Step |
Hyp |
Ref |
Expression |
1 |
|
dihjatcclem.b |
|- B = ( Base ` K ) |
2 |
|
dihjatcclem.l |
|- .<_ = ( le ` K ) |
3 |
|
dihjatcclem.h |
|- H = ( LHyp ` K ) |
4 |
|
dihjatcclem.j |
|- .\/ = ( join ` K ) |
5 |
|
dihjatcclem.m |
|- ./\ = ( meet ` K ) |
6 |
|
dihjatcclem.a |
|- A = ( Atoms ` K ) |
7 |
|
dihjatcclem.u |
|- U = ( ( DVecH ` K ) ` W ) |
8 |
|
dihjatcclem.s |
|- .(+) = ( LSSum ` U ) |
9 |
|
dihjatcclem.i |
|- I = ( ( DIsoH ` K ) ` W ) |
10 |
|
dihjatcclem.v |
|- V = ( ( P .\/ Q ) ./\ W ) |
11 |
|
dihjatcclem.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
12 |
|
dihjatcclem.p |
|- ( ph -> ( P e. A /\ -. P .<_ W ) ) |
13 |
|
dihjatcclem.q |
|- ( ph -> ( Q e. A /\ -. Q .<_ W ) ) |
14 |
|
dihjatcc.w |
|- C = ( ( oc ` K ) ` W ) |
15 |
|
dihjatcc.t |
|- T = ( ( LTrn ` K ) ` W ) |
16 |
|
dihjatcc.r |
|- R = ( ( trL ` K ) ` W ) |
17 |
|
dihjatcc.e |
|- E = ( ( TEndo ` K ) ` W ) |
18 |
|
dihjatcc.g |
|- G = ( iota_ d e. T ( d ` C ) = P ) |
19 |
|
dihjatcc.dd |
|- D = ( iota_ d e. T ( d ` C ) = Q ) |
20 |
|
dihjatcc.n |
|- N = ( a e. E |-> ( d e. T |-> `' ( a ` d ) ) ) |
21 |
|
dihjatcc.o |
|- .0. = ( d e. T |-> ( _I |` B ) ) |
22 |
|
dihjatcc.d |
|- J = ( a e. E , b e. E |-> ( d e. T |-> ( ( a ` d ) o. ( b ` d ) ) ) ) |
23 |
3 9
|
dihvalrel |
|- ( ( K e. HL /\ W e. H ) -> Rel ( I ` V ) ) |
24 |
11 23
|
syl |
|- ( ph -> Rel ( I ` V ) ) |
25 |
11
|
adantr |
|- ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) -> ( K e. HL /\ W e. H ) ) |
26 |
2 6 3 14
|
lhpocnel2 |
|- ( ( K e. HL /\ W e. H ) -> ( C e. A /\ -. C .<_ W ) ) |
27 |
11 26
|
syl |
|- ( ph -> ( C e. A /\ -. C .<_ W ) ) |
28 |
2 6 3 15 18
|
ltrniotacl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( C e. A /\ -. C .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) -> G e. T ) |
29 |
11 27 12 28
|
syl3anc |
|- ( ph -> G e. T ) |
30 |
2 6 3 15 19
|
ltrniotacl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( C e. A /\ -. C .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> D e. T ) |
31 |
11 27 13 30
|
syl3anc |
|- ( ph -> D e. T ) |
32 |
3 15
|
ltrncnv |
|- ( ( ( K e. HL /\ W e. H ) /\ D e. T ) -> `' D e. T ) |
33 |
11 31 32
|
syl2anc |
|- ( ph -> `' D e. T ) |
34 |
3 15
|
ltrnco |
|- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ `' D e. T ) -> ( G o. `' D ) e. T ) |
35 |
11 29 33 34
|
syl3anc |
|- ( ph -> ( G o. `' D ) e. T ) |
36 |
35
|
adantr |
|- ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) -> ( G o. `' D ) e. T ) |
37 |
|
simprll |
|- ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) -> f e. T ) |
38 |
|
simprlr |
|- ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) -> ( R ` f ) .<_ V ) |
