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Metamath Proof Explorer

Theorem dalemdnee

Description: Lemma for dath . Axis of perspectivity points D and E are different. (Contributed by NM, 10-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 )
dalem3.m 
|- ./\ = ( meet ` K )
dalem3.o 
|- O = ( LPlanes ` K )
dalem3.y 
|- Y = ( ( P .\/ Q ) .\/ R )
dalem3.z 
|- Z = ( ( S .\/ T ) .\/ U )
dalem3.d 
|- D = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
dalem3.e 
|- E = ( ( Q .\/ R ) ./\ ( T .\/ U ) )
Assertion dalemdnee 
|- ( ph -> D =/= E )

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 dalem3.m 
 |-  ./\ = ( meet ` K )
6 dalem3.o 
 |-  O = ( LPlanes ` K )
7 dalem3.y 
 |-  Y = ( ( P .\/ Q ) .\/ R )
8 dalem3.z 
 |-  Z = ( ( S .\/ T ) .\/ U )
9 dalem3.d 
 |-  D = ( ( P .\/ Q ) ./\ ( S .\/ T ) )
10 dalem3.e 
 |-  E = ( ( Q .\/ R ) ./\ ( T .\/ U ) )
11 simpr 
 |-  ( ( ph /\ D = Q ) -> D = Q )
12 1 2 3 4 6 7 dalemqnet 
 |-  ( ph -> Q =/= T )
13 12 adantr 
 |-  ( ( ph /\ D = Q ) -> Q =/= T )
14 11 13 eqnetrd 
 |-  ( ( ph /\ D = Q ) -> D =/= T )
15 1 2 3 4 5 6 7 8 9 10 dalem4 
 |-  ( ( ph /\ D =/= T ) -> D =/= E )
16 14 15 syldan 
 |-  ( ( ph /\ D = Q ) -> D =/= E )
17 1 2 3 4 5 6 7 8 9 10 dalem3 
 |-  ( ( ph /\ D =/= Q ) -> D =/= E )
18 16 17 pm2.61dane 
 |-  ( ph -> D =/= E )