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Metamath Proof Explorer

Theorem ltrniotavalbN

Description: Value of the unique translation specified by a value. (Contributed by NM, 10-Mar-2014) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 ltrniotavalb.l 
 |-  .<_ = ( le ` K )
2 ltrniotavalb.a 
 |-  A = ( Atoms ` K )
3 ltrniotavalb.h 
 |-  H = ( LHyp ` K )
4 ltrniotavalb.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) /\ ( F ` P ) = Q ) -> ( K e. HL /\ W e. H ) )
6 simpl3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) /\ ( F ` P ) = Q ) -> F e. T )
7 simpl2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) /\ ( F ` P ) = Q ) -> ( P e. A /\ -. P .<_ W ) )
8 simpl2r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) /\ ( F ` P ) = Q ) -> ( Q e. A /\ -. Q .<_ W ) )
9 eqid 
 |-  ( iota_ f e. T ( f ` P ) = Q ) = ( iota_ f e. T ( f ` P ) = Q )
10 1 2 3 4 9 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 )
11 5 7 8 10 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) /\ ( F ` P ) = Q ) -> ( iota_ f e. T ( f ` P ) = Q ) e. T )
12 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) /\ ( F ` P ) = Q ) -> ( F ` P ) = Q )
13 1 2 3 4 9 ltrniotaval 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( iota_ f e. T ( f ` P ) = Q ) ` P ) = Q )
14 5 7 8 13 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) /\ ( F ` P ) = Q ) -> ( ( iota_ f e. T ( f ` P ) = Q ) ` P ) = Q )
15 12 14 eqtr4d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) /\ ( F ` P ) = Q ) -> ( F ` P ) = ( ( iota_ f e. T ( f ` P ) = Q ) ` P ) )
16 1 2 3 4 cdlemd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( iota_ f e. T ( f ` P ) = Q ) e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F ` P ) = ( ( iota_ f e. T ( f ` P ) = Q ) ` P ) ) -> F = ( iota_ f e. T ( f ` P ) = Q ) )
17 5 6 11 7 15 16 syl311anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) /\ ( F ` P ) = Q ) -> F = ( iota_ f e. T ( f ` P ) = Q ) )
18 fveq1 
 |-  ( F = ( iota_ f e. T ( f ` P ) = Q ) -> ( F ` P ) = ( ( iota_ f e. T ( f ` P ) = Q ) ` P ) )
19 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) -> ( K e. HL /\ W e. H ) )
20 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) -> ( P e. A /\ -. P .<_ W ) )
21 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) -> ( Q e. A /\ -. Q .<_ W ) )
22 19 20 21 13 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) -> ( ( iota_ f e. T ( f ` P ) = Q ) ` P ) = Q )
23 18 22 sylan9eqr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) /\ F = ( iota_ f e. T ( f ` P ) = Q ) ) -> ( F ` P ) = Q )
24 17 23 impbida 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) -> ( ( F ` P ) = Q <-> F = ( iota_ f e. T ( f ` P ) = Q ) ) )