This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ltrn2ateq

Description: Property of the equality of a lattice translation with its value. (Contributed by NM, 27-May-2012)

Ref Expression
Hypotheses ltrn2eq.l 
|- .<_ = ( le ` K )
ltrn2eq.a 
|- A = ( Atoms ` K )
ltrn2eq.h 
|- H = ( LHyp ` K )
ltrn2eq.t 
|- T = ( ( LTrn ` K ) ` W )
Assertion ltrn2ateq 
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( ( F ` P ) = P <-> ( F ` Q ) = Q ) )

Proof

Step Hyp Ref Expression
1 ltrn2eq.l 
 |-  .<_ = ( le ` K )
2 ltrn2eq.a 
 |-  A = ( Atoms ` K )
3 ltrn2eq.h 
 |-  H = ( LHyp ` K )
4 ltrn2eq.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 eqid 
 |-  ( Base ` K ) = ( Base ` K )
6 5 1 2 3 4 ltrnideq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( F = ( _I |` ( Base ` K ) ) <-> ( F ` P ) = P ) )
7 6 3adant3r3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( F = ( _I |` ( Base ` K ) ) <-> ( F ` P ) = P ) )
8 5 1 2 3 4 ltrnideq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( F = ( _I |` ( Base ` K ) ) <-> ( F ` Q ) = Q ) )
9 8 3adant3r2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( F = ( _I |` ( Base ` K ) ) <-> ( F ` Q ) = Q ) )
10 7 9 bitr3d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( ( F ` P ) = P <-> ( F ` Q ) = Q ) )