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Metamath Proof Explorer

Theorem cdlemg9b

Description: The triples <. P , ( F( GP ) ) , ( FP ) >. and <. Q , ( F( GQ ) ) , ( FQ ) >. are centrally perspective. TODO: FIX COMMENT. (Contributed by NM, 1-May-2013)

Ref Expression
Hypotheses cdlemg8.l 
|- .<_ = ( le ` K )
cdlemg8.j 
|- .\/ = ( join ` K )
cdlemg8.m 
|- ./\ = ( meet ` K )
cdlemg8.a 
|- A = ( Atoms ` K )
cdlemg8.h 
|- H = ( LHyp ` K )
cdlemg8.t 
|- T = ( ( LTrn ` K ) ` W )
Assertion cdlemg9b 
|- ( ( ( 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 ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( P .\/ Q ) ./\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) ) .<_ ( ( G ` P ) .\/ ( G ` Q ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg8.l 
 |-  .<_ = ( le ` K )
2 cdlemg8.j 
 |-  .\/ = ( join ` K )
3 cdlemg8.m 
 |-  ./\ = ( meet ` K )
4 cdlemg8.a 
 |-  A = ( Atoms ` K )
5 cdlemg8.h 
 |-  H = ( LHyp ` K )
6 cdlemg8.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 eqid 
 |-  ( ( P .\/ Q ) ./\ W ) = ( ( P .\/ Q ) ./\ W )
8 1 2 3 4 5 6 7 cdlemg9a 
 |-  ( ( ( 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 ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( P .\/ ( ( P .\/ Q ) ./\ W ) ) ./\ ( ( F ` ( G ` P ) ) .\/ ( ( P .\/ Q ) ./\ W ) ) ) .<_ ( ( G ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) )
9 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 /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> K e. HL )
10 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 /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> W e. H )
11 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 ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( P e. A /\ -. P .<_ W ) )
12 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 /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> Q e. A )
13 1 2 3 4 5 7 cdlemg3a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) -> ( P .\/ Q ) = ( P .\/ ( ( P .\/ Q ) ./\ W ) ) )
14 9 10 11 12 13 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 /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( P .\/ Q ) = ( P .\/ ( ( P .\/ Q ) ./\ W ) ) )
15 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 /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( K e. HL /\ W e. H ) )
16 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 ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
17 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 /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> F e. T )
18 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 /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> G e. T )
19 5 6 1 2 4 3 7 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 ) ) )
20 15 11 16 17 18 19 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 /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) = ( ( F ` ( G ` P ) ) .\/ ( ( P .\/ Q ) ./\ W ) ) )
21 14 20 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 /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( P .\/ Q ) ./\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) ) = ( ( P .\/ ( ( P .\/ Q ) ./\ W ) ) ./\ ( ( F ` ( G ` P ) ) .\/ ( ( P .\/ Q ) ./\ W ) ) ) )
22 5 6 1 2 4 3 7 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 ) ) )
23 15 11 16 18 22 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 /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( G ` P ) .\/ ( G ` Q ) ) = ( ( G ` P ) .\/ ( ( P .\/ Q ) ./\ W ) ) )
24 8 21 23 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 /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) ) ) -> ( ( P .\/ Q ) ./\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) ) .<_ ( ( G ` P ) .\/ ( G ` Q ) ) )