Metamath Proof Explorer

Theorem cdlemg44a

Description: Part of proof of Lemma G of Crawley p. 116, fourth line of third paragraph on p. 117: "so fg(p) = gf(p)." (Contributed by NM, 3-Jun-2013)

Ref Expression
Hypotheses cdlemg44.h 
|- H = ( LHyp ` K )
cdlemg44.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemg44.r 
|- R = ( ( trL ` K ) ` W )
cdlemg44.l 
|- .<_ = ( le ` K )
cdlemg44.a 
|- A = ( Atoms ` K )
Assertion cdlemg44a 
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( F ` ( G ` P ) ) = ( G ` ( F ` P ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg44.h 
 |-  H = ( LHyp ` K )
2 cdlemg44.t 
 |-  T = ( ( LTrn ` K ) ` W )
3 cdlemg44.r 
 |-  R = ( ( trL ` K ) ` W )
4 cdlemg44.l 
 |-  .<_ = ( le ` K )
5 cdlemg44.a 
 |-  A = ( Atoms ` K )
6 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> K e. HL )
7 6 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> K e. Lat )
8 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( K e. HL /\ W e. H ) )
9 simp22 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> G e. T )
10 simp23l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> P e. A )
11 eqid 
 |-  ( Base ` K ) = ( Base ` K )
12 11 5 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
13 10 12 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> P e. ( Base ` K ) )
14 11 1 2 ltrncl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ P e. ( Base ` K ) ) -> ( G ` P ) e. ( Base ` K ) )
15 8 9 13 14 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( G ` P ) e. ( Base ` K ) )
16 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> F e. T )
17 11 1 2 3 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) e. ( Base ` K ) )
18 8 16 17 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` F ) e. ( Base ` K ) )
19 eqid 
 |-  ( join ` K ) = ( join ` K )
20 11 19 latjcl 
 |-  ( ( K e. Lat /\ ( G ` P ) e. ( Base ` K ) /\ ( R ` F ) e. ( Base ` K ) ) -> ( ( G ` P ) ( join ` K ) ( R ` F ) ) e. ( Base ` K ) )
21 7 15 18 20 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( ( G ` P ) ( join ` K ) ( R ` F ) ) e. ( Base ` K ) )
22 11 1 2 ltrncl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ P e. ( Base ` K ) ) -> ( F ` P ) e. ( Base ` K ) )
23 8 16 13 22 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( F ` P ) e. ( Base ` K ) )
24 11 1 2 3 trlcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) e. ( Base ` K ) )
25 8 9 24 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` G ) e. ( Base ` K ) )
26 11 19 latjcl 
 |-  ( ( K e. Lat /\ ( F ` P ) e. ( Base ` K ) /\ ( R ` G ) e. ( Base ` K ) ) -> ( ( F ` P ) ( join ` K ) ( R ` G ) ) e. ( Base ` K ) )
27 7 23 25 26 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( ( F ` P ) ( join ` K ) ( R ` G ) ) e. ( Base ` K ) )
28 eqid 
 |-  ( meet ` K ) = ( meet ` K )
29 11 28 latmcom 
 |-  ( ( K e. Lat /\ ( ( G ` P ) ( join ` K ) ( R ` F ) ) e. ( Base ` K ) /\ ( ( F ` P ) ( join ` K ) ( R ` G ) ) e. ( Base ` K ) ) -> ( ( ( G ` P ) ( join ` K ) ( R ` F ) ) ( meet ` K ) ( ( F ` P ) ( join ` K ) ( R ` G ) ) ) = ( ( ( F ` P ) ( join ` K ) ( R ` G ) ) ( meet ` K ) ( ( G ` P ) ( join ` K ) ( R ` F ) ) ) )
30 7 21 27 29 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( ( ( G ` P ) ( join ` K ) ( R ` F ) ) ( meet ` K ) ( ( F ` P ) ( join ` K ) ( R ` G ) ) ) = ( ( ( F ` P ) ( join ` K ) ( R ` G ) ) ( meet ` K ) ( ( G ` P ) ( join ` K ) ( R ` F ) ) ) )
31 simp23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( P e. A /\ -. P .<_ W ) )
32 simp32 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( G ` P ) =/= P )
33 simp33 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` F ) =/= ( R ` G ) )
34 4 19 5 1 2 3 28 cdlemg43 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( F ` ( G ` P ) ) = ( ( ( G ` P ) ( join ` K ) ( R ` F ) ) ( meet ` K ) ( ( F ` P ) ( join ` K ) ( R ` G ) ) ) )
35 8 16 9 31 32 33 34 syl123anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( F ` ( G ` P ) ) = ( ( ( G ` P ) ( join ` K ) ( R ` F ) ) ( meet ` K ) ( ( F ` P ) ( join ` K ) ( R ` G ) ) ) )
36 simp31 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( F ` P ) =/= P )
37 33 necomd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` G ) =/= ( R ` F ) )
38 4 19 5 1 2 3 28 cdlemg43 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( G e. T /\ F e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( F ` P ) =/= P /\ ( R ` G ) =/= ( R ` F ) ) ) -> ( G ` ( F ` P ) ) = ( ( ( F ` P ) ( join ` K ) ( R ` G ) ) ( meet ` K ) ( ( G ` P ) ( join ` K ) ( R ` F ) ) ) )
39 8 9 16 31 36 37 38 syl123anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( G ` ( F ` P ) ) = ( ( ( F ` P ) ( join ` K ) ( R ` G ) ) ( meet ` K ) ( ( G ` P ) ( join ` K ) ( R ` F ) ) ) )
40 30 35 39 3eqtr4d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) /\ ( ( F ` P ) =/= P /\ ( G ` P ) =/= P /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( F ` ( G ` P ) ) = ( G ` ( F ` P ) ) )