Metamath Proof Explorer

Theorem nmull0

Description: Natural multiplication by zero. (Contributed by Scott Fenton, 10-Jun-2026)

Ref Expression
Assertion nmull0 
|- ( A e. On -> ( (/) .no A ) = (/) )

Proof

Step Hyp Ref Expression
1 0elon 
 |-  (/) e. On
2 nmulcom 
 |-  ( ( A e. On /\ (/) e. On ) -> ( A .no (/) ) = ( (/) .no A ) )
3 1 2 mpan2 
 |-  ( A e. On -> ( A .no (/) ) = ( (/) .no A ) )
4 nmulr0 
 |-  ( A e. On -> ( A .no (/) ) = (/) )
5 3 4 eqtr3d 
 |-  ( A e. On -> ( (/) .no A ) = (/) )