Metamath Proof Explorer
Description: Specialization of breq2d to reals and less than. (Contributed by Stanislas Polu, 9-Mar-2020)
|
|
Ref |
Expression |
|
Hypotheses |
leeq2d.1 |
⊢ ( 𝜑 → 𝐴 ≤ 𝐶 ) |
|
|
leeq2d.2 |
⊢ ( 𝜑 → 𝐶 = 𝐷 ) |
|
|
leeq2d.3 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
|
|
leeq2d.4 |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
|
Assertion |
leeq2d |
⊢ ( 𝜑 → 𝐴 ≤ 𝐷 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
leeq2d.1 |
⊢ ( 𝜑 → 𝐴 ≤ 𝐶 ) |
2 |
|
leeq2d.2 |
⊢ ( 𝜑 → 𝐶 = 𝐷 ) |
3 |
|
leeq2d.3 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
4 |
|
leeq2d.4 |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
5 |
1 2
|
breqtrd |
⊢ ( 𝜑 → 𝐴 ≤ 𝐷 ) |