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Metamath Proof Explorer

Theorem ltsub13d

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

Ref Expression
Hypotheses leidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
ltnegd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
ltadd1d.3 ⊢ ( 𝜑 → 𝐶 ∈ ℝ )
ltsub13d.4 ⊢ ( 𝜑 → 𝐴 < ( 𝐵 − 𝐶 ) )
Assertion ltsub13d ( 𝜑 → 𝐶 < ( 𝐵 − 𝐴 ) )

Proof

Step Hyp Ref Expression
1 leidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 ltnegd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 ltadd1d.3 ⊢ ( 𝜑 → 𝐶 ∈ ℝ )
4 ltsub13d.4 ⊢ ( 𝜑 → 𝐴 < ( 𝐵 − 𝐶 ) )
5 ltsub13 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( 𝐴 < ( 𝐵 − 𝐶 ) ↔ 𝐶 < ( 𝐵 − 𝐴 ) ) )
6 1 2 3 5 syl3anc ⊢ ( 𝜑 → ( 𝐴 < ( 𝐵 − 𝐶 ) ↔ 𝐶 < ( 𝐵 − 𝐴 ) ) )
7 4 6 mpbid ⊢ ( 𝜑 → 𝐶 < ( 𝐵 − 𝐴 ) )