39 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
|
dihjatcclem3 |
|- ( ph -> ( R ` ( G o. `' D ) ) = V ) |
40 |
39
|
adantr |
|- ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) -> ( R ` ( G o. `' D ) ) = V ) |
41 |
38 40
|
breqtrrd |
|- ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) -> ( R ` f ) .<_ ( R ` ( G o. `' D ) ) ) |
42 |
2 3 15 16 17
|
tendoex |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( G o. `' D ) e. T /\ f e. T ) /\ ( R ` f ) .<_ ( R ` ( G o. `' D ) ) ) -> E. t e. E ( t ` ( G o. `' D ) ) = f ) |
43 |
25 36 37 41 42
|
syl121anc |
|- ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) -> E. t e. E ( t ` ( G o. `' D ) ) = f ) |
44 |
|
df-rex |
|- ( E. t e. E ( t ` ( G o. `' D ) ) = f <-> E. t ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) |
45 |
43 44
|
sylib |
|- ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) -> E. t ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) |
46 |
|
eqidd |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> ( t ` G ) = ( t ` G ) ) |
47 |
|
simprl |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> t e. E ) |
48 |
11
|
ad2antrr |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> ( K e. HL /\ W e. H ) ) |
49 |
12
|
ad2antrr |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> ( P e. A /\ -. P .<_ W ) ) |
50 |
|
fvex |
|- ( t ` G ) e. _V |
51 |
|
vex |
|- t e. _V |
52 |
2 6 3 14 15 17 9 18 50 51
|
dihopelvalcqat |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( <. ( t ` G ) , t >. e. ( I ` P ) <-> ( ( t ` G ) = ( t ` G ) /\ t e. E ) ) ) |
53 |
48 49 52
|
syl2anc |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> ( <. ( t ` G ) , t >. e. ( I ` P ) <-> ( ( t ` G ) = ( t ` G ) /\ t e. E ) ) ) |
54 |
46 47 53
|
mpbir2and |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> <. ( t ` G ) , t >. e. ( I ` P ) ) |
55 |
|
eqidd |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> ( ( N ` t ) ` D ) = ( ( N ` t ) ` D ) ) |
56 |
3 15 17 20
|
tendoicl |
|- ( ( ( K e. HL /\ W e. H ) /\ t e. E ) -> ( N ` t ) e. E ) |
57 |
48 47 56
|
syl2anc |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> ( N ` t ) e. E ) |
58 |
13
|
ad2antrr |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> ( Q e. A /\ -. Q .<_ W ) ) |
59 |
|
fvex |
|- ( ( N ` t ) ` D ) e. _V |
60 |
|
fvex |
|- ( N ` t ) e. _V |
61 |
2 6 3 14 15 17 9 19 59 60
|
dihopelvalcqat |
|- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( <. ( ( N ` t ) ` D ) , ( N ` t ) >. e. ( I ` Q ) <-> ( ( ( N ` t ) ` D ) = ( ( N ` t ) ` D ) /\ ( N ` t ) e. E ) ) ) |
62 |
48 58 61
|
syl2anc |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> ( <. ( ( N ` t ) ` D ) , ( N ` t ) >. e. ( I ` Q ) <-> ( ( ( N ` t ) ` D ) = ( ( N ` t ) ` D ) /\ ( N ` t ) e. E ) ) ) |
