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Metamath Proof Explorer

Theorem 3atlem5

Description: Lemma for 3at . (Contributed by NM, 23-Jun-2012)

Ref Expression
Hypotheses 3at.l 
|- .<_ = ( le ` K )
3at.j 
|- .\/ = ( join ` K )
3at.a 
|- A = ( Atoms ` K )
Assertion 3atlem5 
|- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ U ) )

Proof

Step Hyp Ref Expression
1 3at.l 
 |-  .<_ = ( le ` K )
2 3at.j 
 |-  .\/ = ( join ` K )
3 3at.a 
 |-  A = ( Atoms ` K )
4 oveq2 
 |-  ( U = P -> ( ( S .\/ T ) .\/ U ) = ( ( S .\/ T ) .\/ P ) )
5 4 eqcoms 
 |-  ( P = U -> ( ( S .\/ T ) .\/ U ) = ( ( S .\/ T ) .\/ P ) )
6 5 breq2d 
 |-  ( P = U -> ( ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) <-> ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ P ) ) )
7 5 eqeq2d 
 |-  ( P = U -> ( ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ U ) <-> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ P ) ) )
8 6 7 imbi12d 
 |-  ( P = U -> ( ( ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ U ) ) <-> ( ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ P ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ P ) ) ) )
9 simp1l 
 |-  ( ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) ) /\ P =/= U /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) )
10 simp1r1 
 |-  ( ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) ) /\ P =/= U /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> -. R .<_ ( P .\/ Q ) )
11 simp2 
 |-  ( ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) ) /\ P =/= U /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> P =/= U )
12 simp1r3 
 |-  ( ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) ) /\ P =/= U /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> -. Q .<_ ( P .\/ U ) )
13 simp3 
 |-  ( ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) ) /\ P =/= U /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) )
14 1 2 3 3atlem3 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= U /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ U ) )
15 9 10 11 12 13 14 syl131anc 
 |-  ( ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) ) /\ P =/= U /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ U ) )
16 15 3expia 
 |-  ( ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) ) /\ P =/= U ) -> ( ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ U ) ) )
17 simp11 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) ) -> K e. HL )
18 simp123 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) ) -> R e. A )
19 simp122 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) ) -> Q e. A )
20 simp121 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) ) -> P e. A )
21 18 19 20 3jca 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) ) -> ( R e. A /\ Q e. A /\ P e. A ) )
22 simp131 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) ) -> S e. A )
23 simp132 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) ) -> T e. A )
24 22 23 jca 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) ) -> ( S e. A /\ T e. A ) )
25 simp21 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) ) -> -. R .<_ ( P .\/ Q ) )
26 simp22 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) ) -> P =/= Q )
27 1 2 3 hlatexch2 
 |-  ( ( K e. HL /\ ( P e. A /\ R e. A /\ Q e. A ) /\ P =/= Q ) -> ( P .<_ ( R .\/ Q ) -> R .<_ ( P .\/ Q ) ) )
28 17 20 18 19 26 27 syl131anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) ) -> ( P .<_ ( R .\/ Q ) -> R .<_ ( P .\/ Q ) ) )
29 25 28 mtod 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) ) -> -. P .<_ ( R .\/ Q ) )
30 17 hllatd 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) ) -> K e. Lat )
31 eqid 
 |-  ( Base ` K ) = ( Base ` K )
32 31 3 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
33 18 32 syl 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) ) -> R e. ( Base ` K ) )
34 31 3 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
35 20 34 syl 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) ) -> P e. ( Base ` K ) )
36 31 3 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
37 19 36 syl 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) ) -> Q e. ( Base ` K ) )
38 31 1 2 latnlej1r 
 |-  ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) /\ -. R .<_ ( P .\/ Q ) ) -> R =/= Q )
39 30 33 35 37 25 38 syl131anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) ) -> R =/= Q )
40 simp3 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) ) -> ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) )
41 1 2 3 3atlem4 
 |-  ( ( ( K e. HL /\ ( R e. A /\ Q e. A /\ P e. A ) /\ ( S e. A /\ T e. A ) ) /\ ( -. P .<_ ( R .\/ Q ) /\ R =/= Q ) /\ ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) ) -> ( ( R .\/ Q ) .\/ P ) = ( ( S .\/ T ) .\/ P ) )
42 17 21 24 29 39 40 41 syl321anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) ) -> ( ( R .\/ Q ) .\/ P ) = ( ( S .\/ T ) .\/ P ) )
43 42 3expia 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) ) -> ( ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) -> ( ( R .\/ Q ) .\/ P ) = ( ( S .\/ T ) .\/ P ) ) )
44 simpl1 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) ) -> K e. HL )
45 44 hllatd 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) ) -> K e. Lat )
46 simpl21 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) ) -> P e. A )
47 46 34 syl 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) ) -> P e. ( Base ` K ) )
48 simpl22 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) ) -> Q e. A )
49 48 36 syl 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) ) -> Q e. ( Base ` K ) )
50 simpl23 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) ) -> R e. A )
51 50 32 syl 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) ) -> R e. ( Base ` K ) )
52 31 2 latj31 
 |-  ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ Q e. ( Base ` K ) /\ R e. ( Base ` K ) ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( R .\/ Q ) .\/ P ) )
53 45 47 49 51 52 syl13anc 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( R .\/ Q ) .\/ P ) )
54 53 breq1d 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ P ) <-> ( ( R .\/ Q ) .\/ P ) .<_ ( ( S .\/ T ) .\/ P ) ) )
55 53 eqeq1d 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ P ) <-> ( ( R .\/ Q ) .\/ P ) = ( ( S .\/ T ) .\/ P ) ) )
56 43 54 55 3imtr4d 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ P ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ P ) ) )
57 8 16 56 pm2.61ne 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) ) -> ( ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ U ) ) )
58 57 3impia 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( -. R .<_ ( P .\/ Q ) /\ P =/= Q /\ -. Q .<_ ( P .\/ U ) ) /\ ( ( P .\/ Q ) .\/ R ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( S .\/ T ) .\/ U ) )