Metamath Proof Explorer

Theorem dihglblem4

Description: Isomorphism H of a lattice glb. (Contributed by NM, 21-Mar-2014)

Ref Expression
Hypotheses dihglblem.b 
|- B = ( Base ` K )
dihglblem.l 
|- .<_ = ( le ` K )
dihglblem.m 
|- ./\ = ( meet ` K )
dihglblem.g 
|- G = ( glb ` K )
dihglblem.h 
|- H = ( LHyp ` K )
dihglblem.t 
|- T = { u e. B | E. v e. S u = ( v ./\ W ) }
dihglblem.i 
|- J = ( ( DIsoB ` K ) ` W )
dihglblem.ih 
|- I = ( ( DIsoH ` K ) ` W )
Assertion dihglblem4 
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) -> ( I ` ( G ` S ) ) C_ |^|_ x e. S ( I ` x ) )

Proof

Step Hyp Ref Expression
1 dihglblem.b 
 |-  B = ( Base ` K )
2 dihglblem.l 
 |-  .<_ = ( le ` K )
3 dihglblem.m 
 |-  ./\ = ( meet ` K )
4 dihglblem.g 
 |-  G = ( glb ` K )
5 dihglblem.h 
 |-  H = ( LHyp ` K )
6 dihglblem.t 
 |-  T = { u e. B | E. v e. S u = ( v ./\ W ) }
7 dihglblem.i 
 |-  J = ( ( DIsoB ` K ) ` W )
8 dihglblem.ih 
 |-  I = ( ( DIsoH ` K ) ` W )
9 hlclat 
 |-  ( K e. HL -> K e. CLat )
10 9 ad3antrrr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ x e. S ) -> K e. CLat )
11 simplrl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ x e. S ) -> S C_ B )
12 simpr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ x e. S ) -> x e. S )
13 1 2 4 clatglble 
 |-  ( ( K e. CLat /\ S C_ B /\ x e. S ) -> ( G ` S ) .<_ x )
14 10 11 12 13 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ x e. S ) -> ( G ` S ) .<_ x )
15 simpll 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ x e. S ) -> ( K e. HL /\ W e. H ) )
16 1 4 clatglbcl 
 |-  ( ( K e. CLat /\ S C_ B ) -> ( G ` S ) e. B )
17 10 11 16 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ x e. S ) -> ( G ` S ) e. B )
18 11 12 sseldd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ x e. S ) -> x e. B )
19 1 2 5 8 dihord 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( G ` S ) e. B /\ x e. B ) -> ( ( I ` ( G ` S ) ) C_ ( I ` x ) <-> ( G ` S ) .<_ x ) )
20 15 17 18 19 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ x e. S ) -> ( ( I ` ( G ` S ) ) C_ ( I ` x ) <-> ( G ` S ) .<_ x ) )
21 14 20 mpbird 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) /\ x e. S ) -> ( I ` ( G ` S ) ) C_ ( I ` x ) )
22 21 ralrimiva 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) -> A. x e. S ( I ` ( G ` S ) ) C_ ( I ` x ) )
23 ssiin 
 |-  ( ( I ` ( G ` S ) ) C_ |^|_ x e. S ( I ` x ) <-> A. x e. S ( I ` ( G ` S ) ) C_ ( I ` x ) )
24 22 23 sylibr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) -> ( I ` ( G ` S ) ) C_ |^|_ x e. S ( I ` x ) )