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Metamath Proof Explorer

Theorem ltsub1dd

Description: Subtraction from both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016)

Ref Expression
Hypotheses leidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
ltnegd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
ltadd1d.3 ⊢ ( 𝜑 → 𝐶 ∈ ℝ )
ltadd1dd.4 ⊢ ( 𝜑 → 𝐴 < 𝐵 )
Assertion ltsub1dd ( 𝜑 → ( 𝐴 − 𝐶 ) < ( 𝐵 − 𝐶 ) )

Proof

Step Hyp Ref Expression
1 leidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 ltnegd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 ltadd1d.3 ⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4 ltadd1dd.4 ⊢ ( 𝜑 → 𝐴 < 𝐵 )
5 1 2 3 ltsub1d ⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ ( 𝐴 − 𝐶 ) < ( 𝐵 − 𝐶 ) ) )
6 4 5 mpbid ⊢ ( 𝜑 → ( 𝐴 − 𝐶 ) < ( 𝐵 − 𝐶 ) )