Metamath Proof Explorer


Theorem lcmineqlem2

Description: Part of lcm inequality lemma, this part eventually shows that F times the least common multiple of 1 to n is an integer. (Contributed by metakunt, 29-Apr-2024)

Ref Expression
Hypotheses lcmineqlem2.1 𝐹 = ∫ ( 0 [,] 1 ) ( ( 𝑥 ↑ ( 𝑀 − 1 ) ) · ( ( 1 − 𝑥 ) ↑ ( 𝑁𝑀 ) ) ) d 𝑥
lcmineqlem2.2 ( 𝜑𝑁 ∈ ℕ )
lcmineqlem2.3 ( 𝜑𝑀 ∈ ℕ )
lcmineqlem2.4 ( 𝜑𝑀𝑁 )
Assertion lcmineqlem2 ( 𝜑𝐹 = Σ 𝑘 ∈ ( 0 ... ( 𝑁𝑀 ) ) ( ( ( - 1 ↑ 𝑘 ) · ( ( 𝑁𝑀 ) C 𝑘 ) ) · ∫ ( 0 [,] 1 ) ( ( 𝑥 ↑ ( 𝑀 − 1 ) ) · ( 𝑥𝑘 ) ) d 𝑥 ) )

Proof

Step Hyp Ref Expression
1 lcmineqlem2.1 𝐹 = ∫ ( 0 [,] 1 ) ( ( 𝑥 ↑ ( 𝑀 − 1 ) ) · ( ( 1 − 𝑥 ) ↑ ( 𝑁𝑀 ) ) ) d 𝑥
2 lcmineqlem2.2 ( 𝜑𝑁 ∈ ℕ )
3 lcmineqlem2.3 ( 𝜑𝑀 ∈ ℕ )
4 lcmineqlem2.4 ( 𝜑𝑀𝑁 )
5 1 2 3 4 lcmineqlem1 ( 𝜑𝐹 = ∫ ( 0 [,] 1 ) ( ( 𝑥 ↑ ( 𝑀 − 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑁𝑀 ) ) ( ( ( - 1 ↑ 𝑘 ) · ( ( 𝑁𝑀 ) C 𝑘 ) ) · ( 𝑥𝑘 ) ) ) d 𝑥 )
6 eqid ( 0 [,] 1 ) = ( 0 [,] 1 )
7 fzfid ( 𝜑 → ( 0 ... ( 𝑁𝑀 ) ) ∈ 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 ... ( 𝑁𝑀 ) ) → 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 ... ( 𝑁𝑀 ) ) → ( - 1 ↑ 𝑘 ) ∈ ℂ )
25 24 adantl ( ( 𝜑𝑘 ∈ ( 0 ... ( 𝑁𝑀 ) ) ) → ( - 1 ↑ 𝑘 ) ∈ ℂ )
26 3 nnnn0d ( 𝜑𝑀 ∈ ℕ0 )
27 2 nnnn0d ( 𝜑𝑁 ∈ ℕ0 )
28 nn0sub ( ( 𝑀 ∈ ℕ0𝑁 ∈ ℕ0 ) → ( 𝑀𝑁 ↔ ( 𝑁𝑀 ) ∈ ℕ0 ) )
29 26 27 28 syl2anc ( 𝜑 → ( 𝑀𝑁 ↔ ( 𝑁𝑀 ) ∈ ℕ0 ) )
30 4 29 mpbid ( 𝜑 → ( 𝑁𝑀 ) ∈ ℕ0 )
31 nn0z ( 𝑘 ∈ ℕ0𝑘 ∈ ℤ )
32 20 31 syl ( 𝑘 ∈ ( 0 ... ( 𝑁𝑀 ) ) → 𝑘 ∈ ℤ )
33 bccl ( ( ( 𝑁𝑀 ) ∈ ℕ0𝑘 ∈ ℤ ) → ( ( 𝑁𝑀 ) C 𝑘 ) ∈ ℕ0 )
34 32 33 sylan2 ( ( ( 𝑁𝑀 ) ∈ ℕ0𝑘 ∈ ( 0 ... ( 𝑁𝑀 ) ) ) → ( ( 𝑁𝑀 ) C 𝑘 ) ∈ ℕ0 )
