Theorem iblidicc

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

	Ref	Expression
Hypotheses	iblidicc.a	$$⊢\phi \to A\in ℝ$$
	iblidicc.b	$$⊢\phi \to B\in ℝ$$
Assertion	iblidicc	$$⊢\phi \to \left(x\in \left[A,B\right]⟼x\right)\in {𝐿}^1$$

Proof

Step	Hyp	Ref	Expression
1		iblidicc.a	$$⊢\phi \to A\in ℝ$$
2		iblidicc.b	$$⊢\phi \to B\in ℝ$$
3		iccssre	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left[A,B\right]\subseteq ℝ$$
4	1 2 3	syl2anc	$$⊢\phi \to \left[A,B\right]\subseteq ℝ$$
5		ax-resscn	$$⊢ℝ\subseteq ℂ$$
6	4 5	sstrdi	$$⊢\phi \to \left[A,B\right]\subseteq ℂ$$
7		ssid	$$⊢ℂ\subseteq ℂ$$
8		cncfmptid	$$⊢\left(\left[A,B\right]\subseteq ℂ\wedge ℂ\subseteq ℂ\right)\to \left(x\in \left[A,B\right]⟼x\right):\left[A,B\right]\underset{cn}{⟶}ℂ$$
9	6 7 8	sylancl	$$⊢\phi \to \left(x\in \left[A,B\right]⟼x\right):\left[A,B\right]\underset{cn}{⟶}ℂ$$
10		cniccibl	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge \left(x\in \left[A,B\right]⟼x\right):\left[A,B\right]\underset{cn}{⟶}ℂ\right)\to \left(x\in \left[A,B\right]⟼x\right)\in {𝐿}^1$$
11	1 2 9 10	syl3anc	$$⊢\phi \to \left(x\in \left[A,B\right]⟼x\right)\in {𝐿}^1$$