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Metamath Proof Explorer

Theorem dalem2

Description: Lemma for dath . Show the lines P Q and S T form a plane. (Contributed by NM, 11-Aug-2012)

Ref Expression
Hypotheses dalema.ph 
|- ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) )
dalemc.l 
|- .<_ = ( le ` K )
dalemc.j 
|- .\/ = ( join ` K )
dalemc.a 
|- A = ( Atoms ` K )
dalem1.o 
|- O = ( LPlanes ` K )
dalem1.y 
|- Y = ( ( P .\/ Q ) .\/ R )
Assertion dalem2 
|- ( ph -> ( ( P .\/ Q ) .\/ ( S .\/ T ) ) e. O )

Proof

Step Hyp Ref Expression
1 dalema.ph 
 |-  ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) )
2 dalemc.l 
 |-  .<_ = ( le ` K )
3 dalemc.j 
 |-  .\/ = ( join ` K )
4 dalemc.a 
 |-  A = ( Atoms ` K )
5 dalem1.o 
 |-  O = ( LPlanes ` K )
6 dalem1.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
7 1 dalemkehl 
 |-  ( ph -> K e. HL )
8 1 dalempea 
 |-  ( ph -> P e. A )
9 1 dalemqea 
 |-  ( ph -> Q e. A )
10 1 dalemsea 
 |-  ( ph -> S e. A )
11 1 dalemtea 
 |-  ( ph -> T e. A )
12 3 4 hlatj4 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( S e. A /\ T e. A ) ) -> ( ( P .\/ Q ) .\/ ( S .\/ T ) ) = ( ( P .\/ S ) .\/ ( Q .\/ T ) ) )
13 7 8 9 10 11 12 syl122anc 
 |-  ( ph -> ( ( P .\/ Q ) .\/ ( S .\/ T ) ) = ( ( P .\/ S ) .\/ ( Q .\/ T ) ) )
14 1 2 3 4 5 6 dalempjsen 
 |-  ( ph -> ( P .\/ S ) e. ( LLines ` K ) )
15 1 2 3 4 5 6 dalemqnet 
 |-  ( ph -> Q =/= T )
16 eqid 
 |-  ( LLines ` K ) = ( LLines ` K )
17 3 4 16 llni2 
 |-  ( ( ( K e. HL /\ Q e. A /\ T e. A ) /\ Q =/= T ) -> ( Q .\/ T ) e. ( LLines ` K ) )
18 7 9 11 15 17 syl31anc 
 |-  ( ph -> ( Q .\/ T ) e. ( LLines ` K ) )
19 1 2 3 4 5 6 dalem1 
 |-  ( ph -> ( P .\/ S ) =/= ( Q .\/ T ) )
20 1 2 3 4 5 6 dalemcea 
 |-  ( ph -> C e. A )
21 1 dalemclpjs 
 |-  ( ph -> C .<_ ( P .\/ S ) )
22 1 dalemclqjt 
 |-  ( ph -> C .<_ ( Q .\/ T ) )
23 eqid 
 |-  ( meet ` K ) = ( meet ` K )
24 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
25 2 23 24 4 16 2llnm4 
 |-  ( ( K e. HL /\ ( C e. A /\ ( P .\/ S ) e. ( LLines ` K ) /\ ( Q .\/ T ) e. ( LLines ` K ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) ) ) -> ( ( P .\/ S ) ( meet ` K ) ( Q .\/ T ) ) =/= ( 0. ` K ) )
26 7 20 14 18 21 22 25 syl132anc 
 |-  ( ph -> ( ( P .\/ S ) ( meet ` K ) ( Q .\/ T ) ) =/= ( 0. ` K ) )
27 23 24 4 16 2llnmat 
 |-  ( ( ( K e. HL /\ ( P .\/ S ) e. ( LLines ` K ) /\ ( Q .\/ T ) e. ( LLines ` K ) ) /\ ( ( P .\/ S ) =/= ( Q .\/ T ) /\ ( ( P .\/ S ) ( meet ` K ) ( Q .\/ T ) ) =/= ( 0. ` K ) ) ) -> ( ( P .\/ S ) ( meet ` K ) ( Q .\/ T ) ) e. A )
28 7 14 18 19 26 27 syl32anc 
 |-  ( ph -> ( ( P .\/ S ) ( meet ` K ) ( Q .\/ T ) ) e. A )
29 3 23 4 16 5 2llnmj 
 |-  ( ( K e. HL /\ ( P .\/ S ) e. ( LLines ` K ) /\ ( Q .\/ T ) e. ( LLines ` K ) ) -> ( ( ( P .\/ S ) ( meet ` K ) ( Q .\/ T ) ) e. A <-> ( ( P .\/ S ) .\/ ( Q .\/ T ) ) e. O ) )
30 7 14 18 29 syl3anc 
 |-  ( ph -> ( ( ( P .\/ S ) ( meet ` K ) ( Q .\/ T ) ) e. A <-> ( ( P .\/ S ) .\/ ( Q .\/ T ) ) e. O ) )
31 28 30 mpbid 
 |-  ( ph -> ( ( P .\/ S ) .\/ ( Q .\/ T ) ) e. O )
32 13 31 eqeltrd 
 |-  ( ph -> ( ( P .\/ Q ) .\/ ( S .\/ T ) ) e. O )