Step |
Hyp |
Ref |
Expression |
1 |
|
dihglblem5a.b |
|- B = ( Base ` K ) |
2 |
|
dihglblem5a.m |
|- ./\ = ( meet ` K ) |
3 |
|
dihglblem5a.h |
|- H = ( LHyp ` K ) |
4 |
|
dihglblem5a.i |
|- I = ( ( DIsoH ` K ) ` W ) |
5 |
|
dihglblem5a.l |
|- .<_ = ( le ` K ) |
6 |
|
dihglblem5a.j |
|- .\/ = ( join ` K ) |
7 |
|
dihglblem5a.a |
|- A = ( Atoms ` K ) |
8 |
|
dihglblem5a.p |
|- P = ( ( oc ` K ) ` W ) |
9 |
|
dihglblem5a.t |
|- T = ( ( LTrn ` K ) ` W ) |
10 |
|
dihglblem5a.r |
|- R = ( ( trL ` K ) ` W ) |
11 |
|
dihglblem5a.e |
|- E = ( ( TEndo ` K ) ` W ) |
12 |
|
dihglblem5a.g |
|- G = ( iota_ h e. T ( h ` P ) = q ) |
13 |
|
dihglblem5a.o |
|- .0. = ( h e. T |-> ( _I |` B ) ) |
14 |
|
hllat |
|- ( K e. HL -> K e. Lat ) |
15 |
14
|
ad2antrr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> K e. Lat ) |
16 |
|
simprl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> X e. B ) |
17 |
1 3
|
lhpbase |
|- ( W e. H -> W e. B ) |
18 |
17
|
ad2antlr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> W e. B ) |
19 |
1 5 2
|
latmle1 |
|- ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ./\ W ) .<_ X ) |
20 |
15 16 18 19
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( X ./\ W ) .<_ X ) |
21 |
|
simpl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( K e. HL /\ W e. H ) ) |
22 |
1 2
|
latmcl |
|- ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ./\ W ) e. B ) |
23 |
15 16 18 22
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( X ./\ W ) e. B ) |
24 |
1 5 3 4
|
dihord |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X ./\ W ) e. B /\ X e. B ) -> ( ( I ` ( X ./\ W ) ) C_ ( I ` X ) <-> ( X ./\ W ) .<_ X ) ) |
25 |
21 23 16 24
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( ( I ` ( X ./\ W ) ) C_ ( I ` X ) <-> ( X ./\ W ) .<_ X ) ) |
26 |
20 25
|
mpbird |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( I ` ( X ./\ W ) ) C_ ( I ` X ) ) |
27 |
1 5 2
|
latmle2 |
|- ( ( K e. Lat /\ X e. B /\ W e. B ) -> ( X ./\ W ) .<_ W ) |
28 |
15 16 18 27
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( X ./\ W ) .<_ W ) |
29 |
1 5 3 4
|
dihord |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X ./\ W ) e. B /\ W e. B ) -> ( ( I ` ( X ./\ W ) ) C_ ( I ` W ) <-> ( X ./\ W ) .<_ W ) ) |
30 |
21 23 18 29
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( ( I ` ( X ./\ W ) ) C_ ( I ` W ) <-> ( X ./\ W ) .<_ W ) ) |
31 |
28 30
|
mpbird |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( I ` ( X ./\ W ) ) C_ ( I ` W ) ) |
32 |
26 31
|
ssind |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( I ` ( X ./\ W ) ) C_ ( ( I ` X ) i^i ( I ` W ) ) ) |
33 |
3 4
|
dihvalrel |
|- ( ( K e. HL /\ W e. H ) -> Rel ( I ` X ) ) |
34 |
|
relin1 |
|- ( Rel ( I ` X ) -> Rel ( ( I ` X ) i^i ( I ` W ) ) ) |
35 |
33 34
|
syl |
|- ( ( K e. HL /\ W e. H ) -> Rel ( ( I ` X ) i^i ( I ` W ) ) ) |
36 |
35
|
adantr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> Rel ( ( I ` X ) i^i ( I ` W ) ) ) |
37 |
|
elin |
|- ( <. f , s >. e. ( ( I ` X ) i^i ( I ` W ) ) <-> ( <. f , s >. e. ( I ` X ) /\ <. f , s >. e. ( I ` W ) ) ) |
38 |
1 5 6 2 7 3
|
lhpmcvr2 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> E. q e. A ( -. q .<_ W /\ ( q .\/ ( X ./\ W ) ) = X ) ) |
39 |
|
vex |
|- f e. _V |
40 |
|
vex |
|- s e. _V |
41 |
1 5 6 2 7 3 8 9 10 11 4 12 39 40
|
dihopelvalc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) -> ( <. f , s >. e. ( I ` X ) <-> ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) ) ) |
42 |
|
id |
|- ( ( K e. HL /\ W e. H ) -> ( K e. HL /\ W e. H ) ) |
43 |
17
|
adantl |
|- ( ( K e. HL /\ W e. H ) -> W e. B ) |
44 |
