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Metamath Proof Explorer

Theorem cdlemg6d

Description: TODO: FIX COMMENT. (Contributed by NM, 27-Apr-2013)

Ref Expression
Hypotheses cdlemg4.l 
|- .<_ = ( le ` K )
cdlemg4.a 
|- A = ( Atoms ` K )
cdlemg4.h 
|- H = ( LHyp ` K )
cdlemg4.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemg4.r 
|- R = ( ( trL ` K ) ` W )
cdlemg4.j 
|- .\/ = ( join ` K )
cdlemg4b.v 
|- V = ( R ` G )
Assertion cdlemg6d 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ Q .<_ ( P .\/ V ) /\ ( F ` ( G ` P ) ) = P ) ) -> ( ( ( r e. A /\ -. r .<_ W ) /\ -. r .<_ ( P .\/ ( G ` P ) ) ) -> ( F ` ( G ` Q ) ) = Q ) )

Proof

Step Hyp Ref Expression
1 cdlemg4.l 
 |-  .<_ = ( le ` K )
2 cdlemg4.a 
 |-  A = ( Atoms ` K )
3 cdlemg4.h 
 |-  H = ( LHyp ` K )
4 cdlemg4.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 cdlemg4.r 
 |-  R = ( ( trL ` K ) ` W )
6 cdlemg4.j 
 |-  .\/ = ( join ` K )
7 cdlemg4b.v 
 |-  V = ( R ` G )
8 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ Q .<_ ( P .\/ V ) /\ ( F ` ( G ` P ) ) = P ) ) -> ( K e. HL /\ W e. H ) )
9 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ Q .<_ ( P .\/ V ) /\ ( F ` ( G ` P ) ) = P ) ) -> ( P e. A /\ -. P .<_ W ) )
10 simp31 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ Q .<_ ( P .\/ V ) /\ ( F ` ( G ` P ) ) = P ) ) -> G e. T )
11 1 2 3 4 5 6 7 cdlemg4b1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ G e. T ) -> ( P .\/ V ) = ( P .\/ ( G ` P ) ) )
12 8 9 10 11 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ Q .<_ ( P .\/ V ) /\ ( F ` ( G ` P ) ) = P ) ) -> ( P .\/ V ) = ( P .\/ ( G ` P ) ) )
13 12 breq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ Q .<_ ( P .\/ V ) /\ ( F ` ( G ` P ) ) = P ) ) -> ( r .<_ ( P .\/ V ) <-> r .<_ ( P .\/ ( G ` P ) ) ) )
14 13 notbid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ Q .<_ ( P .\/ V ) /\ ( F ` ( G ` P ) ) = P ) ) -> ( -. r .<_ ( P .\/ V ) <-> -. r .<_ ( P .\/ ( G ` P ) ) ) )
15 14 anbi2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ Q .<_ ( P .\/ V ) /\ ( F ` ( G ` P ) ) = P ) ) -> ( ( ( r e. A /\ -. r .<_ W ) /\ -. r .<_ ( P .\/ V ) ) <-> ( ( r e. A /\ -. r .<_ W ) /\ -. r .<_ ( P .\/ ( G ` P ) ) ) ) )
16 1 2 3 4 5 6 7 cdlemg6c 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ Q .<_ ( P .\/ V ) /\ ( F ` ( G ` P ) ) = P ) ) -> ( ( ( r e. A /\ -. r .<_ W ) /\ -. r .<_ ( P .\/ V ) ) -> ( F ` ( G ` Q ) ) = Q ) )
17 15 16 sylbird 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( G e. T /\ Q .<_ ( P .\/ V ) /\ ( F ` ( G ` P ) ) = P ) ) -> ( ( ( r e. A /\ -. r .<_ W ) /\ -. r .<_ ( P .\/ ( G ` P ) ) ) -> ( F ` ( G ` Q ) ) = Q ) )