Metamath Proof Explorer
Description: Adding a positive number to another number increases it. (Contributed by Mario Carneiro, 27-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
leidd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
|
|
ltnegd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
|
Assertion |
ltaddpos2d |
⊢ ( 𝜑 → ( 0 < 𝐴 ↔ 𝐵 < ( 𝐴 + 𝐵 ) ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
leidd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
ltnegd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
ltaddpos2 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 < 𝐴 ↔ 𝐵 < ( 𝐴 + 𝐵 ) ) ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → ( 0 < 𝐴 ↔ 𝐵 < ( 𝐴 + 𝐵 ) ) ) |