Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemef47.b |
|- B = ( Base ` K ) |
2 |
|
cdlemef47.l |
|- .<_ = ( le ` K ) |
3 |
|
cdlemef47.j |
|- .\/ = ( join ` K ) |
4 |
|
cdlemef47.m |
|- ./\ = ( meet ` K ) |
5 |
|
cdlemef47.a |
|- A = ( Atoms ` K ) |
6 |
|
cdlemef47.h |
|- H = ( LHyp ` K ) |
7 |
|
cdlemef47.v |
|- V = ( ( Q .\/ P ) ./\ W ) |
8 |
|
cdlemef47.n |
|- N = ( ( v .\/ V ) ./\ ( P .\/ ( ( Q .\/ v ) ./\ W ) ) ) |
9 |
|
cdlemefs47.o |
|- O = ( ( Q .\/ P ) ./\ ( N .\/ ( ( u .\/ v ) ./\ W ) ) ) |
10 |
|
cdlemef47.g |
|- G = ( a e. B |-> if ( ( Q =/= P /\ -. a .<_ W ) , ( iota_ c e. B A. u e. A ( ( -. u .<_ W /\ ( u .\/ ( a ./\ W ) ) = a ) -> c = ( if ( u .<_ ( Q .\/ P ) , ( iota_ b e. B A. v e. A ( ( -. v .<_ W /\ -. v .<_ ( Q .\/ P ) ) -> b = O ) ) , [_ u / v ]_ N ) .\/ ( a ./\ W ) ) ) ) , a ) ) |
11 |
3 5
|
cdleme46f2g2 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( Q =/= P /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( Q .\/ P ) ) ) |
12 |
1 2 3 4 5 6 7 8 10
|
cdlemefr45 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) /\ ( Q =/= P /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( Q .\/ P ) ) -> ( G ` S ) = [_ S / v ]_ N ) |
13 |
11 12
|
syl |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( G ` S ) = [_ S / v ]_ N ) |