Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemk1.b |
|- B = ( Base ` K ) |
2 |
|
cdlemk1.l |
|- .<_ = ( le ` K ) |
3 |
|
cdlemk1.j |
|- .\/ = ( join ` K ) |
4 |
|
cdlemk1.m |
|- ./\ = ( meet ` K ) |
5 |
|
cdlemk1.a |
|- A = ( Atoms ` K ) |
6 |
|
cdlemk1.h |
|- H = ( LHyp ` K ) |
7 |
|
cdlemk1.t |
|- T = ( ( LTrn ` K ) ` W ) |
8 |
|
cdlemk1.r |
|- R = ( ( trL ` K ) ` W ) |
9 |
|
cdlemk1.s |
|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) ) |
10 |
|
cdlemk1.o |
|- O = ( S ` D ) |
11 |
|
cdlemk1.u |
|- U = ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( O ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) ) |
12 |
|
simp11 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( K e. HL /\ W e. H ) ) |
13 |
|
simp21r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> G e. T ) |
14 |
|
simp22 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
15 |
2 3 5 6 7 8
|
trljat1 |
|- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( P .\/ ( R ` G ) ) = ( P .\/ ( G ` P ) ) ) |
16 |
12 13 14 15
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( P .\/ ( R ` G ) ) = ( P .\/ ( G ` P ) ) ) |
17 |
10
|
fveq1i |
|- ( O ` P ) = ( ( S ` D ) ` P ) |
18 |
17
|
a1i |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( O ` P ) = ( ( S ` D ) ` P ) ) |
19 |
|
simp13 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> D e. T ) |
20 |
6 7 8
|
trlcocnv |
|- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ D e. T ) -> ( R ` ( G o. `' D ) ) = ( R ` ( D o. `' G ) ) ) |
21 |
12 13 19 20
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( R ` ( G o. `' D ) ) = ( R ` ( D o. `' G ) ) ) |
22 |
18 21
|
oveq12d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) = ( ( ( S ` D ) ` P ) .\/ ( R ` ( D o. `' G ) ) ) ) |
23 |
16 22
|
oveq12d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) = ( ( P .\/ ( G ` P ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( D o. `' G ) ) ) ) ) |
24 |
|
simp23 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( R ` F ) = ( R ` N ) ) |
25 |
|
simp12 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> F e. T ) |
26 |
|
simp21l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> N e. T ) |
27 |
|
simp3r1 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( R ` D ) =/= ( R ` F ) ) |
28 |
|
simp3r2 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( R ` G ) =/= ( R ` D ) ) |
29 |
28
|
necomd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( R ` D ) =/= ( R ` G ) ) |
30 |
27 29
|
jca |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) ) |
31 |
|
simp3l1 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> F =/= ( _I |` B ) ) |
32 |
|
simp3l3 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> G =/= ( _I |` B ) ) |
33 |
|
simp3l2 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> D =/= ( _I |` B ) ) |
34 |
31 32 33
|
3jca |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) ) |
35 |
1 2 3 4 5 6 7 8 9 10 11
|
cdlemkuv2 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) /\ G e. T ) /\ ( F e. T /\ D e. T /\ N e. T ) /\ ( ( ( R ` D ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` G ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( U ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) |
36 |
12 24 13 25 19 26 30 34 14 35
|
syl333anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( U ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) |
37 |
26 19
|
jca |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( N e. T /\ D e. T ) ) |
38 |
|
simp3r3 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( R ` G ) =/= ( R ` F ) ) |
39 |
38 27
|
jca |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( R ` G ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) ) |
40 |
1 2 3 5 6 7 8 4 9
|
cdlemk12 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( N e. T /\ D e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ D =/= ( _I |` B ) ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( R ` D ) =/= ( R ` F ) ) /\ ( R ` G ) =/= ( R ` D ) ) ) -> ( ( S ` G ) ` P ) = ( ( P .\/ ( G ` P ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( D o. `' G ) ) ) ) ) |
41 |
12 25 13 37 14 24 34 39 28 40
|
syl333anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( S ` G ) ` P ) = ( ( P .\/ ( G ` P ) ) ./\ ( ( ( S ` D ) ` P ) .\/ ( R ` ( D o. `' G ) ) ) ) ) |
42 |
23 36 41
|
3eqtr4rd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ D e. T ) /\ ( ( N e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ D =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( ( R ` D ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` D ) /\ ( R ` G ) =/= ( R ` F ) ) ) ) -> ( ( S ` G ) ` P ) = ( ( U ` G ) ` P ) ) |