Metamath Proof Explorer

Theorem cdlemg39

Description: Eliminate =/= conditions from cdlemg38 . TODO: Would this better be done at cdlemg35 ? 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 cdlemg39 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) -> ( ( 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 ) ) /\ ( R ` F ) = ( R ` G ) ) -> ( K e. HL /\ W e. H ) )
9 simpl2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) = ( R ` G ) ) -> ( P e. A /\ -. P .<_ W ) )
10 simpl2r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) = ( R ` G ) ) -> ( Q e. A /\ -. Q .<_ W ) )
11 simpl31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) = ( R ` G ) ) -> F e. T )
12 simpl32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) = ( R ` G ) ) -> G e. T )
13 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 ) ) /\ ( R ` F ) = ( R ` G ) ) -> ( R ` F ) = ( R ` G ) )
14 1 2 3 4 5 6 7 cdlemg15 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) /\ ( R ` F ) = ( R ` G ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
15 8 9 10 11 12 13 14 syl321anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) = ( R ` G ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
16 simpll1 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) =/= ( R ` G ) ) /\ ( F ` P ) = P ) -> ( K e. HL /\ W e. H ) )
17 simpll2 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) =/= ( R ` G ) ) /\ ( F ` P ) = P ) -> ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
18 simpl31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) =/= ( R ` G ) ) -> F e. T )
19 18 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) =/= ( R ` G ) ) /\ ( F ` P ) = P ) -> F e. T )
20 simpl32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) =/= ( R ` G ) ) -> G e. T )
21 20 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) =/= ( R ` G ) ) /\ ( F ` P ) = P ) -> G e. T )
22 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 ) ) /\ ( R ` F ) =/= ( R ` G ) ) /\ ( F ` P ) = P ) -> ( F ` P ) = P )
23 1 2 3 4 5 6 7 cdlemg14f 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( F ` P ) = P ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
24 16 17 19 21 22 23 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 ) ) /\ ( R ` F ) =/= ( R ` G ) ) /\ ( F ` P ) = P ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
25 simpll1 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) =/= ( R ` G ) ) /\ ( G ` P ) = P ) -> ( K e. HL /\ W e. H ) )
26 simpll2 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) =/= ( R ` G ) ) /\ ( G ` P ) = P ) -> ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
27 18 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) =/= ( R ` G ) ) /\ ( G ` P ) = P ) -> F e. T )
28 20 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) =/= ( R ` G ) ) /\ ( G ` P ) = P ) -> G e. T )
29 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 ) ) /\ ( R ` F ) =/= ( R ` G ) ) /\ ( G ` P ) = P ) -> ( G ` P ) = P )
30 1 2 3 4 5 6 7 cdlemg14g 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ ( G ` P ) = P ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
31 25 26 27 28 29 30 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 ) ) /\ ( R ` F ) =/= ( R ` G ) ) /\ ( G ` P ) = P ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
32 simpll1 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) -> ( K e. HL /\ W e. H ) )
33 simpl2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( P e. A /\ -. P .<_ W ) )
34 33 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) -> ( P e. A /\ -. P .<_ W ) )
35 simpl2r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( Q e. A /\ -. Q .<_ W ) )
36 35 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) -> ( Q e. A /\ -. Q .<_ W ) )
37 simpll3 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) -> ( F e. T /\ G e. T /\ P =/= Q ) )
38 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 ) ) /\ ( R ` F ) =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) -> ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) )
39 simplr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) -> ( R ` F ) =/= ( R ` G ) )
40 1 2 3 4 5 6 7 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 ) )
41 32 34 36 37 38 39 40 syl312anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) =/= ( R ` G ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
42 24 31 41 pm2.61da2ne 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) /\ ( R ` F ) =/= ( R ` G ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )
43 15 42 pm2.61dane 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ W ) = ( ( Q .\/ ( F ` ( G ` Q ) ) ) ./\ W ) )