Step |
Hyp |
Ref |
Expression |
1 |
|
djaj.k |
|- .\/ = ( join ` K ) |
2 |
|
djaj.h |
|- H = ( LHyp ` K ) |
3 |
|
djaj.i |
|- I = ( ( DIsoA ` K ) ` W ) |
4 |
|
djaj.j |
|- J = ( ( vA ` K ) ` W ) |
5 |
|
hllat |
|- ( K e. HL -> K e. Lat ) |
6 |
5
|
ad2antrr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> K e. Lat ) |
7 |
|
hlop |
|- ( K e. HL -> K e. OP ) |
8 |
7
|
ad2antrr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> K e. OP ) |
9 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
10 |
9 2 3
|
diadmclN |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. dom I ) -> X e. ( Base ` K ) ) |
11 |
10
|
adantrr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> X e. ( Base ` K ) ) |
12 |
|
eqid |
|- ( oc ` K ) = ( oc ` K ) |
13 |
9 12
|
opoccl |
|- ( ( K e. OP /\ X e. ( Base ` K ) ) -> ( ( oc ` K ) ` X ) e. ( Base ` K ) ) |
14 |
8 11 13
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( oc ` K ) ` X ) e. ( Base ` K ) ) |
15 |
9 2
|
lhpbase |
|- ( W e. H -> W e. ( Base ` K ) ) |
16 |
15
|
ad2antlr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> W e. ( Base ` K ) ) |
17 |
9 12
|
opoccl |
|- ( ( K e. OP /\ W e. ( Base ` K ) ) -> ( ( oc ` K ) ` W ) e. ( Base ` K ) ) |
18 |
8 16 17
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( oc ` K ) ` W ) e. ( Base ` K ) ) |
19 |
9 1
|
latjcl |
|- ( ( K e. Lat /\ ( ( oc ` K ) ` X ) e. ( Base ` K ) /\ ( ( oc ` K ) ` W ) e. ( Base ` K ) ) -> ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) e. ( Base ` K ) ) |
20 |
6 14 18 19
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) e. ( Base ` K ) ) |
21 |
|
eqid |
|- ( meet ` K ) = ( meet ` K ) |
22 |
9 21
|
latmcl |
|- ( ( K e. Lat /\ ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. ( Base ` K ) ) |
23 |
6 20 16 22
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. ( Base ` K ) ) |
24 |
9 2 3
|
diadmclN |
|- ( ( ( K e. HL /\ W e. H ) /\ Y e. dom I ) -> Y e. ( Base ` K ) ) |
25 |
24
|
adantrl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> Y e. ( Base ` K ) ) |
26 |
9 12
|
opoccl |
|- ( ( K e. OP /\ Y e. ( Base ` K ) ) -> ( ( oc ` K ) ` Y ) e. ( Base ` K ) ) |
27 |
8 25 26
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( oc ` K ) ` Y ) e. ( Base ` K ) ) |
28 |
9 1
|
latjcl |
|- ( ( K e. Lat /\ ( ( oc ` K ) ` Y ) e. ( Base ` K ) /\ ( ( oc ` K ) ` W ) e. ( Base ` K ) ) -> ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) e. ( Base ` K ) ) |
29 |
6 27 18 28
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) e. ( Base ` K ) ) |
30 |
9 21
|
latmcl |
|- ( ( K e. Lat /\ ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. ( Base ` K ) ) |
31 |
6 29 16 30
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. ( Base ` K ) ) |
32 |
9 21
|
latmcl |
|- ( ( K e. Lat /\ ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. ( Base ` K ) /\ ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. ( Base ` K ) ) -> ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) e. ( Base ` K ) ) |
33 |
6 23 31 32
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) e. ( Base ` K ) ) |
34 |
|
eqid |
|- ( le ` K ) = ( le ` K ) |
35 |
9 34 21
|
latmle2 |
|- ( ( K e. Lat /\ ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. ( Base ` K ) /\ ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. ( Base ` K ) ) -> ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ( le ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) |
36 |
6 23 31 35
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ( le ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) |
37 |
9 34 21
|
latmle2 |
|- ( ( K e. Lat /\ ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( le ` K ) W ) |
38 |
6 29 16 37
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( le ` K ) W ) |
39 |
