Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme17.l |
|- .<_ = ( le ` K ) |
2 |
|
cdleme17.j |
|- .\/ = ( join ` K ) |
3 |
|
cdleme17.m |
|- ./\ = ( meet ` K ) |
4 |
|
cdleme17.a |
|- A = ( Atoms ` K ) |
5 |
|
cdleme17.h |
|- H = ( LHyp ` K ) |
6 |
|
cdleme17.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
7 |
|
cdleme17.f |
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) |
8 |
|
cdleme17.g |
|- G = ( ( P .\/ Q ) ./\ ( F .\/ ( ( P .\/ S ) ./\ W ) ) ) |
9 |
|
cdleme17.c |
|- C = ( ( P .\/ S ) ./\ W ) |
10 |
1 2 3 4 5 6 7 8 9
|
cdleme7a |
|- G = ( ( P .\/ Q ) ./\ ( F .\/ C ) ) |
11 |
1 2 3 4 5 6 7 9
|
cdleme9 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( F .\/ C ) = ( Q .\/ C ) ) |
12 |
11
|
oveq2d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( P .\/ Q ) ./\ ( F .\/ C ) ) = ( ( P .\/ Q ) ./\ ( Q .\/ C ) ) ) |
13 |
10 12
|
syl5eq |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> G = ( ( P .\/ Q ) ./\ ( Q .\/ C ) ) ) |