Description: Split the first element of a finite set of sequential nonnegative integers. (Contributed by AV, 12-Sep-2025)
Ref | Expression | ||
---|---|---|---|
Assertion | fz0dif1 | ⊢ ( 𝑁 ∈ ℕ0 → ( ( 0 ... 𝑁 ) ∖ { 0 } ) = ( 1 ... 𝑁 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0uz | ⊢ ( 𝑁 ∈ ℕ0 ↔ 𝑁 ∈ ( ℤ≥ ‘ 0 ) ) | |
2 | fzdif1 | ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 0 ) → ( ( 0 ... 𝑁 ) ∖ { 0 } ) = ( ( 0 + 1 ) ... 𝑁 ) ) | |
3 | 1 2 | sylbi | ⊢ ( 𝑁 ∈ ℕ0 → ( ( 0 ... 𝑁 ) ∖ { 0 } ) = ( ( 0 + 1 ) ... 𝑁 ) ) |
4 | 0p1e1 | ⊢ ( 0 + 1 ) = 1 | |
5 | 4 | oveq1i | ⊢ ( ( 0 + 1 ) ... 𝑁 ) = ( 1 ... 𝑁 ) |
6 | 3 5 | eqtrdi | ⊢ ( 𝑁 ∈ ℕ0 → ( ( 0 ... 𝑁 ) ∖ { 0 } ) = ( 1 ... 𝑁 ) ) |