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Metamath Proof Explorer

Theorem lautcnv

Description: The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013)

Ref Expression
Hypothesis lautcnv.i 
|- I = ( LAut ` K )
Assertion lautcnv 
|- ( ( K e. V /\ F e. I ) -> `' F e. I )

Proof

Step Hyp Ref Expression
1 lautcnv.i 
 |-  I = ( LAut ` K )
2 eqid 
 |-  ( Base ` K ) = ( Base ` K )
3 2 1 laut1o 
 |-  ( ( K e. V /\ F e. I ) -> F : ( Base ` K ) -1-1-onto-> ( Base ` K ) )
4 f1ocnv 
 |-  ( F : ( Base ` K ) -1-1-onto-> ( Base ` K ) -> `' F : ( Base ` K ) -1-1-onto-> ( Base ` K ) )
5 3 4 syl 
 |-  ( ( K e. V /\ F e. I ) -> `' F : ( Base ` K ) -1-1-onto-> ( Base ` K ) )
6 eqid 
 |-  ( le ` K ) = ( le ` K )
7 2 6 1 lautcnvle 
 |-  ( ( ( K e. V /\ F e. I ) /\ ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) ) -> ( x ( le ` K ) y <-> ( `' F ` x ) ( le ` K ) ( `' F ` y ) ) )
8 7 ralrimivva 
 |-  ( ( K e. V /\ F e. I ) -> A. x e. ( Base ` K ) A. y e. ( Base ` K ) ( x ( le ` K ) y <-> ( `' F ` x ) ( le ` K ) ( `' F ` y ) ) )
9 2 6 1 islaut 
 |-  ( K e. V -> ( `' F e. I <-> ( `' F : ( Base ` K ) -1-1-onto-> ( Base ` K ) /\ A. x e. ( Base ` K ) A. y e. ( Base ` K ) ( x ( le ` K ) y <-> ( `' F ` x ) ( le ` K ) ( `' F ` y ) ) ) ) )
10 9 adantr 
 |-  ( ( K e. V /\ F e. I ) -> ( `' F e. I <-> ( `' F : ( Base ` K ) -1-1-onto-> ( Base ` K ) /\ A. x e. ( Base ` K ) A. y e. ( Base ` K ) ( x ( le ` K ) y <-> ( `' F ` x ) ( le ` K ) ( `' F ` y ) ) ) ) )
11 5 8 10 mpbir2and 
 |-  ( ( K e. V /\ F e. I ) -> `' F e. I )