Metamath Proof Explorer

Theorem dalem17

Description: Lemma for dath . When planes Y and Z are equal, the center of perspectivity C is in Y . (Contributed by NM, 1-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 )
dalem17.o 
|- O = ( LPlanes ` K )
dalem17.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem17.z 
|- Z = ( ( S .\/ T ) .\/ U )
Assertion dalem17 
|- ( ( ph /\ Y = Z ) -> C .<_ Y )

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 dalem17.o 
 |-  O = ( LPlanes ` K )
6 dalem17.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
7 dalem17.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
8 1 dalemclrju 
 |-  ( ph -> C .<_ ( R .\/ U ) )
9 8 adantr 
 |-  ( ( ph /\ Y = Z ) -> C .<_ ( R .\/ U ) )
10 1 dalemkelat 
 |-  ( ph -> K e. Lat )
11 1 3 4 dalempjqeb 
 |-  ( ph -> ( P .\/ Q ) e. ( Base ` K ) )
12 1 4 dalemreb 
 |-  ( ph -> R e. ( Base ` K ) )
13 eqid 
 |-  ( Base ` K ) = ( Base ` K )
14 13 2 3 latlej2 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> R .<_ ( ( P .\/ Q ) .\/ R ) )
15 10 11 12 14 syl3anc 
 |-  ( ph -> R .<_ ( ( P .\/ Q ) .\/ R ) )
16 15 6 breqtrrdi 
 |-  ( ph -> R .<_ Y )
17 16 adantr 
 |-  ( ( ph /\ Y = Z ) -> R .<_ Y )
18 1 3 4 dalemsjteb 
 |-  ( ph -> ( S .\/ T ) e. ( Base ` K ) )
19 1 4 dalemueb 
 |-  ( ph -> U e. ( Base ` K ) )
20 13 2 3 latlej2 
 |-  ( ( K e. Lat /\ ( S .\/ T ) e. ( Base ` K ) /\ U e. ( Base ` K ) ) -> U .<_ ( ( S .\/ T ) .\/ U ) )
21 10 18 19 20 syl3anc 
 |-  ( ph -> U .<_ ( ( S .\/ T ) .\/ U ) )
22 21 7 breqtrrdi 
 |-  ( ph -> U .<_ Z )
23 22 adantr 
 |-  ( ( ph /\ Y = Z ) -> U .<_ Z )
24 simpr 
 |-  ( ( ph /\ Y = Z ) -> Y = Z )
25 23 24 breqtrrd 
 |-  ( ( ph /\ Y = Z ) -> U .<_ Y )
26 1 5 dalemyeb 
 |-  ( ph -> Y e. ( Base ` K ) )
27 13 2 3 latjle12 
 |-  ( ( K e. Lat /\ ( R e. ( Base ` K ) /\ U e. ( Base ` K ) /\ Y e. ( Base ` K ) ) ) -> ( ( R .<_ Y /\ U .<_ Y ) <-> ( R .\/ U ) .<_ Y ) )
28 10 12 19 26 27 syl13anc 
 |-  ( ph -> ( ( R .<_ Y /\ U .<_ Y ) <-> ( R .\/ U ) .<_ Y ) )
29 28 adantr 
 |-  ( ( ph /\ Y = Z ) -> ( ( R .<_ Y /\ U .<_ Y ) <-> ( R .\/ U ) .<_ Y ) )
30 17 25 29 mpbi2and 
 |-  ( ( ph /\ Y = Z ) -> ( R .\/ U ) .<_ Y )
31 1 4 dalemceb 
 |-  ( ph -> C e. ( Base ` K ) )
32 1 dalemkehl 
 |-  ( ph -> K e. HL )
33 1 dalemrea 
 |-  ( ph -> R e. A )
34 1 dalemuea 
 |-  ( ph -> U e. A )
35 13 3 4 hlatjcl 
 |-  ( ( K e. HL /\ R e. A /\ U e. A ) -> ( R .\/ U ) e. ( Base ` K ) )
36 32 33 34 35 syl3anc 
 |-  ( ph -> ( R .\/ U ) e. ( Base ` K ) )
37 13 2 lattr 
 |-  ( ( K e. Lat /\ ( C e. ( Base ` K ) /\ ( R .\/ U ) e. ( Base ` K ) /\ Y e. ( Base ` K ) ) ) -> ( ( C .<_ ( R .\/ U ) /\ ( R .\/ U ) .<_ Y ) -> C .<_ Y ) )
38 10 31 36 26 37 syl13anc 
 |-  ( ph -> ( ( C .<_ ( R .\/ U ) /\ ( R .\/ U ) .<_ Y ) -> C .<_ Y ) )
39 38 adantr 
 |-  ( ( ph /\ Y = Z ) -> ( ( C .<_ ( R .\/ U ) /\ ( R .\/ U ) .<_ Y ) -> C .<_ Y ) )
40 9 30 39 mp2and 
 |-  ( ( ph /\ Y = Z ) -> C .<_ Y )