Metamath Proof Explorer

Theorem smu01

Description: Multiplication of a sequence by 0 on the right. (Contributed by Mario Carneiro, 19-Sep-2016)

Ref Expression
Assertion smu01 
|- ( A C_ NN0 -> ( A smul (/) ) = (/) )

Proof

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