Metamath Proof Explorer

Theorem dalemdea

Description: Lemma for dath . Frequently-used utility lemma. (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 )
dalemdea.m 
|- ./\ = ( meet ` K )
dalemdea.o 
|- O = ( LPlanes ` K )
dalemdea.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalemdea.z 
|- Z = ( ( S .\/ T ) .\/ U )
dalemdea.d 
|- D = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
Assertion dalemdea 
|- ( ph -> D e. A )

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 dalemdea.m 
 |-  ./\ = ( meet ` K )
6 dalemdea.o 
 |-  O = ( LPlanes ` K )
7 dalemdea.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
8 dalemdea.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
9 dalemdea.d 
 |-  D = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
10 1 2 3 4 6 7 dalem2 
 |-  ( ph -> ( ( P .\/ Q ) .\/ ( S .\/ T ) ) e. O )
11 1 dalemkehl 
 |-  ( ph -> K e. HL )
12 1 dalempea 
 |-  ( ph -> P e. A )
13 1 dalemqea 
 |-  ( ph -> Q e. A )
14 1 dalemrea 
 |-  ( ph -> R e. A )
15 1 dalemyeo 
 |-  ( ph -> Y e. O )
16 3 4 6 7 lplnri1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ Y e. O ) -> P =/= Q )
17 11 12 13 14 15 16 syl131anc 
 |-  ( ph -> P =/= Q )
18 eqid 
 |-  ( LLines ` K ) = ( LLines ` K )
19 3 4 18 llni2 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> ( P .\/ Q ) e. ( LLines ` K ) )
20 11 12 13 17 19 syl31anc 
 |-  ( ph -> ( P .\/ Q ) e. ( LLines ` K ) )
21 1 dalemsea 
 |-  ( ph -> S e. A )
22 1 dalemtea 
 |-  ( ph -> T e. A )
23 1 dalemuea 
 |-  ( ph -> U e. A )
24 1 dalemzeo 
 |-  ( ph -> Z e. O )
25 3 4 6 8 lplnri1 
 |-  ( ( K e. HL /\ ( S e. A /\ T e. A /\ U e. A ) /\ Z e. O ) -> S =/= T )
26 11 21 22 23 24 25 syl131anc 
 |-  ( ph -> S =/= T )
27 3 4 18 llni2 
 |-  ( ( ( K e. HL /\ S e. A /\ T e. A ) /\ S =/= T ) -> ( S .\/ T ) e. ( LLines ` K ) )
28 11 21 22 26 27 syl31anc 
 |-  ( ph -> ( S .\/ T ) e. ( LLines ` K ) )
29 3 5 4 18 6 2llnmj 
 |-  ( ( K e. HL /\ ( P .\/ Q ) e. ( LLines ` K ) /\ ( S .\/ T ) e. ( LLines ` K ) ) -> ( ( ( P .\/ Q ) ./\ ( S .\/ T ) ) e. A <-> ( ( P .\/ Q ) .\/ ( S .\/ T ) ) e. O ) )
30 11 20 28 29 syl3anc 
 |-  ( ph -> ( ( ( P .\/ Q ) ./\ ( S .\/ T ) ) e. A <-> ( ( P .\/ Q ) .\/ ( S .\/ T ) ) e. O ) )
31 10 30 mpbird 
 |-  ( ph -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) e. A )
32 9 31 eqeltrid 
 |-  ( ph -> D e. A )