Metamath Proof Explorer

Theorem dalem10

Description: Lemma for dath . Atom D belongs to the axis of perspectivity X . (Contributed by NM, 19-Jul-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 )
dalem10.m 
|- ./\ = ( meet ` K )
dalem10.o 
|- O = ( LPlanes ` K )
dalem10.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem10.z 
|- Z = ( ( S .\/ T ) .\/ U )
dalem10.x 
|- X = ( Y ./\ Z )
dalem10.d 
|- D = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
Assertion dalem10 
|- ( ph -> D .<_ X )

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 dalem10.m 
 |-  ./\ = ( meet ` K )
6 dalem10.o 
 |-  O = ( LPlanes ` K )
7 dalem10.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
8 dalem10.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
9 dalem10.x 
 |-  X = ( Y ./\ Z )
10 dalem10.d 
 |-  D = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
11 1 dalemkelat 
 |-  ( ph -> K e. Lat )
12 1 3 4 dalempjqeb 
 |-  ( ph -> ( P .\/ Q ) e. ( Base ` K ) )
13 1 4 dalemreb 
 |-  ( ph -> R e. ( Base ` K ) )
14 eqid 
 |-  ( Base ` K ) = ( Base ` K )
15 14 2 3 latlej1 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> ( P .\/ Q ) .<_ ( ( P .\/ Q ) .\/ R ) )
16 11 12 13 15 syl3anc 
 |-  ( ph -> ( P .\/ Q ) .<_ ( ( P .\/ Q ) .\/ R ) )
17 1 3 4 dalemsjteb 
 |-  ( ph -> ( S .\/ T ) e. ( Base ` K ) )
18 1 4 dalemueb 
 |-  ( ph -> U e. ( Base ` K ) )
19 14 2 3 latlej1 
 |-  ( ( K e. Lat /\ ( S .\/ T ) e. ( Base ` K ) /\ U e. ( Base ` K ) ) -> ( S .\/ T ) .<_ ( ( S .\/ T ) .\/ U ) )
20 11 17 18 19 syl3anc 
 |-  ( ph -> ( S .\/ T ) .<_ ( ( S .\/ T ) .\/ U ) )
21 1 6 dalemyeb 
 |-  ( ph -> Y e. ( Base ` K ) )
22 7 21 eqeltrrid 
 |-  ( ph -> ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) )
23 1 dalemzeo 
 |-  ( ph -> Z e. O )
24 14 6 lplnbase 
 |-  ( Z e. O -> Z e. ( Base ` K ) )
25 23 24 syl 
 |-  ( ph -> Z e. ( Base ` K ) )
26 8 25 eqeltrrid 
 |-  ( ph -> ( ( S .\/ T ) .\/ U ) e. ( Base ` K ) )
27 14 2 5 latmlem12 
 |-  ( ( K e. Lat /\ ( ( P .\/ Q ) e. ( Base ` K ) /\ ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) ) /\ ( ( S .\/ T ) e. ( Base ` K ) /\ ( ( S .\/ T ) .\/ U ) e. ( Base ` K ) ) ) -> ( ( ( P .\/ Q ) .<_ ( ( P .\/ Q ) .\/ R ) /\ ( S .\/ T ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( ( ( P .\/ Q ) .\/ R ) ./\ ( ( S .\/ T ) .\/ U ) ) ) )
28 11 12 22 17 26 27 syl122anc 
 |-  ( ph -> ( ( ( P .\/ Q ) .<_ ( ( P .\/ Q ) .\/ R ) /\ ( S .\/ T ) .<_ ( ( S .\/ T ) .\/ U ) ) -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( ( ( P .\/ Q ) .\/ R ) ./\ ( ( S .\/ T ) .\/ U ) ) ) )
29 16 20 28 mp2and 
 |-  ( ph -> ( ( P .\/ Q ) ./\ ( S .\/ T ) ) .<_ ( ( ( P .\/ Q ) .\/ R ) ./\ ( ( S .\/ T ) .\/ U ) ) )
30 7 8 oveq12i 
 |-  ( Y ./\ Z ) = ( ( ( P .\/ Q ) .\/ R ) ./\ ( ( S .\/ T ) .\/ U ) )
31 9 30 eqtri 
 |-  X = ( ( ( P .\/ Q ) .\/ R ) ./\ ( ( S .\/ T ) .\/ U ) )
32 29 10 31 3brtr4g 
 |-  ( ph -> D .<_ X )