Theorem sqneg

Description: The square of the negative of a number. (Contributed by NM, 15-Jan-2006)

	Ref	Expression
Assertion	sqneg	⊢ ( 𝐴 ∈ ℂ → ( - 𝐴 ↑ 2 ) = ( 𝐴 ↑ 2 ) )

Proof

Step	Hyp	Ref	Expression
1		mul2neg	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( - 𝐴 · - 𝐴 ) = ( 𝐴 · 𝐴 ) )
2	1	anidms	⊢ ( 𝐴 ∈ ℂ → ( - 𝐴 · - 𝐴 ) = ( 𝐴 · 𝐴 ) )
3		negcl	⊢ ( 𝐴 ∈ ℂ → - 𝐴 ∈ ℂ )
4		sqval	⊢ ( - 𝐴 ∈ ℂ → ( - 𝐴 ↑ 2 ) = ( - 𝐴 · - 𝐴 ) )
5	3 4	syl	⊢ ( 𝐴 ∈ ℂ → ( - 𝐴 ↑ 2 ) = ( - 𝐴 · - 𝐴 ) )
6		sqval	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ↑ 2 ) = ( 𝐴 · 𝐴 ) )
7	2 5 6	3eqtr4d	⊢ ( 𝐴 ∈ ℂ → ( - 𝐴 ↑ 2 ) = ( 𝐴 ↑ 2 ) )