Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemk6.b |
|- B = ( Base ` K ) |
2 |
|
cdlemk6.j |
|- .\/ = ( join ` K ) |
3 |
|
cdlemk6.m |
|- ./\ = ( meet ` K ) |
4 |
|
cdlemk6.o |
|- ._|_ = ( oc ` K ) |
5 |
|
cdlemk6.a |
|- A = ( Atoms ` K ) |
6 |
|
cdlemk6.h |
|- H = ( LHyp ` K ) |
7 |
|
cdlemk6.t |
|- T = ( ( LTrn ` K ) ` W ) |
8 |
|
cdlemk6.r |
|- R = ( ( trL ` K ) ` W ) |
9 |
|
cdlemk6.p |
|- P = ( ._|_ ` W ) |
10 |
|
cdlemk6.z |
|- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) ) |
11 |
|
cdlemk6.y |
|- Y = ( ( P .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) ) |
12 |
|
cdlemk6.x |
|- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` P ) = Y ) ) |
13 |
|
cdlemk6.u |
|- U = ( g e. T |-> if ( F = N , g , X ) ) |
14 |
|
3simpb |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( R ` F ) = ( R ` N ) ) -> ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) ) |
15 |
|
simp2 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( R ` F ) = ( R ` N ) ) -> ( F e. T /\ N e. T ) ) |
16 |
|
eqid |
|- ( le ` K ) = ( le ` K ) |
17 |
16 4 5 6
|
lhpocnel |
|- ( ( K e. HL /\ W e. H ) -> ( ( ._|_ ` W ) e. A /\ -. ( ._|_ ` W ) ( le ` K ) W ) ) |
18 |
17
|
3ad2ant1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( R ` F ) = ( R ` N ) ) -> ( ( ._|_ ` W ) e. A /\ -. ( ._|_ ` W ) ( le ` K ) W ) ) |
19 |
9
|
eleq1i |
|- ( P e. A <-> ( ._|_ ` W ) e. A ) |
20 |
9
|
breq1i |
|- ( P ( le ` K ) W <-> ( ._|_ ` W ) ( le ` K ) W ) |
21 |
20
|
notbii |
|- ( -. P ( le ` K ) W <-> -. ( ._|_ ` W ) ( le ` K ) W ) |
22 |
19 21
|
anbi12i |
|- ( ( P e. A /\ -. P ( le ` K ) W ) <-> ( ( ._|_ ` W ) e. A /\ -. ( ._|_ ` W ) ( le ` K ) W ) ) |
23 |
18 22
|
sylibr |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( R ` F ) = ( R ` N ) ) -> ( P e. A /\ -. P ( le ` K ) W ) ) |
24 |
1 16 2 3 5 6 7 8 10 11 12 13
|
cdlemk19u |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( R ` F ) = ( R ` N ) ) /\ ( F e. T /\ N e. T ) /\ ( P e. A /\ -. P ( le ` K ) W ) ) -> ( U ` F ) = N ) |
25 |
14 15 23 24
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ N e. T ) /\ ( R ` F ) = ( R ` N ) ) -> ( U ` F ) = N ) |