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Metamath Proof Explorer

Theorem cssval

Description: The set of closed subspaces of a pre-Hilbert space. (Contributed by NM, 7-Oct-2011) (Revised by Mario Carneiro, 13-Oct-2015)

Ref Expression
Hypotheses cssval.o 
|- ._|_ = ( ocv ` W )
cssval.c 
|- C = ( ClSubSp ` W )
Assertion cssval 
|- ( W e. X -> C = { s | s = ( ._|_ ` ( ._|_ ` s ) ) } )

Proof

Step Hyp Ref Expression
1 cssval.o 
 |-  ._|_ = ( ocv ` W )
2 cssval.c 
 |-  C = ( ClSubSp ` W )
3 elex 
 |-  ( W e. X -> W e. _V )
4 fveq2 
 |-  ( w = W -> ( ocv ` w ) = ( ocv ` W ) )
5 4 1 eqtr4di 
 |-  ( w = W -> ( ocv ` w ) = ._|_ )
6 5 fveq1d 
 |-  ( w = W -> ( ( ocv ` w ) ` s ) = ( ._|_ ` s ) )
7 5 6 fveq12d 
 |-  ( w = W -> ( ( ocv ` w ) ` ( ( ocv ` w ) ` s ) ) = ( ._|_ ` ( ._|_ ` s ) ) )
8 7 eqeq2d 
 |-  ( w = W -> ( s = ( ( ocv ` w ) ` ( ( ocv ` w ) ` s ) ) <-> s = ( ._|_ ` ( ._|_ ` s ) ) ) )
9 8 abbidv 
 |-  ( w = W -> { s | s = ( ( ocv ` w ) ` ( ( ocv ` w ) ` s ) ) } = { s | s = ( ._|_ ` ( ._|_ ` s ) ) } )
10 df-css 
 |-  ClSubSp = ( w e. _V |-> { s | s = ( ( ocv ` w ) ` ( ( ocv ` w ) ` s ) ) } )
11 fvex 
 |-  ( Base ` W ) e. _V
12 11 pwex 
 |-  ~P ( Base ` W ) e. _V
13 id 
 |-  ( s = ( ._|_ ` ( ._|_ ` s ) ) -> s = ( ._|_ ` ( ._|_ ` s ) ) )
14 eqid 
 |-  ( Base ` W ) = ( Base ` W )
15 14 1 ocvss 
 |-  ( ._|_ ` ( ._|_ ` s ) ) C_ ( Base ` W )
16 fvex 
 |-  ( ._|_ ` ( ._|_ ` s ) ) e. _V
17 16 elpw 
 |-  ( ( ._|_ ` ( ._|_ ` s ) ) e. ~P ( Base ` W ) <-> ( ._|_ ` ( ._|_ ` s ) ) C_ ( Base ` W ) )
18 15 17 mpbir 
 |-  ( ._|_ ` ( ._|_ ` s ) ) e. ~P ( Base ` W )
19 13 18 eqeltrdi 
 |-  ( s = ( ._|_ ` ( ._|_ ` s ) ) -> s e. ~P ( Base ` W ) )
20 19 abssi 
 |-  { s | s = ( ._|_ ` ( ._|_ ` s ) ) } C_ ~P ( Base ` W )
21 12 20 ssexi 
 |-  { s | s = ( ._|_ ` ( ._|_ ` s ) ) } e. _V
22 9 10 21 fvmpt 
 |-  ( W e. _V -> ( ClSubSp ` W ) = { s | s = ( ._|_ ` ( ._|_ ` s ) ) } )
23 2 22 eqtrid 
 |-  ( W e. _V -> C = { s | s = ( ._|_ ` ( ._|_ ` s ) ) } )
24 3 23 syl 
 |-  ( W e. X -> C = { s | s = ( ._|_ ` ( ._|_ ` s ) ) } )