Metamath Proof Explorer


Theorem ltaddpos2

Description: Adding a positive number to another number increases it. (Contributed by NM, 8-Apr-2005)

Ref Expression
Assertion ltaddpos2 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 < 𝐴𝐵 < ( 𝐴 + 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 ltaddpos ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 < 𝐴𝐵 < ( 𝐵 + 𝐴 ) ) )
2 recn ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
3 recn ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ )
4 addcom ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) )
5 2 3 4 syl2an ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 + 𝐵 ) = ( 𝐵 + 𝐴 ) )
6 5 breq2d ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐵 < ( 𝐴 + 𝐵 ) ↔ 𝐵 < ( 𝐵 + 𝐴 ) ) )
7 1 6 bitr4d ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 < 𝐴𝐵 < ( 𝐴 + 𝐵 ) ) )