Metamath Proof Explorer
Description: Difference of absolute values. (Contributed by Mario Carneiro, 29-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
abscld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
|
abssubd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
|
Assertion |
abs2dif2d |
⊢ ( 𝜑 → ( abs ‘ ( 𝐴 − 𝐵 ) ) ≤ ( ( abs ‘ 𝐴 ) + ( abs ‘ 𝐵 ) ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
abscld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
abssubd.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
3 |
|
abs2dif2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( abs ‘ ( 𝐴 − 𝐵 ) ) ≤ ( ( abs ‘ 𝐴 ) + ( abs ‘ 𝐵 ) ) ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → ( abs ‘ ( 𝐴 − 𝐵 ) ) ≤ ( ( abs ‘ 𝐴 ) + ( abs ‘ 𝐵 ) ) ) |