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Metamath Proof Explorer

Theorem riotaocN

Description: The orthocomplement of the unique poset element such that ps . ( riotaneg analog.) (Contributed by NM, 16-Jan-2012) (New usage is discouraged.)

Ref Expression
Hypotheses riotaoc.b 
|- B = ( Base ` K )
riotaoc.o 
|- ._|_ = ( oc ` K )
riotaoc.a 
|- ( x = ( ._|_ ` y ) -> ( ph <-> ps ) )
Assertion riotaocN 
|- ( ( K e. OP /\ E! x e. B ph ) -> ( iota_ x e. B ph ) = ( ._|_ ` ( iota_ y e. B ps ) ) )

Proof

Step Hyp Ref Expression
1 riotaoc.b 
 |-  B = ( Base ` K )
2 riotaoc.o 
 |-  ._|_ = ( oc ` K )
3 riotaoc.a 
 |-  ( x = ( ._|_ ` y ) -> ( ph <-> ps ) )
4 nfcv 
 |-  F/_ y ._|_
5 nfriota1 
 |-  F/_ y ( iota_ y e. B ps )
6 4 5 nffv 
 |-  F/_ y ( ._|_ ` ( iota_ y e. B ps ) )
7 1 2 opoccl 
 |-  ( ( K e. OP /\ y e. B ) -> ( ._|_ ` y ) e. B )
8 1 2 opoccl 
 |-  ( ( K e. OP /\ ( iota_ y e. B ps ) e. B ) -> ( ._|_ ` ( iota_ y e. B ps ) ) e. B )
9 fveq2 
 |-  ( y = ( iota_ y e. B ps ) -> ( ._|_ ` y ) = ( ._|_ ` ( iota_ y e. B ps ) ) )
10 1 2 opoccl 
 |-  ( ( K e. OP /\ x e. B ) -> ( ._|_ ` x ) e. B )
11 1 2 opcon2b 
 |-  ( ( K e. OP /\ x e. B /\ y e. B ) -> ( x = ( ._|_ ` y ) <-> y = ( ._|_ ` x ) ) )
12 10 11 reuhypd 
 |-  ( ( K e. OP /\ x e. B ) -> E! y e. B x = ( ._|_ ` y ) )
13 6 7 8 3 9 12 riotaxfrd 
 |-  ( ( K e. OP /\ E! x e. B ph ) -> ( iota_ x e. B ph ) = ( ._|_ ` ( iota_ y e. B ps ) ) )