Metamath Proof Explorer
Description: Adding a positive number to another number increases it. (Contributed by NM, 25-Aug-1999)
|
|
Ref |
Expression |
|
Hypotheses |
lt2.1 |
⊢ 𝐴 ∈ ℝ |
|
|
lt2.2 |
⊢ 𝐵 ∈ ℝ |
|
Assertion |
ltaddposi |
⊢ ( 0 < 𝐴 ↔ 𝐵 < ( 𝐵 + 𝐴 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
lt2.1 |
⊢ 𝐴 ∈ ℝ |
2 |
|
lt2.2 |
⊢ 𝐵 ∈ ℝ |
3 |
|
ltaddpos |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 0 < 𝐴 ↔ 𝐵 < ( 𝐵 + 𝐴 ) ) ) |
4 |
1 2 3
|
mp2an |
⊢ ( 0 < 𝐴 ↔ 𝐵 < ( 𝐵 + 𝐴 ) ) |