Metamath Proof Explorer
Description: IneqMoveTerm generator rule. (Contributed by Stanislas Polu, 7-Apr-2020)
|
|
Ref |
Expression |
|
Hypotheses |
int-ineqmvtd.1 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
|
|
int-ineqmvtd.2 |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
|
|
int-ineqmvtd.3 |
⊢ ( 𝜑 → 𝐷 ∈ ℝ ) |
|
|
int-ineqmvtd.4 |
⊢ ( 𝜑 → 𝐵 ≤ 𝐴 ) |
|
|
int-ineqmvtd.5 |
⊢ ( 𝜑 → 𝐴 = ( 𝐶 + 𝐷 ) ) |
|
Assertion |
int-ineqmvtd |
⊢ ( 𝜑 → ( 𝐵 − 𝐷 ) ≤ 𝐶 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
int-ineqmvtd.1 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
2 |
|
int-ineqmvtd.2 |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
3 |
|
int-ineqmvtd.3 |
⊢ ( 𝜑 → 𝐷 ∈ ℝ ) |
4 |
|
int-ineqmvtd.4 |
⊢ ( 𝜑 → 𝐵 ≤ 𝐴 ) |
5 |
|
int-ineqmvtd.5 |
⊢ ( 𝜑 → 𝐴 = ( 𝐶 + 𝐷 ) ) |
6 |
4 5
|
breqtrd |
⊢ ( 𝜑 → 𝐵 ≤ ( 𝐶 + 𝐷 ) ) |
7 |
1 3 2
|
lesubaddd |
⊢ ( 𝜑 → ( ( 𝐵 − 𝐷 ) ≤ 𝐶 ↔ 𝐵 ≤ ( 𝐶 + 𝐷 ) ) ) |
8 |
6 7
|
mpbird |
⊢ ( 𝜑 → ( 𝐵 − 𝐷 ) ≤ 𝐶 ) |