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Metamath Proof Explorer

Theorem dihglblem3N

Description: Isomorphism H of a lattice glb. (Contributed by NM, 20-Mar-2014) (New usage is discouraged.)

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 dihglblem3N 
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( I ` ( G ` T ) ) = |^|_ x e. T ( 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 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( K e. HL /\ W e. H ) )
10 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ u e. B /\ v e. S ) -> K e. HL )
11 10 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ u e. B /\ v e. S ) -> K e. Lat )
12 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ u e. B /\ v e. S ) -> S C_ B )
13 simp3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ u e. B /\ v e. S ) -> v e. S )
14 12 13 sseldd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ u e. B /\ v e. S ) -> v e. B )
15 simp11r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ u e. B /\ v e. S ) -> W e. H )
16 1 5 lhpbase 
 |-  ( W e. H -> W e. B )
17 15 16 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ u e. B /\ v e. S ) -> W e. B )
18 1 2 3 latmle2 
 |-  ( ( K e. Lat /\ v e. B /\ W e. B ) -> ( v ./\ W ) .<_ W )
19 11 14 17 18 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ u e. B /\ v e. S ) -> ( v ./\ W ) .<_ W )
20 19 3expia 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ u e. B ) -> ( v e. S -> ( v ./\ W ) .<_ W ) )
21 breq1 
 |-  ( u = ( v ./\ W ) -> ( u .<_ W <-> ( v ./\ W ) .<_ W ) )
22 21 biimprcd 
 |-  ( ( v ./\ W ) .<_ W -> ( u = ( v ./\ W ) -> u .<_ W ) )
23 20 22 syl6 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ u e. B ) -> ( v e. S -> ( u = ( v ./\ W ) -> u .<_ W ) ) )
24 23 rexlimdv 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ u e. B ) -> ( E. v e. S u = ( v ./\ W ) -> u .<_ W ) )
25 24 ss2rabdv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> { u e. B | E. v e. S u = ( v ./\ W ) } C_ { u e. B | u .<_ W } )
26 6 25 eqsstrid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> T C_ { u e. B | u .<_ W } )
27 1 2 5 7 dibdmN 
 |-  ( ( K e. HL /\ W e. H ) -> dom J = { u e. B | u .<_ W } )
28 27 3ad2ant1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> dom J = { u e. B | u .<_ W } )
29 26 28 sseqtrrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> T C_ dom J )
30 1 2 3 4 5 6 dihglblem2aN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) -> T =/= (/) )
31 30 3adant3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> T =/= (/) )
32 4 5 7 dibglbN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( T C_ dom J /\ T =/= (/) ) ) -> ( J ` ( G ` T ) ) = |^|_ x e. T ( J ` x ) )
33 9 29 31 32 syl12anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( J ` ( G ` T ) ) = |^|_ x e. T ( J ` x ) )
34 1 2 3 4 5 6 dihglblem2N 
 |-  ( ( ( K e. HL /\ W e. H ) /\ S C_ B /\ ( G ` S ) .<_ W ) -> ( G ` S ) = ( G ` T ) )
35 34 3adant2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( G ` S ) = ( G ` T ) )
36 35 fveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( J ` ( G ` S ) ) = ( J ` ( G ` T ) ) )
37 simpl1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ x e. T ) -> ( K e. HL /\ W e. H ) )
38 26 sselda 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ x e. T ) -> x e. { u e. B | u .<_ W } )
39 breq1 
 |-  ( u = x -> ( u .<_ W <-> x .<_ W ) )
40 39 elrab 
 |-  ( x e. { u e. B | u .<_ W } <-> ( x e. B /\ x .<_ W ) )
41 38 40 sylib 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ x e. T ) -> ( x e. B /\ x .<_ W ) )
42 1 2 5 8 7 dihvalb 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( x e. B /\ x .<_ W ) ) -> ( I ` x ) = ( J ` x ) )
43 37 41 42 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) /\ x e. T ) -> ( I ` x ) = ( J ` x ) )
44 43 iineq2dv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> |^|_ x e. T ( I ` x ) = |^|_ x e. T ( J ` x ) )
45 33 36 44 3eqtr4rd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> |^|_ x e. T ( I ` x ) = ( J ` ( G ` S ) ) )
46 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> K e. HL )
47 hlclat 
 |-  ( K e. HL -> K e. CLat )
48 46 47 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> K e. CLat )
49 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> S C_ B )
50 1 4 clatglbcl 
 |-  ( ( K e. CLat /\ S C_ B ) -> ( G ` S ) e. B )
51 48 49 50 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( G ` S ) e. B )
52 simp3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( G ` S ) .<_ W )
53 1 2 5 8 7 dihvalb 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( G ` S ) e. B /\ ( G ` S ) .<_ W ) ) -> ( I ` ( G ` S ) ) = ( J ` ( G ` S ) ) )
54 9 51 52 53 syl12anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( I ` ( G ` S ) ) = ( J ` ( G ` S ) ) )
55 35 fveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( I ` ( G ` S ) ) = ( I ` ( G ` T ) ) )
56 45 54 55 3eqtr2rd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ ( G ` S ) .<_ W ) -> ( I ` ( G ` T ) ) = |^|_ x e. T ( I ` x ) )