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Metamath Proof Explorer

Theorem ltrniotaidvalN

Description: Value of the unique translation specified by identity value. (Contributed by NM, 25-Aug-2014) (New usage is discouraged.)

Ref Expression
Hypotheses ltrniotaidval.b 
|- B = ( Base ` K )
ltrniotaidval.l 
|- .<_ = ( le ` K )
ltrniotaidval.a 
|- A = ( Atoms ` K )
ltrniotaidval.h 
|- H = ( LHyp ` K )
ltrniotaidval.t 
|- T = ( ( LTrn ` K ) ` W )
ltrniotaidval.f 
|- F = ( iota_ f e. T ( f ` P ) = P )
Assertion ltrniotaidvalN 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> F = ( _I |` B ) )

Proof

Step Hyp Ref Expression
1 ltrniotaidval.b 
 |-  B = ( Base ` K )
2 ltrniotaidval.l 
 |-  .<_ = ( le ` K )
3 ltrniotaidval.a 
 |-  A = ( Atoms ` K )
4 ltrniotaidval.h 
 |-  H = ( LHyp ` K )
5 ltrniotaidval.t 
 |-  T = ( ( LTrn ` K ) ` W )
6 ltrniotaidval.f 
 |-  F = ( iota_ f e. T ( f ` P ) = P )
7 2 3 4 5 6 ltrniotaval 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( F ` P ) = P )
8 7 3anidm23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( F ` P ) = P )
9 simpl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( K e. HL /\ W e. H ) )
10 2 3 4 5 6 ltrniotacl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) -> F e. T )
11 10 3anidm23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> F e. T )
12 simpr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( P e. A /\ -. P .<_ W ) )
13 1 2 3 4 5 ltrnideq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( F = ( _I |` B ) <-> ( F ` P ) = P ) )
14 9 11 12 13 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( F = ( _I |` B ) <-> ( F ` P ) = P ) )
15 8 14 mpbird 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> F = ( _I |` B ) )