Metamath Proof Explorer
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 ) |
|
|
ltsub23d.4 |
⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) < 𝐶 ) |
|
Assertion |
ltsub23d |
⊢ ( 𝜑 → ( 𝐴 − 𝐶 ) < 𝐵 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
leidd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
ltnegd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
ltadd1d.3 |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
4 |
|
ltsub23d.4 |
⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) < 𝐶 ) |
5 |
|
ltsub23 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 − 𝐵 ) < 𝐶 ↔ ( 𝐴 − 𝐶 ) < 𝐵 ) ) |
6 |
1 2 3 5
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) < 𝐶 ↔ ( 𝐴 − 𝐶 ) < 𝐵 ) ) |
7 |
4 6
|
mpbid |
⊢ ( 𝜑 → ( 𝐴 − 𝐶 ) < 𝐵 ) |