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	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
Assertion	releabsd	⊢ ( 𝜑 → ( ℜ ‘ 𝐴 ) ≤ ( abs ‘ 𝐴 ) )

Proof

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