Theorem lcmineqlem15

Description: F times the least common multiple of 1 to n is a natural number. (Contributed by metakunt, 10-May-2024)

	Ref	Expression
Hypotheses	lcmineqlem15.1	⊢ 𝐹 = ∫ ( 0 [,] 1 ) ( ( 𝑥 ↑ ( 𝑀 − 1 ) ) · ( ( 1 − 𝑥 ) ↑ ( 𝑁 − 𝑀 ) ) ) d 𝑥
	lcmineqlem15.2	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
	lcmineqlem15.3	⊢ ( 𝜑 → 𝑀 ∈ ℕ )
	lcmineqlem15.4	⊢ ( 𝜑 → 𝑀 ≤ 𝑁 )
Assertion	lcmineqlem15	⊢ ( 𝜑 → ( ( lcm ‘ ( 1 ... 𝑁 ) ) · 𝐹 ) ∈ ℕ )

Proof

Step	Hyp	Ref	Expression
1		lcmineqlem15.1	⊢ 𝐹 = ∫ ( 0 [,] 1 ) ( ( 𝑥 ↑ ( 𝑀 − 1 ) ) · ( ( 1 − 𝑥 ) ↑ ( 𝑁 − 𝑀 ) ) ) d 𝑥
2		lcmineqlem15.2	⊢ ( 𝜑 → 𝑁 ∈ ℕ )
3		lcmineqlem15.3	⊢ ( 𝜑 → 𝑀 ∈ ℕ )
4		lcmineqlem15.4	⊢ ( 𝜑 → 𝑀 ≤ 𝑁 )
5	1 2 3 4	lcmineqlem6	⊢ ( 𝜑 → ( ( lcm ‘ ( 1 ... 𝑁 ) ) · 𝐹 ) ∈ ℤ )
6		fz1ssnn	⊢ ( 1 ... 𝑁 ) ⊆ ℕ
7		fzfi	⊢ ( 1 ... 𝑁 ) ∈ Fin
8		lcmfnncl	⊢ ( ( ( 1 ... 𝑁 ) ⊆ ℕ ∧ ( 1 ... 𝑁 ) ∈ Fin ) → ( lcm ‘ ( 1 ... 𝑁 ) ) ∈ ℕ )
9	6 7 8	mp2an	⊢ ( lcm ‘ ( 1 ... 𝑁 ) ) ∈ ℕ
10	9	a1i	⊢ ( 𝜑 → ( lcm ‘ ( 1 ... 𝑁 ) ) ∈ ℕ )
11	10	nnred	⊢ ( 𝜑 → ( lcm ‘ ( 1 ... 𝑁 ) ) ∈ ℝ )
12	1 3 2 4	lcmineqlem13	⊢ ( 𝜑 → 𝐹 = ( 1 / ( 𝑀 · ( 𝑁 C 𝑀 ) ) ) )
13		1red	⊢ ( 𝜑 → 1 ∈ ℝ )
14	3	nnnn0d	⊢ ( 𝜑 → 𝑀 ∈ ℕ0 )
15	2 14 4	bccl2d	⊢ ( 𝜑 → ( 𝑁 C 𝑀 ) ∈ ℕ )
16	3 15	nnmulcld	⊢ ( 𝜑 → ( 𝑀 · ( 𝑁 C 𝑀 ) ) ∈ ℕ )
17	16	nnred	⊢ ( 𝜑 → ( 𝑀 · ( 𝑁 C 𝑀 ) ) ∈ ℝ )
18	16	nnne0d	⊢ ( 𝜑 → ( 𝑀 · ( 𝑁 C 𝑀 ) ) ≠ 0 )
19	13 17 18	redivcld	⊢ ( 𝜑 → ( 1 / ( 𝑀 · ( 𝑁 C 𝑀 ) ) ) ∈ ℝ )
20	12 19	eqeltrd	⊢ ( 𝜑 → 𝐹 ∈ ℝ )
21	10	nngt0d	⊢ ( 𝜑 → 0 < ( lcm ‘ ( 1 ... 𝑁 ) ) )
22		nnrecgt0	⊢ ( ( 𝑀 · ( 𝑁 C 𝑀 ) ) ∈ ℕ → 0 < ( 1 / ( 𝑀 · ( 𝑁 C 𝑀 ) ) ) )
23	16 22	syl	⊢ ( 𝜑 → 0 < ( 1 / ( 𝑀 · ( 𝑁 C 𝑀 ) ) ) )
24	23 12	breqtrrd	⊢ ( 𝜑 → 0 < 𝐹 )
25	11 20 21 24	mulgt0d	⊢ ( 𝜑 → 0 < ( ( lcm ‘ ( 1 ... 𝑁 ) ) · 𝐹 ) )
26		elnnz	⊢ ( ( ( lcm ‘ ( 1 ... 𝑁 ) ) · 𝐹 ) ∈ ℕ ↔ ( ( ( lcm ‘ ( 1 ... 𝑁 ) ) · 𝐹 ) ∈ ℤ ∧ 0 < ( ( lcm ‘ ( 1 ... 𝑁 ) ) · 𝐹 ) ) )
27	5 25 26	sylanbrc	⊢ ( 𝜑 → ( ( lcm ‘ ( 1 ... 𝑁 ) ) · 𝐹 ) ∈ ℕ )