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Metamath Proof Explorer

Theorem diaf1oN

Description: The partial isomorphism A for a lattice K is a one-to-one, onto function. Part of Lemma M of Crawley p. 121 line 12, with closed subspaces rather than subspaces. See diadm for the domain. (Contributed by NM, 17-Jan-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dvadia.h 
|- H = ( LHyp ` K )
dvadia.u 
|- U = ( ( DVecA ` K ) ` W )
dvadia.i 
|- I = ( ( DIsoA ` K ) ` W )
dvadia.n 
|- ._|_ = ( ( ocA ` K ) ` W )
dvadia.s 
|- S = ( LSubSp ` U )
Assertion diaf1oN 
|- ( ( K e. HL /\ W e. H ) -> I : dom I -1-1-onto-> { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } )

Proof

Step Hyp Ref Expression
1 dvadia.h 
 |-  H = ( LHyp ` K )
2 dvadia.u 
 |-  U = ( ( DVecA ` K ) ` W )
3 dvadia.i 
 |-  I = ( ( DIsoA ` K ) ` W )
4 dvadia.n 
 |-  ._|_ = ( ( ocA ` K ) ` W )
5 dvadia.s 
 |-  S = ( LSubSp ` U )
6 1 3 diaf11N 
 |-  ( ( K e. HL /\ W e. H ) -> I : dom I -1-1-onto-> ran I )
7 f1of1 
 |-  ( I : dom I -1-1-onto-> ran I -> I : dom I -1-1-> ran I )
8 6 7 syl 
 |-  ( ( K e. HL /\ W e. H ) -> I : dom I -1-1-> ran I )
9 1 2 3 4 5 diarnN 
 |-  ( ( K e. HL /\ W e. H ) -> ran I = { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } )
10 f1eq3 
 |-  ( ran I = { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } -> ( I : dom I -1-1-> ran I <-> I : dom I -1-1-> { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } ) )
11 9 10 syl 
 |-  ( ( K e. HL /\ W e. H ) -> ( I : dom I -1-1-> ran I <-> I : dom I -1-1-> { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } ) )
12 8 11 mpbid 
 |-  ( ( K e. HL /\ W e. H ) -> I : dom I -1-1-> { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } )
13 dff1o5 
 |-  ( I : dom I -1-1-onto-> { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } <-> ( I : dom I -1-1-> { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } /\ ran I = { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } ) )
14 12 9 13 sylanbrc 
 |-  ( ( K e. HL /\ W e. H ) -> I : dom I -1-1-onto-> { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } )