Metamath Proof Explorer


Theorem ltsub23d

Description: 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses leidd.1 ( 𝜑𝐴 ∈ ℝ )
ltnegd.2 ( 𝜑𝐵 ∈ ℝ )
ltadd1d.3 ( 𝜑𝐶 ∈ ℝ )
ltsub23d.4 ( 𝜑 → ( 𝐴𝐵 ) < 𝐶 )
Assertion ltsub23d ( 𝜑 → ( 𝐴𝐶 ) < 𝐵 )

Proof

Step Hyp Ref Expression
1 leidd.1 ( 𝜑𝐴 ∈ ℝ )
2 ltnegd.2 ( 𝜑𝐵 ∈ ℝ )
3 ltadd1d.3 ( 𝜑𝐶 ∈ ℝ )
4 ltsub23d.4 ( 𝜑 → ( 𝐴𝐵 ) < 𝐶 )
5 ltsub23 ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴𝐵 ) < 𝐶 ↔ ( 𝐴𝐶 ) < 𝐵 ) )
6 1 2 3 5 syl3anc ( 𝜑 → ( ( 𝐴𝐵 ) < 𝐶 ↔ ( 𝐴𝐶 ) < 𝐵 ) )
7 4 6 mpbid ( 𝜑 → ( 𝐴𝐶 ) < 𝐵 )