Step |
Hyp |
Ref |
Expression |
1 |
|
diaintcl.h |
|- H = ( LHyp ` K ) |
2 |
|
diaintcl.i |
|- I = ( ( DIsoA ` K ) ` W ) |
3 |
1 2
|
diaf11N |
|- ( ( K e. HL /\ W e. H ) -> I : dom I -1-1-onto-> ran I ) |
4 |
3
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> I : dom I -1-1-onto-> ran I ) |
5 |
|
f1ofn |
|- ( I : dom I -1-1-onto-> ran I -> I Fn dom I ) |
6 |
4 5
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> I Fn dom I ) |
7 |
|
cnvimass |
|- ( `' I " S ) C_ dom I |
8 |
|
fnssres |
|- ( ( I Fn dom I /\ ( `' I " S ) C_ dom I ) -> ( I |` ( `' I " S ) ) Fn ( `' I " S ) ) |
9 |
6 7 8
|
sylancl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( I |` ( `' I " S ) ) Fn ( `' I " S ) ) |
10 |
|
fniinfv |
|- ( ( I |` ( `' I " S ) ) Fn ( `' I " S ) -> |^|_ y e. ( `' I " S ) ( ( I |` ( `' I " S ) ) ` y ) = |^| ran ( I |` ( `' I " S ) ) ) |
11 |
9 10
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> |^|_ y e. ( `' I " S ) ( ( I |` ( `' I " S ) ) ` y ) = |^| ran ( I |` ( `' I " S ) ) ) |
12 |
|
df-ima |
|- ( I " ( `' I " S ) ) = ran ( I |` ( `' I " S ) ) |
13 |
|
f1ofo |
|- ( I : dom I -1-1-onto-> ran I -> I : dom I -onto-> ran I ) |
14 |
3 13
|
syl |
|- ( ( K e. HL /\ W e. H ) -> I : dom I -onto-> ran I ) |
15 |
14
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> I : dom I -onto-> ran I ) |
16 |
|
simprl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> S C_ ran I ) |
17 |
|
foimacnv |
|- ( ( I : dom I -onto-> ran I /\ S C_ ran I ) -> ( I " ( `' I " S ) ) = S ) |
18 |
15 16 17
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( I " ( `' I " S ) ) = S ) |
19 |
12 18
|
syl5eqr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ran ( I |` ( `' I " S ) ) = S ) |
20 |
19
|
inteqd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> |^| ran ( I |` ( `' I " S ) ) = |^| S ) |
21 |
11 20
|
eqtrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> |^|_ y e. ( `' I " S ) ( ( I |` ( `' I " S ) ) ` y ) = |^| S ) |
22 |
|
simpl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( K e. HL /\ W e. H ) ) |
23 |
7
|
a1i |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( `' I " S ) C_ dom I ) |
24 |
|
simprr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> S =/= (/) ) |
25 |
|
n0 |
|- ( S =/= (/) <-> E. y y e. S ) |
26 |
24 25
|
sylib |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> E. y y e. S ) |
27 |
16
|
sselda |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> y e. ran I ) |
28 |
3
|
ad2antrr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> I : dom I -1-1-onto-> ran I ) |
29 |
28 5
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> I Fn dom I ) |
30 |
|
fvelrnb |
|- ( I Fn dom I -> ( y e. ran I <-> E. x e. dom I ( I ` x ) = y ) ) |
31 |
29 30
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> ( y e. ran I <-> E. x e. dom I ( I ` x ) = y ) ) |
32 |
27 31
|
mpbid |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> E. x e. dom I ( I ` x ) = y ) |
33 |
|
f1ofun |
|- ( I : dom I -1-1-onto-> ran I -> Fun I ) |
34 |
3 33
|
syl |
|- ( ( K e. HL /\ W e. H ) -> Fun I ) |
35 |
34
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> Fun I ) |
36 |
|
fvimacnv |
|- ( ( Fun I /\ x e. dom I ) -> ( ( I ` x ) e. S <-> x e. ( `' I " S ) ) ) |
37 |
35 36
|
sylan |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ x e. dom I ) -> ( ( I ` x ) e. S <-> x e. ( `' I " S ) ) ) |
