Metamath Proof Explorer

Theorem cdleme35b

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT. (Contributed by NM, 10-Mar-2013)

Ref Expression
Hypotheses cdleme35.l 
|- .<_ = ( le ` K )
cdleme35.j 
|- .\/ = ( join ` K )
cdleme35.m 
|- ./\ = ( meet ` K )
cdleme35.a 
|- A = ( Atoms ` K )
cdleme35.h 
|- H = ( LHyp ` K )
cdleme35.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme35.f 
|- F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
Assertion cdleme35b 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( Q .\/ ( ( P .\/ R ) ./\ W ) ) .<_ ( Q .\/ ( R .\/ U ) ) )

Proof

Step Hyp Ref Expression
1 cdleme35.l 
 |-  .<_ = ( le ` K )
2 cdleme35.j 
 |-  .\/ = ( join ` K )
3 cdleme35.m 
 |-  ./\ = ( meet ` K )
4 cdleme35.a 
 |-  A = ( Atoms ` K )
5 cdleme35.h 
 |-  H = ( LHyp ` K )
6 cdleme35.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme35.f 
 |-  F = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
8 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> K e. HL )
9 8 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> K e. Lat )
10 simp13l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> Q e. A )
11 eqid 
 |-  ( Base ` K ) = ( Base ` K )
12 11 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
13 10 12 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> Q e. ( Base ` K ) )
14 simp2rl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> R e. A )
15 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( K e. HL /\ W e. H ) )
16 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( P e. A /\ -. P .<_ W ) )
17 simp2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> P =/= Q )
18 1 2 3 4 5 6 cdleme0a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> U e. A )
19 15 16 10 17 18 syl112anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> U e. A )
20 11 2 4 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ U e. A ) -> ( R .\/ U ) e. ( Base ` K ) )
21 8 14 19 20 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( R .\/ U ) e. ( Base ` K ) )
22 11 1 2 latlej1 
 |-  ( ( K e. Lat /\ Q e. ( Base ` K ) /\ ( R .\/ U ) e. ( Base ` K ) ) -> Q .<_ ( Q .\/ ( R .\/ U ) ) )
23 9 13 21 22 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> Q .<_ ( Q .\/ ( R .\/ U ) ) )
24 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> P e. A )
25 11 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
26 24 25 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> P e. ( Base ` K ) )
27 11 4 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
28 14 27 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> R e. ( Base ` K ) )
29 11 2 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> ( P .\/ R ) e. ( Base ` K ) )
30 9 26 28 29 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( P .\/ R ) e. ( Base ` K ) )
31 simp11r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> W e. H )
32 11 5 lhpbase 
 |-  ( W e. H -> W e. ( Base ` K ) )
33 31 32 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> W e. ( Base ` K ) )
34 11 3 latmcl 
 |-  ( ( K e. Lat /\ ( P .\/ R ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ R ) ./\ W ) e. ( Base ` K ) )
35 9 30 33 34 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( P .\/ R ) ./\ W ) e. ( Base ` K ) )
36 11 2 latjcl 
 |-  ( ( K e. Lat /\ ( P .\/ R ) e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( ( P .\/ R ) .\/ Q ) e. ( Base ` K ) )
37 9 30 13 36 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( P .\/ R ) .\/ Q ) e. ( Base ` K ) )
38 11 1 3 latmle1 
 |-  ( ( K e. Lat /\ ( P .\/ R ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ R ) ./\ W ) .<_ ( P .\/ R ) )
39 9 30 33 38 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( P .\/ R ) ./\ W ) .<_ ( P .\/ R ) )
40 11 1 2 latlej1 
 |-  ( ( K e. Lat /\ ( P .\/ R ) e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( P .\/ R ) .<_ ( ( P .\/ R ) .\/ Q ) )
41 9 30 13 40 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( P .\/ R ) .<_ ( ( P .\/ R ) .\/ Q ) )
