Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemg35.l |
|- .<_ = ( le ` K ) |
2 |
|
cdlemg35.j |
|- .\/ = ( join ` K ) |
3 |
|
cdlemg35.m |
|- ./\ = ( meet ` K ) |
4 |
|
cdlemg35.a |
|- A = ( Atoms ` K ) |
5 |
|
cdlemg35.h |
|- H = ( LHyp ` K ) |
6 |
|
cdlemg35.t |
|- T = ( ( LTrn ` K ) ` W ) |
7 |
|
cdlemg35.r |
|- R = ( ( trL ` K ) ` W ) |
8 |
|
simp11 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) /\ ( R ` F ) =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( K e. HL /\ W e. H ) ) |
9 |
|
simp12 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) /\ ( R ` F ) =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( P e. A /\ -. P .<_ W ) ) |
10 |
|
simp21 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) /\ ( R ` F ) =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> F e. T ) |
11 |
|
simp22 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) /\ ( R ` F ) =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> G e. T ) |
12 |
|
simp31l |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) /\ ( R ` F ) =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( F ` P ) =/= P ) |
13 |
|
simp31r |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) /\ ( R ` F ) =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( G ` P ) =/= P ) |
14 |
|
simp32 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) /\ ( R ` F ) =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( R ` F ) =/= ( R ` G ) ) |
15 |
1 2 3 4 5 6 7
|
cdlemg35 |
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> E. v e. A ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) |
16 |
8 9 10 11 12 13 14 15
|
syl133anc |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) /\ ( R ` F ) =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. v e. A ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) |
17 |
|
simp11 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) /\ ( R ` F ) =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ v e. A /\ ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) |
18 |
|
simp2 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) /\ ( R ` F ) =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ v e. A /\ ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) -> v e. A ) |
19 |
|
simp3l |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) /\ ( R ` F ) =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ v e. A /\ ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) -> v .<_ W ) |
20 |
18 19
|
jca |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) /\ ( R ` F ) =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ v e. A /\ ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) -> ( v e. A /\ v .<_ W ) ) |
21 |
|
simp121 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) /\ ( R ` F ) =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ v e. A /\ ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) -> F e. T ) |
22 |
|
simp122 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) /\ ( R ` F ) =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ v e. A /\ ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) -> G e. T ) |
23 |
21 22
|
jca |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) /\ ( R ` F ) =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ v e. A /\ ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) -> ( F e. T /\ G e. T ) ) |
24 |
|
simp123 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) /\ ( R ` F ) =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ v e. A /\ ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) -> P =/= Q ) |
25 |
|
simp3rl |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) /\ ( R ` F ) =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ v e. A /\ ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) -> v =/= ( R ` F ) ) |
26 |
|
simp3rr |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) /\ ( R ` F ) =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ v e. A /\ ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) -> v =/= ( R ` G ) ) |
27 |
|
simp133 |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) /\ ( R ` F ) =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ v e. A /\ ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) -> E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) |
28 |
|
eqid |
|- ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) |
29 |
|
eqid |
|- ( ( P .\/ v ) ./\ ( Q .\/ ( R ` G ) ) ) = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` G ) ) ) |
30 |
1 2 3 4 5 6 7 28 29
|
cdlemg34 |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( v e. A /\ v .<_ W ) /\ ( F e. T /\ G e. T ) /\ P =/= Q ) /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) ) |
31 |
17 20 23 24 25 26 27 30
|
syl133anc |
|- ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) /\ ( R ` F ) =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) /\ v e. A /\ ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) ) |
32 |
31
|
rexlimdv3a |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) /\ ( R ` F ) =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( E. v e. A ( v .<_ W /\ ( v =/= ( R ` F ) /\ v =/= ( R ` G ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) ) ) |
33 |
16 32
|
mpd |
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) /\ ( R ` F ) =/= ( R ` G ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) ) |