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Metamath Proof Explorer

Theorem 4atlem3

Description: Lemma for 4at . Break inequality into 4 cases. (Contributed by NM, 8-Jul-2012)

Ref Expression
Hypotheses 4at.l 
|- .<_ = ( le ` K )
4at.j 
|- .\/ = ( join ` K )
4at.a 
|- A = ( Atoms ` K )
Assertion 4atlem3 
|- ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( -. P .<_ ( ( T .\/ U ) .\/ V ) \/ -. Q .<_ ( ( T .\/ U ) .\/ V ) ) \/ ( -. R .<_ ( ( T .\/ U ) .\/ V ) \/ -. S .<_ ( ( T .\/ U ) .\/ V ) ) ) )

Proof

Step Hyp Ref Expression
1 4at.l 
 |-  .<_ = ( le ` K )
2 4at.j 
 |-  .\/ = ( join ` K )
3 4at.a 
 |-  A = ( Atoms ` K )
4 simpl11 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> K e. HL )
5 simpl1 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( K e. HL /\ P e. A /\ Q e. A ) )
6 simpl21 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> R e. A )
7 simpl22 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> S e. A )
8 simpr 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) )
9 eqid 
 |-  ( LVols ` K ) = ( LVols ` K )
10 1 2 3 9 lvoli2 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) e. ( LVols ` K ) )
11 5 6 7 8 10 syl121anc 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) e. ( LVols ` K ) )
12 simpl23 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> T e. A )
13 simpl3l 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> U e. A )
14 simpl3r 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> V e. A )
15 1 2 3 9 lvolnle3at 
 |-  ( ( ( K e. HL /\ ( ( ( P .\/ Q ) .\/ R ) .\/ S ) e. ( LVols ` K ) ) /\ ( T e. A /\ U e. A /\ V e. A ) ) -> -. ( ( ( P .\/ Q ) .\/ R ) .\/ S ) .<_ ( ( T .\/ U ) .\/ V ) )
16 4 11 12 13 14 15 syl23anc 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> -. ( ( ( P .\/ Q ) .\/ R ) .\/ S ) .<_ ( ( T .\/ U ) .\/ V ) )
17 4 hllatd 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> K e. Lat )
18 eqid 
 |-  ( Base ` K ) = ( Base ` K )
19 18 2 3 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
20 5 19 syl 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
21 18 2 3 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R .\/ S ) e. ( Base ` K ) )
22 4 6 7 21 syl3anc 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( R .\/ S ) e. ( Base ` K ) )
23 18 2 3 hlatjcl 
 |-  ( ( K e. HL /\ T e. A /\ U e. A ) -> ( T .\/ U ) e. ( Base ` K ) )
24 4 12 13 23 syl3anc 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( T .\/ U ) e. ( Base ` K ) )
25 18 3 atbase 
 |-  ( V e. A -> V e. ( Base ` K ) )
26 14 25 syl 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> V e. ( Base ` K ) )
27 18 2 latjcl 
 |-  ( ( K e. Lat /\ ( T .\/ U ) e. ( Base ` K ) /\ V e. ( Base ` K ) ) -> ( ( T .\/ U ) .\/ V ) e. ( Base ` K ) )
28 17 24 26 27 syl3anc 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( T .\/ U ) .\/ V ) e. ( Base ` K ) )
29 18 1 2 latjle12 
 |-  ( ( K e. Lat /\ ( ( P .\/ Q ) e. ( Base ` K ) /\ ( R .\/ S ) e. ( Base ` K ) /\ ( ( T .\/ U ) .\/ V ) e. ( Base ` K ) ) ) -> ( ( ( P .\/ Q ) .<_ ( ( T .\/ U ) .\/ V ) /\ ( R .\/ S ) .<_ ( ( T .\/ U ) .\/ V ) ) <-> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) .<_ ( ( T .\/ U ) .\/ V ) ) )
30 17 20 22 28 29 syl13anc 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( ( P .\/ Q ) .<_ ( ( T .\/ U ) .\/ V ) /\ ( R .\/ S ) .<_ ( ( T .\/ U ) .\/ V ) ) <-> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) .<_ ( ( T .\/ U ) .\/ V ) ) )
31 simpl12 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> P e. A )
32 18 3 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
33 31 32 syl 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> P e. ( Base ` K ) )
34 simpl13 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> Q e. A )
35 18 3 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
36 34 35 syl 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> Q e. ( Base ` K ) )
37 18 1 2 latjle12 