35 30 34 sylan ( ( 𝜑𝑘 ∈ ( 0 ... ( 𝑁𝑀 ) ) ) → ( ( 𝑁𝑀 ) C 𝑘 ) ∈ ℕ0 )
36 35 nn0cnd ( ( 𝜑𝑘 ∈ ( 0 ... ( 𝑁𝑀 ) ) ) → ( ( 𝑁𝑀 ) C 𝑘 ) ∈ ℂ )
37 25 36 mulcld ( ( 𝜑𝑘 ∈ ( 0 ... ( 𝑁𝑀 ) ) ) → ( ( - 1 ↑ 𝑘 ) · ( ( 𝑁𝑀 ) C 𝑘 ) ) ∈ ℂ )
38 resmpt ( ( 0 [,] 1 ) ⊆ ℂ → ( ( 𝑥 ∈ ℂ ↦ ( 𝑥𝑘 ) ) ↾ ( 0 [,] 1 ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝑥𝑘 ) ) )
39 10 38 ax-mp ( ( 𝑥 ∈ ℂ ↦ ( 𝑥𝑘 ) ) ↾ ( 0 [,] 1 ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝑥𝑘 ) )
40 expcncf ( 𝑘 ∈ ℕ0 → ( 𝑥 ∈ ℂ ↦ ( 𝑥𝑘 ) ) ∈ ( ℂ –cn→ ℂ ) )
41 20 40 syl ( 𝑘 ∈ ( 0 ... ( 𝑁𝑀 ) ) → ( 𝑥 ∈ ℂ ↦ ( 𝑥𝑘 ) ) ∈ ( ℂ –cn→ ℂ ) )
42 rescncf ( ( 0 [,] 1 ) ⊆ ℂ → ( ( 𝑥 ∈ ℂ ↦ ( 𝑥𝑘 ) ) ∈ ( ℂ –cn→ ℂ ) → ( ( 𝑥 ∈ ℂ ↦ ( 𝑥𝑘 ) ) ↾ ( 0 [,] 1 ) ) ∈ ( ( 0 [,] 1 ) –cn→ ℂ ) ) )
43 10 42 ax-mp ( ( 𝑥 ∈ ℂ ↦ ( 𝑥𝑘 ) ) ∈ ( ℂ –cn→ ℂ ) → ( ( 𝑥 ∈ ℂ ↦ ( 𝑥𝑘 ) ) ↾ ( 0 [,] 1 ) ) ∈ ( ( 0 [,] 1 ) –cn→ ℂ ) )
44 41 43 syl ( 𝑘 ∈ ( 0 ... ( 𝑁𝑀 ) ) → ( ( 𝑥 ∈ ℂ ↦ ( 𝑥𝑘 ) ) ↾ ( 0 [,] 1 ) ) ∈ ( ( 0 [,] 1 ) –cn→ ℂ ) )
45 39 44 eqeltrrid ( 𝑘 ∈ ( 0 ... ( 𝑁𝑀 ) ) → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝑥𝑘 ) ) ∈ ( ( 0 [,] 1 ) –cn→ ℂ ) )
46 45 adantl ( ( 𝜑𝑘 ∈ ( 0 ... ( 𝑁𝑀 ) ) ) → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( 𝑥𝑘 ) ) ∈ ( ( 0 [,] 1 ) –cn→ ℂ ) )
47 6 7 8 9 19 37 46 3factsumint ( 𝜑 → ∫ ( 0 [,] 1 ) ( ( 𝑥 ↑ ( 𝑀 − 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑁𝑀 ) ) ( ( ( - 1 ↑ 𝑘 ) · ( ( 𝑁𝑀 ) C 𝑘 ) ) · ( 𝑥𝑘 ) ) ) d 𝑥 = Σ 𝑘 ∈ ( 0 ... ( 𝑁𝑀 ) ) ( ( ( - 1 ↑ 𝑘 ) · ( ( 𝑁𝑀 ) C 𝑘 ) ) · ∫ ( 0 [,] 1 ) ( ( 𝑥 ↑ ( 𝑀 − 1 ) ) · ( 𝑥𝑘 ) ) d 𝑥 ) )
48 5 47 eqtrd ( 𝜑𝐹 = Σ 𝑘 ∈ ( 0 ... ( 𝑁𝑀 ) ) ( ( ( - 1 ↑ 𝑘 ) · ( ( 𝑁𝑀 ) C 𝑘 ) ) · ∫ ( 0 [,] 1 ) ( ( 𝑥 ↑ ( 𝑀 − 1 ) ) · ( 𝑥𝑘 ) ) d 𝑥 ) )