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Metamath Proof Explorer

Theorem abs2difabsd

Description: Absolute value of 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 abs2difabsd ⊢ φ → A − B ≤ A − B

Proof

Step Hyp Ref Expression
1 abscld.1 ⊢ φ → A ∈ ℂ
2 abssubd.2 ⊢ φ → B ∈ ℂ
3 abs2difabs ⊢ A ∈ ℂ ∧ B ∈ ℂ → A − B ≤ A − B
4 1 2 3 syl2anc ⊢ φ → A − B ≤ A − B