Metamath Proof Explorer


Theorem releabsd

Description: The real part of a number is less than or equal to its absolute value. Proposition 10-3.7(d) of Gleason p. 133. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypothesis abscld.1 ( 𝜑𝐴 ∈ ℂ )
Assertion releabsd ( 𝜑 → ( ℜ ‘ 𝐴 ) ≤ ( abs ‘ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 abscld.1 ( 𝜑𝐴 ∈ ℂ )
2 releabs ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ≤ ( abs ‘ 𝐴 ) )
3 1 2 syl ( 𝜑 → ( ℜ ‘ 𝐴 ) ≤ ( abs ‘ 𝐴 ) )