Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemk2.b |
|- B = ( Base ` K ) |
2 |
|
cdlemk2.l |
|- .<_ = ( le ` K ) |
3 |
|
cdlemk2.j |
|- .\/ = ( join ` K ) |
4 |
|
cdlemk2.m |
|- ./\ = ( meet ` K ) |
5 |
|
cdlemk2.a |
|- A = ( Atoms ` K ) |
6 |
|
cdlemk2.h |
|- H = ( LHyp ` K ) |
7 |
|
cdlemk2.t |
|- T = ( ( LTrn ` K ) ` W ) |
8 |
|
cdlemk2.r |
|- R = ( ( trL ` K ) ` W ) |
9 |
|
cdlemk2.s |
|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) ) |
10 |
|
cdlemk2.q |
|- Q = ( S ` C ) |
11 |
|
cdlemk2.v |
|- V = ( d e. T |-> ( iota_ k e. T ( k ` P ) = ( ( P .\/ ( R ` d ) ) ./\ ( ( Q ` P ) .\/ ( R ` ( d o. `' C ) ) ) ) ) ) |
12 |
|
simp11 |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( ( R ` G ) =/= ( R ` X ) /\ F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> K e. HL ) |
13 |
|
simp12 |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( ( R ` G ) =/= ( R ` X ) /\ F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> W e. H ) |
14 |
12 13
|
jca |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( ( R ` G ) =/= ( R ` X ) /\ F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( K e. HL /\ W e. H ) ) |
15 |
|
simp211 |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( ( R ` G ) =/= ( R ` X ) /\ F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F e. T ) |
16 |
|
simp212 |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( ( R ` G ) =/= ( R ` X ) /\ F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> C e. T ) |
17 |
|
simp213 |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( ( R ` G ) =/= ( R ` X ) /\ F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> N e. T ) |
18 |
|
simp22l |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( ( R ` G ) =/= ( R ` X ) /\ F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> G e. T ) |
19 |
|
simp23l |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( ( R ` G ) =/= ( R ` X ) /\ F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> X e. T ) |
20 |
17 18 19
|
3jca |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( ( R ` G ) =/= ( R ` X ) /\ F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( N e. T /\ G e. T /\ X e. T ) ) |
21 |
|
simp33 |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( ( R ` G ) =/= ( R ` X ) /\ F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
22 |
|
simp13 |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( ( R ` G ) =/= ( R ` X ) /\ F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` F ) = ( R ` N ) ) |
23 |
|
simp322 |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( ( R ` G ) =/= ( R ` X ) /\ F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F =/= ( _I |` B ) ) |
24 |
|
simp323 |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( ( R ` G ) =/= ( R ` X ) /\ F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> C =/= ( _I |` B ) ) |
25 |
|
simp22r |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( ( R ` G ) =/= ( R ` X ) /\ F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> G =/= ( _I |` B ) ) |
26 |
23 24 25
|
3jca |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( ( R ` G ) =/= ( R ` X ) /\ F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) ) |
27 |
|
simp23r |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( ( R ` G ) =/= ( R ` X ) /\ F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> X =/= ( _I |` B ) ) |
28 |
|
simp321 |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( ( R ` G ) =/= ( R ` X ) /\ F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` G ) =/= ( R ` X ) ) |
29 |
27 28
|
jca |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( ( R ` G ) =/= ( R ` X ) /\ F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( X =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` X ) ) ) |
30 |
|
simp31 |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( ( R ` G ) =/= ( R ` X ) /\ F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) ) |
31 |
1 2 3 4 5 6 7 8 9 10 11
|
cdlemk12u |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ C e. T ) /\ ( ( N e. T /\ G e. T /\ X e. T ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F =/= ( _I |` B ) /\ C =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( X =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` X ) ) /\ ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) ) ) -> ( ( V ` G ) ` P ) = ( ( P .\/ ( G ` P ) ) ./\ ( ( ( V ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) ) ) |
32 |
14 15 16 20 21 22 26 29 30 31
|
syl333anc |
|- ( ( ( K e. HL /\ W e. H /\ ( R ` F ) = ( R ` N ) ) /\ ( ( F e. T /\ C e. T /\ N e. T ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ ( X e. T /\ X =/= ( _I |` B ) ) ) /\ ( ( ( R ` C ) =/= ( R ` F ) /\ ( R ` G ) =/= ( R ` C ) /\ ( R ` X ) =/= ( R ` C ) ) /\ ( ( R ` G ) =/= ( R ` X ) /\ F =/= ( _I |` B ) /\ C =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( V ` G ) ` P ) = ( ( P .\/ ( G ` P ) ) ./\ ( ( ( V ` X ) ` P ) .\/ ( R ` ( X o. `' G ) ) ) ) ) |