Theorem ltaddrp

Description: Adding a positive number to another number increases it. (Contributed by FL, 27-Dec-2007)

	Ref	Expression
Assertion	ltaddrp	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → 𝐴 < ( 𝐴 + 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		elrp	⊢ ( 𝐵 ∈ ℝ+ ↔ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) )
2		ltaddpos	⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 0 < 𝐵 ↔ 𝐴 < ( 𝐴 + 𝐵 ) ) )
3	2	biimpd	⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 0 < 𝐵 → 𝐴 < ( 𝐴 + 𝐵 ) ) )
4	3	expcom	⊢ ( 𝐴 ∈ ℝ → ( 𝐵 ∈ ℝ → ( 0 < 𝐵 → 𝐴 < ( 𝐴 + 𝐵 ) ) ) )
5	4	imp32	⊢ ( ( 𝐴 ∈ ℝ ∧ ( 𝐵 ∈ ℝ ∧ 0 < 𝐵 ) ) → 𝐴 < ( 𝐴 + 𝐵 ) )
6	1 5	sylan2b	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → 𝐴 < ( 𝐴 + 𝐵 ) )