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Metamath Proof Explorer

Theorem cdlemg4a

Description: TODO: FIX COMMENT If fg(p) = p, then tr f = tr g. (Contributed by NM, 23-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 )
Assertion 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 ) )

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 simp3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( F ` ( G ` P ) ) = P )
7 6 oveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( ( G ` P ) ( join ` K ) ( F ` ( G ` P ) ) ) = ( ( G ` P ) ( join ` K ) P ) )
8 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> K e. HL )
9 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( K e. HL /\ W e. H ) )
10 simp23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> G e. T )
11 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( P e. A /\ -. P .<_ W ) )
12 1 2 3 4 ltrnel 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) )
13 12 simpld 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( G ` P ) e. A )
14 9 10 11 13 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( G ` P ) e. A )
15 simp21l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> P e. A )
16 eqid 
 |-  ( join ` K ) = ( join ` K )
17 16 2 hlatjcom 
 |-  ( ( K e. HL /\ ( G ` P ) e. A /\ P e. A ) -> ( ( G ` P ) ( join ` K ) P ) = ( P ( join ` K ) ( G ` P ) ) )
18 8 14 15 17 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( ( G ` P ) ( join ` K ) P ) = ( P ( join ` K ) ( G ` P ) ) )
19 7 18 eqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( ( G ` P ) ( join ` K ) ( F ` ( G ` P ) ) ) = ( P ( join ` K ) ( G ` P ) ) )
20 19 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( ( ( G ` P ) ( join ` K ) ( F ` ( G ` P ) ) ) ( meet ` K ) W ) = ( ( P ( join ` K ) ( G ` P ) ) ( meet ` K ) W ) )
21 simp22 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> F e. T )
22 9 10 11 12 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) )
23 eqid 
 |-  ( meet ` K ) = ( meet ` K )
24 1 16 23 2 3 4 5 trlval2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) ) -> ( R ` F ) = ( ( ( G ` P ) ( join ` K ) ( F ` ( G ` P ) ) ) ( meet ` K ) W ) )
25 9 21 22 24 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( R ` F ) = ( ( ( G ` P ) ( join ` K ) ( F ` ( G ` P ) ) ) ( meet ` K ) W ) )
26 1 16 23 2 3 4 5 trlval2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( R ` G ) = ( ( P ( join ` K ) ( G ` P ) ) ( meet ` K ) W ) )
27 9 10 11 26 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ F e. T /\ G e. T ) /\ ( F ` ( G ` P ) ) = P ) -> ( R ` G ) = ( ( P ( join ` K ) ( G ` P ) ) ( meet ` K ) W ) )
28 20 25 27 3eqtr4d 
 |-  ( ( ( 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 ) )