Metamath Proof Explorer


Theorem releabsi

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 NM, 2-Oct-1999)

Ref Expression
Hypothesis absvalsqi.1 𝐴 ∈ ℂ
Assertion releabsi ( ℜ ‘ 𝐴 ) ≤ ( abs ‘ 𝐴 )

Proof

Step Hyp Ref Expression
1 absvalsqi.1 𝐴 ∈ ℂ
2 releabs ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ≤ ( abs ‘ 𝐴 ) )
3 1 2 ax-mp ( ℜ ‘ 𝐴 ) ≤ ( abs ‘ 𝐴 )