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Metamath Proof Explorer

Theorem cdlemk3

Description: Part of proof of Lemma K of Crawley p. 118. (Contributed by NM, 3-Jul-2013)

Ref Expression
Hypotheses cdlemk.b 
|- B = ( Base ` K )
cdlemk.l 
|- .<_ = ( le ` K )
cdlemk.j 
|- .\/ = ( join ` K )
cdlemk.a 
|- A = ( Atoms ` K )
cdlemk.h 
|- H = ( LHyp ` K )
cdlemk.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemk.r 
|- R = ( ( trL ` K ) ` W )
cdlemk.m 
|- ./\ = ( meet ` K )
Assertion cdlemk3 
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( ( F ` P ) .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) = ( F ` P ) )

Proof

Step Hyp Ref Expression
1 cdlemk.b 
 |-  B = ( Base ` K )
2 cdlemk.l 
 |-  .<_ = ( le ` K )
3 cdlemk.j 
 |-  .\/ = ( join ` K )
4 cdlemk.a 
 |-  A = ( Atoms ` K )
5 cdlemk.h 
 |-  H = ( LHyp ` K )
6 cdlemk.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 cdlemk.r 
 |-  R = ( ( trL ` K ) ` W )
8 cdlemk.m 
 |-  ./\ = ( meet ` K )
9 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> K e. HL )
10 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( K e. HL /\ W e. H ) )
11 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F e. T )
12 simp32l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> F =/= ( _I |` B ) )
13 1 4 5 6 7 trlnidat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> ( R ` F ) e. A )
14 10 11 12 13 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` F ) e. A )
15 simp2r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> G e. T )
16 simp31 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` G ) =/= ( R ` F ) )
17 4 5 6 7 trlcocnvat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( G e. T /\ F e. T ) /\ ( R ` G ) =/= ( R ` F ) ) -> ( R ` ( G o. `' F ) ) e. A )
18 10 15 11 16 17 syl121anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` ( G o. `' F ) ) e. A )
19 simp33l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> P e. A )
20 2 4 5 6 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ P e. A ) -> ( F ` P ) e. A )
21 10 11 19 20 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( F ` P ) e. A )
22 5 6 ltrncnv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> `' F e. T )
23 10 11 22 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> `' F e. T )
24 5 6 7 trlcnv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` `' F ) = ( R ` F ) )
25 10 11 24 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` `' F ) = ( R ` F ) )
26 16 necomd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` F ) =/= ( R ` G ) )
27 25 26 eqnetrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` `' F ) =/= ( R ` G ) )
28 simp32r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> G =/= ( _I |` B ) )
29 1 5 6 7 trlcone 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( `' F e. T /\ G e. T ) /\ ( ( R ` `' F ) =/= ( R ` G ) /\ G =/= ( _I |` B ) ) ) -> ( R ` `' F ) =/= ( R ` ( `' F o. G ) ) )
30 10 23 15 27 28 29 syl122anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` `' F ) =/= ( R ` ( `' F o. G ) ) )
31 5 6 ltrncom 
 |-  ( ( ( K e. HL /\ W e. H ) /\ `' F e. T /\ G e. T ) -> ( `' F o. G ) = ( G o. `' F ) )
32 10 23 15 31 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( `' F o. G ) = ( G o. `' F ) )
33 32 fveq2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` ( `' F o. G ) ) = ( R ` ( G o. `' F ) ) )
34 30 25 33 3netr3d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` F ) =/= ( R ` ( G o. `' F ) ) )
35 simp33 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( P e. A /\ -. P .<_ W ) )
