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Metamath Proof Explorer

Theorem cdlemg1ci2

Description: Any function of the form of the function constructed for cdleme is a translation. TODO: Fix comment. (Contributed by NM, 18-Apr-2013)

Ref Expression
Hypotheses cdlemg1c.l 
|- .<_ = ( le ` K )
cdlemg1c.a 
|- A = ( Atoms ` K )
cdlemg1c.h 
|- H = ( LHyp ` K )
cdlemg1c.t 
|- T = ( ( LTrn ` K ) ` W )
Assertion cdlemg1ci2 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F = ( iota_ f e. T ( f ` P ) = Q ) ) -> F e. T )

Proof

Step Hyp Ref Expression
1 cdlemg1c.l 
 |-  .<_ = ( le ` K )
2 cdlemg1c.a 
 |-  A = ( Atoms ` K )
3 cdlemg1c.h 
 |-  H = ( LHyp ` K )
4 cdlemg1c.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F = ( iota_ f e. T ( f ` P ) = Q ) ) -> F = ( iota_ f e. T ( f ` P ) = Q ) )
6 eqid 
 |-  ( iota_ f e. T ( f ` P ) = Q ) = ( iota_ f e. T ( f ` P ) = Q )
7 1 2 3 4 6 ltrniotacl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( iota_ f e. T ( f ` P ) = Q ) e. T )
8 7 adantr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F = ( iota_ f e. T ( f ` P ) = Q ) ) -> ( iota_ f e. T ( f ` P ) = Q ) e. T )
9 5 8 eqeltrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F = ( iota_ f e. T ( f ` P ) = Q ) ) -> F e. T )