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Metamath Proof Explorer

Theorem abstrii

Description: Triangle inequality for absolute value. Proposition 10-3.7(h) of Gleason p. 133. This is Metamath 100 proof #91. (Contributed by NM, 2-Oct-1999)

Ref Expression
Hypotheses absvalsqi.1 ⊢ A ∈ ℂ
abssub.2 ⊢ B ∈ ℂ
Assertion abstrii ⊢ A + B ≤ A + B

Proof

Step Hyp Ref Expression
1 absvalsqi.1 ⊢ A ∈ ℂ
2 abssub.2 ⊢ B ∈ ℂ
3 abstri ⊢ A ∈ ℂ ∧ B ∈ ℂ → A + B ≤ A + B
4 1 2 3 mp2an ⊢ A + B ≤ A + B