Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme43.b |
|- B = ( Base ` K ) |
2 |
|
cdleme43.l |
|- .<_ = ( le ` K ) |
3 |
|
cdleme43.j |
|- .\/ = ( join ` K ) |
4 |
|
cdleme43.m |
|- ./\ = ( meet ` K ) |
5 |
|
cdleme43.a |
|- A = ( Atoms ` K ) |
6 |
|
cdleme43.h |
|- H = ( LHyp ` K ) |
7 |
|
cdleme43.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
8 |
|
cdleme43.x |
|- X = ( ( Q .\/ P ) ./\ W ) |
9 |
|
cdleme43.c |
|- C = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) |
10 |
|
cdleme43.f |
|- Z = ( ( P .\/ Q ) ./\ ( C .\/ ( ( R .\/ S ) ./\ W ) ) ) |
11 |
|
cdleme43.d |
|- D = ( ( S .\/ X ) ./\ ( P .\/ ( ( Q .\/ S ) ./\ W ) ) ) |
12 |
|
cdleme43.g |
|- G = ( ( Q .\/ P ) ./\ ( D .\/ ( ( Z .\/ S ) ./\ W ) ) ) |
13 |
|
cdleme43.e |
|- E = ( ( D .\/ U ) ./\ ( Q .\/ ( ( P .\/ D ) ./\ W ) ) ) |
14 |
|
cdleme43.v |
|- V = ( ( Z .\/ S ) ./\ W ) |
15 |
|
cdleme43.y |
|- Y = ( ( R .\/ D ) ./\ W ) |
16 |
|
simp11 |
|- ( ( ( ( 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 ) ) -> ( K e. HL /\ W e. H ) ) |
17 |
|
simp13 |
|- ( ( ( ( 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 ) ) -> ( Q e. A /\ -. Q .<_ W ) ) |
18 |
|
simp12 |
|- ( ( ( ( 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 ) ) -> ( P e. A /\ -. P .<_ W ) ) |
19 |
|
simp2r |
|- ( ( ( ( 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 ) ) -> ( S e. A /\ -. S .<_ W ) ) |
20 |
|
simp2l |
|- ( ( ( ( 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 ) ) -> P =/= Q ) |
21 |
20
|
necomd |
|- ( ( ( ( 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 ) ) -> Q =/= P ) |
22 |
|
simp3 |
|- ( ( ( ( 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 ) ) -> -. S .<_ ( P .\/ Q ) ) |
23 |
|
simp11l |
|- ( ( ( ( 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 ) ) -> K e. HL ) |
24 |
|
simp12l |
|- ( ( ( ( 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 ) ) -> P e. A ) |
25 |
|
simp13l |
|- ( ( ( ( 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 ) ) -> Q e. A ) |
26 |
3 5
|
hlatjcom |
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) ) |
27 |
23 24 25 26
|
syl3anc |
|- ( ( ( ( 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 ) ) -> ( P .\/ Q ) = ( Q .\/ P ) ) |
28 |
27
|
breq2d |
|- ( ( ( ( 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 ) ) -> ( S .<_ ( P .\/ Q ) <-> S .<_ ( Q .\/ P ) ) ) |
29 |
22 28
|
mtbid |
|- ( ( ( ( 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 ) ) -> -. S .<_ ( Q .\/ P ) ) |
30 |
2 3 4 5 6 8 11
|
cdleme3fa |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( P e. A /\ -. P .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( Q =/= P /\ -. S .<_ ( Q .\/ P ) ) ) -> D e. A ) |
31 |
16 17 18 19 21 29 30
|
syl132anc |
|- ( ( ( ( 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 ) ) -> D e. A ) |
32 |
2 3 4 5 6 8 11
|
cdleme3 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( Q e. A /\ -. Q .<_ W ) /\ ( P e. A /\ -. P .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( Q =/= P /\ -. S .<_ ( Q .\/ P ) ) ) -> -. D .<_ W ) |
33 |
16 17 18 19 21 29 32
|
syl132anc |
|- ( ( ( ( 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 ) ) -> -. D .<_ W ) |
34 |
31 33
|
jca |
|- ( ( ( ( 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 ) ) -> ( D e. A /\ -. D .<_ W ) ) |