36 2 4 5 6 ltrnel 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( F ` P ) e. A /\ -. ( F ` P ) .<_ W ) )
37 36 simprd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> -. ( F ` P ) .<_ W )
38 10 11 35 37 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> -. ( F ` P ) .<_ W )
39 2 5 6 7 trlle 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) .<_ W )
40 10 11 39 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` F ) .<_ W )
41 5 6 ltrnco 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ `' F e. T ) -> ( G o. `' F ) e. T )
42 10 15 23 41 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( G o. `' F ) e. T )
43 2 5 6 7 trlle 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( G o. `' F ) e. T ) -> ( R ` ( G o. `' F ) ) .<_ W )
44 10 42 43 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` ( G o. `' F ) ) .<_ W )
45 9 hllatd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> K e. Lat )
46 1 4 atbase 
 |-  ( ( R ` F ) e. A -> ( R ` F ) e. B )
47 14 46 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` F ) e. B )
48 1 4 atbase 
 |-  ( ( R ` ( G o. `' F ) ) e. A -> ( R ` ( G o. `' F ) ) e. B )
49 18 48 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( R ` ( G o. `' F ) ) e. B )
50 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> W e. H )
51 1 5 lhpbase 
 |-  ( W e. H -> W e. B )
52 50 51 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> W e. B )
53 1 2 3 latjle12 
 |-  ( ( K e. Lat /\ ( ( R ` F ) e. B /\ ( R ` ( G o. `' F ) ) e. B /\ W e. B ) ) -> ( ( ( R ` F ) .<_ W /\ ( R ` ( G o. `' F ) ) .<_ W ) <-> ( ( R ` F ) .\/ ( R ` ( G o. `' F ) ) ) .<_ W ) )
54 45 47 49 52 53 syl13anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( ( R ` F ) .<_ W /\ ( R ` ( G o. `' F ) ) .<_ W ) <-> ( ( R ` F ) .\/ ( R ` ( G o. `' F ) ) ) .<_ W ) )
55 40 44 54 mpbi2and 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( R ` F ) .\/ ( R ` ( G o. `' F ) ) ) .<_ W )
56 1 4 atbase 
 |-  ( ( F ` P ) e. A -> ( F ` P ) e. B )
57 21 56 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( F ` P ) e. B )
58 1 3 4 hlatjcl 
 |-  ( ( K e. HL /\ ( R ` F ) e. A /\ ( R ` ( G o. `' F ) ) e. A ) -> ( ( R ` F ) .\/ ( R ` ( G o. `' F ) ) ) e. B )
59 9 14 18 58 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( R ` F ) .\/ ( R ` ( G o. `' F ) ) ) e. B )
60 1 2 lattr 
 |-  ( ( K e. Lat /\ ( ( F ` P ) e. B /\ ( ( R ` F ) .\/ ( R ` ( G o. `' F ) ) ) e. B /\ W e. B ) ) -> ( ( ( F ` P ) .<_ ( ( R ` F ) .\/ ( R ` ( G o. `' F ) ) ) /\ ( ( R ` F ) .\/ ( R ` ( G o. `' F ) ) ) .<_ W ) -> ( F ` P ) .<_ W ) )
61 45 57 59 52 60 syl13anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( ( F ` P ) .<_ ( ( R ` F ) .\/ ( R ` ( G o. `' F ) ) ) /\ ( ( R ` F ) .\/ ( R ` ( G o. `' F ) ) ) .<_ W ) -> ( F ` P ) .<_ W ) )
62 55 61 mpan2d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( F ` P ) .<_ ( ( R ` F ) .\/ ( R ` ( G o. `' F ) ) ) -> ( F ` P ) .<_ W ) )
63 38 62 mtod 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> -. ( F ` P ) .<_ ( ( R ` F ) .\/ ( R ` ( G o. `' F ) ) ) )
64 2 3 8 4 2llnma2 
 |-  ( ( K e. HL /\ ( ( R ` F ) e. A /\ ( R ` ( G o. `' F ) ) e. A /\ ( F ` P ) e. A ) /\ ( ( R ` F ) =/= ( R ` ( G o. `' F ) ) /\ -. ( F ` P ) .<_ ( ( R ` F ) .\/ ( R ` ( G o. `' F ) ) ) ) ) -> ( ( ( F ` P ) .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) = ( F ` P ) )
65 9 14 18 21 34 63 64 syl132anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( ( R ` G ) =/= ( R ` F ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) ) /\ ( P e. A /\ -. P .<_ W ) ) ) -> ( ( ( F ` P ) .\/ ( R ` F ) ) ./\ ( ( F ` P ) .\/ ( R ` ( G o. `' F ) ) ) ) = ( F ` P ) )