Theorem 1fzopredsuc

Description: Join 0 and a successor to the beginning and the end of an open integer interval starting at 1. (Contributed by AV, 14-Jul-2020)

	Ref	Expression
Assertion	1fzopredsuc	$$⊢N\in {\mathbb{N}}_0\to \left(0\dots N\right)=\left(\left\{0\right\}\cup \left(1..^N\right)\right)\cup \left\{N\right\}$$

Proof

Step	Hyp	Ref	Expression
1		elnn0uz	$$⊢N\in {\mathbb{N}}_0\leftrightarrow N\in {\mathbb{Z}}_{\ge 0}$$
2		fzopredsuc	$$⊢N\in {\mathbb{Z}}_{\ge 0}\to \left(0\dots N\right)=\left(\left\{0\right\}\cup \left(0+1..^N\right)\right)\cup \left\{N\right\}$$
3		0p1e1	$$⊢0+1=1$$
4	3	oveq1i	$$⊢\left(0+1..^N\right)=\left(1..^N\right)$$
5	4	uneq2i	$$⊢\left\{0\right\}\cup \left(0+1..^N\right)=\left\{0\right\}\cup \left(1..^N\right)$$
6	5	uneq1i	$$⊢\left(\left\{0\right\}\cup \left(0+1..^N\right)\right)\cup \left\{N\right\}=\left(\left\{0\right\}\cup \left(1..^N\right)\right)\cup \left\{N\right\}$$
7	2 6	eqtrdi	$$⊢N\in {\mathbb{Z}}_{\ge 0}\to \left(0\dots N\right)=\left(\left\{0\right\}\cup \left(1..^N\right)\right)\cup \left\{N\right\}$$
8	1 7	sylbi	$$⊢N\in {\mathbb{N}}_0\to \left(0\dots N\right)=\left(\left\{0\right\}\cup \left(1..^N\right)\right)\cup \left\{N\right\}$$