 |-  ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ Q e. ( Base ` K ) /\ ( ( T .\/ U ) .\/ V ) e. ( Base ` K ) ) ) -> ( ( P .<_ ( ( T .\/ U ) .\/ V ) /\ Q .<_ ( ( T .\/ U ) .\/ V ) ) <-> ( P .\/ Q ) .<_ ( ( T .\/ U ) .\/ V ) ) )
38 17 33 36 28 37 syl13anc 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( P .<_ ( ( T .\/ U ) .\/ V ) /\ Q .<_ ( ( T .\/ U ) .\/ V ) ) <-> ( P .\/ Q ) .<_ ( ( T .\/ U ) .\/ V ) ) )
39 18 3 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
40 6 39 syl 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> R e. ( Base ` K ) )
41 18 3 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
42 7 41 syl 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> S e. ( Base ` K ) )
43 18 1 2 latjle12 
 |-  ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ S e. ( Base ` K ) /\ ( ( T .\/ U ) .\/ V ) e. ( Base ` K ) ) ) -> ( ( R .<_ ( ( T .\/ U ) .\/ V ) /\ S .<_ ( ( T .\/ U ) .\/ V ) ) <-> ( R .\/ S ) .<_ ( ( T .\/ U ) .\/ V ) ) )
44 17 40 42 28 43 syl13anc 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( R .<_ ( ( T .\/ U ) .\/ V ) /\ S .<_ ( ( T .\/ U ) .\/ V ) ) <-> ( R .\/ S ) .<_ ( ( T .\/ U ) .\/ V ) ) )
45 38 44 anbi12d 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( ( P .<_ ( ( T .\/ U ) .\/ V ) /\ Q .<_ ( ( T .\/ U ) .\/ V ) ) /\ ( R .<_ ( ( T .\/ U ) .\/ V ) /\ S .<_ ( ( T .\/ U ) .\/ V ) ) ) <-> ( ( P .\/ Q ) .<_ ( ( T .\/ U ) .\/ V ) /\ ( R .\/ S ) .<_ ( ( T .\/ U ) .\/ V ) ) ) )
46 18 2 latjass 
 |-  ( ( K e. Lat /\ ( ( P .\/ Q ) e. ( Base ` K ) /\ R e. ( Base ` K ) /\ S e. ( Base ` K ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) = ( ( P .\/ Q ) .\/ ( R .\/ S ) ) )
47 17 20 40 42 46 syl13anc 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) = ( ( P .\/ Q ) .\/ ( R .\/ S ) ) )
48 47 breq1d 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( ( ( P .\/ Q ) .\/ R ) .\/ S ) .<_ ( ( T .\/ U ) .\/ V ) <-> ( ( P .\/ Q ) .\/ ( R .\/ S ) ) .<_ ( ( T .\/ U ) .\/ V ) ) )
49 30 45 48 3bitr4d 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( ( P .<_ ( ( T .\/ U ) .\/ V ) /\ Q .<_ ( ( T .\/ U ) .\/ V ) ) /\ ( R .<_ ( ( T .\/ U ) .\/ V ) /\ S .<_ ( ( T .\/ U ) .\/ V ) ) ) <-> ( ( ( P .\/ Q ) .\/ R ) .\/ S ) .<_ ( ( T .\/ U ) .\/ V ) ) )
50 16 49 mtbird 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> -. ( ( P .<_ ( ( T .\/ U ) .\/ V ) /\ Q .<_ ( ( T .\/ U ) .\/ V ) ) /\ ( R .<_ ( ( T .\/ U ) .\/ V ) /\ S .<_ ( ( T .\/ U ) .\/ V ) ) ) )
51 ianor 
 |-  ( -. ( ( P .<_ ( ( T .\/ U ) .\/ V ) /\ Q .<_ ( ( T .\/ U ) .\/ V ) ) /\ ( R .<_ ( ( T .\/ U ) .\/ V ) /\ S .<_ ( ( T .\/ U ) .\/ V ) ) ) <-> ( -. ( P .<_ ( ( T .\/ U ) .\/ V ) /\ Q .<_ ( ( T .\/ U ) .\/ V ) ) \/ -. ( R .<_ ( ( T .\/ U ) .\/ V ) /\ S .<_ ( ( T .\/ U ) .\/ V ) ) ) )
52 ianor 
 |-  ( -. ( P .<_ ( ( T .\/ U ) .\/ V ) /\ Q .<_ ( ( T .\/ U ) .\/ V ) ) <-> ( -. P .<_ ( ( T .\/ U ) .\/ V ) \/ -. Q .<_ ( ( T .\/ U ) .\/ V ) ) )
53 ianor 
 |-  ( -. ( R .<_ ( ( T .\/ U ) .\/ V ) /\ S .<_ ( ( T .\/ U ) .\/ V ) ) <-> ( -. R .<_ ( ( T .\/ U ) .\/ V ) \/ -. S .<_ ( ( T .\/ U ) .\/ V ) ) )
54 52 53 orbi12i 
 |-  ( ( -. ( P .<_ ( ( T .\/ U ) .\/ V ) /\ Q .<_ ( ( T .\/ U ) .\/ V ) ) \/ -. ( R .<_ ( ( T .\/ U ) .\/ V ) /\ S .<_ ( ( T .\/ U ) .\/ V ) ) ) <-> ( ( -. P .<_ ( ( T .\/ U ) .\/ V ) \/ -. Q .<_ ( ( T .\/ U ) .\/ V ) ) \/ ( -. R .<_ ( ( T .\/ U ) .\/ V ) \/ -. S .<_ ( ( T .\/ U ) .\/ V ) ) ) )
55 51 54 bitri 
 |-  ( -. ( ( P .<_ ( ( T .\/ U ) .\/ V ) /\ Q .<_ ( ( T .\/ U ) .\/ V ) ) /\ ( R .<_ ( ( T .\/ U ) .\/ V ) /\ S .<_ ( ( T .\/ U ) .\/ V ) ) ) <-> ( ( -. P .<_ ( ( T .\/ U ) .\/ V ) \/ -. Q .<_ ( ( T .\/ U ) .\/ V ) ) \/ ( -. R .<_ ( ( T .\/ U ) .\/ V ) \/ -. S .<_ ( ( T .\/ U ) .\/ V ) ) ) )
56 50 55 sylib 
 |-  ( ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( R e. A /\ S e. A /\ T e. A ) /\ ( U e. A /\ V e. A ) ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) /\ -. S .<_ ( ( P .\/ Q ) .\/ R ) ) ) -> ( ( -. P .<_ ( ( T .\/ U ) .\/ V ) \/ -. Q .<_ ( ( T .\/ U ) .\/ V ) ) \/ ( -. R .<_ ( ( T .\/ U ) .\/ V ) \/ -. S .<_ ( ( T .\/ U ) .\/ V ) ) ) )