Metamath Proof Explorer

Theorem dalem61

Description: Lemma for dath . Show that atoms D , E , and F lie on the same line (axis of perspectivity). Eliminate hypotheses containing dummy atoms c and d . (Contributed by NM, 11-Aug-2012)

Ref Expression
Hypotheses dalem.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 ) ) ) ) )
dalem.l 
|- .<_ = ( le ` K )
dalem.j 
|- .\/ = ( join ` K )
dalem.a 
|- A = ( Atoms ` K )
dalem.ps 
|- ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) )
dalem61.m 
|- ./\ = ( meet ` K )
dalem61.o 
|- O = ( LPlanes ` K )
dalem61.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem61.z 
|- Z = ( ( S .\/ T ) .\/ U )
dalem61.d 
|- D = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
dalem61.e 
|- E = ( ( Q .\/ R ) ./\ ( T .\/ U ) )
dalem61.f 
|- F = ( ( R .\/ P ) ./\ ( U .\/ S ) )
Assertion dalem61 
|- ( ( ph /\ Y = Z /\ ps ) -> F .<_ ( D .\/ E ) )

Proof

Step Hyp Ref Expression
1 dalem.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 dalem.l 
 |-  .<_ = ( le ` K )
3 dalem.j 
 |-  .\/ = ( join ` K )
4 dalem.a 
 |-  A = ( Atoms ` K )
5 dalem.ps 
 |-  ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) )
6 dalem61.m 
 |-  ./\ = ( meet ` K )
7 dalem61.o 
 |-  O = ( LPlanes ` K )
8 dalem61.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
9 dalem61.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
10 dalem61.d 
 |-  D = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
11 dalem61.e 
 |-  E = ( ( Q .\/ R ) ./\ ( T .\/ U ) )
12 dalem61.f 
 |-  F = ( ( R .\/ P ) ./\ ( U .\/ S ) )
13 eqid 
 |-  ( ( c .\/ P ) ./\ ( d .\/ S ) ) = ( ( c .\/ P ) ./\ ( d .\/ S ) )
14 eqid 
 |-  ( ( c .\/ Q ) ./\ ( d .\/ T ) ) = ( ( c .\/ Q ) ./\ ( d .\/ T ) )
15 eqid 
 |-  ( ( c .\/ R ) ./\ ( d .\/ U ) ) = ( ( c .\/ R ) ./\ ( d .\/ U ) )
16 eqid 
 |-  ( ( ( ( ( c .\/ P ) ./\ ( d .\/ S ) ) .\/ ( ( c .\/ Q ) ./\ ( d .\/ T ) ) ) .\/ ( ( c .\/ R ) ./\ ( d .\/ U ) ) ) ./\ Y ) = ( ( ( ( ( c .\/ P ) ./\ ( d .\/ S ) ) .\/ ( ( c .\/ Q ) ./\ ( d .\/ T ) ) ) .\/ ( ( c .\/ R ) ./\ ( d .\/ U ) ) ) ./\ Y )
17 1 2 3 4 5 6 7 8 9 12 13 14 15 16 dalem59 
 |-  ( ( ph /\ Y = Z /\ ps ) -> F .<_ ( ( ( ( ( c .\/ P ) ./\ ( d .\/ S ) ) .\/ ( ( c .\/ Q ) ./\ ( d .\/ T ) ) ) .\/ ( ( c .\/ R ) ./\ ( d .\/ U ) ) ) ./\ Y ) )
18 1 2 3 4 5 6 7 8 9 10 11 13 14 15 16 dalem60 
 |-  ( ( ph /\ Y = Z /\ ps ) -> ( D .\/ E ) = ( ( ( ( ( c .\/ P ) ./\ ( d .\/ S ) ) .\/ ( ( c .\/ Q ) ./\ ( d .\/ T ) ) ) .\/ ( ( c .\/ R ) ./\ ( d .\/ U ) ) ) ./\ Y ) )
19 17 18 breqtrrd 
 |-  ( ( ph /\ Y = Z /\ ps ) -> F .<_ ( D .\/ E ) )