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 )

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 )

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