38 |
|
ne0i |
|- ( x e. ( `' I " S ) -> ( `' I " S ) =/= (/) ) |
39 |
37 38
|
syl6bi |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ x e. dom I ) -> ( ( I ` x ) e. S -> ( `' I " S ) =/= (/) ) ) |
40 |
39
|
ex |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( x e. dom I -> ( ( I ` x ) e. S -> ( `' I " S ) =/= (/) ) ) ) |
41 |
|
eleq1 |
|- ( ( I ` x ) = y -> ( ( I ` x ) e. S <-> y e. S ) ) |
42 |
41
|
biimprd |
|- ( ( I ` x ) = y -> ( y e. S -> ( I ` x ) e. S ) ) |
43 |
42
|
imim1d |
|- ( ( I ` x ) = y -> ( ( ( I ` x ) e. S -> ( `' I " S ) =/= (/) ) -> ( y e. S -> ( `' I " S ) =/= (/) ) ) ) |
44 |
40 43
|
syl9 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( ( I ` x ) = y -> ( x e. dom I -> ( y e. S -> ( `' I " S ) =/= (/) ) ) ) ) |
45 |
44
|
com24 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( y e. S -> ( x e. dom I -> ( ( I ` x ) = y -> ( `' I " S ) =/= (/) ) ) ) ) |
46 |
45
|
imp |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> ( x e. dom I -> ( ( I ` x ) = y -> ( `' I " S ) =/= (/) ) ) ) |
47 |
46
|
rexlimdv |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> ( E. x e. dom I ( I ` x ) = y -> ( `' I " S ) =/= (/) ) ) |
48 |
32 47
|
mpd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. S ) -> ( `' I " S ) =/= (/) ) |
49 |
26 48
|
exlimddv |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( `' I " S ) =/= (/) ) |
50 |
|
eqid |
|- ( glb ` K ) = ( glb ` K ) |
51 |
50 1 2
|
diaglbN |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( `' I " S ) C_ dom I /\ ( `' I " S ) =/= (/) ) ) -> ( I ` ( ( glb ` K ) ` ( `' I " S ) ) ) = |^|_ y e. ( `' I " S ) ( I ` y ) ) |
52 |
22 23 49 51
|
syl12anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( I ` ( ( glb ` K ) ` ( `' I " S ) ) ) = |^|_ y e. ( `' I " S ) ( I ` y ) ) |
53 |
|
fvres |
|- ( y e. ( `' I " S ) -> ( ( I |` ( `' I " S ) ) ` y ) = ( I ` y ) ) |
54 |
53
|
iineq2i |
|- |^|_ y e. ( `' I " S ) ( ( I |` ( `' I " S ) ) ` y ) = |^|_ y e. ( `' I " S ) ( I ` y ) |
55 |
52 54
|
eqtr4di |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( I ` ( ( glb ` K ) ` ( `' I " S ) ) ) = |^|_ y e. ( `' I " S ) ( ( I |` ( `' I " S ) ) ` y ) ) |
56 |
|
hlclat |
|- ( K e. HL -> K e. CLat ) |
57 |
56
|
ad2antrr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> K e. CLat ) |
58 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
59 |
|
eqid |
|- ( le ` K ) = ( le ` K ) |
60 |
58 59 1 2
|
diadm |
|- ( ( K e. HL /\ W e. H ) -> dom I = { x e. ( Base ` K ) | x ( le ` K ) W } ) |
61 |
|
ssrab2 |
|- { x e. ( Base ` K ) | x ( le ` K ) W } C_ ( Base ` K ) |
62 |
60 61
|
eqsstrdi |
|- ( ( K e. HL /\ W e. H ) -> dom I C_ ( Base ` K ) ) |
63 |
62
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> dom I C_ ( Base ` K ) ) |
64 |
7 63
|
sstrid |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( `' I " S ) C_ ( Base ` K ) ) |
65 |
58 50
|
clatglbcl |
|- ( ( K e. CLat /\ ( `' I " S ) C_ ( Base ` K ) ) -> ( ( glb ` K ) ` ( `' I " S ) ) e. ( Base ` K ) ) |
66 |
57 64 65
|
syl2anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( ( glb ` K ) ` ( `' I " S ) ) e. ( Base ` K ) ) |
67 |
|
n0 |
|- ( ( `' I " S ) =/= (/) <-> E. y y e. ( `' I " S ) ) |
68 |
49 67
|
sylib |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> E. y y e. ( `' I " S ) ) |
69 |
|
hllat |
|- ( K e. HL -> K e. Lat ) |
70 |
69
