Metamath Proof Explorer

Theorem cdlemh

Description: Lemma H of Crawley p. 118. (Contributed by NM, 17-Jun-2013)

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

Proof

Step Hyp Ref Expression
1 cdlemh.b 
 |-  B = ( Base ` K )
2 cdlemh.l 
 |-  .<_ = ( le ` K )
3 cdlemh.j 
 |-  .\/ = ( join ` K )
4 cdlemh.m 
 |-  ./\ = ( meet ` K )
5 cdlemh.a 
 |-  A = ( Atoms ` K )
6 cdlemh.h 
 |-  H = ( LHyp ` K )
7 cdlemh.t 
 |-  T = ( ( LTrn ` K ) ` W )
8 cdlemh.r 
 |-  R = ( ( trL ` K ) ` W )
9 cdlemh.s 
 |-  S = ( ( P .\/ ( R ` G ) ) ./\ ( Q .\/ ( R ` ( G o. `' F ) ) ) )
10 simp1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) )
11 simp21l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> P e. A )
12 simp22l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> Q e. A )
13 simp23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> Q .<_ ( P .\/ ( R ` F ) ) )
14 simp33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` F ) =/= ( R ` G ) )
15 1 2 3 4 5 6 7 8 9 cdlemh1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( P e. A /\ Q e. A ) /\ ( Q .<_ ( P .\/ ( R ` F ) ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( S .\/ ( R ` ( G o. `' F ) ) ) = ( Q .\/ ( R ` ( G o. `' F ) ) ) )
16 10 11 12 13 14 15 syl122anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( S .\/ ( R ` ( G o. `' F ) ) ) = ( Q .\/ ( R ` ( G o. `' F ) ) ) )
17 oveq1 
 |-  ( S = ( 0. ` K ) -> ( S .\/ ( R ` ( G o. `' F ) ) ) = ( ( 0. ` K ) .\/ ( R ` ( G o. `' F ) ) ) )
18 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> K e. HL )
19 hlol 
 |-  ( K e. HL -> K e. OL )
20 18 19 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> K e. OL )
21 simp11r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> W e. H )
22 18 21 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( K e. HL /\ W e. H ) )
23 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> G e. T )
24 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> F e. T )
25 6 7 ltrncnv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> `' F e. T )
26 22 24 25 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> `' F e. T )
27 23 26 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( G e. T /\ `' F e. T ) )
28 14 necomd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` G ) =/= ( R ` F ) )
29 6 7 8 trlcnv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` `' F ) = ( R ` F ) )
30 22 24 29 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` `' F ) = ( R ` F ) )
31 28 30 neeqtrrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` G ) =/= ( R ` `' F ) )
32 5 6 7 8 trlcoat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( G e. T /\ `' F e. T ) /\ ( R ` G ) =/= ( R ` `' F ) ) -> ( R ` ( G o. `' F ) ) e. A )
33 22 27 31 32 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` ( G o. `' F ) ) e. A )
34 1 5 atbase 
 |-  ( ( R ` ( G o. `' F ) ) e. A -> ( R ` ( G o. `' F ) ) e. B )
35 33 34 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` ( G o. `' F ) ) e. B )
36 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
37 1 3 36 olj02 
 |-  ( ( K e. OL /\ ( R ` ( G o. `' F ) ) e. B ) -> ( ( 0. ` K ) .\/ ( R ` ( G o. `' F ) ) ) = ( R ` ( G o. `' F ) ) )
38 20 35 37 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( ( 0. ` K ) .\/ ( R ` ( G o. `' F ) ) ) = ( R ` ( G o. `' F ) ) )
39 17 38 sylan9eqr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ S = ( 0. ` K ) ) -> ( S .\/ ( R ` ( G o. `' F ) ) ) = ( R ` ( G o. `' F ) ) )
