Theorem ltsubrpd

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

	Ref	Expression
Hypotheses	rpgecld.1	$$⊢\phi \to A\in ℝ$$
	rpgecld.2	$$⊢\phi \to B\in {ℝ}^{+}$$
Assertion	ltsubrpd	$$⊢\phi \to A-B<A$$

Proof

Step	Hyp	Ref	Expression
1		rpgecld.1	$$⊢\phi \to A\in ℝ$$
2		rpgecld.2	$$⊢\phi \to B\in {ℝ}^{+}$$
3		ltsubrp	$$⊢\left(A\in ℝ\wedge B\in {ℝ}^{+}\right)\to A-B<A$$
4	1 2 3	syl2anc	$$⊢\phi \to A-B<A$$