Metamath Proof Explorer

Theorem ltrneq

Description: The equality of two translations is determined by their equality at atoms not under co-atom W . (Contributed by NM, 20-Jun-2013)

Ref Expression
Hypotheses ltrne.l 
|- .<_ = ( le ` K )
ltrne.a 
|- A = ( Atoms ` K )
ltrne.h 
|- H = ( LHyp ` K )
ltrne.t 
|- T = ( ( LTrn ` K ) ` W )
Assertion ltrneq 
|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( A. p e. A ( -. p .<_ W -> ( F ` p ) = ( G ` p ) ) <-> F = G ) )

Proof

Step Hyp Ref Expression
1 ltrne.l 
 |-  .<_ = ( le ` K )
2 ltrne.a 
 |-  A = ( Atoms ` K )
3 ltrne.h 
 |-  H = ( LHyp ` K )
4 ltrne.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ p e. A /\ p .<_ W ) -> ( K e. HL /\ W e. H ) )
6 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ p e. A /\ p .<_ W ) -> F e. T )
7 eqid 
 |-  ( Base ` K ) = ( Base ` K )
8 7 2 atbase 
 |-  ( p e. A -> p e. ( Base ` K ) )
9 8 3ad2ant2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ p e. A /\ p .<_ W ) -> p e. ( Base ` K ) )
10 simp3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ p e. A /\ p .<_ W ) -> p .<_ W )
11 7 1 3 4 ltrnval1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( p e. ( Base ` K ) /\ p .<_ W ) ) -> ( F ` p ) = p )
12 5 6 9 10 11 syl112anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ p e. A /\ p .<_ W ) -> ( F ` p ) = p )
13 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ p e. A /\ p .<_ W ) -> G e. T )
14 7 1 3 4 ltrnval1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( p e. ( Base ` K ) /\ p .<_ W ) ) -> ( G ` p ) = p )
15 5 13 9 10 14 syl112anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ p e. A /\ p .<_ W ) -> ( G ` p ) = p )
16 12 15 eqtr4d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ p e. A /\ p .<_ W ) -> ( F ` p ) = ( G ` p ) )
17 16 3expia 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ p e. A ) -> ( p .<_ W -> ( F ` p ) = ( G ` p ) ) )
18 pm2.61 
 |-  ( ( p .<_ W -> ( F ` p ) = ( G ` p ) ) -> ( ( -. p .<_ W -> ( F ` p ) = ( G ` p ) ) -> ( F ` p ) = ( G ` p ) ) )
19 17 18 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ p e. A ) -> ( ( -. p .<_ W -> ( F ` p ) = ( G ` p ) ) -> ( F ` p ) = ( G ` p ) ) )
20 re1tbw2 
 |-  ( ( F ` p ) = ( G ` p ) -> ( -. p .<_ W -> ( F ` p ) = ( G ` p ) ) )
21 19 20 impbid1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ p e. A ) -> ( ( -. p .<_ W -> ( F ` p ) = ( G ` p ) ) <-> ( F ` p ) = ( G ` p ) ) )
22 21 ralbidva 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( A. p e. A ( -. p .<_ W -> ( F ` p ) = ( G ` p ) ) <-> A. p e. A ( F ` p ) = ( G ` p ) ) )
23 2 3 4 ltrneq2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( A. p e. A ( F ` p ) = ( G ` p ) <-> F = G ) )
24 22 23 bitrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) -> ( A. p e. A ( -. p .<_ W -> ( F ` p ) = ( G ` p ) ) <-> F = G ) )