Metamath Proof Explorer

Theorem mul01

Description: Multiplication by 0 . Theorem I.6 of Apostol p. 18. (Contributed by NM, 15-May-1999) (Revised by Scott Fenton, 3-Jan-2013)

Ref Expression
Assertion mul01 
|- ( A e. CC -> ( A x. 0 ) = 0 )

Proof

Step Hyp Ref Expression
1 0cn 
 |-  0 e. CC
2 mulcom 
 |-  ( ( A e. CC /\ 0 e. CC ) -> ( A x. 0 ) = ( 0 x. A ) )
3 1 2 mpan2 
 |-  ( A e. CC -> ( A x. 0 ) = ( 0 x. A ) )
4 mul02 
 |-  ( A e. CC -> ( 0 x. A ) = 0 )
5 3 4 eqtrd 
 |-  ( A e. CC -> ( A x. 0 ) = 0 )