Metamath Proof Explorer


Theorem ige2m1fz1

Description: Membership of an integer greater than 1 decreased by 1 in a 1-based finite set of sequential integers. (Contributed by Alexander van der Vekens, 14-Sep-2018)

Ref Expression
Assertion ige2m1fz1 ( 𝑁 ∈ ( ℤ ‘ 2 ) → ( 𝑁 − 1 ) ∈ ( 1 ... 𝑁 ) )

Proof

Step Hyp Ref Expression
1 1e2m1 1 = ( 2 − 1 )
2 1 a1i ( 𝑁 ∈ ( ℤ ‘ 2 ) → 1 = ( 2 − 1 ) )
3 2 oveq2d ( 𝑁 ∈ ( ℤ ‘ 2 ) → ( 𝑁 − 1 ) = ( 𝑁 − ( 2 − 1 ) ) )
4 2nn 2 ∈ ℕ
5 uzsubsubfz1 ( ( 2 ∈ ℕ ∧ 𝑁 ∈ ( ℤ ‘ 2 ) ) → ( 𝑁 − ( 2 − 1 ) ) ∈ ( 1 ... 𝑁 ) )
6 4 5 mpan ( 𝑁 ∈ ( ℤ ‘ 2 ) → ( 𝑁 − ( 2 − 1 ) ) ∈ ( 1 ... 𝑁 ) )
7 3 6 eqeltrd ( 𝑁 ∈ ( ℤ ‘ 2 ) → ( 𝑁 − 1 ) ∈ ( 1 ... 𝑁 ) )