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Metamath Proof Explorer

Theorem cdlemeiota

Description: A translation is uniquely determined by one of its values. (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 cdlemeiota 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ F e. T ) -> F = ( iota_ f e. T ( f ` P ) = ( F ` P ) ) )

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 eqidd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ F e. T ) -> ( F ` P ) = ( F ` P ) )
6 simp3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ F e. T ) -> F e. T )
7 1 2 3 4 ltrnel 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( F ` P ) e. A /\ -. ( F ` P ) .<_ W ) )
8 7 3com23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ F e. T ) -> ( ( F ` P ) e. A /\ -. ( F ` P ) .<_ W ) )
9 1 2 3 4 cdleme 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( F ` P ) e. A /\ -. ( F ` P ) .<_ W ) ) -> E! f e. T ( f ` P ) = ( F ` P ) )
10 8 9 syld3an3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ F e. T ) -> E! f e. T ( f ` P ) = ( F ` P ) )
11 fveq1 
 |-  ( f = F -> ( f ` P ) = ( F ` P ) )
12 11 eqeq1d 
 |-  ( f = F -> ( ( f ` P ) = ( F ` P ) <-> ( F ` P ) = ( F ` P ) ) )
13 12 riota2 
 |-  ( ( F e. T /\ E! f e. T ( f ` P ) = ( F ` P ) ) -> ( ( F ` P ) = ( F ` P ) <-> ( iota_ f e. T ( f ` P ) = ( F ` P ) ) = F ) )
14 6 10 13 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ F e. T ) -> ( ( F ` P ) = ( F ` P ) <-> ( iota_ f e. T ( f ` P ) = ( F ` P ) ) = F ) )
15 5 14 mpbid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ F e. T ) -> ( iota_ f e. T ( f ` P ) = ( F ` P ) ) = F )
16 15 eqcomd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ F e. T ) -> F = ( iota_ f e. T ( f ` P ) = ( F ` P ) ) )