Metamath Proof Explorer

Theorem 4atlem11b

Description: Lemma for 4at . Substitute U for Q (cont.). (Contributed by NM, 10-Jul-2012)

Ref Expression
Hypotheses 4at.l 
|- .<_ = ( le ` K )
4at.j 
|- .\/ = ( join ` K )
4at.a 
|- A = ( Atoms ` K )
Assertion 4atlem11b 
|- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( P .\/ U ) .\/ ( V .\/ W ) ) )

Proof

Step Hyp Ref Expression
1 4at.l 
 |-  .<_ = ( le ` K )
2 4at.j 
 |-  .\/ = ( join ` K )
3 4at.a 
 |-  A = ( Atoms ` K )
4 simp11 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> ( K e. HL /\ P e. A /\ Q e. A ) )
5 simp12 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> ( R e. A /\ S e. A ) )
6 simp132 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> V e. A )
7 simp133 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> W e. A )
8 5 6 7 3jca 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> ( ( R e. A /\ S e. A ) /\ V e. A /\ W e. A ) )
9 simp2l 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) )
10 4 8 9 3jca 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( ( R e. A /\ S e. A ) /\ V e. A /\ W e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) )
11 simp32 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) )
12 simp33 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) )
13 simp111 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> K e. HL )
14 13 hllatd 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> K e. Lat )
15 simp12l 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> R e. A )
16 eqid 
 |-  ( Base ` K ) = ( Base ` K )
17 16 3 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
18 15 17 syl 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> R e. ( Base ` K ) )
19 simp12r 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> S e. A )
20 16 3 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
21 19 20 syl 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> S e. ( Base ` K ) )
22 simp112 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> P e. A )
23 simp131 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> U e. A )
24 16 2 3 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ U e. A ) -> ( P .\/ U ) e. ( Base ` K ) )
25 13 22 23 24 syl3anc 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> ( P .\/ U ) e. ( Base ` K ) )
26 16 2 3 hlatjcl 
 |-  ( ( K e. HL /\ V e. A /\ W e. A ) -> ( V .\/ W ) e. ( Base ` K ) )
27 13 6 7 26 syl3anc 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> ( V .\/ W ) e. ( Base ` K ) )
28 16 2 latjcl 
 |-  ( ( K e. Lat /\ ( P .\/ U ) e. ( Base ` K ) /\ ( V .\/ W ) e. ( Base ` K ) ) -> ( ( P .\/ U ) .\/ ( V .\/ W ) ) e. ( Base ` K ) )
29 14 25 27 28 syl3anc 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> ( ( P .\/ U ) .\/ ( V .\/ W ) ) e. ( Base ` K ) )
30 16 1 2 latjle12 
 |-  ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ S e. ( Base ` K ) /\ ( ( P .\/ U ) .\/ ( V .\/ W ) ) e. ( Base ` K ) ) ) -> ( ( R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) <-> ( R .\/ S ) .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) )
31 14 18 21 29 30 syl13anc 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> ( ( R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) <-> ( R .\/ S ) .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) )
32 11 12 31 mpbi2and 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> ( R .\/ S ) .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) )
33 simp31 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) )
34 simp13 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> ( U e. A /\ V e. A /\ W e. A ) )
35 simp2r 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> -. Q .<_ ( ( P .\/ V ) .\/ W ) )
36 1 2 3 4atlem11a 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) -> ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) <-> ( ( P .\/ Q ) .\/ ( V .\/ W ) ) = ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) )
37 4 34 35 36 syl3anc 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) <-> ( ( P .\/ Q ) .\/ ( V .\/ W ) ) = ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) )
38 33 37 mpbid 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> ( ( P .\/ Q ) .\/ ( V .\/ W ) ) = ( ( P .\/ U ) .\/ ( V .\/ W ) ) )
39 32 38 breqtrrd 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> ( R .\/ S ) .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) )
40 1 2 3 4atlem10 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( ( R e. A /\ S e. A ) /\ V e. A /\ W e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( R .\/ S ) .<_ ( ( P .\/ Q ) .\/ ( V .\/ W ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( P .\/ Q ) .\/ ( V .\/ W ) ) ) )
41 10 39 40 sylc 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( P .\/ Q ) .\/ ( V .\/ W ) ) )
42 41 38 eqtrd 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) ) /\ ( ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) /\ ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ R .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) /\ S .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) ) -> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) = ( ( P .\/ U ) .\/ ( V .\/ W ) ) )