|
ad3antrrr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> K e. Lat ) |
71 |
66
|
adantr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> ( ( glb ` K ) ` ( `' I " S ) ) e. ( Base ` K ) ) |
72 |
64
|
sselda |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> y e. ( Base ` K ) ) |
73 |
58 1
|
lhpbase |
|- ( W e. H -> W e. ( Base ` K ) ) |
74 |
73
|
ad3antlr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> W e. ( Base ` K ) ) |
75 |
56
|
ad3antrrr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> K e. CLat ) |
76 |
60
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> dom I = { x e. ( Base ` K ) | x ( le ` K ) W } ) |
77 |
7 76
|
sseqtrid |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( `' I " S ) C_ { x e. ( Base ` K ) | x ( le ` K ) W } ) |
78 |
77 61
|
sstrdi |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( `' I " S ) C_ ( Base ` K ) ) |
79 |
78
|
adantr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> ( `' I " S ) C_ ( Base ` K ) ) |
80 |
|
simpr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> y e. ( `' I " S ) ) |
81 |
58 59 50
|
clatglble |
|- ( ( K e. CLat /\ ( `' I " S ) C_ ( Base ` K ) /\ y e. ( `' I " S ) ) -> ( ( glb ` K ) ` ( `' I " S ) ) ( le ` K ) y ) |
82 |
75 79 80 81
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> ( ( glb ` K ) ` ( `' I " S ) ) ( le ` K ) y ) |
83 |
7
|
sseli |
|- ( y e. ( `' I " S ) -> y e. dom I ) |
84 |
83
|
adantl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> y e. dom I ) |
85 |
58 59 1 2
|
diaeldm |
|- ( ( K e. HL /\ W e. H ) -> ( y e. dom I <-> ( y e. ( Base ` K ) /\ y ( le ` K ) W ) ) ) |
86 |
85
|
ad2antrr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> ( y e. dom I <-> ( y e. ( Base ` K ) /\ y ( le ` K ) W ) ) ) |
87 |
84 86
|
mpbid |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> ( y e. ( Base ` K ) /\ y ( le ` K ) W ) ) |
88 |
87
|
simprd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> y ( le ` K ) W ) |
89 |
58 59 70 71 72 74 82 88
|
lattrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) /\ y e. ( `' I " S ) ) -> ( ( glb ` K ) ` ( `' I " S ) ) ( le ` K ) W ) |
90 |
68 89
|
exlimddv |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( ( glb ` K ) ` ( `' I " S ) ) ( le ` K ) W ) |
91 |
58 59 1 2
|
diaeldm |
|- ( ( K e. HL /\ W e. H ) -> ( ( ( glb ` K ) ` ( `' I " S ) ) e. dom I <-> ( ( ( glb ` K ) ` ( `' I " S ) ) e. ( Base ` K ) /\ ( ( glb ` K ) ` ( `' I " S ) ) ( le ` K ) W ) ) ) |
92 |
91
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( ( ( glb ` K ) ` ( `' I " S ) ) e. dom I <-> ( ( ( glb ` K ) ` ( `' I " S ) ) e. ( Base ` K ) /\ ( ( glb ` K ) ` ( `' I " S ) ) ( le ` K ) W ) ) ) |
93 |
66 90 92
|
mpbir2and |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( ( glb ` K ) ` ( `' I " S ) ) e. dom I ) |
94 |
1 2
|
diaclN |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( glb ` K ) ` ( `' I " S ) ) e. dom I ) -> ( I ` ( ( glb ` K ) ` ( `' I " S ) ) ) e. ran I ) |
95 |
93 94
|
syldan |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> ( I ` ( ( glb ` K ) ` ( `' I " S ) ) ) e. ran I ) |
96 |
55 95
|
eqeltrrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> |^|_ y e. ( `' I " S ) ( ( I |` ( `' I " S ) ) ` y ) e. ran I ) |
97 |
21 96
|
eqeltrrd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ ran I /\ S =/= (/) ) ) -> |^| S e. ran I ) |