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 |
|
simp13 |
|- ( ( ( ( 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 ) ) ) -> G e. T ) |
13 |
1 2 3 5 6 7 8 4 11
|
cdlemksv |
|- ( G e. T -> ( U ` G ) = ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) ) |
14 |
12 13
|
syl |
|- ( ( ( ( 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 ) = ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) ) |
15 |
14
|
eqcomd |
|- ( ( ( ( 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 ) ) ) -> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) = ( U ` G ) ) |
16 |
1 2 3 4 5 6 7 8 9 10 11
|
cdlemkuel |
|- ( ( ( ( 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 ) e. T ) |
17 |
|
simp11l |
|- ( ( ( ( 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 ) ) ) -> K e. HL ) |
18 |
|
simp11r |
|- ( ( ( ( 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 ) ) ) -> W e. H ) |
19 |
|
simp33 |
|- ( ( ( ( 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 ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
20 |
1 2 3 4 5 6 7 8 9 10
|
cdlemk16a |
|- ( ( ( ( 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 ) ) ) -> ( ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) e. A /\ -. ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ W ) ) |
21 |
2 5 6 7
|
cdleme |
|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) e. A /\ -. ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) .<_ W ) ) -> E! j e. T ( j ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) |
22 |
17 18 19 20 21
|
syl211anc |
|- ( ( ( ( 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 ) ) ) -> E! j e. T ( j ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) |
23 |
|
nfcv |
|- F/_ j T |
24 |
|
nfriota1 |
|- F/_ j ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( O ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) |
25 |
23 24
|
nfmpt |
|- F/_ j ( e e. T |-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` e ) ) ./\ ( ( O ` P ) .\/ ( R ` ( e o. `' D ) ) ) ) ) ) |
26 |
11 25
|
nfcxfr |
|- F/_ j U |
27 |
|
nfcv |
|- F/_ j G |
28 |
26 27
|
nffv |
|- F/_ j ( U ` G ) |
29 |
|
nfcv |
|- F/_ j P |
30 |
28 29
|
nffv |
|- F/_ j ( ( U ` G ) ` P ) |
31 |
30
|
nfeq1 |
|- F/ j ( ( U ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) |
32 |
|
fveq1 |
|- ( j = ( U ` G ) -> ( j ` P ) = ( ( U ` G ) ` P ) ) |
33 |
32
|
eqeq1d |
|- ( j = ( U ` G ) -> ( ( j ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) <-> ( ( U ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) ) |
34 |
28 31 33
|
riota2f |
|- ( ( ( U ` G ) e. T /\ E! j e. T ( j ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) -> ( ( ( U ` G ) ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) <-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) = ( U ` G ) ) ) |
35 |
16 22 34
|
syl2anc |
|- ( ( ( ( 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 ) ) ) ) <-> ( iota_ j e. T ( j ` P ) = ( ( P .\/ ( R ` G ) ) ./\ ( ( O ` P ) .\/ ( R ` ( G o. `' D ) ) ) ) ) = ( U ` G ) ) ) |
36 |
15 35
|
mpbird |
|- ( ( ( ( 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 ) ) ) ) ) |