Theorem absneg

Description: Absolute value of the opposite. (Contributed by NM, 27-Feb-2005)

	Ref	Expression
Assertion	absneg	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ - 𝐴 ) = ( abs ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		cjneg	⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ - 𝐴 ) = - ( ∗ ‘ 𝐴 ) )
2	1	oveq2d	⊢ ( 𝐴 ∈ ℂ → ( - 𝐴 · ( ∗ ‘ - 𝐴 ) ) = ( - 𝐴 · - ( ∗ ‘ 𝐴 ) ) )
3		cjcl	⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ 𝐴 ) ∈ ℂ )
4		mul2neg	⊢ ( ( 𝐴 ∈ ℂ ∧ ( ∗ ‘ 𝐴 ) ∈ ℂ ) → ( - 𝐴 · - ( ∗ ‘ 𝐴 ) ) = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) )
5	3 4	mpdan	⊢ ( 𝐴 ∈ ℂ → ( - 𝐴 · - ( ∗ ‘ 𝐴 ) ) = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) )
6	2 5	eqtrd	⊢ ( 𝐴 ∈ ℂ → ( - 𝐴 · ( ∗ ‘ - 𝐴 ) ) = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) )
7	6	fveq2d	⊢ ( 𝐴 ∈ ℂ → ( √ ‘ ( - 𝐴 · ( ∗ ‘ - 𝐴 ) ) ) = ( √ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) )
8		negcl	⊢ ( 𝐴 ∈ ℂ → - 𝐴 ∈ ℂ )
9		absval	⊢ ( - 𝐴 ∈ ℂ → ( abs ‘ - 𝐴 ) = ( √ ‘ ( - 𝐴 · ( ∗ ‘ - 𝐴 ) ) ) )
10	8 9	syl	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ - 𝐴 ) = ( √ ‘ ( - 𝐴 · ( ∗ ‘ - 𝐴 ) ) ) )
11		absval	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ 𝐴 ) = ( √ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) )
12	7 10 11	3eqtr4d	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ - 𝐴 ) = ( abs ‘ 𝐴 ) )