Metamath Proof Explorer

Theorem smu02

Description: Multiplication of a sequence by 0 on the left. (Contributed by Mario Carneiro, 9-Sep-2016)

Ref Expression
Assertion smu02 
|- ( A C_ NN0 -> ( (/) smul A ) = (/) )

Proof

Step Hyp Ref Expression
1 0ss 
 |-  (/) C_ NN0
2 1 a1i 
 |-  ( A C_ NN0 -> (/) C_ NN0 )
3 id 
 |-  ( A C_ NN0 -> A C_ NN0 )
4 noel 
 |-  -. k e. (/)
5 4 intnanr 
 |-  -. ( k e. (/) /\ ( n - k ) e. A )
6 5 a1i 
 |-  ( ( A C_ NN0 /\ ( k e. NN0 /\ n e. NN0 ) ) -> -. ( k e. (/) /\ ( n - k ) e. A ) )
7 2 3 6 smu01lem 
 |-  ( A C_ NN0 -> ( (/) smul A ) = (/) )