Theorem iblidicc

Description: The identity function is integrable on any closed interval. (Contributed by Thierry Arnoux, 13-Dec-2021)

	Ref	Expression
Hypotheses	iblidicc.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	iblidicc.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
Assertion	iblidicc	⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ 𝑥 ) ∈ 𝐿1 )

Proof

Step	Hyp	Ref	Expression
1		iblidicc.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		iblidicc.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3		iccssre	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 [,] 𝐵 ) ⊆ ℝ )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ ℝ )
5		ax-resscn	⊢ ℝ ⊆ ℂ
6	4 5	sstrdi	⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ ℂ )
7		ssid	⊢ ℂ ⊆ ℂ
8		cncfmptid	⊢ ( ( ( 𝐴 [,] 𝐵 ) ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ 𝑥 ) ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) )
9	6 7 8	sylancl	⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ 𝑥 ) ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) )
10		cniccibl	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ 𝑥 ) ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ 𝑥 ) ∈ 𝐿1 )
11	1 2 9 10	syl3anc	⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ 𝑥 ) ∈ 𝐿1 )