Theorem abs0

Description: The absolute value of 0. (Contributed by NM, 26-Mar-2005) (Revised by Mario Carneiro, 29-May-2016)

	Ref	Expression
Assertion	abs0	⊢ ( abs ‘ 0 ) = 0

Proof

Step	Hyp	Ref	Expression
1		0cn	⊢ 0 ∈ ℂ
2		absval	⊢ ( 0 ∈ ℂ → ( abs ‘ 0 ) = ( √ ‘ ( 0 · ( ∗ ‘ 0 ) ) ) )
3	1 2	ax-mp	⊢ ( abs ‘ 0 ) = ( √ ‘ ( 0 · ( ∗ ‘ 0 ) ) )
4	1	cjcli	⊢ ( ∗ ‘ 0 ) ∈ ℂ
5	4	mul02i	⊢ ( 0 · ( ∗ ‘ 0 ) ) = 0
6	5	fveq2i	⊢ ( √ ‘ ( 0 · ( ∗ ‘ 0 ) ) ) = ( √ ‘ 0 )
7		sqrt0	⊢ ( √ ‘ 0 ) = 0
8	3 6 7	3eqtri	⊢ ( abs ‘ 0 ) = 0