63 |
55 57 62
|
mpbir2and |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> <. ( ( N ` t ) ` D ) , ( N ` t ) >. e. ( I ` Q ) ) |
64 |
29
|
ad2antrr |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> G e. T ) |
65 |
33
|
ad2antrr |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> `' D e. T ) |
66 |
3 15 17
|
tendospdi1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( t e. E /\ G e. T /\ `' D e. T ) ) -> ( t ` ( G o. `' D ) ) = ( ( t ` G ) o. ( t ` `' D ) ) ) |
67 |
48 47 64 65 66
|
syl13anc |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> ( t ` ( G o. `' D ) ) = ( ( t ` G ) o. ( t ` `' D ) ) ) |
68 |
|
simprr |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> ( t ` ( G o. `' D ) ) = f ) |
69 |
31
|
ad2antrr |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> D e. T ) |
70 |
20 15
|
tendoi2 |
|- ( ( t e. E /\ D e. T ) -> ( ( N ` t ) ` D ) = `' ( t ` D ) ) |
71 |
47 69 70
|
syl2anc |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> ( ( N ` t ) ` D ) = `' ( t ` D ) ) |
72 |
3 15 17
|
tendocnv |
|- ( ( ( K e. HL /\ W e. H ) /\ t e. E /\ D e. T ) -> `' ( t ` D ) = ( t ` `' D ) ) |
73 |
48 47 69 72
|
syl3anc |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> `' ( t ` D ) = ( t ` `' D ) ) |
74 |
71 73
|
eqtr2d |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> ( t ` `' D ) = ( ( N ` t ) ` D ) ) |
75 |
74
|
coeq2d |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> ( ( t ` G ) o. ( t ` `' D ) ) = ( ( t ` G ) o. ( ( N ` t ) ` D ) ) ) |
76 |
67 68 75
|
3eqtr3d |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> f = ( ( t ` G ) o. ( ( N ` t ) ` D ) ) ) |
77 |
|
simplrr |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> s = .0. ) |
78 |
3 15 17 20 1 22 21
|
tendoipl2 |
|- ( ( ( K e. HL /\ W e. H ) /\ t e. E ) -> ( t J ( N ` t ) ) = .0. ) |
79 |
48 47 78
|
syl2anc |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> ( t J ( N ` t ) ) = .0. ) |
80 |
77 79
|
eqtr4d |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> s = ( t J ( N ` t ) ) ) |
81 |
|
opeq1 |
|- ( g = ( t ` G ) -> <. g , t >. = <. ( t ` G ) , t >. ) |
82 |
81
|
eleq1d |
|- ( g = ( t ` G ) -> ( <. g , t >. e. ( I ` P ) <-> <. ( t ` G ) , t >. e. ( I ` P ) ) ) |
83 |
82
|
anbi1d |
|- ( g = ( t ` G ) -> ( ( <. g , t >. e. ( I ` P ) /\ <. h , u >. e. ( I ` Q ) ) <-> ( <. ( t ` G ) , t >. e. ( I ` P ) /\ <. h , u >. e. ( I ` Q ) ) ) ) |
84 |
|
coeq1 |
|- ( g = ( t ` G ) -> ( g o. h ) = ( ( t ` G ) o. h ) ) |
85 |
84
|
eqeq2d |
|- ( g = ( t ` G ) -> ( f = ( g o. h ) <-> f = ( ( t ` G ) o. h ) ) ) |
86 |
85
|
anbi1d |
