Metamath Proof Explorer


Theorem absred

Description: Absolute value of a real number. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypothesis resqrcld.1
|- ( ph -> A e. RR )
Assertion absred
|- ( ph -> ( abs ` A ) = ( sqrt ` ( A ^ 2 ) ) )

Proof

Step Hyp Ref Expression
1 resqrcld.1
 |-  ( ph -> A e. RR )
2 absre
 |-  ( A e. RR -> ( abs ` A ) = ( sqrt ` ( A ^ 2 ) ) )
3 1 2 syl
 |-  ( ph -> ( abs ` A ) = ( sqrt ` ( A ^ 2 ) ) )