Theorem absval2d

Description: Value of absolute value function. Definition 10.36 of Gleason p. 133. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypothesis	abscld.1	|- ( ph -> A e. CC )
Assertion	absval2d	|- ( ph -> ( abs ` A ) = ( sqrt ` ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		abscld.1	|- ( ph -> A e. CC )
2		absval2	|- ( A e. CC -> ( abs ` A ) = ( sqrt ` ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) ) )
3	1 2	syl	|- ( ph -> ( abs ` A ) = ( sqrt ` ( ( ( Re ` A ) ^ 2 ) + ( ( Im ` A ) ^ 2 ) ) ) )