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	$$⊢\phi \to A\in ℂ$$
Assertion	releabsd	$$⊢\phi \to \Re \left(A\right)\le \left|A\right|$$

Proof

Step	Hyp	Ref	Expression
1		abscld.1	$$⊢\phi \to A\in ℂ$$
2		releabs	$$⊢A\in ℂ\to \Re \left(A\right)\le \left|A\right|$$
3	1 2	syl	$$⊢\phi \to \Re \left(A\right)\le \left|A\right|$$