Metamath Proof Explorer
Description: A sum is less than the whole if each term is less than half.
(Contributed by Thierry Arnoux, 29-Nov-2017)
|
|
Ref |
Expression |
|
Hypotheses |
le2halvesd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
|
|
le2halvesd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
|
|
le2halvesd.3 |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
|
|
le2halvesd.4 |
⊢ ( 𝜑 → 𝐴 ≤ ( 𝐶 / 2 ) ) |
|
|
le2halvesd.5 |
⊢ ( 𝜑 → 𝐵 ≤ ( 𝐶 / 2 ) ) |
|
Assertion |
le2halvesd |
⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) ≤ 𝐶 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
le2halvesd.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
le2halvesd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
le2halvesd.3 |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
4 |
|
le2halvesd.4 |
⊢ ( 𝜑 → 𝐴 ≤ ( 𝐶 / 2 ) ) |
5 |
|
le2halvesd.5 |
⊢ ( 𝜑 → 𝐵 ≤ ( 𝐶 / 2 ) ) |
6 |
3
|
rehalfcld |
⊢ ( 𝜑 → ( 𝐶 / 2 ) ∈ ℝ ) |
7 |
1 2 6 6 4 5
|
le2addd |
⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) ≤ ( ( 𝐶 / 2 ) + ( 𝐶 / 2 ) ) ) |
8 |
3
|
recnd |
⊢ ( 𝜑 → 𝐶 ∈ ℂ ) |
9 |
8
|
2halvesd |
⊢ ( 𝜑 → ( ( 𝐶 / 2 ) + ( 𝐶 / 2 ) ) = 𝐶 ) |
10 |
7 9
|
breqtrd |
⊢ ( 𝜑 → ( 𝐴 + 𝐵 ) ≤ 𝐶 ) |