This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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