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Metamath Proof Explorer

Theorem zzlesq

Description: An integer is less than or equal to its square. (Contributed by BJ, 6-Feb-2025)

Ref Expression
Assertion zzlesq ( 𝑁 ∈ ℤ → 𝑁 ≤ ( 𝑁 ↑ 2 ) )

Proof

Step Hyp Ref Expression
1 elznn ⊢ ( 𝑁 ∈ ℤ ↔ ( 𝑁 ∈ ℝ ∧ ( 𝑁 ∈ ℕ ∨ - 𝑁 ∈ ℕ0 ) ) )
2 animorrl ⊢ ( ( 𝑁 ∈ ℝ ∧ 𝑁 ∈ ℕ ) → ( 𝑁 ∈ ℕ ∨ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ0 ) ) )
3 olc ⊢ ( ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ0 ) → ( 𝑁 ∈ ℕ ∨ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ0 ) ) )
4 2 3 jaodan ⊢ ( ( 𝑁 ∈ ℝ ∧ ( 𝑁 ∈ ℕ ∨ - 𝑁 ∈ ℕ0 ) ) → ( 𝑁 ∈ ℕ ∨ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ0 ) ) )
5 1 4 sylbi ⊢ ( 𝑁 ∈ ℤ → ( 𝑁 ∈ ℕ ∨ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ0 ) ) )
6 nnlesq ⊢ ( 𝑁 ∈ ℕ → 𝑁 ≤ ( 𝑁 ↑ 2 ) )
7 simpl ⊢ ( ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ0 ) → 𝑁 ∈ ℝ )
8 0red ⊢ ( ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ0 ) → 0 ∈ ℝ )
9 7 resqcld ⊢ ( ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ0 ) → ( 𝑁 ↑ 2 ) ∈ ℝ )
10 nn0ge0 ⊢ ( - 𝑁 ∈ ℕ0 → 0 ≤ - 𝑁 )
11 le0neg1 ⊢ ( 𝑁 ∈ ℝ → ( 𝑁 ≤ 0 ↔ 0 ≤ - 𝑁 ) )
12 11 biimpar ⊢ ( ( 𝑁 ∈ ℝ ∧ 0 ≤ - 𝑁 ) → 𝑁 ≤ 0 )
13 10 12 sylan2 ⊢ ( ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ0 ) → 𝑁 ≤ 0 )
14 7 sqge0d ⊢ ( ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ0 ) → 0 ≤ ( 𝑁 ↑ 2 ) )
15 7 8 9 13 14 letrd ⊢ ( ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ0 ) → 𝑁 ≤ ( 𝑁 ↑ 2 ) )
16 6 15 jaoi ⊢ ( ( 𝑁 ∈ ℕ ∨ ( 𝑁 ∈ ℝ ∧ - 𝑁 ∈ ℕ0 ) ) → 𝑁 ≤ ( 𝑁 ↑ 2 ) )
17 5 16 syl ⊢ ( 𝑁 ∈ ℤ → 𝑁 ≤ ( 𝑁 ↑ 2 ) )