Metamath Proof Explorer


Theorem addge01

Description: A number is less than or equal to itself plus a nonnegative number. (Contributed by NM, 21-Feb-2005)

Ref Expression
Assertion addge01 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 ≤ 𝐵𝐴 ≤ ( 𝐴 + 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 0re 0 ∈ ℝ
2 leadd2 ( ( 0 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 0 ≤ 𝐵 ↔ ( 𝐴 + 0 ) ≤ ( 𝐴 + 𝐵 ) ) )
3 1 2 mp3an1 ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 0 ≤ 𝐵 ↔ ( 𝐴 + 0 ) ≤ ( 𝐴 + 𝐵 ) ) )
4 3 ancoms ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 ≤ 𝐵 ↔ ( 𝐴 + 0 ) ≤ ( 𝐴 + 𝐵 ) ) )
5 recn ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
6 5 addid1d ( 𝐴 ∈ ℝ → ( 𝐴 + 0 ) = 𝐴 )
7 6 adantr ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 + 0 ) = 𝐴 )
8 7 breq1d ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐴 + 0 ) ≤ ( 𝐴 + 𝐵 ) ↔ 𝐴 ≤ ( 𝐴 + 𝐵 ) ) )
9 4 8 bitrd ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 ≤ 𝐵𝐴 ≤ ( 𝐴 + 𝐵 ) ) )