9 34 6 33 31 16 36 38
|
lattrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ( le ` K ) W ) |
40 |
9 34 2 3
|
diaeldm |
|- ( ( K e. HL /\ W e. H ) -> ( ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) e. dom I <-> ( ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) e. ( Base ` K ) /\ ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ( le ` K ) W ) ) ) |
41 |
40
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) e. dom I <-> ( ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) e. ( Base ` K ) /\ ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ( le ` K ) W ) ) ) |
42 |
33 39 41
|
mpbir2and |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) e. dom I ) |
43 |
|
eqid |
|- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W ) |
44 |
|
eqid |
|- ( ( ocA ` K ) ` W ) = ( ( ocA ` K ) ` W ) |
45 |
1 21 12 2 43 3 44
|
diaocN |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) e. dom I ) -> ( I ` ( ( ( ( oc ` K ) ` ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) = ( ( ( ocA ` K ) ` W ) ` ( I ` ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) ) ) |
46 |
42 45
|
syldan |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( I ` ( ( ( ( oc ` K ) ` ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) = ( ( ( ocA ` K ) ` W ) ` ( I ` ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) ) ) |
47 |
|
hloml |
|- ( K e. HL -> K e. OML ) |
48 |
47
|
ad2antrr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> K e. OML ) |
49 |
9 1
|
latjcl |
|- ( ( K e. Lat /\ X e. ( Base ` K ) /\ Y e. ( Base ` K ) ) -> ( X .\/ Y ) e. ( Base ` K ) ) |
50 |
6 11 25 49
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( X .\/ Y ) e. ( Base ` K ) ) |
51 |
34 2 3
|
diadmleN |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. dom I ) -> X ( le ` K ) W ) |
52 |
51
|
adantrr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> X ( le ` K ) W ) |
53 |
34 2 3
|
diadmleN |
|- ( ( ( K e. HL /\ W e. H ) /\ Y e. dom I ) -> Y ( le ` K ) W ) |
54 |
53
|
adantrl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> Y ( le ` K ) W ) |
55 |
9 34 1
|
latjle12 |
|- ( ( K e. Lat /\ ( X e. ( Base ` K ) /\ Y e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( X ( le ` K ) W /\ Y ( le ` K ) W ) <-> ( X .\/ Y ) ( le ` K ) W ) ) |
56 |
6 11 25 16 55
|
syl13anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( X ( le ` K ) W /\ Y ( le ` K ) W ) <-> ( X .\/ Y ) ( le ` K ) W ) ) |
57 |
52 54 56
|
mpbi2and |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( X .\/ Y ) ( le ` K ) W ) |
58 |
9 34 1 21 12
|
omlspjN |
|- ( ( K e. OML /\ ( ( X .\/ Y ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) /\ ( X .\/ Y ) ( le ` K ) W ) -> ( ( ( X .\/ Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) = ( X .\/ Y ) ) |
59 |
48 50 16 57 58
|
syl121anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( X .\/ Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) = ( X .\/ Y ) ) |
60 |
9 1
|
latjidm |
|- ( ( K e. Lat /\ ( ( oc ` K ) ` W ) e. ( Base ` K ) ) -> ( ( ( oc ` K ) ` W ) .\/ ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) |
61 |
6 18 60
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( oc ` K ) ` W ) .\/ ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` W ) ) |
62 |
61
|
oveq2d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( X .\/ Y ) .\/ ( ( ( oc ` K ) ` W ) .\/ ( ( oc ` K ) ` W ) ) ) = ( ( X .\/ Y ) .\/ ( ( oc ` K ) ` W ) ) ) |
63 |
9 1
|
latjass |
|- ( ( K e. Lat /\ ( ( X .\/ Y ) e. ( Base ` K ) /\ ( ( oc ` K ) ` W ) e. ( Base ` K ) /\ ( ( oc ` K ) ` W ) e. ( Base ` K ) ) ) -> ( ( ( X .\/ Y ) .\/ ( ( oc ` K ) ` W ) ) .\/ ( ( oc ` K ) ` W ) ) = ( ( X .\/ Y ) .\/ ( ( ( oc ` K ) ` W ) .\/ ( ( oc ` K ) ` W ) ) ) ) |
64 |
6 50 18 18 63
|
