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Metamath Proof Explorer

Theorem ltrniotacnvval

Description: Converse value of the unique translation specified by a value. (Contributed by NM, 21-Feb-2014)

Ref Expression
Hypotheses ltrniotaval.l 
|- .<_ = ( le ` K )
ltrniotaval.a 
|- A = ( Atoms ` K )
ltrniotaval.h 
|- H = ( LHyp ` K )
ltrniotaval.t 
|- T = ( ( LTrn ` K ) ` W )
ltrniotaval.f 
|- F = ( iota_ f e. T ( f ` P ) = Q )
Assertion ltrniotacnvval 
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( `' F ` Q ) = P )

Proof

Step Hyp Ref Expression
1 ltrniotaval.l 
 |-  .<_ = ( le ` K )
2 ltrniotaval.a 
 |-  A = ( Atoms ` K )
3 ltrniotaval.h 
 |-  H = ( LHyp ` K )
4 ltrniotaval.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 ltrniotaval.f 
 |-  F = ( iota_ f e. T ( f ` P ) = Q )
6 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( K e. HL /\ W e. H ) )
7 1 2 3 4 5 ltrniotacl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> F e. T )
8 eqid 
 |-  ( Base ` K ) = ( Base ` K )
9 8 3 4 ltrn1o 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> F : ( Base ` K ) -1-1-onto-> ( Base ` K ) )
10 6 7 9 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> F : ( Base ` K ) -1-1-onto-> ( Base ` K ) )
11 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> P e. A )
12 8 2 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
13 11 12 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> P e. ( Base ` K ) )
14 10 13 jca 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( F : ( Base ` K ) -1-1-onto-> ( Base ` K ) /\ P e. ( Base ` K ) ) )
15 1 2 3 4 5 ltrniotaval 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( F ` P ) = Q )
16 f1ocnvfv 
 |-  ( ( F : ( Base ` K ) -1-1-onto-> ( Base ` K ) /\ P e. ( Base ` K ) ) -> ( ( F ` P ) = Q -> ( `' F ` Q ) = P ) )
17 14 15 16 sylc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( `' F ` Q ) = P )