 ) ) ∈ ( ( 0 [,] 1 ) –cn→ ℂ ) ) |
19 |
12 18
|
eqeltrrid |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 ↑ ( 𝑀 − 1 ) ) ) ∈ ( ( 0 [,] 1 ) –cn→ ℂ ) ) |
20 |
|
elfznn0 |
⊢ ( 𝑘 ∈ ( 0 ... ( 𝑁 − 𝑀 ) ) → 𝑘 ∈ ℕ0 ) |
21 |
|
neg1cn |
⊢ - 1 ∈ ℂ |
22 |
|
expcl |
⊢ ( ( - 1 ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( - 1 ↑ 𝑘 ) ∈ ℂ ) |
23 |
21 22
|
mpan |
⊢ ( 𝑘 ∈ ℕ0 → ( - 1 ↑ 𝑘 ) ∈ ℂ ) |
24 |
20 23
|
syl |
⊢ ( 𝑘 ∈ ( 0 ... ( 𝑁 − 𝑀 ) ) → ( - 1 ↑ 𝑘 ) ∈ ℂ ) |
25 |
24
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 0 ... ( 𝑁 − 𝑀 ) ) ) → ( - 1 ↑ 𝑘 ) ∈ ℂ ) |
26 |
3
|
nnnn0d |
⊢ ( 𝜑 → 𝑀 ∈ ℕ0 ) |
27 |
2
|
nnnn0d |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
28 |
|
nn0sub |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0 ) → ( 𝑀 ≤ 𝑁 ↔ ( 𝑁 − 𝑀 ) ∈ ℕ0 ) ) |
29 |
26 27 28
|
syl2anc |
⊢ ( 𝜑 → ( 𝑀 ≤ 𝑁 ↔ ( 𝑁 − 𝑀 ) ∈ ℕ0 ) ) |
30 |
4 29
|
mpbid |
⊢ ( 𝜑 → ( 𝑁 − 𝑀 ) ∈ ℕ0 ) |
31 |
|
nn0z |
⊢ ( 𝑘 ∈ ℕ0 → 𝑘 ∈ ℤ ) |
32 |
20 31
|
syl |
⊢ ( 𝑘 ∈ ( 0 ... ( 𝑁 − 𝑀 ) ) → 𝑘 ∈ ℤ ) |
33 |
|
bccl |
⊢ ( ( ( 𝑁 − 𝑀 ) ∈ ℕ0 ∧ 𝑘 ∈ ℤ ) → ( ( 𝑁 − 𝑀 ) C 𝑘 ) ∈ ℕ0 ) |
34 |
32 33
|
sylan2 |
⊢ ( ( ( 𝑁 − 𝑀 ) ∈ ℕ0 ∧ 𝑘 ∈ ( 0 ... ( 𝑁 − 𝑀 ) ) ) → ( ( 𝑁 − 𝑀 ) C 𝑘 ) ∈ ℕ0 ) |
35 |
30 34
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 0 ... ( 𝑁 − 𝑀 ) ) ) → ( ( 𝑁 − 𝑀 ) C 𝑘 ) ∈ ℕ0 ) |
36 |
35
|
nn0cnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 0 ... ( 𝑁 − 𝑀 ) ) ) → ( ( 𝑁 − 𝑀 ) C 𝑘 ) ∈ ℂ ) |
37 |
25 36
|
mulcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 0 ... ( 𝑁 − 𝑀 ) ) ) → ( ( - 1 ↑ 𝑘 ) · ( ( 𝑁 − 𝑀 ) C 𝑘 ) ) ∈ ℂ ) |
38 |
|
resmpt |
⊢ ( ( 0 [,] 1 ) ⊆ ℂ → ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 ↑ 𝑘 ) ) ↾ ( 0 [,] 1 ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 ↑ 𝑘 ) ) ) |
39 |
10 38
|
ax-mp |
⊢ ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 ↑ 𝑘 ) ) ↾ ( 0 [,] 1 ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 ↑ 𝑘 ) ) |
40 |
|
expcncf |
⊢ ( 𝑘 ∈ ℕ0 → ( 𝑥 ∈ ℂ ↦ ( 𝑥 ↑ 𝑘 ) ) ∈ ( ℂ –cn→ ℂ ) ) |
41 |
20 40
|
syl |
⊢ ( 𝑘 ∈ ( 0 ... ( 𝑁 − 𝑀 ) ) → ( 𝑥 ∈ ℂ ↦ ( 𝑥 ↑ 𝑘 ) ) ∈ ( ℂ –cn→ ℂ ) ) |
42 |
|
rescncf |
⊢ ( ( 0 [,] 1 ) ⊆ ℂ → ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 ↑ 𝑘 ) ) ∈ ( ℂ –cn→ ℂ ) → ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 ↑ 𝑘 ) ) ↾ ( 0 [,] 1 ) ) ∈ ( ( 0 [,] 1 ) –cn→ ℂ ) ) ) |
43 |
10 42
|
ax-mp |
⊢ ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 ↑ 𝑘 ) ) ∈ ( ℂ –cn→ ℂ ) → ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 ↑ 𝑘 ) ) ↾ ( 0 [,] 1 ) ) ∈ ( ( 0 [,] 1 ) –cn→ ℂ ) ) |
44 |
41 43
|
syl |
⊢ ( 𝑘 ∈ ( 0 ... ( 𝑁 − 𝑀 ) ) → ( ( 𝑥 ∈ ℂ ↦ ( 𝑥 ↑ 𝑘 ) ) ↾ ( 0 [,] 1 ) ) ∈ ( ( 0 [,] 1 ) –cn→ ℂ ) ) |
45 |
39 44
|
eqeltrrid |
⊢ ( 𝑘 ∈ ( 0 ... ( 𝑁 − 𝑀 ) ) → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 ↑ 𝑘 ) ) ∈ ( ( 0 [,] 1 ) –cn→ ℂ ) ) |
46 |
45
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 0 ... ( 𝑁 − 𝑀 ) ) ) → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝑥 ↑ 𝑘 ) ) ∈ ( ( 0 [,] 1 ) –cn→ ℂ ) ) |
47 |
6 7 8 9 19 37 46
|
3factsumint |
⊢ ( 𝜑 → ∫ ( 0 [,] 1 ) ( ( 𝑥 ↑ ( 𝑀 − 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑁 − 𝑀 ) ) ( ( ( - 1 ↑ 𝑘 ) · ( ( 𝑁 − 𝑀 ) C 𝑘 ) ) · ( 𝑥 ↑ 𝑘 ) ) ) d 𝑥 = Σ 𝑘 ∈ ( 0 ... ( 𝑁 − 𝑀 ) ) ( ( ( - 1 ↑ 𝑘 ) · ( ( 𝑁 − 𝑀 ) C 𝑘 ) ) · ∫ ( 0 [,] 1 ) ( ( 𝑥 ↑ ( 𝑀 − 1 ) ) · ( 𝑥 ↑ 𝑘 ) ) d 𝑥 ) ) |
48 |
5 47
|
eqtrd |
⊢ ( 𝜑 → 𝐹 = Σ 𝑘 ∈ ( 0 ... ( 𝑁 − 𝑀 ) ) ( ( ( - 1 ↑ 𝑘 ) · ( ( 𝑁 − 𝑀 ) C 𝑘 ) ) · ∫ ( 0 [,] 1 ) ( ( 𝑥 ↑ ( 𝑀 − 1 ) ) · ( 𝑥 ↑ 𝑘 ) ) d 𝑥 ) ) |