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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 )