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Metamath Proof Explorer

Theorem cdlemg27b

Description: TODO: Fix comment. (Contributed by NM, 28-May-2013)

Ref Expression
Hypotheses cdlemg12.l 
|- .<_ = ( le ` K )
cdlemg12.j 
|- .\/ = ( join ` K )
cdlemg12.m 
|- ./\ = ( meet ` K )
cdlemg12.a 
|- A = ( Atoms ` K )
cdlemg12.h 
|- H = ( LHyp ` K )
cdlemg12.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemg12b.r 
|- R = ( ( trL ` K ) ` W )
cdlemg31.n 
|- N = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) )
Assertion cdlemg27b 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> -. ( R ` F ) .<_ ( Q .\/ z ) )

Proof

Step Hyp Ref Expression
1 cdlemg12.l 
 |-  .<_ = ( le ` K )
2 cdlemg12.j 
 |-  .\/ = ( join ` K )
3 cdlemg12.m 
 |-  ./\ = ( meet ` K )
4 cdlemg12.a 
 |-  A = ( Atoms ` K )
5 cdlemg12.h 
 |-  H = ( LHyp ` K )
6 cdlemg12.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 cdlemg12b.r 
 |-  R = ( ( trL ` K ) ` W )
8 cdlemg31.n 
 |-  N = ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) )
9 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( K e. HL /\ W e. H ) )
10 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( P e. A /\ -. P .<_ W ) )
11 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( Q e. A /\ -. Q .<_ W ) )
12 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( v e. A /\ v .<_ W ) )
13 simp23l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> F e. T )
14 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> v =/= ( R ` F ) )
15 1 2 3 4 5 6 7 8 cdlemg31b0a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( v e. A /\ v .<_ W ) ) /\ ( F e. T /\ v =/= ( R ` F ) ) ) -> ( N e. A \/ N = ( 0. ` K ) ) )
16 9 10 11 12 13 14 15 syl132anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( N e. A \/ N = ( 0. ` K ) ) )
17 simp23r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> z =/= N )
18 17 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ ( N e. A \/ N = ( 0. ` K ) ) ) -> z =/= N )
19 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> K e. HL )
20 19 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N e. A ) -> K e. HL )
21 hlatl 
 |-  ( K e. HL -> K e. AtLat )
22 20 21 syl 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N e. A ) -> K e. AtLat )
23 simpl21 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N e. A ) -> z e. A )
24 simpr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N e. A ) -> N e. A )
25 1 4 atcmp 
 |-  ( ( K e. AtLat /\ z e. A /\ N e. A ) -> ( z .<_ N <-> z = N ) )
26 22 23 24 25 syl3anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N e. A ) -> ( z .<_ N <-> z = N ) )
27 26 necon3bbid 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N e. A ) -> ( -. z .<_ N <-> z =/= N ) )
28 19 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N = ( 0. ` K ) ) -> K e. HL )
29 28 21 syl 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N = ( 0. ` K ) ) -> K e. AtLat )
30 simpl21 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N = ( 0. ` K ) ) -> z e. A )
31 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
32 1 31 4 atnle0 
 |-  ( ( K e. AtLat /\ z e. A ) -> -. z .<_ ( 0. ` K ) )
33 29 30 32 syl2anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N = ( 0. ` K ) ) -> -. z .<_ ( 0. ` K ) )
34 simpr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N = ( 0. ` K ) ) -> N = ( 0. ` K ) )
35 34 breq2d 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N = ( 0. ` K ) ) -> ( z .<_ N <-> z .<_ ( 0. ` K ) ) )
36 33 35 mtbird 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N = ( 0. ` K ) ) -> -. z .<_ N )
37 17 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N = ( 0. ` K ) ) -> z =/= N )
