Theorem mulneg1

Description: Product with negative is negative of product. Theorem I.12 of Apostol p. 18. (Contributed by NM, 14-May-1999) (Proof shortened by Mario Carneiro, 27-May-2016)

	Ref	Expression
Assertion	mulneg1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 · 𝐵 ) = - ( 𝐴 · 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		0cn	⊢ 0 ∈ ℂ
2		subdir	⊢ ( ( 0 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 0 − 𝐴 ) · 𝐵 ) = ( ( 0 · 𝐵 ) − ( 𝐴 · 𝐵 ) ) )
3	1 2	mp3an1	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 0 − 𝐴 ) · 𝐵 ) = ( ( 0 · 𝐵 ) − ( 𝐴 · 𝐵 ) ) )
4		simpr	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → 𝐵 ∈ ℂ )
5	4	mul02d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 0 · 𝐵 ) = 0 )
6	5	oveq1d	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 0 · 𝐵 ) − ( 𝐴 · 𝐵 ) ) = ( 0 − ( 𝐴 · 𝐵 ) ) )
7	3 6	eqtrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ( 0 − 𝐴 ) · 𝐵 ) = ( 0 − ( 𝐴 · 𝐵 ) ) )
8		df-neg	⊢ - 𝐴 = ( 0 − 𝐴 )
9	8	oveq1i	⊢ ( - 𝐴 · 𝐵 ) = ( ( 0 − 𝐴 ) · 𝐵 )
10		df-neg	⊢ - ( 𝐴 · 𝐵 ) = ( 0 − ( 𝐴 · 𝐵 ) )
11	7 9 10	3eqtr4g	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 · 𝐵 ) = - ( 𝐴 · 𝐵 ) )