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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 ≤ 𝐵 ↔ 𝐴 ≤ ( 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 addridd ⊢ ( 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 ≤ 𝐵 ↔ 𝐴 ≤ ( 𝐴 + 𝐵 ) ) )