Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme38.l |
|- .<_ = ( le ` K ) |
2 |
|
cdleme38.j |
|- .\/ = ( join ` K ) |
3 |
|
cdleme38.m |
|- ./\ = ( meet ` K ) |
4 |
|
cdleme38.a |
|- A = ( Atoms ` K ) |
5 |
|
cdleme38.h |
|- H = ( LHyp ` K ) |
6 |
|
cdleme38.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
7 |
|
cdleme38.e |
|- E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) |
8 |
|
cdleme38.d |
|- D = ( ( u .\/ U ) ./\ ( Q .\/ ( ( P .\/ u ) ./\ W ) ) ) |
9 |
|
cdleme38.v |
|- V = ( ( t .\/ E ) ./\ W ) |
10 |
|
cdleme38.x |
|- X = ( ( u .\/ D ) ./\ W ) |
11 |
|
cdleme38.f |
|- F = ( ( R .\/ V ) ./\ ( E .\/ ( ( t .\/ R ) ./\ W ) ) ) |
12 |
|
cdleme38.g |
|- G = ( ( S .\/ X ) ./\ ( D .\/ ( ( u .\/ S ) ./\ W ) ) ) |
13 |
|
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 ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) |
14 |
|
simp2 |
|- ( ( ( ( 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 ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) ) |
15 |
|
simp311 |
|- ( ( ( ( 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 ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> R .<_ ( P .\/ Q ) ) |
16 |
|
simp312 |
|- ( ( ( ( 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 ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> S .<_ ( P .\/ Q ) ) |
17 |
|
simp313 |
|- ( ( ( ( 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 ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> F = G ) |
18 |
15 16
|
jca |
|- ( ( ( ( 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 ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) ) ) |
19 |
|
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 ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) ) |
20 |
|
simp33 |
|- ( ( ( ( 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 ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) |
21 |
|
eqid |
|- ( ( S .\/ V ) ./\ ( E .\/ ( ( t .\/ S ) ./\ W ) ) ) = ( ( S .\/ V ) ./\ ( E .\/ ( ( t .\/ S ) ./\ W ) ) ) |
22 |
1 2 3 4 5 6 7 8 9 10 21 12
|
cdleme37m |
|- ( ( ( ( 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 ) ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> ( ( S .\/ V ) ./\ ( E .\/ ( ( t .\/ S ) ./\ W ) ) ) = G ) |
23 |
13 14 18 19 20 22
|
syl113anc |
|- ( ( ( ( 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 ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> ( ( S .\/ V ) ./\ ( E .\/ ( ( t .\/ S ) ./\ W ) ) ) = G ) |
24 |
17 23
|
eqtr4d |
|- ( ( ( ( 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 ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> F = ( ( S .\/ V ) ./\ ( E .\/ ( ( t .\/ S ) ./\ W ) ) ) ) |
25 |
15 16 24
|
3jca |
|- ( ( ( ( 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 ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ F = ( ( S .\/ V ) ./\ ( E .\/ ( ( t .\/ S ) ./\ W ) ) ) ) ) |
26 |
|
eqid |
|- ( Base ` K ) = ( Base ` K ) |
27 |
26 1 2 3 4 5 6 7 9 11 21
|
cdleme36m |
|- ( ( ( ( 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 = ( ( S .\/ V ) ./\ ( E .\/ ( ( t .\/ S ) ./\ W ) ) ) ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) ) ) -> R = S ) |
28 |
13 14 25 19 27
|
syl112anc |
|- ( ( ( ( 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 ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> R = S ) |