syl13anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( X .\/ Y ) .\/ ( ( oc ` K ) ` W ) ) .\/ ( ( oc ` K ) ` W ) ) = ( ( X .\/ Y ) .\/ ( ( ( oc ` K ) ` W ) .\/ ( ( oc ` K ) ` W ) ) ) ) |
65 |
|
hlol |
|- ( K e. HL -> K e. OL ) |
66 |
65
|
ad2antrr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> K e. OL ) |
67 |
9 1 21 12
|
oldmm2 |
|- ( ( K e. OL /\ ( X .\/ Y ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( oc ` K ) ` ( ( ( oc ` K ) ` ( X .\/ Y ) ) ( meet ` K ) W ) ) = ( ( X .\/ Y ) .\/ ( ( oc ` K ) ` W ) ) ) |
68 |
66 50 16 67
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( oc ` K ) ` ( ( ( oc ` K ) ` ( X .\/ Y ) ) ( meet ` K ) W ) ) = ( ( X .\/ Y ) .\/ ( ( oc ` K ) ` W ) ) ) |
69 |
9 1 21 12
|
oldmj1 |
|- ( ( K e. OL /\ X e. ( Base ` K ) /\ Y e. ( Base ` K ) ) -> ( ( oc ` K ) ` ( X .\/ Y ) ) = ( ( ( oc ` K ) ` X ) ( meet ` K ) ( ( oc ` K ) ` Y ) ) ) |
70 |
66 11 25 69
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( oc ` K ) ` ( X .\/ Y ) ) = ( ( ( oc ` K ) ` X ) ( meet ` K ) ( ( oc ` K ) ` Y ) ) ) |
71 |
9 34 21
|
latleeqm1 |
|- ( ( K e. Lat /\ X e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( X ( le ` K ) W <-> ( X ( meet ` K ) W ) = X ) ) |
72 |
6 11 16 71
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( X ( le ` K ) W <-> ( X ( meet ` K ) W ) = X ) ) |
73 |
52 72
|
mpbid |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( X ( meet ` K ) W ) = X ) |
74 |
73
|
fveq2d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( oc ` K ) ` ( X ( meet ` K ) W ) ) = ( ( oc ` K ) ` X ) ) |
75 |
9 1 21 12
|
oldmm1 |
|- ( ( K e. OL /\ X e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( oc ` K ) ` ( X ( meet ` K ) W ) ) = ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ) |
76 |
66 11 16 75
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( oc ` K ) ` ( X ( meet ` K ) W ) ) = ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ) |
77 |
74 76
|
eqtr3d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( oc ` K ) ` X ) = ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ) |
78 |
9 34 21
|
latleeqm1 |
|- ( ( K e. Lat /\ Y e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( Y ( le ` K ) W <-> ( Y ( meet ` K ) W ) = Y ) ) |
79 |
6 25 16 78
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( Y ( le ` K ) W <-> ( Y ( meet ` K ) W ) = Y ) ) |
80 |
54 79
|
mpbid |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( Y ( meet ` K ) W ) = Y ) |
81 |
80
|
fveq2d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( oc ` K ) ` ( Y ( meet ` K ) W ) ) = ( ( oc ` K ) ` Y ) ) |
82 |
9 1 21 12
|
oldmm1 |
|- ( ( K e. OL /\ Y e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( oc ` K ) ` ( Y ( meet ` K ) W ) ) = ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ) |
83 |
66 25 16 82
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( oc ` K ) ` ( Y ( meet ` K ) W ) ) = ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ) |
84 |
81 83
|
eqtr3d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( oc ` K ) ` Y ) = ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ) |
85 |
77 84
|
oveq12d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( oc ` K ) ` X ) ( meet ` K ) ( ( oc ` K ) ` Y ) ) = ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ) ) |
86 |
70 85
|
eqtrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( oc ` K ) ` ( X .\/ Y ) ) = ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ) ) |
87 |
86
|
oveq1d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( oc ` K ) ` ( X .\/ Y ) ) ( meet ` K ) W ) = ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ) ( meet ` K ) W ) ) |
88 |
9 21
|
latmmdir |
|- ( ( K e. OL /\ ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) e. ( Base ` K ) /\ ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) ) -> ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ) ( meet ` K ) W ) = ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) |
