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 < 𝐴 ↔ 𝐵 < ( 𝐴 + 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltaddpos	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 < 𝐴 ↔ 𝐵 < ( 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 < 𝐴 ↔ 𝐵 < ( 𝐴 + 𝐵 ) ) )