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Metamath Proof Explorer

Theorem cdleme11c

Description: Part of proof of Lemma E in Crawley p. 113. Lemma leading to cdleme11 . (Contributed by NM, 13-Jun-2012)

Ref Expression
Hypotheses cdleme11.l 
|- .<_ = ( le ` K )
cdleme11.j 
|- .\/ = ( join ` K )
cdleme11.m 
|- ./\ = ( meet ` K )
cdleme11.a 
|- A = ( Atoms ` K )
cdleme11.h 
|- H = ( LHyp ` K )
cdleme11.u 
|- U = ( ( P .\/ Q ) ./\ W )
Assertion cdleme11c 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> -. P .<_ ( S .\/ T ) )

Proof

Step Hyp Ref Expression
1 cdleme11.l 
 |-  .<_ = ( le ` K )
2 cdleme11.j 
 |-  .\/ = ( join ` K )
3 cdleme11.m 
 |-  ./\ = ( meet ` K )
4 cdleme11.a 
 |-  A = ( Atoms ` K )
5 cdleme11.h 
 |-  H = ( LHyp ` K )
6 cdleme11.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 simp3l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> -. S .<_ ( P .\/ Q ) )
8 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> K e. HL )
9 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> P e. A )
10 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( K e. HL /\ W e. H ) )
11 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( P e. A /\ -. P .<_ W ) )
12 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> Q e. A )
13 simp23 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> P =/= Q )
14 1 2 3 4 5 6 lhpat2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> U e. A )
15 10 11 12 13 14 syl112anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> U e. A )
16 1 2 4 hlatlej1 
 |-  ( ( K e. HL /\ P e. A /\ U e. A ) -> P .<_ ( P .\/ U ) )
17 8 9 15 16 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> P .<_ ( P .\/ U ) )
18 17 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) /\ P .<_ ( S .\/ T ) ) -> P .<_ ( P .\/ U ) )
19 12 13 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( Q e. A /\ P =/= Q ) )
20 simp21 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( S e. A /\ -. S .<_ W ) )
21 simp22 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> T e. A )
22 simp3r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> U .<_ ( S .\/ T ) )
23 21 22 jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( T e. A /\ U .<_ ( S .\/ T ) ) )
24 1 2 3 4 5 6 cdleme11a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ U .<_ ( S .\/ T ) ) ) ) -> ( S .\/ U ) = ( S .\/ T ) )
25 10 11 19 20 23 24 syl122anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( S .\/ U ) = ( S .\/ T ) )
26 25 breq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( P .<_ ( S .\/ U ) <-> P .<_ ( S .\/ T ) ) )
27 simp21l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> S e. A )
28 1 2 3 4 5 6 cdleme0b 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) -> U =/= P )
29 10 11 12 28 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> U =/= P )
30 29 necomd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> P =/= U )
31 1 2 4 hlatexch2 
 |-  ( ( K e. HL /\ ( P e. A /\ S e. A /\ U e. A ) /\ P =/= U ) -> ( P .<_ ( S .\/ U ) -> S .<_ ( P .\/ U ) ) )
32 8 9 27 15 30 31 syl131anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( P .<_ ( S .\/ U ) -> S .<_ ( P .\/ U ) ) )
33 26 32 sylbird 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( P .<_ ( S .\/ T ) -> S .<_ ( P .\/ U ) ) )
34 33 imp 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) /\ P .<_ ( S .\/ T ) ) -> S .<_ ( P .\/ U ) )
35 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> Q .<_ ( P .\/ Q ) )
36 8 9 12 35 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> Q .<_ ( P .\/ Q ) )
37 1 2 3 4 5 6 cdleme0cp 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) ) -> ( P .\/ U ) = ( P .\/ Q ) )
38 10 11 12 37 syl12anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( P .\/ U ) = ( P .\/ Q ) )
39 36 38 breqtrrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> Q .<_ ( P .\/ U ) )
40 39 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) /\ P .<_ ( S .\/ T ) ) -> Q .<_ ( P .\/ U ) )
41 8 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> K e. Lat )
42 eqid 
 |-  ( Base ` K ) = ( Base ` K )
43 42 4 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
44 27 43 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> S e. ( Base ` K ) )
45 42 4 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
46 12 45 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> Q e. ( Base ` K ) )
47 42 2 4 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ U e. A ) -> ( P .\/ U ) e. ( Base ` K ) )
48 8 9 15 47 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( P .\/ U ) e. ( Base ` K ) )
49 42 1 2 latjle12 
 |-  ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ Q e. ( Base ` K ) /\ ( P .\/ U ) e. ( Base ` K ) ) ) -> ( ( S .<_ ( P .\/ U ) /\ Q .<_ ( P .\/ U ) ) <-> ( S .\/ Q ) .<_ ( P .\/ U ) ) )
50 41 44 46 48 49 syl13anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( ( S .<_ ( P .\/ U ) /\ Q .<_ ( P .\/ U ) ) <-> ( S .\/ Q ) .<_ ( P .\/ U ) ) )
51 50 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) /\ P .<_ ( S .\/ T ) ) -> ( ( S .<_ ( P .\/ U ) /\ Q .<_ ( P .\/ U ) ) <-> ( S .\/ Q ) .<_ ( P .\/ U ) ) )
52 34 40 51 mpbi2and 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) /\ P .<_ ( S .\/ T ) ) -> ( S .\/ Q ) .<_ ( P .\/ U ) )
53 42 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
54 9 53 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> P e. ( Base ` K ) )
55 42 1 2 latnlej1r 
 |-  ( ( K e. Lat /\ ( S e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. S .<_ ( P .\/ Q ) ) -> S =/= Q )
56 41 44 54 46 7 55 syl131anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> S =/= Q )
57 1 2 4 ps-1 
 |-  ( ( K e. HL /\ ( S e. A /\ Q e. A /\ S =/= Q ) /\ ( P e. A /\ U e. A ) ) -> ( ( S .\/ Q ) .<_ ( P .\/ U ) <-> ( S .\/ Q ) = ( P .\/ U ) ) )
58 8 27 12 56 9 15 57 syl132anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( ( S .\/ Q ) .<_ ( P .\/ U ) <-> ( S .\/ Q ) = ( P .\/ U ) ) )
59 58 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) /\ P .<_ ( S .\/ T ) ) -> ( ( S .\/ Q ) .<_ ( P .\/ U ) <-> ( S .\/ Q ) = ( P .\/ U ) ) )
60 52 59 mpbid 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) /\ P .<_ ( S .\/ T ) ) -> ( S .\/ Q ) = ( P .\/ U ) )
61 18 60 breqtrrd 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) /\ P .<_ ( S .\/ T ) ) -> P .<_ ( S .\/ Q ) )
62 61 ex 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( P .<_ ( S .\/ T ) -> P .<_ ( S .\/ Q ) ) )
63 1 2 4 hlatexch2 
 |-  ( ( K e. HL /\ ( P e. A /\ S e. A /\ Q e. A ) /\ P =/= Q ) -> ( P .<_ ( S .\/ Q ) -> S .<_ ( P .\/ Q ) ) )
64 8 9 27 12 13 63 syl131anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( P .<_ ( S .\/ Q ) -> S .<_ ( P .\/ Q ) ) )
65 62 64 syld 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( P .<_ ( S .\/ T ) -> S .<_ ( P .\/ Q ) ) )
66 7 65 mtod 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> -. P .<_ ( S .\/ T ) )