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 )