Theorem mulm1

Description: Product with minus one is negative. (Contributed by NM, 16-Nov-1999)

	Ref	Expression
Assertion	mulm1	⊢ ( 𝐴 ∈ ℂ → ( - 1 · 𝐴 ) = - 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		ax-1cn	⊢ 1 ∈ ℂ
2		mulneg1	⊢ ( ( 1 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( - 1 · 𝐴 ) = - ( 1 · 𝐴 ) )
3	1 2	mpan	⊢ ( 𝐴 ∈ ℂ → ( - 1 · 𝐴 ) = - ( 1 · 𝐴 ) )
4		mulid2	⊢ ( 𝐴 ∈ ℂ → ( 1 · 𝐴 ) = 𝐴 )
5	4	negeqd	⊢ ( 𝐴 ∈ ℂ → - ( 1 · 𝐴 ) = - 𝐴 )
6	3 5	eqtrd	⊢ ( 𝐴 ∈ ℂ → ( - 1 · 𝐴 ) = - 𝐴 )