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 |
|
cdleme35.g |
|- G = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) |
9 |
|
oveq1 |
|- ( F = G -> ( F .\/ U ) = ( G .\/ U ) ) |
10 |
|
oveq2 |
|- ( F = G -> ( Q .\/ F ) = ( Q .\/ G ) ) |
11 |
10
|
oveq1d |
|- ( F = G -> ( ( Q .\/ F ) ./\ W ) = ( ( Q .\/ G ) ./\ W ) ) |
12 |
11
|
oveq2d |
|- ( F = G -> ( P .\/ ( ( Q .\/ F ) ./\ W ) ) = ( P .\/ ( ( Q .\/ G ) ./\ W ) ) ) |
13 |
9 12
|
oveq12d |
|- ( F = G -> ( ( F .\/ U ) ./\ ( P .\/ ( ( Q .\/ F ) ./\ W ) ) ) = ( ( G .\/ U ) ./\ ( P .\/ ( ( Q .\/ G ) ./\ W ) ) ) ) |
14 |
13
|
3ad2ant3 |
|- ( ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) -> ( ( F .\/ U ) ./\ ( P .\/ ( ( Q .\/ F ) ./\ W ) ) ) = ( ( G .\/ U ) ./\ ( P .\/ ( ( Q .\/ G ) ./\ W ) ) ) ) |
15 |
14
|
3ad2ant3 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) ) -> ( ( F .\/ U ) ./\ ( P .\/ ( ( Q .\/ F ) ./\ W ) ) ) = ( ( G .\/ U ) ./\ ( P .\/ ( ( Q .\/ G ) ./\ W ) ) ) ) |
16 |
|
simp1 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) |
17 |
|
simp21 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) ) -> P =/= Q ) |
18 |
|
simp22 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) ) -> ( R e. A /\ -. R .<_ W ) ) |
19 |
|
simp31 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) ) -> -. R .<_ ( P .\/ Q ) ) |
20 |
1 2 3 4 5 6 7
|
cdleme35g |
|- ( ( ( ( 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 ) |
21 |
16 17 18 19 20
|
syl121anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) ) -> ( ( F .\/ U ) ./\ ( P .\/ ( ( Q .\/ F ) ./\ W ) ) ) = R ) |
22 |
|
simp23 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) ) -> ( S e. A /\ -. S .<_ W ) ) |
23 |
|
simp32 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) ) -> -. S .<_ ( P .\/ Q ) ) |
24 |
1 2 3 4 5 6 8
|
cdleme35g |
|- ( ( ( ( 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 .\/ U ) ./\ ( P .\/ ( ( Q .\/ G ) ./\ W ) ) ) = S ) |
25 |
16 17 22 23 24
|
syl121anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) ) -> ( ( G .\/ U ) ./\ ( P .\/ ( ( Q .\/ G ) ./\ W ) ) ) = S ) |
26 |
15 21 25
|
3eqtr3d |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ F = G ) ) -> R = S ) |