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Metamath Proof Explorer

Theorem cdlemg36

Description: Use cdlemg35 to eliminate v from cdlemg34 . 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 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 ) )

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 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 ) )