Theorem absvalsqd

Description: Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypothesis	abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
Assertion	absvalsqd	⊢ ( 𝜑 → ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		absvalsq	⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) )
3	1 2	syl	⊢ ( 𝜑 → ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) )