42 11 1 9 35 30 37 39 41 lattrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( P .\/ R ) ./\ W ) .<_ ( ( P .\/ R ) .\/ Q ) )
43 6 oveq2i 
 |-  ( Q .\/ U ) = ( Q .\/ ( ( P .\/ Q ) ./\ W ) )
44 11 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
45 8 24 10 44 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
46 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> Q .<_ ( P .\/ Q ) )
47 8 24 10 46 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> Q .<_ ( P .\/ Q ) )
48 11 1 2 3 4 atmod3i1 
 |-  ( ( K e. HL /\ ( Q e. A /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) /\ Q .<_ ( P .\/ Q ) ) -> ( Q .\/ ( ( P .\/ Q ) ./\ W ) ) = ( ( P .\/ Q ) ./\ ( Q .\/ W ) ) )
49 8 10 45 33 47 48 syl131anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( Q .\/ ( ( P .\/ Q ) ./\ W ) ) = ( ( P .\/ Q ) ./\ ( Q .\/ W ) ) )
50 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( Q e. A /\ -. Q .<_ W ) )
51 eqid 
 |-  ( 1. ` K ) = ( 1. ` K )
52 1 2 51 4 5 lhpjat2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( Q .\/ W ) = ( 1. ` K ) )
53 15 50 52 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( Q .\/ W ) = ( 1. ` K ) )
54 53 oveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( P .\/ Q ) ./\ ( Q .\/ W ) ) = ( ( P .\/ Q ) ./\ ( 1. ` K ) ) )
55 hlol 
 |-  ( K e. HL -> K e. OL )
56 8 55 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> K e. OL )
57 11 3 51 olm11 
 |-  ( ( K e. OL /\ ( P .\/ Q ) e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ ( 1. ` K ) ) = ( P .\/ Q ) )
58 56 45 57 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( P .\/ Q ) ./\ ( 1. ` K ) ) = ( P .\/ Q ) )
59 49 54 58 3eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( Q .\/ ( ( P .\/ Q ) ./\ W ) ) = ( P .\/ Q ) )
60 43 59 syl5eq 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( Q .\/ U ) = ( P .\/ Q ) )
61 60 oveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( R .\/ ( Q .\/ U ) ) = ( R .\/ ( P .\/ Q ) ) )
62 2 4 hlatj12 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ U e. A ) ) -> ( Q .\/ ( R .\/ U ) ) = ( R .\/ ( Q .\/ U ) ) )
63 8 10 14 19 62 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( Q .\/ ( R .\/ U ) ) = ( R .\/ ( Q .\/ U ) ) )
64 2 4 hlatjcom 
 |-  ( ( K e. HL /\ P e. A /\ R e. A ) -> ( P .\/ R ) = ( R .\/ P ) )
65 8 24 14 64 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( P .\/ R ) = ( R .\/ P ) )
66 65 oveq1d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( P .\/ R ) .\/ Q ) = ( ( R .\/ P ) .\/ Q ) )
67 2 4 hlatjass 
 |-  ( ( K e. HL /\ ( R e. A /\ P e. A /\ Q e. A ) ) -> ( ( R .\/ P ) .\/ Q ) = ( R .\/ ( P .\/ Q ) ) )
68 8 14 24 10 67 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( R .\/ P ) .\/ Q ) = ( R .\/ ( P .\/ Q ) ) )
69 66 68 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( P .\/ R ) .\/ Q ) = ( R .\/ ( P .\/ Q ) ) )
70 61 63 69 3eqtr4rd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( P .\/ R ) .\/ Q ) = ( Q .\/ ( R .\/ U ) ) )
71 42 70 breqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( P .\/ R ) ./\ W ) .<_ ( Q .\/ ( R .\/ U ) ) )
72 11 2 latjcl 
 |-  ( ( K e. Lat /\ Q e. ( Base ` K ) /\ ( R .\/ U ) e. ( Base ` K ) ) -> ( Q .\/ ( R .\/ U ) ) e. ( Base ` K ) )
73 9 13 21 72 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( Q .\/ ( R .\/ U ) ) e. ( Base ` K ) )
74 11 1 2 latjle12 
 |-  ( ( K e. Lat /\ ( Q e. ( Base ` K ) /\ ( ( P .\/ R ) ./\ W ) e. ( Base ` K ) /\ ( Q .\/ ( R .\/ U ) ) e. ( Base ` K ) ) ) -> ( ( Q .<_ ( Q .\/ ( R .\/ U ) ) /\ ( ( P .\/ R ) ./\ W ) .<_ ( Q .\/ ( R .\/ U ) ) ) <-> ( Q .\/ ( ( P .\/ R ) ./\ W ) ) .<_ ( Q .\/ ( R .\/ U ) ) ) )
75 9 13 35 73 74 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( Q .<_ ( Q .\/ ( R .\/ U ) ) /\ ( ( P .\/ R ) ./\ W ) .<_ ( Q .\/ ( R .\/ U ) ) ) <-> ( Q .\/ ( ( P .\/ R ) ./\ W ) ) .<_ ( Q .\/ ( R .\/ U ) ) ) )
76 23 71 75 mpbi2and 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ -. R .<_ ( P .\/ Q ) ) -> ( Q .\/ ( ( P .\/ R ) ./\ W ) ) .<_ ( Q .\/ ( R .\/ U ) ) )