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 e. CC
mulneg.2
|- B e. CC
Assertion mulneg1i
|- ( -u A x. B ) = -u ( A x. B )

Proof

Step Hyp Ref Expression
1 mulm1.1
 |-  A e. CC
2 mulneg.2
 |-  B e. CC
3 mulneg1
 |-  ( ( A e. CC /\ B e. CC ) -> ( -u A x. B ) = -u ( A x. B ) )
4 1 2 3 mp2an
 |-  ( -u A x. B ) = -u ( A x. B )