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Metamath Proof Explorer

Theorem mulneg2

Description: The product with a negative is the negative of the product. (Contributed by NM, 30-Jul-2004)

Ref Expression
Assertion mulneg2 
|- ( ( A e. CC /\ B e. CC ) -> ( A x. -u B ) = -u ( A x. B ) )

Proof

Step Hyp Ref Expression
1 mulneg1 
 |-  ( ( B e. CC /\ A e. CC ) -> ( -u B x. A ) = -u ( B x. A ) )
2 1 ancoms 
 |-  ( ( A e. CC /\ B e. CC ) -> ( -u B x. A ) = -u ( B x. A ) )
3 negcl 
 |-  ( B e. CC -> -u B e. CC )
4 mulcom 
 |-  ( ( A e. CC /\ -u B e. CC ) -> ( A x. -u B ) = ( -u B x. A ) )
5 3 4 sylan2 
 |-  ( ( A e. CC /\ B e. CC ) -> ( A x. -u B ) = ( -u B x. A ) )
6 mulcom 
 |-  ( ( A e. CC /\ B e. CC ) -> ( A x. B ) = ( B x. A ) )
7 6 negeqd 
 |-  ( ( A e. CC /\ B e. CC ) -> -u ( A x. B ) = -u ( B x. A ) )
8 2 5 7 3eqtr4d 
 |-  ( ( A e. CC /\ B e. CC ) -> ( A x. -u B ) = -u ( A x. B ) )