Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemk.b |
|- B = ( Base ` K ) |
2 |
|
cdlemk.l |
|- .<_ = ( le ` K ) |
3 |
|
cdlemk.j |
|- .\/ = ( join ` K ) |
4 |
|
cdlemk.a |
|- A = ( Atoms ` K ) |
5 |
|
cdlemk.h |
|- H = ( LHyp ` K ) |
6 |
|
cdlemk.t |
|- T = ( ( LTrn ` K ) ` W ) |
7 |
|
cdlemk.r |
|- R = ( ( trL ` K ) ` W ) |
8 |
|
cdlemk.m |
|- ./\ = ( meet ` K ) |
9 |
|
cdlemk.s |
|- S = ( f e. T |-> ( iota_ i e. T ( i ` P ) = ( ( P .\/ ( R ` f ) ) ./\ ( ( N ` P ) .\/ ( R ` ( f o. `' F ) ) ) ) ) ) |
10 |
|
simp11 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( K e. HL /\ W e. H ) ) |
11 |
1 2 3 4 5 6 7 8 9
|
cdlemksel |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( S ` G ) e. T ) |
12 |
|
simp22l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> P e. A ) |
13 |
2 4 5 6
|
ltrnat |
|- ( ( ( K e. HL /\ W e. H ) /\ ( S ` G ) e. T /\ P e. A ) -> ( ( S ` G ) ` P ) e. A ) |
14 |
10 11 12 13
|
syl3anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( N e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( ( S ` G ) ` P ) e. A ) |