Metamath Proof Explorer

Theorem matbas0

Description: There is no matrix for a not finite dimension or a proper class as the underlying ring. (Contributed by AV, 28-Dec-2018)

Ref Expression
Assertion matbas0 
|- ( -. ( N e. Fin /\ R e. _V ) -> ( Base ` ( N Mat R ) ) = (/) )

Proof

Step Hyp Ref Expression
1 df-mat 
 |-  Mat = ( n e. Fin , r e. _V |-> ( ( r freeLMod ( n X. n ) ) sSet <. ( .r ` ndx ) , ( r maMul <. n , n , n >. ) >. ) )
2 1 mpondm0 
 |-  ( -. ( N e. Fin /\ R e. _V ) -> ( N Mat R ) = (/) )
3 2 fveq2d 
 |-  ( -. ( N e. Fin /\ R e. _V ) -> ( Base ` ( N Mat R ) ) = ( Base ` (/) ) )
4 base0 
 |-  (/) = ( Base ` (/) )
5 3 4 eqtr4di 
 |-  ( -. ( N e. Fin /\ R e. _V ) -> ( Base ` ( N Mat R ) ) = (/) )