|- ( g = ( t ` G ) -> ( ( f = ( g o. h ) /\ s = ( t J u ) ) <-> ( f = ( ( t ` G ) o. h ) /\ s = ( t J u ) ) ) ) |
87 |
83 86
|
anbi12d |
|- ( g = ( t ` G ) -> ( ( ( <. g , t >. e. ( I ` P ) /\ <. h , u >. e. ( I ` Q ) ) /\ ( f = ( g o. h ) /\ s = ( t J u ) ) ) <-> ( ( <. ( t ` G ) , t >. e. ( I ` P ) /\ <. h , u >. e. ( I ` Q ) ) /\ ( f = ( ( t ` G ) o. h ) /\ s = ( t J u ) ) ) ) ) |
88 |
|
opeq1 |
|- ( h = ( ( N ` t ) ` D ) -> <. h , u >. = <. ( ( N ` t ) ` D ) , u >. ) |
89 |
88
|
eleq1d |
|- ( h = ( ( N ` t ) ` D ) -> ( <. h , u >. e. ( I ` Q ) <-> <. ( ( N ` t ) ` D ) , u >. e. ( I ` Q ) ) ) |
90 |
89
|
anbi2d |
|- ( h = ( ( N ` t ) ` D ) -> ( ( <. ( t ` G ) , t >. e. ( I ` P ) /\ <. h , u >. e. ( I ` Q ) ) <-> ( <. ( t ` G ) , t >. e. ( I ` P ) /\ <. ( ( N ` t ) ` D ) , u >. e. ( I ` Q ) ) ) ) |
91 |
|
coeq2 |
|- ( h = ( ( N ` t ) ` D ) -> ( ( t ` G ) o. h ) = ( ( t ` G ) o. ( ( N ` t ) ` D ) ) ) |
92 |
91
|
eqeq2d |
|- ( h = ( ( N ` t ) ` D ) -> ( f = ( ( t ` G ) o. h ) <-> f = ( ( t ` G ) o. ( ( N ` t ) ` D ) ) ) ) |
93 |
92
|
anbi1d |
|- ( h = ( ( N ` t ) ` D ) -> ( ( f = ( ( t ` G ) o. h ) /\ s = ( t J u ) ) <-> ( f = ( ( t ` G ) o. ( ( N ` t ) ` D ) ) /\ s = ( t J u ) ) ) ) |
94 |
90 93
|
anbi12d |
|- ( h = ( ( N ` t ) ` D ) -> ( ( ( <. ( t ` G ) , t >. e. ( I ` P ) /\ <. h , u >. e. ( I ` Q ) ) /\ ( f = ( ( t ` G ) o. h ) /\ s = ( t J u ) ) ) <-> ( ( <. ( t ` G ) , t >. e. ( I ` P ) /\ <. ( ( N ` t ) ` D ) , u >. e. ( I ` Q ) ) /\ ( f = ( ( t ` G ) o. ( ( N ` t ) ` D ) ) /\ s = ( t J u ) ) ) ) ) |
95 |
|
opeq2 |
|- ( u = ( N ` t ) -> <. ( ( N ` t ) ` D ) , u >. = <. ( ( N ` t ) ` D ) , ( N ` t ) >. ) |
96 |
95
|
eleq1d |
|- ( u = ( N ` t ) -> ( <. ( ( N ` t ) ` D ) , u >. e. ( I ` Q ) <-> <. ( ( N ` t ) ` D ) , ( N ` t ) >. e. ( I ` Q ) ) ) |
97 |
96
|
anbi2d |
|- ( u = ( N ` t ) -> ( ( <. ( t ` G ) , t >. e. ( I ` P ) /\ <. ( ( N ` t ) ` D ) , u >. e. ( I ` Q ) ) <-> ( <. ( t ` G ) , t >. e. ( I ` P ) /\ <. ( ( N ` t ) ` D ) , ( N ` t ) >. e. ( I ` Q ) ) ) ) |
98 |
|
oveq2 |
|- ( u = ( N ` t ) -> ( t J u ) = ( t J ( N ` t ) ) ) |
99 |
98
|
eqeq2d |
|- ( u = ( N ` t ) -> ( s = ( t J u ) <-> s = ( t J ( N ` t ) ) ) ) |
100 |
99
|
anbi2d |
|- ( u = ( N ` t ) -> ( ( f = ( ( t ` G ) o. ( ( N ` t ) ` D ) ) /\ s = ( t J u ) ) <-> ( f = ( ( t ` G ) o. ( ( N ` t ) ` D ) ) /\ s = ( t J ( N ` t ) ) ) ) ) |
101 |
97 100
|
anbi12d |
|- ( u = ( N ` t ) -> ( ( ( <. ( t ` G ) , t >. e. ( I ` P ) /\ <. ( ( N ` t ) ` D ) , u >. e. ( I ` Q ) ) /\ ( f = ( ( t ` G ) o. ( ( N ` t ) ` D ) ) /\ s = ( t J u ) ) ) <-> ( ( <. ( t ` G ) , t >. e. ( I ` P ) /\ <. ( ( N ` t ) ` D ) , ( N ` t ) >. e. ( I ` Q ) ) /\ ( f = ( ( t ` G ) o. ( ( N ` t ) ` D ) ) /\ s = ( t J ( N ` t ) ) ) ) ) ) |
