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 ) ) |
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 ) ) |