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Metamath Proof Explorer

Theorem islpln2a

Description: The predicate "is a lattice plane" for join of atoms. (Contributed by NM, 16-Jul-2012)

Ref Expression
Hypotheses islpln2a.l 
|- .<_ = ( le ` K )
islpln2a.j 
|- .\/ = ( join ` K )
islpln2a.a 
|- A = ( Atoms ` K )
islpln2a.p 
|- P = ( LPlanes ` K )
Assertion islpln2a 
|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( ( ( Q .\/ R ) .\/ S ) e. P <-> ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) )

Proof

Step Hyp Ref Expression
1 islpln2a.l 
 |-  .<_ = ( le ` K )
2 islpln2a.j 
 |-  .\/ = ( join ` K )
3 islpln2a.a 
 |-  A = ( Atoms ` K )
4 islpln2a.p 
 |-  P = ( LPlanes ` K )
5 oveq1 
 |-  ( Q = R -> ( Q .\/ R ) = ( R .\/ R ) )
6 2 3 hlatjidm 
 |-  ( ( K e. HL /\ R e. A ) -> ( R .\/ R ) = R )
7 6 3ad2antr2 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( R .\/ R ) = R )
8 5 7 sylan9eqr 
 |-  ( ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) /\ Q = R ) -> ( Q .\/ R ) = R )
9 8 oveq1d 
 |-  ( ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) /\ Q = R ) -> ( ( Q .\/ R ) .\/ S ) = ( R .\/ S ) )
10 simpll 
 |-  ( ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) /\ Q = R ) -> K e. HL )
11 simplr2 
 |-  ( ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) /\ Q = R ) -> R e. A )
12 simplr3 
 |-  ( ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) /\ Q = R ) -> S e. A )
13 2 3 4 2atnelpln 
 |-  ( ( K e. HL /\ R e. A /\ S e. A ) -> -. ( R .\/ S ) e. P )
14 10 11 12 13 syl3anc 
 |-  ( ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) /\ Q = R ) -> -. ( R .\/ S ) e. P )
15 9 14 eqneltrd 
 |-  ( ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) /\ Q = R ) -> -. ( ( Q .\/ R ) .\/ S ) e. P )
16 15 ex 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( Q = R -> -. ( ( Q .\/ R ) .\/ S ) e. P ) )
17 16 necon2ad 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( ( ( Q .\/ R ) .\/ S ) e. P -> Q =/= R ) )
18 hllat 
 |-  ( K e. HL -> K e. Lat )
19 18 adantr 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> K e. Lat )
20 simpr3 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> S e. A )
21 eqid 
 |-  ( Base ` K ) = ( Base ` K )
22 21 3 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
23 20 22 syl 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> S e. ( Base ` K ) )
24 21 2 3 hlatjcl 
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> ( Q .\/ R ) e. ( Base ` K ) )
25 24 3adant3r3 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( Q .\/ R ) e. ( Base ` K ) )
26 21 1 2 latleeqj2 
 |-  ( ( K e. Lat /\ S e. ( Base ` K ) /\ ( Q .\/ R ) e. ( Base ` K ) ) -> ( S .<_ ( Q .\/ R ) <-> ( ( Q .\/ R ) .\/ S ) = ( Q .\/ R ) ) )
27 19 23 25 26 syl3anc 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( S .<_ ( Q .\/ R ) <-> ( ( Q .\/ R ) .\/ S ) = ( Q .\/ R ) ) )
28 2 3 4 2atnelpln 
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> -. ( Q .\/ R ) e. P )
29 28 3adant3r3 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> -. ( Q .\/ R ) e. P )
30 eleq1 
 |-  ( ( ( Q .\/ R ) .\/ S ) = ( Q .\/ R ) -> ( ( ( Q .\/ R ) .\/ S ) e. P <-> ( Q .\/ R ) e. P ) )
31 30 notbid 
 |-  ( ( ( Q .\/ R ) .\/ S ) = ( Q .\/ R ) -> ( -. ( ( Q .\/ R ) .\/ S ) e. P <-> -. ( Q .\/ R ) e. P ) )
32 29 31 syl5ibrcom 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( ( ( Q .\/ R ) .\/ S ) = ( Q .\/ R ) -> -. ( ( Q .\/ R ) .\/ S ) e. P ) )
33 27 32 sylbid 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( S .<_ ( Q .\/ R ) -> -. ( ( Q .\/ R ) .\/ S ) e. P ) )
34 33 con2d 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( ( ( Q .\/ R ) .\/ S ) e. P -> -. S .<_ ( Q .\/ R ) ) )
35 17 34 jcad 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( ( ( Q .\/ R ) .\/ S ) e. P -> ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) )
36 1 2 3 4 lplni2 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> ( ( Q .\/ R ) .\/ S ) e. P )
37 36 3expia 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) -> ( ( Q .\/ R ) .\/ S ) e. P ) )
38 35 37 impbid 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( ( ( Q .\/ R ) .\/ S ) e. P <-> ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) )