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Metamath Proof Explorer

Theorem css0

Description: The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015)

Ref Expression
Hypotheses css0.c 
|- C = ( ClSubSp ` W )
css0.z 
|- .0. = ( 0g ` W )
Assertion css0 
|- ( W e. PreHil -> { .0. } e. C )

Proof

Step Hyp Ref Expression
1 css0.c 
 |-  C = ( ClSubSp ` W )
2 css0.z 
 |-  .0. = ( 0g ` W )
3 eqid 
 |-  ( Base ` W ) = ( Base ` W )
4 eqid 
 |-  ( ocv ` W ) = ( ocv ` W )
5 3 4 2 ocv1 
 |-  ( W e. PreHil -> ( ( ocv ` W ) ` ( Base ` W ) ) = { .0. } )
6 ssid 
 |-  ( Base ` W ) C_ ( Base ` W )
7 3 1 4 ocvcss 
 |-  ( ( W e. PreHil /\ ( Base ` W ) C_ ( Base ` W ) ) -> ( ( ocv ` W ) ` ( Base ` W ) ) e. C )
8 6 7 mpan2 
 |-  ( W e. PreHil -> ( ( ocv ` W ) ` ( Base ` W ) ) e. C )
9 5 8 eqeltrrd 
 |-  ( W e. PreHil -> { .0. } e. C )