40 6 7 ltrnco 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ `' F e. T ) -> ( G o. `' F ) e. T )
41 22 23 26 40 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( G o. `' F ) e. T )
42 2 6 7 8 trlle 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( G o. `' F ) e. T ) -> ( R ` ( G o. `' F ) ) .<_ W )
43 22 41 42 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` ( G o. `' F ) ) .<_ W )
44 simp22r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> -. Q .<_ W )
45 nbrne2 
 |-  ( ( ( R ` ( G o. `' F ) ) .<_ W /\ -. Q .<_ W ) -> ( R ` ( G o. `' F ) ) =/= Q )
46 45 necomd 
 |-  ( ( ( R ` ( G o. `' F ) ) .<_ W /\ -. Q .<_ W ) -> Q =/= ( R ` ( G o. `' F ) ) )
47 43 44 46 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> Q =/= ( R ` ( G o. `' F ) ) )
48 eqid 
 |-  ( LLines ` K ) = ( LLines ` K )
49 3 5 48 llni2 
 |-  ( ( ( K e. HL /\ Q e. A /\ ( R ` ( G o. `' F ) ) e. A ) /\ Q =/= ( R ` ( G o. `' F ) ) ) -> ( Q .\/ ( R ` ( G o. `' F ) ) ) e. ( LLines ` K ) )
50 18 12 33 47 49 syl31anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( Q .\/ ( R ` ( G o. `' F ) ) ) e. ( LLines ` K ) )
51 5 48 llnneat 
 |-  ( ( K e. HL /\ ( Q .\/ ( R ` ( G o. `' F ) ) ) e. ( LLines ` K ) ) -> -. ( Q .\/ ( R ` ( G o. `' F ) ) ) e. A )
52 18 50 51 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> -. ( Q .\/ ( R ` ( G o. `' F ) ) ) e. A )
53 nelne2 
 |-  ( ( ( R ` ( G o. `' F ) ) e. A /\ -. ( Q .\/ ( R ` ( G o. `' F ) ) ) e. A ) -> ( R ` ( G o. `' F ) ) =/= ( Q .\/ ( R ` ( G o. `' F ) ) ) )
54 33 52 53 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` ( G o. `' F ) ) =/= ( Q .\/ ( R ` ( G o. `' F ) ) ) )
55 54 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ S = ( 0. ` K ) ) -> ( R ` ( G o. `' F ) ) =/= ( Q .\/ ( R ` ( G o. `' F ) ) ) )
56 39 55 eqnetrd 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) /\ S = ( 0. ` K ) ) -> ( S .\/ ( R ` ( G o. `' F ) ) ) =/= ( Q .\/ ( R ` ( G o. `' F ) ) ) )
57 56 ex 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( S = ( 0. ` K ) -> ( S .\/ ( R ` ( G o. `' F ) ) ) =/= ( Q .\/ ( R ` ( G o. `' F ) ) ) ) )
58 57 necon2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( ( S .\/ ( R ` ( G o. `' F ) ) ) = ( Q .\/ ( R ` ( G o. `' F ) ) ) -> S =/= ( 0. ` K ) ) )
59 16 58 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> S =/= ( 0. ` K ) )
60 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> G =/= ( _I |` B ) )
61 1 5 6 7 8 trlnidat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ G =/= ( _I |` B ) ) -> ( R ` G ) e. A )
62 22 23 60 61 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` G ) e. A )
63 2 3 5 hlatlej2 
 |-  ( ( K e. HL /\ P e. A /\ ( R ` G ) e. A ) -> ( R ` G ) .<_ ( P .\/ ( R ` G ) ) )
64 18 11 62 63 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` G ) .<_ ( P .\/ ( R ` G ) ) )
65 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
66 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> F =/= ( _I |` B ) )
67 1 6 7 ltrncnvnid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ F =/= ( _I |` B ) ) -> `' F =/= ( _I |` B ) )
68 22 24 66 67 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> `' F =/= ( _I |` B ) )
69 1 6 7 8 trlcone 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( G e. T /\ `' F e. T ) /\ ( ( R ` G ) =/= ( R ` `' F ) /\ `' F =/= ( _I |` B ) ) ) -> ( R ` G ) =/= ( R ` ( G o. `' F ) ) )
