This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem trlset

Description: The set of traces of lattice translations for a fiducial co-atom W . (Contributed by NM, 20-May-2012)

Ref Expression
Hypotheses trlset.b 
|- B = ( Base ` K )
trlset.l 
|- .<_ = ( le ` K )
trlset.j 
|- .\/ = ( join ` K )
trlset.m 
|- ./\ = ( meet ` K )
trlset.a 
|- A = ( Atoms ` K )
trlset.h 
|- H = ( LHyp ` K )
trlset.t 
|- T = ( ( LTrn ` K ) ` W )
trlset.r 
|- R = ( ( trL ` K ) ` W )
Assertion trlset 
|- ( ( K e. C /\ W e. H ) -> R = ( f e. T |-> ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) ) )

Proof

Step Hyp Ref Expression
1 trlset.b 
 |-  B = ( Base ` K )
2 trlset.l 
 |-  .<_ = ( le ` K )
3 trlset.j 
 |-  .\/ = ( join ` K )
4 trlset.m 
 |-  ./\ = ( meet ` K )
5 trlset.a 
 |-  A = ( Atoms ` K )
6 trlset.h 
 |-  H = ( LHyp ` K )
7 trlset.t 
 |-  T = ( ( LTrn ` K ) ` W )
8 trlset.r 
 |-  R = ( ( trL ` K ) ` W )
9 1 2 3 4 5 6 trlfset 
 |-  ( K e. C -> ( trL ` K ) = ( w e. H |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) ) )
10 9 fveq1d 
 |-  ( K e. C -> ( ( trL ` K ) ` W ) = ( ( w e. H |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) ) ` W ) )
11 8 10 eqtrid 
 |-  ( K e. C -> R = ( ( w e. H |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) ) ` W ) )
12 fveq2 
 |-  ( w = W -> ( ( LTrn ` K ) ` w ) = ( ( LTrn ` K ) ` W ) )
13 breq2 
 |-  ( w = W -> ( p .<_ w <-> p .<_ W ) )
14 13 notbid 
 |-  ( w = W -> ( -. p .<_ w <-> -. p .<_ W ) )
15 oveq2 
 |-  ( w = W -> ( ( p .\/ ( f ` p ) ) ./\ w ) = ( ( p .\/ ( f ` p ) ) ./\ W ) )
16 15 eqeq2d 
 |-  ( w = W -> ( x = ( ( p .\/ ( f ` p ) ) ./\ w ) <-> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) )
17 14 16 imbi12d 
 |-  ( w = W -> ( ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) <-> ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) )
18 17 ralbidv 
 |-  ( w = W -> ( A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) <-> A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) )
19 18 riotabidv 
 |-  ( w = W -> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) = ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) )
20 12 19 mpteq12dv 
 |-  ( w = W -> ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) = ( f e. ( ( LTrn ` K ) ` W ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) ) )
21 eqid 
 |-  ( w e. H |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) ) = ( w e. H |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) )
22 fvex 
 |-  ( ( LTrn ` K ) ` W ) e. _V
23 22 mptex 
 |-  ( f e. ( ( LTrn ` K ) ` W ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) ) e. _V
24 20 21 23 fvmpt 
 |-  ( W e. H -> ( ( w e. H |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) ) ` W ) = ( f e. ( ( LTrn ` K ) ` W ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) ) )
25 7 mpteq1i 
 |-  ( f e. T |-> ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) ) = ( f e. ( ( LTrn ` K ) ` W ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) )
26 24 25 eqtr4di 
 |-  ( W e. H -> ( ( w e. H |-> ( f e. ( ( LTrn ` K ) ` w ) |-> ( iota_ x e. B A. p e. A ( -. p .<_ w -> x = ( ( p .\/ ( f ` p ) ) ./\ w ) ) ) ) ) ` W ) = ( f e. T |-> ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) ) )
27 11 26 sylan9eq 
 |-  ( ( K e. C /\ W e. H ) -> R = ( f e. T |-> ( iota_ x e. B A. p e. A ( -. p .<_ W -> x = ( ( p .\/ ( f ` p ) ) ./\ W ) ) ) ) )