Description: A number is less than or equal to itself plus a nonnegative integer. (Contributed by NM, 10-Mar-2005)
Ref | Expression | ||
---|---|---|---|
Assertion | nn0addge2 | ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ) → 𝐴 ≤ ( 𝑁 + 𝐴 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0re | ⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ ) | |
2 | nn0ge0 | ⊢ ( 𝑁 ∈ ℕ0 → 0 ≤ 𝑁 ) | |
3 | 1 2 | jca | ⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 ∈ ℝ ∧ 0 ≤ 𝑁 ) ) |
4 | addge02 | ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( 0 ≤ 𝑁 ↔ 𝐴 ≤ ( 𝑁 + 𝐴 ) ) ) | |
5 | 4 | biimp3a | ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 0 ≤ 𝑁 ) → 𝐴 ≤ ( 𝑁 + 𝐴 ) ) |
6 | 5 | 3expb | ⊢ ( ( 𝐴 ∈ ℝ ∧ ( 𝑁 ∈ ℝ ∧ 0 ≤ 𝑁 ) ) → 𝐴 ≤ ( 𝑁 + 𝐴 ) ) |
7 | 3 6 | sylan2 | ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ) → 𝐴 ≤ ( 𝑁 + 𝐴 ) ) |