70 69 necomd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( G e. T /\ `' F e. T ) /\ ( ( R ` G ) =/= ( R ` `' F ) /\ `' F =/= ( _I |` B ) ) ) -> ( R ` ( G o. `' F ) ) =/= ( R ` G ) )
71 22 23 26 31 68 70 syl122anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` ( G o. `' F ) ) =/= ( R ` G ) )
72 2 6 7 8 trlle 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T ) -> ( R ` G ) .<_ W )
73 22 23 72 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( R ` G ) .<_ W )
74 2 3 5 6 lhp2atnle 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( R ` ( G o. `' F ) ) =/= ( R ` G ) ) /\ ( ( R ` ( G o. `' F ) ) e. A /\ ( R ` ( G o. `' F ) ) .<_ W ) /\ ( ( R ` G ) e. A /\ ( R ` G ) .<_ W ) ) -> -. ( R ` G ) .<_ ( Q .\/ ( R ` ( G o. `' F ) ) ) )
75 22 65 71 33 43 62 73 74 syl322anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> -. ( R ` G ) .<_ ( Q .\/ ( R ` ( G o. `' F ) ) ) )
76 nbrne1 
 |-  ( ( ( R ` G ) .<_ ( P .\/ ( R ` G ) ) /\ -. ( R ` G ) .<_ ( Q .\/ ( R ` ( G o. `' F ) ) ) ) -> ( P .\/ ( R ` G ) ) =/= ( Q .\/ ( R ` ( G o. `' F ) ) ) )
77 64 75 76 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( P .\/ ( R ` G ) ) =/= ( Q .\/ ( R ` ( G o. `' F ) ) ) )
78 3 4 36 5 2atmat0 
 |-  ( ( ( K e. HL /\ P e. A /\ ( R ` G ) e. A ) /\ ( Q e. A /\ ( R ` ( G o. `' F ) ) e. A /\ ( P .\/ ( R ` G ) ) =/= ( Q .\/ ( R ` ( G o. `' F ) ) ) ) ) -> ( ( ( P .\/ ( R ` G ) ) ./\ ( Q .\/ ( R ` ( G o. `' F ) ) ) ) e. A \/ ( ( P .\/ ( R ` G ) ) ./\ ( Q .\/ ( R ` ( G o. `' F ) ) ) ) = ( 0. ` K ) ) )
79 18 11 62 12 33 77 78 syl33anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( ( ( P .\/ ( R ` G ) ) ./\ ( Q .\/ ( R ` ( G o. `' F ) ) ) ) e. A \/ ( ( P .\/ ( R ` G ) ) ./\ ( Q .\/ ( R ` ( G o. `' F ) ) ) ) = ( 0. ` K ) ) )
80 9 eleq1i 
 |-  ( S e. A <-> ( ( P .\/ ( R ` G ) ) ./\ ( Q .\/ ( R ` ( G o. `' F ) ) ) ) e. A )
81 9 eqeq1i 
 |-  ( S = ( 0. ` K ) <-> ( ( P .\/ ( R ` G ) ) ./\ ( Q .\/ ( R ` ( G o. `' F ) ) ) ) = ( 0. ` K ) )
82 80 81 orbi12i 
 |-  ( ( S e. A \/ S = ( 0. ` K ) ) <-> ( ( ( P .\/ ( R ` G ) ) ./\ ( Q .\/ ( R ` ( G o. `' F ) ) ) ) e. A \/ ( ( P .\/ ( R ` G ) ) ./\ ( Q .\/ ( R ` ( G o. `' F ) ) ) ) = ( 0. ` K ) ) )
83 79 82 sylibr 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( S e. A \/ S = ( 0. ` K ) ) )
84 83 ord 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( -. S e. A -> S = ( 0. ` K ) ) )
85 84 necon1ad 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( S =/= ( 0. ` K ) -> S e. A ) )
86 59 85 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> S e. A )
87 simp21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( P e. A /\ -. P .<_ W ) )
88 87 65 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
89 1 2 3 4 5 6 7 8 9 36 cdlemh2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( S ./\ W ) = ( 0. ` K ) )
90 88 89 syld3an2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( S ./\ W ) = ( 0. ` K ) )
91 2 4 36 5 6 lhpmatb 
 |-  ( ( ( K e. HL /\ W e. H ) /\ S e. A ) -> ( -. S .<_ W <-> ( S ./\ W ) = ( 0. ` K ) ) )
92 18 21 86 91 syl21anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( -. S .<_ W <-> ( S ./\ W ) = ( 0. ` K ) ) )
93 90 92 mpbird 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> -. S .<_ W )
94 86 93 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ G e. T ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Q .<_ ( P .\/ ( R ` F ) ) ) /\ ( F =/= ( _I |` B ) /\ G =/= ( _I |` B ) /\ ( R ` F ) =/= ( R ` G ) ) ) -> ( S e. A /\ -. S .<_ W ) )