Metamath Proof Explorer

Theorem cdleme11

Description: Part of proof of Lemma E in Crawley p. 113, 1st sentence of 3rd paragraph on p. 114. F and G represent f(s) and f(t) respectively. Their proof provides no details of our cdleme11a through cdleme11 , so there may be a simpler proof that we have overlooked. (Contributed by NM, 15-Jun-2012)

Ref Expression
Hypotheses cdleme12.l 
|- .<_ = ( le ` K )
cdleme12.j 
|- .\/ = ( join ` K )
cdleme12.m 
|- ./\ = ( meet ` K )
cdleme12.a 
|- A = ( Atoms ` K )
cdleme12.h 
|- H = ( LHyp ` K )
cdleme12.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme12.f 
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme12.g 
|- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
Assertion cdleme11 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( F .\/ G ) = ( S .\/ T ) )

Proof

Step Hyp Ref Expression
1 cdleme12.l 
 |-  .<_ = ( le ` K )
2 cdleme12.j 
 |-  .\/ = ( join ` K )
3 cdleme12.m 
 |-  ./\ = ( meet ` K )
4 cdleme12.a 
 |-  A = ( Atoms ` K )
5 cdleme12.h 
 |-  H = ( LHyp ` K )
6 cdleme12.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme12.f 
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8 cdleme12.g 
 |-  G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
9 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> K e. HL )
10 9 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> K e. Lat )
11 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( K e. HL /\ W e. H ) )
12 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> P e. A )
13 simp13l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> Q e. A )
14 eqid 
 |-  ( Base ` K ) = ( Base ` K )
15 1 2 3 4 5 6 14 cdleme0aa 
 |-  ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) -> U e. ( Base ` K ) )
16 11 12 13 15 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> U e. ( Base ` K ) )
17 14 2 latjidm 
 |-  ( ( K e. Lat /\ U e. ( Base ` K ) ) -> ( U .\/ U ) = U )
18 10 16 17 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( U .\/ U ) = U )
19 18 oveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( ( S .\/ T ) .\/ ( U .\/ U ) ) = ( ( S .\/ T ) .\/ U ) )
20 simp33 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> U .<_ ( S .\/ T ) )
21 simp21l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> S e. A )
22 14 4 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
23 21 22 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> S e. ( Base ` K ) )
24 simp22l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> T e. A )
25 14 4 atbase 
 |-  ( T e. A -> T e. ( Base ` K ) )
26 24 25 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> T e. ( Base ` K ) )
27 14 2 latjcl 
 |-  ( ( K e. Lat /\ S e. ( Base ` K ) /\ T e. ( Base ` K ) ) -> ( S .\/ T ) e. ( Base ` K ) )
28 10 23 26 27 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( S .\/ T ) e. ( Base ` K ) )
29 14 1 2 latleeqj2 
 |-  ( ( K e. Lat /\ U e. ( Base ` K ) /\ ( S .\/ T ) e. ( Base ` K ) ) -> ( U .<_ ( S .\/ T ) <-> ( ( S .\/ T ) .\/ U ) = ( S .\/ T ) ) )
30 10 16 28 29 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( U .<_ ( S .\/ T ) <-> ( ( S .\/ T ) .\/ U ) = ( S .\/ T ) ) )
31 20 30 mpbid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( ( S .\/ T ) .\/ U ) = ( S .\/ T ) )
32 19 31 eqtr2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( S .\/ T ) = ( ( S .\/ T ) .\/ ( U .\/ U ) ) )
33 simp21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( S e. A /\ -. S .<_ W ) )
34 1 2 3 4 5 6 7 cdleme1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) ) -> ( S .\/ F ) = ( S .\/ U ) )
35 11 12 13 33 34 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( S .\/ F ) = ( S .\/ U ) )
36 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( T e. A /\ -. T .<_ W ) )
