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 ) |