Description: A strictly-decreasing nonnegative integer sequence with initial term N reaches zero by the N th term. Inference version. (Contributed by Paul Chapman, 31-Mar-2011)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | nn0seqcvg.1 | ⊢ 𝐹 : ℕ0 ⟶ ℕ0 | |
| nn0seqcvg.2 | ⊢ 𝑁 = ( 𝐹 ‘ 0 ) | ||
| nn0seqcvg.3 | ⊢ ( 𝑘 ∈ ℕ0 → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) ≠ 0 → ( 𝐹 ‘ ( 𝑘 + 1 ) ) < ( 𝐹 ‘ 𝑘 ) ) ) | ||
| Assertion | nn0seqcvg | ⊢ ( 𝐹 ‘ 𝑁 ) = 0 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nn0seqcvg.1 | ⊢ 𝐹 : ℕ0 ⟶ ℕ0 | |
| 2 | nn0seqcvg.2 | ⊢ 𝑁 = ( 𝐹 ‘ 0 ) | |
| 3 | nn0seqcvg.3 | ⊢ ( 𝑘 ∈ ℕ0 → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) ≠ 0 → ( 𝐹 ‘ ( 𝑘 + 1 ) ) < ( 𝐹 ‘ 𝑘 ) ) ) | |
| 4 | eqid | ⊢ 1 = 1 | |
| 5 | 1 | a1i | ⊢ ( 1 = 1 → 𝐹 : ℕ0 ⟶ ℕ0 ) |
| 6 | 2 | a1i | ⊢ ( 1 = 1 → 𝑁 = ( 𝐹 ‘ 0 ) ) |
| 7 | 3 | adantl | ⊢ ( ( 1 = 1 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) ≠ 0 → ( 𝐹 ‘ ( 𝑘 + 1 ) ) < ( 𝐹 ‘ 𝑘 ) ) ) |
| 8 | 5 6 7 | nn0seqcvgd | ⊢ ( 1 = 1 → ( 𝐹 ‘ 𝑁 ) = 0 ) |
| 9 | 4 8 | ax-mp | ⊢ ( 𝐹 ‘ 𝑁 ) = 0 |