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Metamath Proof Explorer

Theorem cdlemg38

Description: Use cdlemg37 to eliminate E. r e. A from cdlemg36 . TODO: Fix comment. (Contributed by NM, 31-May-2013)

Ref Expression
Hypotheses cdlemg35.l 
|- .<_ = ( le ` K )
cdlemg35.j 
|- .\/ = ( join ` K )
cdlemg35.m 
|- ./\ = ( meet ` K )
cdlemg35.a 
|- A = ( Atoms ` K )
cdlemg35.h 
|- H = ( LHyp ` K )
cdlemg35.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemg35.r 
|- R = ( ( trL ` K ) ` W )
Assertion cdlemg38 
|- ( ( ( ( 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 ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )

Proof

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 simpl1 
 |-  ( ( ( ( ( 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 ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
9 simpl2 
 |-  ( ( ( ( ( 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 /\ G e. T /\ P =/= Q ) )
10 simpl3l 
 |-  ( ( ( ( ( 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 /\ ( G ` P ) =/= P ) )
11 simpl3r 
 |-  ( ( ( ( ( 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 ) )
12 simpr 
 |-  ( ( ( ( ( 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. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) )
13 1 2 3 4 5 6 7 cdlemg36 
 |-  ( ( ( ( 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 ) )
14 8 9 10 11 12 13 syl113anc 
 |-  ( ( ( ( ( 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 ) )
15 simpl11 
 |-  ( ( ( ( ( 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 ) )
16 simpl12 
 |-  ( ( ( ( ( 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 ) )
17 simpl13 
 |-  ( ( ( ( ( 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 ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
18 simpl21 
 |-  ( ( ( ( ( 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 )
19 simpl22 
 |-  ( ( ( ( ( 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 )
20 simpl23 
 |-  ( ( ( ( ( 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 =/= Q )
21 simpr 
 |-  ( ( ( ( ( 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. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) )
22 1 2 3 4 5 6 7 cdlemg37 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ P =/= Q /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
23 15 16 17 18 19 20 21 22 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 ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
24 14 23 pm2.61dan 
 |-  ( ( ( ( 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 ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )