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Metamath Proof Explorer

Theorem cdlemg12b

Description: The triples <. P , ( FP ) , ( F( GP ) ) >. and <. Q , ( FQ ) , ( F( GQ ) ) >. are centrally perspective. TODO: FIX COMMENT. (Contributed by NM, 5-May-2013)

Ref Expression
Hypotheses cdlemg12.l 
|- .<_ = ( le ` K )
cdlemg12.j 
|- .\/ = ( join ` K )
cdlemg12.m 
|- ./\ = ( meet ` K )
cdlemg12.a 
|- A = ( Atoms ` K )
cdlemg12.h 
|- H = ( LHyp ` K )
cdlemg12.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemg12b.r 
|- R = ( ( trL ` K ) ` W )
Assertion cdlemg12b 
|- ( ( ( 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 ` G ) .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ Q ) ./\ ( ( G ` P ) .\/ ( G ` Q ) ) ) .<_ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg12.l 
 |-  .<_ = ( le ` K )
2 cdlemg12.j 
 |-  .\/ = ( join ` K )
3 cdlemg12.m 
 |-  ./\ = ( meet ` K )
4 cdlemg12.a 
 |-  A = ( Atoms ` K )
5 cdlemg12.h 
 |-  H = ( LHyp ` K )
6 cdlemg12.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 cdlemg12b.r 
 |-  R = ( ( trL ` K ) ` W )
8 simp1 
 |-  ( ( ( 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 ` G ) .<_ ( P .\/ Q ) ) ) -> ( K e. HL /\ W e. H ) )
9 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 /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) )
10 simp31 
 |-  ( ( ( 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 ` G ) .<_ ( P .\/ Q ) ) ) -> G e. T )
11 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 /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> P =/= Q )
12 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 /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> ( P e. A /\ -. P .<_ W ) )
13 simp22l 
 |-  ( ( ( 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 ` G ) .<_ ( P .\/ Q ) ) ) -> Q e. A )
14 simp33 
 |-  ( ( ( 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 ` G ) .<_ ( P .\/ Q ) ) ) -> -. ( R ` G ) .<_ ( P .\/ Q ) )
15 1 2 3 4 5 6 7 cdlemg11b 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( G e. T /\ P =/= Q /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> ( P .\/ Q ) =/= ( ( G ` P ) .\/ ( G ` Q ) ) )
16 8 12 13 10 11 14 15 syl123anc 
 |-  ( ( ( 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 ` G ) .<_ ( P .\/ Q ) ) ) -> ( P .\/ Q ) =/= ( ( G ` P ) .\/ ( G ` Q ) ) )
17 simp1l 
 |-  ( ( ( 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 ` G ) .<_ ( P .\/ Q ) ) ) -> K e. HL )
18 simp1r 
 |-  ( ( ( 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 ` G ) .<_ ( P .\/ Q ) ) ) -> W e. H )
19 eqid 
 |-  ( ( P .\/ Q ) ./\ W ) = ( ( P .\/ Q ) ./\ W )
20 1 2 3 4 5 19 cdlemg3a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) -> ( P .\/ Q ) = ( P .\/ ( ( P .\/ Q ) ./\ W ) ) )
21 17 18 12 13 20 syl211anc 
 |-  ( ( ( 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 ` G ) .<_ ( P .\/ Q ) ) ) -> ( P .\/ Q ) = ( P .\/ ( ( P .\/ Q ) ./\ W ) ) )
22 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 /\ -. ( R ` G ) .<_ ( P .\/ Q ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
23 5 6 1 2 4 3 19 cdlemg2k 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ G e. T ) -> ( ( G ` P ) .\/ ( G ` Q ) ) = ( ( G ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) )
24 8 12 22 10 23 syl121anc 
 |-  ( ( ( 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 ` G ) .<_ ( P .\/ Q ) ) ) -> ( ( G ` P ) .\/ ( G ` Q ) ) = ( ( G ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) )
25 16 21 24 3netr3d 
 |-  ( ( ( 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 ` G ) .<_ ( P .\/ Q ) ) ) -> ( P .\/ ( ( P .\/ Q ) ./\ W ) ) =/= ( ( G ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) )
26 1 2 3 4 5 6 19 cdlemg12a 
 |-  ( ( ( 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 .\/ ( ( P .\/ Q ) ./\ W ) ) =/= ( ( G ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) ) ) -> ( ( P .\/ ( ( P .\/ Q ) ./\ W ) ) ./\ ( ( G ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) ) .<_ ( ( F ` ( G ` P ) ) .\/ ( ( P .\/ Q ) ./\ W ) ) )
27 8 9 10 11 25 26 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 ` G ) .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ ( ( P .\/ Q ) ./\ W ) ) ./\ ( ( G ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) ) .<_ ( ( F ` ( G ` P ) ) .\/ ( ( P .\/ Q ) ./\ W ) ) )
28 21 24 oveq12d 
 |-  ( ( ( 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 ` G ) .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ Q ) ./\ ( ( G ` P ) .\/ ( G ` Q ) ) ) = ( ( P .\/ ( ( P .\/ Q ) ./\ W ) ) ./\ ( ( G ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) ) )
29 simp23 
 |-  ( ( ( 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 ` G ) .<_ ( P .\/ Q ) ) ) -> F e. T )
30 5 6 1 2 4 3 19 cdlemg2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T ) ) -> ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) = ( ( F ` ( G ` P ) ) .\/ ( ( P .\/ Q ) ./\ W ) ) )
31 8 12 22 29 10 30 syl122anc 
 |-  ( ( ( 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 ` G ) .<_ ( P .\/ Q ) ) ) -> ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) = ( ( F ` ( G ` P ) ) .\/ ( ( P .\/ Q ) ./\ W ) ) )
32 27 28 31 3brtr4d 
 |-  ( ( ( 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 ` G ) .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ Q ) ./\ ( ( G ` P ) .\/ ( G ` Q ) ) ) .<_ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) )