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Metamath Proof Explorer

Theorem abstrid

Description: Triangle inequality for absolute value. Proposition 10-3.7(h) of Gleason p. 133. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses abscld.1 ⊢ φ → A ∈ ℂ
abssubd.2 ⊢ φ → B ∈ ℂ
Assertion abstrid ⊢ φ → A + B ≤ A + B

Proof

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