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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 )