Metamath Proof Explorer


Theorem ldil1o

Description: A lattice dilation is a one-to-one onto function. (Contributed by NM, 19-Apr-2013)

Ref Expression
Hypotheses ldil1o.b
|- B = ( Base ` K )
ldil1o.h
|- H = ( LHyp ` K )
ldil1o.d
|- D = ( ( LDil ` K ) ` W )
Assertion ldil1o
|- ( ( ( K e. V /\ W e. H ) /\ F e. D ) -> F : B -1-1-onto-> B )

Proof

Step Hyp Ref Expression
1 ldil1o.b
 |-  B = ( Base ` K )
2 ldil1o.h
 |-  H = ( LHyp ` K )
3 ldil1o.d
 |-  D = ( ( LDil ` K ) ` W )
4 simpll
 |-  ( ( ( K e. V /\ W e. H ) /\ F e. D ) -> K e. V )
5 eqid
 |-  ( LAut ` K ) = ( LAut ` K )
6 2 5 3 ldillaut
 |-  ( ( ( K e. V /\ W e. H ) /\ F e. D ) -> F e. ( LAut ` K ) )
7 1 5 laut1o
 |-  ( ( K e. V /\ F e. ( LAut ` K ) ) -> F : B -1-1-onto-> B )
8 4 6 7 syl2anc
 |-  ( ( ( K e. V /\ W e. H ) /\ F e. D ) -> F : B -1-1-onto-> B )