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Metamath Proof Explorer

Theorem cdlemg4f

Description: TODO: FIX COMMENT. (Contributed by NM, 25-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 )
cdlemg4.m 
|- ./\ = ( meet ` K )
Assertion cdlemg4f 
|- ( ( ( 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 ) ) -> ( F ` ( G ` Q ) ) = ( ( Q .\/ V ) ./\ ( P .\/ ( ( P .\/ Q ) ./\ W ) ) ) )

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 cdlemg4.m 
 |-  ./\ = ( meet ` K )
9 1 2 3 4 5 6 7 8 cdlemg4e 
 |-  ( ( ( 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 ) ) -> ( F ` ( G ` Q ) ) = ( ( ( G ` Q ) .\/ ( R ` F ) ) ./\ ( ( F ` ( G ` P ) ) .\/ ( ( ( G ` P ) .\/ ( G ` Q ) ) ./\ W ) ) ) )
10 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 ) )
11 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 ) )
12 simp23 
 |-  ( ( ( 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 ) ) -> F e. T )
13 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 )
14 simp33 
 |-  ( ( ( 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 ) ) -> ( F ` ( G ` P ) ) = P )
15 1 2 3 4 5 cdlemg4a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( R ` F ) = ( R ` G ) )
16 10 11 12 13 14 15 syl131anc 
 |-  ( ( ( 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 ` F ) = ( R ` G ) )
17 7 16 eqtr4id 
 |-  ( ( ( 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 ) ) -> V = ( R ` F ) )
18 17 oveq2d 
 |-  ( ( ( 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 ` Q ) .\/ V ) = ( ( G ` Q ) .\/ ( R ` F ) ) )
19 simp22 
 |-  ( ( ( 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 ) ) -> ( Q e. A /\ -. Q .<_ W ) )
20 1 2 3 4 5 6 7 cdlemg4b12 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ G e. T ) -> ( ( G ` Q ) .\/ V ) = ( Q .\/ V ) )
21 10 19 13 20 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 ) ) -> ( ( G ` Q ) .\/ V ) = ( Q .\/ V ) )
22 18 21 eqtr3d 
 |-  ( ( ( 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 ` Q ) .\/ ( R ` F ) ) = ( Q .\/ V ) )
23 eqid 
 |-  ( ( P .\/ Q ) ./\ W ) = ( ( P .\/ Q ) ./\ W )
24 3 4 1 6 2 8 23 cdlemg2m 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ G e. T ) -> ( ( ( G ` P ) .\/ ( G ` Q ) ) ./\ W ) = ( ( P .\/ Q ) ./\ W ) )
25 10 11 19 13 24 syl121anc 
 |-  ( ( ( 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 ` P ) .\/ ( G ` Q ) ) ./\ W ) = ( ( P .\/ Q ) ./\ W ) )
26 14 25 oveq12d 
 |-  ( ( ( 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 ) ) -> ( ( F ` ( G ` P ) ) .\/ ( ( ( G ` P ) .\/ ( G ` Q ) ) ./\ W ) ) = ( P .\/ ( ( P .\/ Q ) ./\ W ) ) )
27 22 26 oveq12d 
 |-  ( ( ( 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 ` Q ) .\/ ( R ` F ) ) ./\ ( ( F ` ( G ` P ) ) .\/ ( ( ( G ` P ) .\/ ( G ` Q ) ) ./\ W ) ) ) = ( ( Q .\/ V ) ./\ ( P .\/ ( ( P .\/ Q ) ./\ W ) ) ) )
28 9 27 eqtrd 
 |-  ( ( ( 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 ) ) -> ( F ` ( G ` Q ) ) = ( ( Q .\/ V ) ./\ ( P .\/ ( ( P .\/ Q ) ./\ W ) ) ) )