102 |
87 94 101
|
syl3an9b |
|- ( ( g = ( t ` G ) /\ h = ( ( N ` t ) ` D ) /\ u = ( N ` t ) ) -> ( ( ( <. g , t >. e. ( I ` P ) /\ <. h , u >. e. ( I ` Q ) ) /\ ( f = ( g o. h ) /\ s = ( t J u ) ) ) <-> ( ( <. ( t ` G ) , t >. e. ( I ` P ) /\ <. ( ( N ` t ) ` D ) , ( N ` t ) >. e. ( I ` Q ) ) /\ ( f = ( ( t ` G ) o. ( ( N ` t ) ` D ) ) /\ s = ( t J ( N ` t ) ) ) ) ) ) |
103 |
102
|
spc3egv |
|- ( ( ( t ` G ) e. _V /\ ( ( N ` t ) ` D ) e. _V /\ ( N ` t ) e. _V ) -> ( ( ( <. ( t ` G ) , t >. e. ( I ` P ) /\ <. ( ( N ` t ) ` D ) , ( N ` t ) >. e. ( I ` Q ) ) /\ ( f = ( ( t ` G ) o. ( ( N ` t ) ` D ) ) /\ s = ( t J ( N ` t ) ) ) ) -> E. g E. h E. u ( ( <. g , t >. e. ( I ` P ) /\ <. h , u >. e. ( I ` Q ) ) /\ ( f = ( g o. h ) /\ s = ( t J u ) ) ) ) ) |
104 |
50 59 60 103
|
mp3an |
|- ( ( ( <. ( t ` G ) , t >. e. ( I ` P ) /\ <. ( ( N ` t ) ` D ) , ( N ` t ) >. e. ( I ` Q ) ) /\ ( f = ( ( t ` G ) o. ( ( N ` t ) ` D ) ) /\ s = ( t J ( N ` t ) ) ) ) -> E. g E. h E. u ( ( <. g , t >. e. ( I ` P ) /\ <. h , u >. e. ( I ` Q ) ) /\ ( f = ( g o. h ) /\ s = ( t J u ) ) ) ) |
105 |
54 63 76 80 104
|
syl22anc |
|- ( ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) /\ ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) ) -> E. g E. h E. u ( ( <. g , t >. e. ( I ` P ) /\ <. h , u >. e. ( I ` Q ) ) /\ ( f = ( g o. h ) /\ s = ( t J u ) ) ) ) |
106 |
105
|
ex |
|- ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) -> ( ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) -> E. g E. h E. u ( ( <. g , t >. e. ( I ` P ) /\ <. h , u >. e. ( I ` Q ) ) /\ ( f = ( g o. h ) /\ s = ( t J u ) ) ) ) ) |
107 |
106
|
eximdv |
|- ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) -> ( E. t ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) -> E. t E. g E. h E. u ( ( <. g , t >. e. ( I ` P ) /\ <. h , u >. e. ( I ` Q ) ) /\ ( f = ( g o. h ) /\ s = ( t J u ) ) ) ) ) |
108 |
|
excom |
|- ( E. t E. g E. h E. u ( ( <. g , t >. e. ( I ` P ) /\ <. h , u >. e. ( I ` Q ) ) /\ ( f = ( g o. h ) /\ s = ( t J u ) ) ) <-> E. g E. t E. h E. u ( ( <. g , t >. e. ( I ` P ) /\ <. h , u >. e. ( I ` Q ) ) /\ ( f = ( g o. h ) /\ s = ( t J u ) ) ) ) |
109 |
107 108
|
syl6ib |
|- ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) -> ( E. t ( t e. E /\ ( t ` ( G o. `' D ) ) = f ) -> E. g E. t E. h E. u ( ( <. g , t >. e. ( I ` P ) /\ <. h , u >. e. ( I ` Q ) ) /\ ( f = ( g o. h ) /\ s = ( t J u ) ) ) ) ) |
110 |
45 109
|
mpd |
|- ( ( ph /\ ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) -> E. g E. t E. h E. u ( ( <. g , t >. e. ( I ` P ) /\ <. h , u >. e. ( I ` Q ) ) /\ ( f = ( g o. h ) /\ s = ( t J u ) ) ) ) |
111 |
110
|
ex |
