Metamath Proof Explorer


Theorem mreclatBAD

Description: A Moore space is a complete lattice under inclusion. (Contributed by Stefan O'Rear, 31-Jan-2015) TODO ( df-riota update): Reprove using isclat instead of the isclatBAD. hypothesis. See commented-out mreclat above. See mreclat for a good version.

Ref Expression
Hypotheses mreclat.i
|- I = ( toInc ` C )
isclatBAD.
|- ( I e. CLat <-> ( I e. Poset /\ A. x ( x C_ ( Base ` I ) -> ( ( ( lub ` I ) ` x ) e. ( Base ` I ) /\ ( ( glb ` I ) ` x ) e. ( Base ` I ) ) ) ) )
Assertion mreclatBAD
|- ( C e. ( Moore ` X ) -> I e. CLat )

Proof

Step Hyp Ref Expression
1 mreclat.i
 |-  I = ( toInc ` C )
2 isclatBAD.
 |-  ( I e. CLat <-> ( I e. Poset /\ A. x ( x C_ ( Base ` I ) -> ( ( ( lub ` I ) ` x ) e. ( Base ` I ) /\ ( ( glb ` I ) ` x ) e. ( Base ` I ) ) ) ) )
3 1 ipopos
 |-  I e. Poset
4 3 a1i
 |-  ( C e. ( Moore ` X ) -> I e. Poset )
5 eqid
 |-  ( mrCls ` C ) = ( mrCls ` C )
6 eqid
 |-  ( lub ` I ) = ( lub ` I )
7 1 5 6 mrelatlub
 |-  ( ( C e. ( Moore ` X ) /\ x C_ C ) -> ( ( lub ` I ) ` x ) = ( ( mrCls ` C ) ` U. x ) )
8 uniss
 |-  ( x C_ C -> U. x C_ U. C )
9 8 adantl
 |-  ( ( C e. ( Moore ` X ) /\ x C_ C ) -> U. x C_ U. C )
10 mreuni
 |-  ( C e. ( Moore ` X ) -> U. C = X )
11 10 adantr
 |-  ( ( C e. ( Moore ` X ) /\ x C_ C ) -> U. C = X )
12 9 11 sseqtrd
 |-  ( ( C e. ( Moore ` X ) /\ x C_ C ) -> U. x C_ X )
13 5 mrccl
 |-  ( ( C e. ( Moore ` X ) /\ U. x C_ X ) -> ( ( mrCls ` C ) ` U. x ) e. C )
14 12 13 syldan
 |-  ( ( C e. ( Moore ` X ) /\ x C_ C ) -> ( ( mrCls ` C ) ` U. x ) e. C )
15 7 14 eqeltrd
 |-  ( ( C e. ( Moore ` X ) /\ x C_ C ) -> ( ( lub ` I ) ` x ) e. C )
16 fveq2
 |-  ( x = (/) -> ( ( glb ` I ) ` x ) = ( ( glb ` I ) ` (/) ) )
17 16 adantl
 |-  ( ( ( C e. ( Moore ` X ) /\ x C_ C ) /\ x = (/) ) -> ( ( glb ` I ) ` x ) = ( ( glb ` I ) ` (/) ) )
18 eqid
 |-  ( glb ` I ) = ( glb ` I )
19 1 18 mrelatglb0
 |-  ( C e. ( Moore ` X ) -> ( ( glb ` I ) ` (/) ) = X )
20 19 ad2antrr
 |-  ( ( ( C e. ( Moore ` X ) /\ x C_ C ) /\ x = (/) ) -> ( ( glb ` I ) ` (/) ) = X )
21 17 20 eqtrd
 |-  ( ( ( C e. ( Moore ` X ) /\ x C_ C ) /\ x = (/) ) -> ( ( glb ` I ) ` x ) = X )
22 mre1cl
 |-  ( C e. ( Moore ` X ) -> X e. C )
23 22 ad2antrr