89 |
66 20 29 16 88
|
syl13anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ) ( meet ` K ) W ) = ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) |
90 |
87 89
|
eqtrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( oc ` K ) ` ( X .\/ Y ) ) ( meet ` K ) W ) = ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) |
91 |
90
|
fveq2d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( oc ` K ) ` ( ( ( oc ` K ) ` ( X .\/ Y ) ) ( meet ` K ) W ) ) = ( ( oc ` K ) ` ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) ) |
92 |
68 91
|
eqtr3d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( X .\/ Y ) .\/ ( ( oc ` K ) ` W ) ) = ( ( oc ` K ) ` ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) ) |
93 |
92
|
oveq1d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( X .\/ Y ) .\/ ( ( oc ` K ) ` W ) ) .\/ ( ( oc ` K ) ` W ) ) = ( ( ( oc ` K ) ` ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) .\/ ( ( oc ` K ) ` W ) ) ) |
94 |
64 93
|
eqtr3d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( X .\/ Y ) .\/ ( ( ( oc ` K ) ` W ) .\/ ( ( oc ` K ) ` W ) ) ) = ( ( ( oc ` K ) ` ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) .\/ ( ( oc ` K ) ` W ) ) ) |
95 |
62 94
|
eqtr3d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( X .\/ Y ) .\/ ( ( oc ` K ) ` W ) ) = ( ( ( oc ` K ) ` ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) .\/ ( ( oc ` K ) ` W ) ) ) |
96 |
95
|
oveq1d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( X .\/ Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) = ( ( ( ( oc ` K ) ` ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) |
97 |
59 96
|
eqtr3d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( X .\/ Y ) = ( ( ( ( oc ` K ) ` ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) |
98 |
97
|
fveq2d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( I ` ( X .\/ Y ) ) = ( I ` ( ( ( ( oc ` K ) ` ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) |
99 |
|
simpl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( K e. HL /\ W e. H ) ) |
100 |
2 3
|
diaclN |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. dom I ) -> ( I ` X ) e. ran I ) |
101 |
100
|
adantrr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( I ` X ) e. ran I ) |
102 |
2 43 3
|
diaelrnN |
|- ( ( ( K e. HL /\ W e. H ) /\ ( I ` X ) e. ran I ) -> ( I ` X ) C_ ( ( LTrn ` K ) ` W ) ) |
103 |
101 102
|
syldan |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( I ` X ) C_ ( ( LTrn ` K ) ` W ) ) |
104 |
2 3
|
diaclN |
|- ( ( ( K e. HL /\ W e. H ) /\ Y e. dom I ) -> ( I ` Y ) e. ran I ) |
105 |
104
|
adantrl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( I ` Y ) e. ran I ) |
106 |
2 43 3
|
diaelrnN |
|- ( ( ( K e. HL /\ W e. H ) /\ ( I ` Y ) e. ran I ) -> ( I ` Y ) C_ ( ( LTrn ` K ) ` W ) ) |
107 |
105 106
|
syldan |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( I ` Y ) C_ ( ( LTrn ` K ) ` W ) ) |
108 |
2 43 3 44 4
|
djavalN |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( I ` X ) C_ ( ( LTrn ` K ) ` W ) /\ ( I ` Y ) C_ ( ( LTrn ` K ) ` W ) ) ) -> ( ( I ` X ) J ( I ` Y ) ) = ( ( ( ocA ` K ) ` W ) ` ( ( ( ( ocA ` K ) ` W ) ` ( I ` X ) ) i^i ( ( ( ocA ` K ) ` W ) ` ( I ` Y ) ) ) ) ) |
109 |
99 103 107 108
|
syl12anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( I ` X ) J ( I ` Y ) ) = ( ( ( ocA ` K ) ` W ) ` ( ( ( ( ocA ` K ) ` W ) ` ( I ` X ) ) i^i ( ( ( ocA ` K ) ` W ) ` ( I ` Y ) ) ) ) ) |
110 |
9 34 21
|
latmle2 |
|- ( ( K e. Lat /\ ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( le ` K ) W ) |
111 |
6 20 16 110
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( le ` K ) W ) |