38 36 37 2thd 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ N = ( 0. ` K ) ) -> ( -. z .<_ N <-> z =/= N ) )
39 27 38 jaodan 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ ( N e. A \/ N = ( 0. ` K ) ) ) -> ( -. z .<_ N <-> z =/= N ) )
40 18 39 mpbird 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) /\ ( N e. A \/ N = ( 0. ` K ) ) ) -> -. z .<_ N )
41 16 40 mpdan 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> -. z .<_ N )
42 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> z .<_ ( P .\/ v ) )
43 19 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> K e. Lat )
44 simp21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> z e. A )
45 eqid 
 |-  ( Base ` K ) = ( Base ` K )
46 45 4 atbase 
 |-  ( z e. A -> z e. ( Base ` K ) )
47 44 46 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> z e. ( Base ` K ) )
48 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> P e. A )
49 simp22l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> v e. A )
50 45 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ v e. A ) -> ( P .\/ v ) e. ( Base ` K ) )
51 19 48 49 50 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( P .\/ v ) e. ( Base ` K ) )
52 simp13l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> Q e. A )
53 simp33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( F ` P ) =/= P )
54 1 4 5 6 7 trlat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( F e. T /\ ( F ` P ) =/= P ) ) -> ( R ` F ) e. A )
55 9 10 13 53 54 syl112anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( R ` F ) e. A )
56 45 2 4 hlatjcl 
 |-  ( ( K e. HL /\ Q e. A /\ ( R ` F ) e. A ) -> ( Q .\/ ( R ` F ) ) e. ( Base ` K ) )
57 19 52 55 56 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( Q .\/ ( R ` F ) ) e. ( Base ` K ) )
58 45 1 3 latlem12 
 |-  ( ( K e. Lat /\ ( z e. ( Base ` K ) /\ ( P .\/ v ) e. ( Base ` K ) /\ ( Q .\/ ( R ` F ) ) e. ( Base ` K ) ) ) -> ( ( z .<_ ( P .\/ v ) /\ z .<_ ( Q .\/ ( R ` F ) ) ) <-> z .<_ ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) ) )
59 43 47 51 57 58 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( ( z .<_ ( P .\/ v ) /\ z .<_ ( Q .\/ ( R ` F ) ) ) <-> z .<_ ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) ) )
60 8 breq2i 
 |-  ( z .<_ N <-> z .<_ ( ( P .\/ v ) ./\ ( Q .\/ ( R ` F ) ) ) )
61 59 60 bitr4di 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( ( z .<_ ( P .\/ v ) /\ z .<_ ( Q .\/ ( R ` F ) ) ) <-> z .<_ N ) )
62 61 biimpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( ( z .<_ ( P .\/ v ) /\ z .<_ ( Q .\/ ( R ` F ) ) ) -> z .<_ N ) )
63 42 62 mpand 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( z .<_ ( Q .\/ ( R ` F ) ) -> z .<_ N ) )
64 41 63 mtod 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> -. z .<_ ( Q .\/ ( R ` F ) ) )
65 1 5 6 7 trlle 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T ) -> ( R ` F ) .<_ W )
66 9 13 65 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( R ` F ) .<_ W )
67 simp13r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> -. Q .<_ W )
68 nbrne2 
 |-  ( ( ( R ` F ) .<_ W /\ -. Q .<_ W ) -> ( R ` F ) =/= Q )
69 66 67 68 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( R ` F ) =/= Q )
70 1 2 4 hlatexch1 
 |-  ( ( K e. HL /\ ( ( R ` F ) e. A /\ z e. A /\ Q e. A ) /\ ( R ` F ) =/= Q ) -> ( ( R ` F ) .<_ ( Q .\/ z ) -> z .<_ ( Q .\/ ( R ` F ) ) ) )
71 19 55 44 52 69 70 syl131anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> ( ( R ` F ) .<_ ( Q .\/ z ) -> z .<_ ( Q .\/ ( R ` F ) ) ) )
72 64 71 mtod 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( z e. A /\ ( v e. A /\ v .<_ W ) /\ ( F e. T /\ z =/= N ) ) /\ ( v =/= ( R ` F ) /\ z .<_ ( P .\/ v ) /\ ( F ` P ) =/= P ) ) -> -. ( R ` F ) .<_ ( Q .\/ z ) )