Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme21.l |
|- .<_ = ( le ` K ) |
2 |
|
cdleme21.j |
|- .\/ = ( join ` K ) |
3 |
|
cdleme21.m |
|- ./\ = ( meet ` K ) |
4 |
|
cdleme21.a |
|- A = ( Atoms ` K ) |
5 |
|
cdleme21.h |
|- H = ( LHyp ` K ) |
6 |
|
cdleme21.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
7 |
|
cdleme21.f |
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) |
8 |
|
cdleme21g.g |
|- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) ) |
9 |
|
cdleme21g.d |
|- D = ( ( R .\/ S ) ./\ W ) |
10 |
|
cdleme21g.y |
|- Y = ( ( R .\/ T ) ./\ W ) |
11 |
|
cdleme21g.n |
|- N = ( ( P .\/ Q ) ./\ ( F .\/ D ) ) |
12 |
|
cdleme21g.o |
|- O = ( ( P .\/ Q ) ./\ ( G .\/ Y ) ) |
13 |
|
oveq1 |
|- ( S = T -> ( S .\/ U ) = ( T .\/ U ) ) |
14 |
|
oveq2 |
|- ( S = T -> ( P .\/ S ) = ( P .\/ T ) ) |
15 |
14
|
oveq1d |
|- ( S = T -> ( ( P .\/ S ) ./\ W ) = ( ( P .\/ T ) ./\ W ) ) |
16 |
15
|
oveq2d |
|- ( S = T -> ( Q .\/ ( ( P .\/ S ) ./\ W ) ) = ( Q .\/ ( ( P .\/ T ) ./\ W ) ) ) |
17 |
13 16
|
oveq12d |
|- ( S = T -> ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) ) ) |
18 |
17 7 8
|
3eqtr4g |
|- ( S = T -> F = G ) |
19 |
|
oveq2 |
|- ( S = T -> ( R .\/ S ) = ( R .\/ T ) ) |
20 |
19
|
oveq1d |
|- ( S = T -> ( ( R .\/ S ) ./\ W ) = ( ( R .\/ T ) ./\ W ) ) |
21 |
20 9 10
|
3eqtr4g |
|- ( S = T -> D = Y ) |
22 |
18 21
|
oveq12d |
|- ( S = T -> ( F .\/ D ) = ( G .\/ Y ) ) |
23 |
22
|
oveq2d |
|- ( S = T -> ( ( P .\/ Q ) ./\ ( F .\/ D ) ) = ( ( P .\/ Q ) ./\ ( G .\/ Y ) ) ) |
24 |
23 11 12
|
3eqtr4g |
|- ( S = T -> N = O ) |
25 |
24
|
eqeq1d |
|- ( S = T -> ( N = O <-> O = O ) ) |
26 |
|
simpl11 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ S =/= T ) -> ( K e. HL /\ W e. H ) ) |
27 |
|
simpl12 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ S =/= T ) -> ( P e. A /\ -. P .<_ W ) ) |
28 |
|
simpl13 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ S =/= T ) -> ( Q e. A /\ -. Q .<_ W ) ) |
29 |
|
simpl21 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ S =/= T ) -> ( R e. A /\ -. R .<_ W ) ) |
30 |
|
simpl22 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ S =/= T ) -> ( S e. A /\ -. S .<_ W ) ) |
31 |
|
simpl23 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ S =/= T ) -> ( T e. A /\ -. T .<_ W ) ) |
32 |
|
simpl3l |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ S =/= T ) -> P =/= Q ) |
33 |
|
simpr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ S =/= T ) -> S =/= T ) |
34 |
32 33
|
jca |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ S =/= T ) -> ( P =/= Q /\ S =/= T ) ) |
35 |
|
simpl3r |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ S =/= T ) -> ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) |
36 |
1 2 3 4 5 6 7 8 9 10 11 12
|
cdleme21 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) -> N = O ) |
37 |
26 27 28 29 30 31 34 35 36
|
syl332anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ S =/= T ) -> N = O ) |
38 |
|
eqidd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) -> O = O ) |
39 |
25 37 38
|
pm2.61ne |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) -> N = O ) |