Theorem ltsubrpd

Description: Subtracting a positive real from another number decreases it. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	rpgecld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	rpgecld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
Assertion	ltsubrpd	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) < 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		rpgecld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		rpgecld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
3		ltsubrp	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 − 𝐵 ) < 𝐴 )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) < 𝐴 )