Theorem ltsubrp

Description: Subtracting a positive real from another number decreases it. (Contributed by FL, 27-Dec-2007)

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

Proof

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