Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme41.b |
|- B = ( Base ` K ) |
2 |
|
cdleme41.l |
|- .<_ = ( le ` K ) |
3 |
|
cdleme41.j |
|- .\/ = ( join ` K ) |
4 |
|
cdleme41.m |
|- ./\ = ( meet ` K ) |
5 |
|
cdleme41.a |
|- A = ( Atoms ` K ) |
6 |
|
cdleme41.h |
|- H = ( LHyp ` K ) |
7 |
|
cdleme41.u |
|- U = ( ( P .\/ Q ) ./\ W ) |
8 |
|
cdleme41.d |
|- D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) ) |
9 |
|
cdleme41.e |
|- E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) ) |
10 |
|
cdleme41.g |
|- G = ( ( P .\/ Q ) ./\ ( E .\/ ( ( s .\/ t ) ./\ W ) ) ) |
11 |
|
cdleme41.i |
|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = G ) ) |
12 |
|
cdleme41.n |
|- N = if ( s .<_ ( P .\/ Q ) , I , D ) |
13 |
|
cdleme41.o |
|- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) ) |
14 |
|
cdleme41.f |
|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) ) |
15 |
|
cdleme34e.v |
|- V = ( ( R .\/ S ) ./\ W ) |
16 |
|
simpl1l |
|- ( ( ( ( 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 ) ) ) -> K e. HL ) |
17 |
|
simpr2 |
|- ( ( ( ( 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 e. A /\ -. R .<_ W ) ) |
18 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14
|
cdleme32fvaw |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( ( F ` R ) e. A /\ -. ( F ` R ) .<_ W ) ) |
19 |
17 18
|
syldan |
|- ( ( ( ( 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 ) ) ) -> ( ( F ` R ) e. A /\ -. ( F ` R ) .<_ W ) ) |
20 |
19
|
simpld |
|- ( ( ( ( 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 ) ) ) -> ( F ` R ) e. A ) |
21 |
3 5
|
hlatjidm |
|- ( ( K e. HL /\ ( F ` R ) e. A ) -> ( ( F ` R ) .\/ ( F ` R ) ) = ( F ` R ) ) |
22 |
16 20 21
|
syl2anc |
|- ( ( ( ( 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 ) ) ) -> ( ( F ` R ) .\/ ( F ` R ) ) = ( F ` R ) ) |
23 |
|
fveq2 |
|- ( R = S -> ( F ` R ) = ( F ` S ) ) |
24 |
23
|
oveq2d |
|- ( R = S -> ( ( F ` R ) .\/ ( F ` R ) ) = ( ( F ` R ) .\/ ( F ` S ) ) ) |
25 |
22 24
|
sylan9req |
|- ( ( ( ( ( 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 = S ) -> ( F ` R ) = ( ( F ` R ) .\/ ( F ` S ) ) ) |
26 |
|
simpr2l |
|- ( ( ( ( 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 e. A ) |
27 |
3 5
|
hlatjidm |
|- ( ( K e. HL /\ R e. A ) -> ( R .\/ R ) = R ) |
28 |
16 26 27
|
syl2anc |
|- ( ( ( ( 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 .\/ R ) = R ) |
29 |
28
|
oveq1d |
|- ( ( ( ( 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 .\/ R ) ./\ W ) = ( R ./\ W ) ) |
30 |
|
simpl1 |
|- ( ( ( ( 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 ) ) ) -> ( K e. HL /\ W e. H ) ) |
31 |
|
eqid |
|- ( 0. ` K ) = ( 0. ` K ) |
32 |
2 4 31 5 6
|
lhpmat |
|- ( ( ( K e. HL /\ W e. H ) /\ ( R e. A /\ -. R .<_ W ) ) -> ( R ./\ W ) = ( 0. ` K ) ) |
33 |
30 17 32
|
syl2anc |
|- ( ( ( ( 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 ./\ W ) = ( 0. ` K ) ) |
34 |
29 33
|
eqtrd |
|- ( ( ( ( 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 .\/ R ) ./\ W ) = ( 0. ` K ) ) |
35 |
34
|
oveq2d |
|- ( ( ( ( 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 ) ) ) -> ( ( F ` R ) .\/ ( ( R .\/ R ) ./\ W ) ) = ( ( F ` R ) .\/ ( 0. ` K ) ) ) |
36 |
|
hlol |
|- ( K e. HL -> K e. OL ) |
37 |
16 36
|
syl |
|- ( ( ( ( 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 ) ) ) -> K e. OL ) |
38 |
1 5
|
atbase |
|- ( ( F ` R ) e. A -> ( F ` R ) e. B ) |
39 |
20 38
|
syl |
|- ( ( ( ( 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 ) ) ) -> ( F ` R ) e. B ) |
40 |
1 3 31
|
olj01 |
|- ( ( K e. OL /\ ( F ` R ) e. B ) -> ( ( F ` R ) .\/ ( 0. ` K ) ) = ( F ` R ) ) |
41 |
37 39 40
|
syl2anc |
|- ( ( ( ( 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 ) ) ) -> ( ( F ` R ) .\/ ( 0. ` K ) ) = ( F ` R ) ) |
42 |
35 41
|
eqtrd |
|- ( ( ( ( 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 ) ) ) -> ( ( F ` R ) .\/ ( ( R .\/ R ) ./\ W ) ) = ( F ` R ) ) |
43 |
|
oveq2 |
|- ( R = S -> ( R .\/ R ) = ( R .\/ S ) ) |
44 |
43
|
oveq1d |
|- ( R = S -> ( ( R .\/ R ) ./\ W ) = ( ( R .\/ S ) ./\ W ) ) |
45 |
44 15
|
eqtr4di |
|- ( R = S -> ( ( R .\/ R ) ./\ W ) = V ) |
46 |
45
|
oveq2d |
|- ( R = S -> ( ( F ` R ) .\/ ( ( R .\/ R ) ./\ W ) ) = ( ( F ` R ) .\/ V ) ) |
47 |
42 46
|
sylan9req |
|- ( ( ( ( ( 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 = S ) -> ( F ` R ) = ( ( F ` R ) .\/ V ) ) |
48 |
25 47
|
eqtr3d |
|- ( ( ( ( ( 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 = S ) -> ( ( F ` R ) .\/ ( F ` S ) ) = ( ( F ` R ) .\/ V ) ) |
49 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
|
cdleme42k |
|- ( ( ( ( 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 =/= S ) -> ( ( F ` R ) .\/ ( F ` S ) ) = ( ( F ` R ) .\/ V ) ) |
50 |
49
|
3expa |
|- ( ( ( ( ( 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 =/= S ) -> ( ( F ` R ) .\/ ( F ` S ) ) = ( ( F ` R ) .\/ V ) ) |
51 |
48 50
|
pm2.61dane |
|- ( ( ( ( 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 ) ) ) -> ( ( F ` R ) .\/ ( F ` S ) ) = ( ( F ` R ) .\/ V ) ) |