 |-  ( ( ( C e. ( Moore ` X ) /\ x C_ C ) /\ x = (/) ) -> X e. C )
24 21 23 eqeltrd
 |-  ( ( ( C e. ( Moore ` X ) /\ x C_ C ) /\ x = (/) ) -> ( ( glb ` I ) ` x ) e. C )
25 1 18 mrelatglb
 |-  ( ( C e. ( Moore ` X ) /\ x C_ C /\ x =/= (/) ) -> ( ( glb ` I ) ` x ) = |^| x )
26 mreintcl
 |-  ( ( C e. ( Moore ` X ) /\ x C_ C /\ x =/= (/) ) -> |^| x e. C )
27 25 26 eqeltrd
 |-  ( ( C e. ( Moore ` X ) /\ x C_ C /\ x =/= (/) ) -> ( ( glb ` I ) ` x ) e. C )
28 27 3expa
 |-  ( ( ( C e. ( Moore ` X ) /\ x C_ C ) /\ x =/= (/) ) -> ( ( glb ` I ) ` x ) e. C )
29 24 28 pm2.61dane
 |-  ( ( C e. ( Moore ` X ) /\ x C_ C ) -> ( ( glb ` I ) ` x ) e. C )
30 15 29 jca
 |-  ( ( C e. ( Moore ` X ) /\ x C_ C ) -> ( ( ( lub ` I ) ` x ) e. C /\ ( ( glb ` I ) ` x ) e. C ) )
31 30 ex
 |-  ( C e. ( Moore ` X ) -> ( x C_ C -> ( ( ( lub ` I ) ` x ) e. C /\ ( ( glb ` I ) ` x ) e. C ) ) )
32 1 ipobas
 |-  ( C e. ( Moore ` X ) -> C = ( Base ` I ) )
33 sseq2
 |-  ( C = ( Base ` I ) -> ( x C_ C <-> x C_ ( Base ` I ) ) )
34 eleq2
 |-  ( C = ( Base ` I ) -> ( ( ( lub ` I ) ` x ) e. C <-> ( ( lub ` I ) ` x ) e. ( Base ` I ) ) )
35 eleq2
 |-  ( C = ( Base ` I ) -> ( ( ( glb ` I ) ` x ) e. C <-> ( ( glb ` I ) ` x ) e. ( Base ` I ) ) )
36 34 35 anbi12d
 |-  ( C = ( Base ` I ) -> ( ( ( ( lub ` I ) ` x ) e. C /\ ( ( glb ` I ) ` x ) e. C ) <-> ( ( ( lub ` I ) ` x ) e. ( Base ` I ) /\ ( ( glb ` I ) ` x ) e. ( Base ` I ) ) ) )
37 33 36 imbi12d
 |-  ( C = ( Base ` I ) -> ( ( x C_ C -> ( ( ( lub ` I ) ` x ) e. C /\ ( ( glb ` I ) ` x ) e. C ) ) <-> ( x C_ ( Base ` I ) -> ( ( ( lub ` I ) ` x ) e. ( Base ` I ) /\ ( ( glb ` I ) ` x ) e. ( Base ` I ) ) ) ) )
38 32 37 syl
 |-  ( C e. ( Moore ` X ) -> ( ( x C_ C -> ( ( ( lub ` I ) ` x ) e. C /\ ( ( glb ` I ) ` x ) e. C ) ) <-> ( x C_ ( Base ` I ) -> ( ( ( lub ` I ) ` x ) e. ( Base ` I ) /\ ( ( glb ` I ) ` x ) e. ( Base ` I ) ) ) ) )
39 31 38 mpbid
 |-  ( C e. ( Moore ` X ) -> ( x C_ ( Base ` I ) -> ( ( ( lub ` I ) ` x ) e. ( Base ` I ) /\ ( ( glb ` I ) ` x ) e. ( Base ` I ) ) ) )
40 39 alrimiv
 |-  ( C e. ( Moore ` X ) -> A. x ( x C_ ( Base ` I ) -> ( ( ( lub ` I ) ` x ) e. ( Base ` I ) /\ ( ( glb ` I ) ` x ) e. ( Base ` I ) ) ) )
41 4 40 2 sylanbrc
 |-  ( C e. ( Moore ` X ) -> I e. CLat )