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
|- ( ph -> A e. CC )
mulnegd.2
|- ( ph -> B e. CC )
Assertion mulneg1d
|- ( ph -> ( -u A x. B ) = -u ( A x. B ) )

Proof

Step Hyp Ref Expression
1 mulm1d.1
 |-  ( ph -> A e. CC )
2 mulnegd.2
 |-  ( ph -> B e. CC )
3 mulneg1
 |-  ( ( A e. CC /\ B e. CC ) -> ( -u A x. B ) = -u ( A x. B ) )
4 1 2 3 syl2anc
 |-  ( ph -> ( -u A x. B ) = -u ( A x. B ) )