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Metamath Proof Explorer

Theorem cdlemg1cN

Description: Any translation belongs to the set of functions constructed for cdleme . TODO: Fix comment. (Contributed by NM, 18-Apr-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemg1c.l 
|- .<_ = ( le ` K )
cdlemg1c.a 
|- A = ( Atoms ` K )
cdlemg1c.h 
|- H = ( LHyp ` K )
cdlemg1c.t 
|- T = ( ( LTrn ` K ) ` W )
Assertion cdlemg1cN 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = Q ) -> ( F e. T <-> F = ( iota_ f e. T ( f ` P ) = Q ) ) )

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 simpll1 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = Q ) /\ F e. T ) -> ( K e. HL /\ W e. H ) )
6 simpll2 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = Q ) /\ F e. T ) -> ( P e. A /\ -. P .<_ W ) )
7 simpr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = Q ) /\ F e. T ) -> F e. T )
8 1 2 3 4 cdlemeiota 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ F e. T ) -> F = ( iota_ f e. T ( f ` P ) = ( F ` P ) ) )
9 5 6 7 8 syl3anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = Q ) /\ F e. T ) -> F = ( iota_ f e. T ( f ` P ) = ( F ` P ) ) )
10 simplr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = Q ) /\ F e. T ) -> ( F ` P ) = Q )
11 10 eqeq2d 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = Q ) /\ F e. T ) -> ( ( f ` P ) = ( F ` P ) <-> ( f ` P ) = Q ) )
12 11 riotabidv 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = Q ) /\ F e. T ) -> ( iota_ f e. T ( f ` P ) = ( F ` P ) ) = ( iota_ f e. T ( f ` P ) = Q ) )
13 9 12 eqtrd 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = Q ) /\ F e. T ) -> F = ( iota_ f e. T ( f ` P ) = Q ) )
14 1 2 3 4 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 )
15 14 adantlr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = Q ) /\ F = ( iota_ f e. T ( f ` P ) = Q ) ) -> F e. T )
16 13 15 impbida 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F ` P ) = Q ) -> ( F e. T <-> F = ( iota_ f e. T ( f ` P ) = Q ) ) )