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 ) |
|
Assertion |
ltsubadd2d |
⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) < 𝐶 ↔ 𝐴 < ( 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 |
|
ltsubadd2 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ ) → ( ( 𝐴 − 𝐵 ) < 𝐶 ↔ 𝐴 < ( 𝐵 + 𝐶 ) ) ) |
5 |
1 2 3 4
|
syl3anc |
⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) < 𝐶 ↔ 𝐴 < ( 𝐵 + 𝐶 ) ) ) |