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Metamath Proof Explorer

Theorem mulneg1d

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

Ref Expression
Hypotheses mulm1d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
mulnegd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion mulneg1d ( 𝜑 → ( - 𝐴 · 𝐵 ) = - ( 𝐴 · 𝐵 ) )

Proof

Step Hyp Ref Expression
1 mulm1d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 mulnegd.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3 mulneg1 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( - 𝐴 · 𝐵 ) = - ( 𝐴 · 𝐵 ) )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( - 𝐴 · 𝐵 ) = - ( 𝐴 · 𝐵 ) )