37 1 2 3 4 5 6 8 cdleme1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( T e. A /\ -. T .<_ W ) ) ) -> ( T .\/ G ) = ( T .\/ U ) )
38 11 12 13 36 37 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( T .\/ G ) = ( T .\/ U ) )
39 35 38 oveq12d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( ( S .\/ F ) .\/ ( T .\/ G ) ) = ( ( S .\/ U ) .\/ ( T .\/ U ) ) )
40 14 2 latj4 
 |-  ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ T e. ( Base ` K ) ) /\ ( U e. ( Base ` K ) /\ U e. ( Base ` K ) ) ) -> ( ( S .\/ T ) .\/ ( U .\/ U ) ) = ( ( S .\/ U ) .\/ ( T .\/ U ) ) )
41 10 23 26 16 16 40 syl122anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( ( S .\/ T ) .\/ ( U .\/ U ) ) = ( ( S .\/ U ) .\/ ( T .\/ U ) ) )
42 39 41 eqtr4d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( ( S .\/ F ) .\/ ( T .\/ G ) ) = ( ( S .\/ T ) .\/ ( U .\/ U ) ) )
43 32 42 eqtr4d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( S .\/ T ) = ( ( S .\/ F ) .\/ ( T .\/ G ) ) )
44 1 2 3 4 5 6 7 14 cdleme1b 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ S e. A ) ) -> F e. ( Base ` K ) )
45 11 12 13 21 44 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> F e. ( Base ` K ) )
46 1 2 3 4 5 6 8 14 cdleme1b 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ T e. A ) ) -> G e. ( Base ` K ) )
47 11 12 13 24 46 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> G e. ( Base ` K ) )
48 14 2 latj4 
 |-  ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ F e. ( Base ` K ) ) /\ ( T e. ( Base ` K ) /\ G e. ( Base ` K ) ) ) -> ( ( S .\/ F ) .\/ ( T .\/ G ) ) = ( ( S .\/ T ) .\/ ( F .\/ G ) ) )
49 10 23 45 26 47 48 syl122anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( ( S .\/ F ) .\/ ( T .\/ G ) ) = ( ( S .\/ T ) .\/ ( F .\/ G ) ) )
50 43 49 eqtr2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( ( S .\/ T ) .\/ ( F .\/ G ) ) = ( S .\/ T ) )
51 14 2 latjcl 
 |-  ( ( K e. Lat /\ F e. ( Base ` K ) /\ G e. ( Base ` K ) ) -> ( F .\/ G ) e. ( Base ` K ) )
52 10 45 47 51 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( F .\/ G ) e. ( Base ` K ) )
53 14 1 2 latleeqj2 
 |-  ( ( K e. Lat /\ ( F .\/ G ) e. ( Base ` K ) /\ ( S .\/ T ) e. ( Base ` K ) ) -> ( ( F .\/ G ) .<_ ( S .\/ T ) <-> ( ( S .\/ T ) .\/ ( F .\/ G ) ) = ( S .\/ T ) ) )
54 10 52 28 53 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( ( F .\/ G ) .<_ ( S .\/ T ) <-> ( ( S .\/ T ) .\/ ( F .\/ G ) ) = ( S .\/ T ) ) )
55 50 54 mpbird 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( F .\/ G ) .<_ ( S .\/ T ) )
56 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( P e. A /\ -. P .<_ W ) )
57 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
58 simp23l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> P =/= Q )
59 simp31 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> -. S .<_ ( P .\/ Q ) )
60 1 2 3 4 5 6 7 cdleme3fa 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> F e. A )
61 11 56 57 33 58 59 60 syl132anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> F e. A )
62 simp32 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> -. T .<_ ( P .\/ Q ) )
63 1 2 3 4 5 6 8 cdleme3fa 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ -. T .<_ ( P .\/ Q ) ) ) -> G e. A )
64 11 56 57 36 58 62 63 syl132anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> G e. A )
65 1 2 3 4 5 6 7 8 cdleme11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> F =/= G )
66 1 2 4 ps-1 
 |-  ( ( K e. HL /\ ( F e. A /\ G e. A /\ F =/= G ) /\ ( S e. A /\ T e. A ) ) -> ( ( F .\/ G ) .<_ ( S .\/ T ) <-> ( F .\/ G ) = ( S .\/ T ) ) )
67 9 61 64 65 21 24 66 syl132anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( ( F .\/ G ) .<_ ( S .\/ T ) <-> ( F .\/ G ) = ( S .\/ T ) ) )
68 55 67 mpbid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( F .\/ G ) = ( S .\/ T ) )