1 5
|
latref |
|- ( ( K e. Lat /\ W e. B ) -> W .<_ W ) |
45 |
14 17 44
|
syl2an |
|- ( ( K e. HL /\ W e. H ) -> W .<_ W ) |
46 |
1 5 3 9 10 13 4
|
dihopelvalbN |
|- ( ( ( K e. HL /\ W e. H ) /\ ( W e. B /\ W .<_ W ) ) -> ( <. f , s >. e. ( I ` W ) <-> ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) |
47 |
42 43 45 46
|
syl12anc |
|- ( ( K e. HL /\ W e. H ) -> ( <. f , s >. e. ( I ` W ) <-> ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) |
48 |
47
|
3ad2ant1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) -> ( <. f , s >. e. ( I ` W ) <-> ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) |
49 |
41 48
|
anbi12d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) -> ( ( <. f , s >. e. ( I ` X ) /\ <. f , s >. e. ( I ` W ) ) <-> ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) ) |
50 |
|
simprll |
|- ( ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) -> f e. T ) |
51 |
50
|
adantl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> f e. T ) |
52 |
|
simprrr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> s = .0. ) |
53 |
52
|
fveq1d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> ( s ` G ) = ( .0. ` G ) ) |
54 |
|
simpl1 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> ( K e. HL /\ W e. H ) ) |
55 |
5 7 3 8
|
lhpocnel2 |
|- ( ( K e. HL /\ W e. H ) -> ( P e. A /\ -. P .<_ W ) ) |
56 |
54 55
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
57 |
|
simpl3l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> ( q e. A /\ -. q .<_ W ) ) |
58 |
5 7 3 9 12
|
ltrniotacl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( q e. A /\ -. q .<_ W ) ) -> G e. T ) |
59 |
54 56 57 58
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> G e. T ) |
60 |
13 1
|
tendo02 |
|- ( G e. T -> ( .0. ` G ) = ( _I |` B ) ) |
61 |
59 60
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> ( .0. ` G ) = ( _I |` B ) ) |
62 |
53 61
|
eqtrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> ( s ` G ) = ( _I |` B ) ) |
63 |
62
|
cnveqd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> `' ( s ` G ) = `' ( _I |` B ) ) |
64 |
|
cnvresid |
|- `' ( _I |` B ) = ( _I |` B ) |
65 |
63 64
|
eqtrdi |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> `' ( s ` G ) = ( _I |` B ) ) |
66 |
65
|
coeq2d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> ( f o. `' ( s ` G ) ) = ( f o. ( _I |` B ) ) ) |
67 |
1 3 9
|
ltrn1o |
|- ( ( ( K e. HL /\ W e. H ) /\ f e. T ) -> f : B -1-1-onto-> B ) |
68 |
54 51 67
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> f : B -1-1-onto-> B ) |
69 |
|
f1of |
|- ( f : B -1-1-onto-> B -> f : B --> B ) |
70 |
|
fcoi1 |
|- ( f : B --> B -> ( f o. ( _I |` B ) ) = f ) |
71 |
68 69 70
|
3syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> ( f o. ( _I |` B ) ) = f ) |
72 |
66 71
|
eqtrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> ( f o. `' ( s ` G ) ) = f ) |
73 |
72
|
fveq2d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> ( R ` ( f o. `' ( s ` G ) ) ) = ( R ` f ) ) |
74 |
|
simprlr |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) |
75 |
73 74
|
eqbrtrrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> ( R ` f ) .<_ X ) |
76 |
5 3 9 10
|
trlle |
|- ( ( ( K e. HL /\ W e. H ) /\ f e. T ) -> ( R ` f ) .<_ W ) |
77 |
54 51 76
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> ( R ` f ) .<_ W ) |
78 |
|
simpl1l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> K e. HL ) |
79 |
78
|
hllatd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> K e. Lat ) |
80 |
1 3 9 10
|
trlcl |
|- ( ( ( K e. HL /\ W e. H ) /\ f e. T ) -> ( R ` f ) e. B ) |
81 |
54 51 80
