Metamath Proof Explorer


Theorem nn0addge2

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 ) )

Proof

Step Hyp Ref Expression
1 nn0re ( 𝑁 ∈ ℕ0𝑁 ∈ ℝ )
2 nn0ge0 ( 𝑁 ∈ ℕ0 → 0 ≤ 𝑁 )
3 1 2 jca ( 𝑁 ∈ ℕ0 → ( 𝑁 ∈ ℝ ∧ 0 ≤ 𝑁 ) )
4 addge02 ( ( 𝐴 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( 0 ≤ 𝑁𝐴 ≤ ( 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 ) )