Metamath Proof Explorer


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
|- ( ph -> A e. CC )
Assertion releabsd
|- ( ph -> ( Re ` A ) <_ ( abs ` A ) )

Proof

Step Hyp Ref Expression
1 abscld.1
 |-  ( ph -> A e. CC )
2 releabs
 |-  ( A e. CC -> ( Re ` A ) <_ ( abs ` A ) )
3 1 2 syl
 |-  ( ph -> ( Re ` A ) <_ ( abs ` A ) )