112 |
9 34 2 3
|
diaeldm |
|- ( ( K e. HL /\ W e. H ) -> ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. dom I <-> ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. ( Base ` K ) /\ ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( le ` K ) W ) ) ) |
113 |
112
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. dom I <-> ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. ( Base ` K ) /\ ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( le ` K ) W ) ) ) |
114 |
23 111 113
|
mpbir2and |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. dom I ) |
115 |
9 34 2 3
|
diaeldm |
|- ( ( K e. HL /\ W e. H ) -> ( ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. dom I <-> ( ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. ( Base ` K ) /\ ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( le ` K ) W ) ) ) |
116 |
115
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. dom I <-> ( ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. ( Base ` K ) /\ ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( le ` K ) W ) ) ) |
117 |
31 38 116
|
mpbir2and |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. dom I ) |
118 |
21 2 3
|
diameetN |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. dom I /\ ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) e. dom I ) ) -> ( I ` ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) = ( ( I ` ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) i^i ( I ` ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) ) |
119 |
99 114 117 118
|
syl12anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( I ` ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) = ( ( I ` ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) i^i ( I ` ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) ) |
120 |
1 21 12 2 43 3 44
|
diaocN |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. dom I ) -> ( I ` ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) = ( ( ( ocA ` K ) ` W ) ` ( I ` X ) ) ) |
121 |
120
|
adantrr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( I ` ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) = ( ( ( ocA ` K ) ` W ) ` ( I ` X ) ) ) |
122 |
1 21 12 2 43 3 44
|
diaocN |
|- ( ( ( K e. HL /\ W e. H ) /\ Y e. dom I ) -> ( I ` ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) = ( ( ( ocA ` K ) ` W ) ` ( I ` Y ) ) ) |
123 |
122
|
adantrl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( I ` ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) = ( ( ( ocA ` K ) ` W ) ` ( I ` Y ) ) ) |
124 |
121 123
|
ineq12d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( I ` ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) i^i ( I ` ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) = ( ( ( ( ocA ` K ) ` W ) ` ( I ` X ) ) i^i ( ( ( ocA ` K ) ` W ) ` ( I ` Y ) ) ) ) |
125 |
119 124
|
eqtrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( I ` ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) = ( ( ( ( ocA ` K ) ` W ) ` ( I ` X ) ) i^i ( ( ( ocA ` K ) ` W ) ` ( I ` Y ) ) ) ) |
126 |
125
|
fveq2d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( ( ocA ` K ) ` W ) ` ( I ` ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) ) = ( ( ( ocA ` K ) ` W ) ` ( ( ( ( ocA ` K ) ` W ) ` ( I ` X ) ) i^i ( ( ( ocA ` K ) ` W ) ` ( I ` Y ) ) ) ) ) |
127 |
109 126
|
eqtr4d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( ( I ` X ) J ( I ` Y ) ) = ( ( ( ocA ` K ) ` W ) ` ( I ` ( ( ( ( ( oc ` K ) ` X ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ( meet ` K ) ( ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` W ) ) ( meet ` K ) W ) ) ) ) ) |
128 |
46 98 127
|
3eqtr4d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( I ` ( X .\/ Y ) ) = ( ( I ` X ) J ( I ` Y ) ) ) |