Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemk4.b |
|- B = ( Base ` K ) |
2 |
|
cdlemk4.l |
|- .<_ = ( le ` K ) |
3 |
|
cdlemk4.j |
|- .\/ = ( join ` K ) |
4 |
|
cdlemk4.m |
|- ./\ = ( meet ` K ) |
5 |
|
cdlemk4.a |
|- A = ( Atoms ` K ) |
6 |
|
cdlemk4.h |
|- H = ( LHyp ` K ) |
7 |
|
cdlemk4.t |
|- T = ( ( LTrn ` K ) ` W ) |
8 |
|
cdlemk4.r |
|- R = ( ( trL ` K ) ` W ) |
9 |
|
cdlemk4.z |
|- Z = ( ( P .\/ ( R ` b ) ) ./\ ( ( N ` P ) .\/ ( R ` ( b o. `' F ) ) ) ) |
10 |
|
cdlemk4.y |
|- Y = ( ( P .\/ ( R ` G ) ) ./\ ( Z .\/ ( R ` ( G o. `' b ) ) ) ) |
11 |
|
cdlemk4.x |
|- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` F ) /\ ( R ` b ) =/= ( R ` G ) ) -> ( z ` P ) = Y ) ) |
12 |
|
simp1l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> K e. HL ) |
13 |
|
simp3ll |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> P e. A ) |
14 |
|
simp1 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( K e. HL /\ W e. H ) ) |
15 |
|
simp22l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> G e. T ) |
16 |
|
simp22r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> G =/= ( _I |` B ) ) |
17 |
1 5 6 7 8
|
trlnidat |
|- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ G =/= ( _I |` B ) ) -> ( R ` G ) e. A ) |
18 |
14 15 16 17
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( R ` G ) e. A ) |
19 |
2 3 5
|
hlatlej1 |
|- ( ( K e. HL /\ P e. A /\ ( R ` G ) e. A ) -> P .<_ ( P .\/ ( R ` G ) ) ) |
20 |
12 13 18 19
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> P .<_ ( P .\/ ( R ` G ) ) ) |
21 |
1 2 3 4 5 6 7 8 9 10 11
|
cdlemk38 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( X ` P ) .<_ ( P .\/ ( R ` G ) ) ) |
22 |
12
|
hllatd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> K e. Lat ) |
23 |
1 5
|
atbase |
|- ( P e. A -> P e. B ) |
24 |
13 23
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> P e. B ) |
25 |
1 2 3 4 5 6 7 8 9 10 11
|
cdlemk35 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> X e. T ) |
26 |
2 5 6 7
|
ltrnat |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. T /\ P e. A ) -> ( X ` P ) e. A ) |
27 |
14 25 13 26
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( X ` P ) e. A ) |
28 |
1 5
|
atbase |
|- ( ( X ` P ) e. A -> ( X ` P ) e. B ) |
29 |
27 28
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( X ` P ) e. B ) |
30 |
1 3 5
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ ( R ` G ) e. A ) -> ( P .\/ ( R ` G ) ) e. B ) |
31 |
12 13 18 30
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( P .\/ ( R ` G ) ) e. B ) |
32 |
1 2 3
|
latjle12 |
|- ( ( K e. Lat /\ ( P e. B /\ ( X ` P ) e. B /\ ( P .\/ ( R ` G ) ) e. B ) ) -> ( ( P .<_ ( P .\/ ( R ` G ) ) /\ ( X ` P ) .<_ ( P .\/ ( R ` G ) ) ) <-> ( P .\/ ( X ` P ) ) .<_ ( P .\/ ( R ` G ) ) ) ) |
33 |
22 24 29 31 32
|
syl13anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( ( P .<_ ( P .\/ ( R ` G ) ) /\ ( X ` P ) .<_ ( P .\/ ( R ` G ) ) ) <-> ( P .\/ ( X ` P ) ) .<_ ( P .\/ ( R ` G ) ) ) ) |
34 |
20 21 33
|
mpbi2and |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( P .\/ ( X ` P ) ) .<_ ( P .\/ ( R ` G ) ) ) |
35 |
1 3 5
|
hlatjcl |
|- ( ( K e. HL /\ P e. A /\ ( X ` P ) e. A ) -> ( P .\/ ( X ` P ) ) e. B ) |
36 |
12 13 27 35
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( P .\/ ( X ` P ) ) e. B ) |
37 |
|
simp1r |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> W e. H ) |
38 |
1 6
|
lhpbase |
|- ( W e. H -> W e. B ) |
39 |
37 38
|
syl |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> W e. B ) |
40 |
1 2 4
|
latmlem1 |
|- ( ( K e. Lat /\ ( ( P .\/ ( X ` P ) ) e. B /\ ( P .\/ ( R ` G ) ) e. B /\ W e. B ) ) -> ( ( P .\/ ( X ` P ) ) .<_ ( P .\/ ( R ` G ) ) -> ( ( P .\/ ( X ` P ) ) ./\ W ) .<_ ( ( P .\/ ( R ` G ) ) ./\ W ) ) ) |
41 |
22 36 31 39 40
|
syl13anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( ( P .\/ ( X ` P ) ) .<_ ( P .\/ ( R ` G ) ) -> ( ( P .\/ ( X ` P ) ) ./\ W ) .<_ ( ( P .\/ ( R ` G ) ) ./\ W ) ) ) |
42 |
34 41
|
mpd |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( ( P .\/ ( X ` P ) ) ./\ W ) .<_ ( ( P .\/ ( R ` G ) ) ./\ W ) ) |
43 |
|
simp3l |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
44 |
2 3 4 5 6 7 8
|
trlval2 |
|- ( ( ( K e. HL /\ W e. H ) /\ X e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` X ) = ( ( P .\/ ( X ` P ) ) ./\ W ) ) |
45 |
14 25 43 44
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( R ` X ) = ( ( P .\/ ( X ` P ) ) ./\ W ) ) |
46 |
2 3 4 5 6 7 8
|
trlval5 |
|- ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` G ) = ( ( P .\/ ( R ` G ) ) ./\ W ) ) |
47 |
14 15 43 46
|
syl3anc |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( R ` G ) = ( ( P .\/ ( R ` G ) ) ./\ W ) ) |
48 |
42 45 47
|
3brtr4d |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( F e. T /\ F =/= ( _I |` B ) ) /\ ( G e. T /\ G =/= ( _I |` B ) ) /\ N e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( R ` F ) = ( R ` N ) ) ) -> ( R ` X ) .<_ ( R ` G ) ) |