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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 → ( 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 abstri ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( abs ‘ ( 𝐴 + 𝐵 ) ) ≤ ( ( abs ‘ 𝐴 ) + ( abs ‘ 𝐵 ) ) )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( abs ‘ ( 𝐴 + 𝐵 ) ) ≤ ( ( abs ‘ 𝐴 ) + ( abs ‘ 𝐵 ) ) )