Step |
Hyp |
Ref |
Expression |
1 |
|
docacl.h |
|- H = ( LHyp ` K ) |
2 |
|
docacl.t |
|- T = ( ( LTrn ` K ) ` W ) |
3 |
|
docacl.i |
|- I = ( ( DIsoA ` K ) ` W ) |
4 |
|
docacl.n |
|- ._|_ = ( ( ocA ` K ) ` W ) |
5 |
|
eqid |
|- ( join ` K ) = ( join ` K ) |
6 |
|
eqid |
|- ( meet ` K ) = ( meet ` K ) |
7 |
|
eqid |
|- ( oc ` K ) = ( oc ` K ) |
8 |
5 6 7 1 2 3 4
|
docavalN |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> ( ._|_ ` X ) = ( I ` ( ( ( ( oc ` K ) ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) ( join ` K ) ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) |
9 |
1 3
|
diaf11N |
|- ( ( K e. HL /\ W e. H ) -> I : dom I -1-1-onto-> ran I ) |
10 |
|
f1ofun |
|- ( I : dom I -1-1-onto-> ran I -> Fun I ) |
11 |
9 10
|
syl |
|- ( ( K e. HL /\ W e. H ) -> Fun I ) |
12 |
11
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> Fun I ) |
13 |
|
hllat |
|- ( K e. HL -> K e. Lat ) |
14 |
13
|
ad2antrr |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> K e. Lat ) |
15 |
|
hlop |
|- ( K e. HL -> K e. OP ) |
16 |
15
|
ad2antrr |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> K e. OP ) |
17 |
|
simpl |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> ( K e. HL /\ W e. H ) ) |
18 |
|
ssrab2 |
|- { z e. ran I | X C_ z } C_ ran I |
19 |
18
|
a1i |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> { z e. ran I | X C_ z } C_ ran I ) |
20 |
1 2 3
|
dia1elN |
|- ( ( K e. HL /\ W e. H ) -> T e. ran I ) |
21 |
20
|
anim1i |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> ( T e. ran I /\ X C_ T ) ) |
22 |
|
sseq2 |
|- ( z = T -> ( X C_ z <-> X C_ T ) ) |
23 |
22
|
elrab |
|- ( T e. { z e. ran I | X C_ z } <-> ( T e. ran I /\ X C_ T ) ) |
24 |
21 23
|
sylibr |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> T e. { z e. ran I | X C_ z } ) |
25 |
24
|
ne0d |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> { z e. ran I | X C_ z } =/= (/) ) |
26 |
1 3
|
diaintclN |
|- ( ( ( K e. HL /\ W e. H ) /\ ( { z e. ran I | X C_ z } C_ ran I /\ { z e. ran I | X C_ z } =/= (/) ) ) -> |^| { z e. ran I | X C_ z } e. ran I ) |
27 |
17 19 25 26
|
syl12anc |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> |^| { z e. ran I | X C_ z } e. ran I ) |
28 |
1 3
|
diacnvclN |
|- ( ( ( K e. HL /\ W e. H ) /\ |^| { z e. ran I | X C_ z } e. ran I ) -> ( `' I ` |^| { z e. ran I | X C_ z } ) e. dom I ) |
29 |
27 28
|
syldan |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> ( `' I ` |^| { z e. ran I | X C_ z } ) e. dom I ) |
30 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
31 |
30 1 3
|
diadmclN |
|- ( ( ( K e. HL /\ W e. H ) /\ ( `' I ` |^| { z e. ran I | X C_ z } ) e. dom I ) -> ( `' I ` |^| { z e. ran I | X C_ z } ) e. ( Base ` K ) ) |
32 |
29 31
|
syldan |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> ( `' I ` |^| { z e. ran I | X C_ z } ) e. ( Base ` K ) ) |
33 |
30 7
|
opoccl |
|- ( ( K e. OP /\ ( `' I ` |^| { z e. ran I | X C_ z } ) e. ( Base ` K ) ) -> ( ( oc ` K ) ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) e. ( Base ` K ) ) |
34 |
16 32 33
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> ( ( oc ` K ) ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) e. ( Base ` K ) ) |
35 |
30 1
|
lhpbase |
|- ( W e. H -> W e. ( Base ` K ) ) |
36 |
35
|
ad2antlr |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> W e. ( Base ` K ) ) |
37 |
30 7
|
opoccl |
|- ( ( K e. OP /\ W e. ( Base ` K ) ) -> ( ( oc ` K ) ` W ) e. ( Base ` K ) ) |
38 |
16 36 37
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> ( ( oc ` K ) ` W ) e. ( Base ` K ) ) |
39 |
30 5
|
latjcl |
|- ( ( K e. Lat /\ ( ( oc ` K ) ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) e. ( Base ` K ) /\ ( ( oc ` K ) ` W ) e. ( Base ` K ) ) -> ( ( ( oc ` K ) ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) ( join ` K ) ( ( oc ` K ) ` W ) ) e. ( Base ` K ) ) |
40 |
14 34 38 39
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> ( ( ( oc ` K ) ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) ( join ` K ) ( ( oc ` K ) ` W ) ) e. ( Base ` K ) ) |
41 |
30 6
|
latmcl |
|- ( ( K e. Lat /\ ( ( ( oc ` K ) ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) ( join ` K ) ( ( oc ` K ) ` W ) ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( ( ( oc ` K ) ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) ( join ` K ) ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. ( Base ` K ) ) |
42 |
14 40 36 41
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> ( ( ( ( oc ` K ) ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) ( join ` K ) ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. ( Base ` K ) ) |
43 |
|
eqid |
|- ( le ` K ) = ( le ` K ) |
44 |
30 43 6
|
latmle2 |
|- ( ( K e. Lat /\ ( ( ( oc ` K ) ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) ( join ` K ) ( ( oc ` K ) ` W ) ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( ( ( oc ` K ) ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) ( join ` K ) ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( le ` K ) W ) |
45 |
14 40 36 44
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> ( ( ( ( oc ` K ) ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) ( join ` K ) ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( le ` K ) W ) |
46 |
30 43 1 3
|
diaeldm |
|- ( ( K e. HL /\ W e. H ) -> ( ( ( ( ( oc ` K ) ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) ( join ` K ) ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. dom I <-> ( ( ( ( ( oc ` K ) ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) ( join ` K ) ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. ( Base ` K ) /\ ( ( ( ( oc ` K ) ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) ( join ` K ) ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( le ` K ) W ) ) ) |
47 |
46
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> ( ( ( ( ( oc ` K ) ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) ( join ` K ) ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. dom I <-> ( ( ( ( ( oc ` K ) ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) ( join ` K ) ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. ( Base ` K ) /\ ( ( ( ( oc ` K ) ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) ( join ` K ) ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( le ` K ) W ) ) ) |
48 |
42 45 47
|
mpbir2and |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> ( ( ( ( oc ` K ) ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) ( join ` K ) ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. dom I ) |
49 |
|
fvelrn |
|- ( ( Fun I /\ ( ( ( ( oc ` K ) ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) ( join ` K ) ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. dom I ) -> ( I ` ( ( ( ( oc ` K ) ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) ( join ` K ) ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) e. ran I ) |
50 |
12 48 49
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> ( I ` ( ( ( ( oc ` K ) ` ( `' I ` |^| { z e. ran I | X C_ z } ) ) ( join ` K ) ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) e. ran I ) |
51 |
8 50
|
eqeltrd |
|- ( ( ( K e. HL /\ W e. H ) /\ X C_ T ) -> ( ._|_ ` X ) e. ran I ) |