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Metamath Proof Explorer

Theorem mulneg1i

Description: Product with negative is negative of product. Theorem I.12 of Apostol p. 18. (Contributed by NM, 10-Feb-1995) (Revised by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	mulm1.1	⊢ 𝐴 ∈ ℂ
	mulneg.2	⊢ 𝐵 ∈ ℂ
Assertion	mulneg1i	⊢ ( - 𝐴 · 𝐵 ) = - ( 𝐴 · 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		mulm1.1	⊢ 𝐴 ∈ ℂ
2		mulneg.2	⊢ 𝐵 ∈ ℂ
3		mulneg1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 · 𝐵 ) = - ( 𝐴 · 𝐵 ) )
4	1 2 3	mp2an	⊢ ( - 𝐴 · 𝐵 ) = - ( 𝐴 · 𝐵 )