|
syl2anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> ( R ` f ) e. B ) |
82 |
|
simpl2l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> X e. B ) |
83 |
|
simpl1r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> W e. H ) |
84 |
83 17
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> W e. B ) |
85 |
1 5 2
|
latlem12 |
|- ( ( K e. Lat /\ ( ( R ` f ) e. B /\ X e. B /\ W e. B ) ) -> ( ( ( R ` f ) .<_ X /\ ( R ` f ) .<_ W ) <-> ( R ` f ) .<_ ( X ./\ W ) ) ) |
86 |
79 81 82 84 85
|
syl13anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> ( ( ( R ` f ) .<_ X /\ ( R ` f ) .<_ W ) <-> ( R ` f ) .<_ ( X ./\ W ) ) ) |
87 |
75 77 86
|
mpbi2and |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> ( R ` f ) .<_ ( X ./\ W ) ) |
88 |
51 87
|
jca |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) ) |
89 |
79 82 84 22
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> ( X ./\ W ) e. B ) |
90 |
79 82 84 27
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> ( X ./\ W ) .<_ W ) |
91 |
1 5 3 9 10 13 4
|
dihopelvalbN |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( X ./\ W ) e. B /\ ( X ./\ W ) .<_ W ) ) -> ( <. f , s >. e. ( I ` ( X ./\ W ) ) <-> ( ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) /\ s = .0. ) ) ) |
92 |
54 89 90 91
|
syl12anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> ( <. f , s >. e. ( I ` ( X ./\ W ) ) <-> ( ( f e. T /\ ( R ` f ) .<_ ( X ./\ W ) ) /\ s = .0. ) ) ) |
93 |
88 52 92
|
mpbir2and |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) /\ ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) ) -> <. f , s >. e. ( I ` ( X ./\ W ) ) ) |
94 |
93
|
ex |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) -> ( ( ( ( f e. T /\ s e. E ) /\ ( R ` ( f o. `' ( s ` G ) ) ) .<_ X ) /\ ( ( f e. T /\ ( R ` f ) .<_ W ) /\ s = .0. ) ) -> <. f , s >. e. ( I ` ( X ./\ W ) ) ) ) |
95 |
49 94
|
sylbid |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) ) -> ( ( <. f , s >. e. ( I ` X ) /\ <. f , s >. e. ( I ` W ) ) -> <. f , s >. e. ( I ` ( X ./\ W ) ) ) ) |
96 |
95
|
3expia |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( ( ( q e. A /\ -. q .<_ W ) /\ ( q .\/ ( X ./\ W ) ) = X ) -> ( ( <. f , s >. e. ( I ` X ) /\ <. f , s >. e. ( I ` W ) ) -> <. f , s >. e. ( I ` ( X ./\ W ) ) ) ) ) |
97 |
96
|
exp4c |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( q e. A -> ( -. q .<_ W -> ( ( q .\/ ( X ./\ W ) ) = X -> ( ( <. f , s >. e. ( I ` X ) /\ <. f , s >. e. ( I ` W ) ) -> <. f , s >. e. ( I ` ( X ./\ W ) ) ) ) ) ) ) |
98 |
97
|
imp4a |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( q e. A -> ( ( -. q .<_ W /\ ( q .\/ ( X ./\ W ) ) = X ) -> ( ( <. f , s >. e. ( I ` X ) /\ <. f , s >. e. ( I ` W ) ) -> <. f , s >. e. ( I ` ( X ./\ W ) ) ) ) ) ) |
99 |
98
|
rexlimdv |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( E. q e. A ( -. q .<_ W /\ ( q .\/ ( X ./\ W ) ) = X ) -> ( ( <. f , s >. e. ( I ` X ) /\ <. f , s >. e. ( I ` W ) ) -> <. f , s >. e. ( I ` ( X ./\ W ) ) ) ) ) |
100 |
38 99
|
mpd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( ( <. f , s >. e. ( I ` X ) /\ <. f , s >. e. ( I ` W ) ) -> <. f , s >. e. ( I ` ( X ./\ W ) ) ) ) |
101 |
37 100
|
syl5bi |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( <. f , s >. e. ( ( I ` X ) i^i ( I ` W ) ) -> <. f , s >. e. ( I ` ( X ./\ W ) ) ) ) |
102 |
36 101
|
relssdv |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( ( I ` X ) i^i ( I ` W ) ) C_ ( I ` ( X ./\ W ) ) ) |
103 |
32 102
|
eqssd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) ) -> ( I ` ( X ./\ W ) ) = ( ( I ` X ) i^i ( I ` W ) ) ) |