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 ∈ ℕ0 → ( 0 ... 𝑁 ) = ( ( { 0 } ∪ ( 1 ..^ 𝑁 ) ) ∪ { 𝑁 } ) )

Proof

Step	Hyp	Ref	Expression
1		elnn0uz	⊢ ( 𝑁 ∈ ℕ0 ↔ 𝑁 ∈ ( ℤ≥ ‘ 0 ) )
2		fzopredsuc	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 0 ) → ( 0 ... 𝑁 ) = ( ( { 0 } ∪ ( ( 0 + 1 ) ..^ 𝑁 ) ) ∪ { 𝑁 } ) )
3		0p1e1	⊢ ( 0 + 1 ) = 1
4	3	oveq1i	⊢ ( ( 0 + 1 ) ..^ 𝑁 ) = ( 1 ..^ 𝑁 )
5	4	uneq2i	⊢ ( { 0 } ∪ ( ( 0 + 1 ) ..^ 𝑁 ) ) = ( { 0 } ∪ ( 1 ..^ 𝑁 ) )
6	5	uneq1i	⊢ ( ( { 0 } ∪ ( ( 0 + 1 ) ..^ 𝑁 ) ) ∪ { 𝑁 } ) = ( ( { 0 } ∪ ( 1 ..^ 𝑁 ) ) ∪ { 𝑁 } )
7	2 6	eqtrdi	⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 0 ) → ( 0 ... 𝑁 ) = ( ( { 0 } ∪ ( 1 ..^ 𝑁 ) ) ∪ { 𝑁 } ) )
8	1 7	sylbi	⊢ ( 𝑁 ∈ ℕ0 → ( 0 ... 𝑁 ) = ( ( { 0 } ∪ ( 1 ..^ 𝑁 ) ) ∪ { 𝑁 } ) )