|- ( ph -> ( ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) -> E. g E. t E. h E. u ( ( <. g , t >. e. ( I ` P ) /\ <. h , u >. e. ( I ` Q ) ) /\ ( f = ( g o. h ) /\ s = ( t J u ) ) ) ) ) |
112 |
11
|
simpld |
|- ( ph -> K e. HL ) |
113 |
112
|
hllatd |
|- ( ph -> K e. Lat ) |
114 |
12
|
simpld |
|- ( ph -> P e. A ) |
115 |
13
|
simpld |
|- ( ph -> Q e. A ) |
116 |
1 4 6
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. B ) |
117 |
112 114 115 116
|
syl3anc |
|- ( ph -> ( P .\/ Q ) e. B ) |
118 |
11
|
simprd |
|- ( ph -> W e. H ) |
119 |
1 3
|
lhpbase |
|- ( W e. H -> W e. B ) |
120 |
118 119
|
syl |
|- ( ph -> W e. B ) |
121 |
1 5
|
latmcl |
|- ( ( K e. Lat /\ ( P .\/ Q ) e. B /\ W e. B ) -> ( ( P .\/ Q ) ./\ W ) e. B ) |
122 |
113 117 120 121
|
syl3anc |
|- ( ph -> ( ( P .\/ Q ) ./\ W ) e. B ) |
123 |
10 122
|
eqeltrid |
|- ( ph -> V e. B ) |
124 |
1 2 5
|
latmle2 |
|- ( ( K e. Lat /\ ( P .\/ Q ) e. B /\ W e. B ) -> ( ( P .\/ Q ) ./\ W ) .<_ W ) |
125 |
113 117 120 124
|
syl3anc |
|- ( ph -> ( ( P .\/ Q ) ./\ W ) .<_ W ) |
126 |
10 125
|
eqbrtrid |
|- ( ph -> V .<_ W ) |
127 |
|
eqid |
|- ( ( DIsoB ` K ) ` W ) = ( ( DIsoB ` K ) ` W ) |
128 |
1 2 3 9 127
|
dihvalb |
|- ( ( ( K e. HL /\ W e. H ) /\ ( V e. B /\ V .<_ W ) ) -> ( I ` V ) = ( ( ( DIsoB ` K ) ` W ) ` V ) ) |
129 |
11 123 126 128
|
syl12anc |
|- ( ph -> ( I ` V ) = ( ( ( DIsoB ` K ) ` W ) ` V ) ) |
130 |
129
|
eleq2d |
|- ( ph -> ( <. f , s >. e. ( I ` V ) <-> <. f , s >. e. ( ( ( DIsoB ` K ) ` W ) ` V ) ) ) |
131 |
1 2 3 15 16 21 127
|
dibopelval3 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( V e. B /\ V .<_ W ) ) -> ( <. f , s >. e. ( ( ( DIsoB ` K ) ` W ) ` V ) <-> ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) ) |
132 |
11 123 126 131
|
syl12anc |
|- ( ph -> ( <. f , s >. e. ( ( ( DIsoB ` K ) ` W ) ` V ) <-> ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) ) |
133 |
130 132
|
bitrd |
|- ( ph -> ( <. f , s >. e. ( I ` V ) <-> ( ( f e. T /\ ( R ` f ) .<_ V ) /\ s = .0. ) ) ) |
134 |
|
eqid |
|- ( LSubSp ` U ) = ( LSubSp ` U ) |
135 |
1 6
|
atbase |
|- ( P e. A -> P e. B ) |
136 |
114 135
|
syl |
|- ( ph -> P e. B ) |
137 |
1 6
|
atbase |
|- ( Q e. A -> Q e. B ) |
138 |
115 137
|
syl |
|- ( ph -> Q e. B ) |
139 |
1 3 15 17 22 7 134 8 9 11 136 138
|
dihopellsm |
|- ( ph -> ( <. f , s >. e. ( ( I ` P ) .(+) ( I ` Q ) ) <-> E. g E. t E. h E. u ( ( <. g , t >. e. ( I ` P ) /\ <. h , u >. e. ( I ` Q ) ) /\ ( f = ( g o. h ) /\ s = ( t J u ) ) ) ) ) |
140 |
111 133 139
|
3imtr4d |
|- ( ph -> ( <. f , s >. e. ( I ` V ) -> <. f , s >. e. ( ( I ` P ) .(+) ( I ` Q ) ) ) ) |
141 |
24 140
|
relssdv |
|- ( ph -> ( I ` V ) C_ ( ( I ` P ) .(+) ( I ` Q ) ) ) |