Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme35.l |
|- .<_ = ( le ` K ) |
2 |
|
cdleme35.j |
|- .\/ = ( join ` K ) |
3 |
|
cdleme35.m |
|- ./\ = ( meet ` K ) |
4 |
|
cdleme35.a |
|- A = ( Atoms ` K ) |
5 |
|
cdleme35.h |
|- H = ( LHyp ` K ) |
6 |
|
cdleme35.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
7 |
|
cdleme35.f |
|- F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) |
8 |
1 2 3 4 5 6 7
|
cdleme35a |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( F .\/ U ) = ( R .\/ U ) ) |
9 |
1 2 3 4 5 6 7
|
cdleme35e |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( P .\/ ( ( Q .\/ F ) ./\ W ) ) = ( P .\/ R ) ) |
10 |
8 9
|
oveq12d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( F .\/ U ) ./\ ( P .\/ ( ( Q .\/ F ) ./\ W ) ) ) = ( ( R .\/ U ) ./\ ( P .\/ R ) ) ) |
11 |
1 2 3 4 5 6 7
|
cdleme35f |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( R .\/ U ) ./\ ( P .\/ R ) ) = R ) |
12 |
10 11
|
eqtrd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( F .\/ U ) ./\ ( P .\/ ( ( Q .\/ F ) ./\ W ) ) ) = R ) |