Metamath Proof Explorer


Theorem dalem63

Description: Lemma for dath . Combine the cases where Y and Z are different planes with the case where Y and Z are the same plane. (Contributed by NM, 11-Aug-2012)

Ref Expression
Hypotheses dalem62.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 ) ) ) ) )
dalem62.l
|- .<_ = ( le ` K )
dalem62.j
|- .\/ = ( join ` K )
dalem62.a
|- A = ( Atoms ` K )
dalem62.m
|- ./\ = ( meet ` K )
dalem62.o
|- O = ( LPlanes ` K )
dalem62.y
|- Y = ( ( P .\/ Q ) .\/ R )
dalem62.z
|- Z = ( ( S .\/ T ) .\/ U )
dalem62.d
|- D = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
dalem62.e
|- E = ( ( Q .\/ R ) ./\ ( T .\/ U ) )
dalem62.f
|- F = ( ( R .\/ P ) ./\ ( U .\/ S ) )
Assertion dalem63
|- ( ph -> F .<_ ( D .\/ E ) )

Proof

Step Hyp Ref Expression
1 dalem62.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 dalem62.l
 |-  .<_ = ( le ` K )
3 dalem62.j
 |-  .\/ = ( join ` K )
4 dalem62.a
 |-  A = ( Atoms ` K )
5 dalem62.m
 |-  ./\ = ( meet ` K )
6 dalem62.o
 |-  O = ( LPlanes ` K )
7 dalem62.y
 |-  Y = ( ( P .\/ Q ) .\/ R )
8 dalem62.z
 |-  Z = ( ( S .\/ T ) .\/ U )
9 dalem62.d
 |-  D = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
10 dalem62.e
 |-  E = ( ( Q .\/ R ) ./\ ( T .\/ U ) )
11 dalem62.f
 |-  F = ( ( R .\/ P ) ./\ ( U .\/ S ) )
12 1 2 3 4 5 6 7 8 9 10 11 dalem62
 |-  ( ( ph /\ Y = Z ) -> F .<_ ( D .\/ E ) )
13 1 2 3 4 5 6 7 8 9 10 11 dalem16
 |-  ( ( ph /\ Y =/= Z ) -> F .<_ ( D .\/ E ) )
14 12 13 pm2.61dane
 |-  ( ph -> F .<_ ( D .\/ E ) )