Theorem dalem18

Description: Lemma for dath . Show that a dummy atom c exists outside of the Y and Z planes (when those planes are equal). This requires that the projective space be 3-dimensional. (Desargues's theorem does not always hold in 2 dimensions.) (Contributed by NM, 29-Jul-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 )
	dalem18.y	|- Y = ( ( P .\/ Q ) .\/ R )
Assertion	dalem18	|- ( ph -> E. c e. A -. 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		dalem18.y	|- Y = ( ( P .\/ Q ) .\/ R )
6	1	dalemkehl	|- ( ph -> K e. HL )
7	1	dalempea	|- ( ph -> P e. A )
8	1	dalemqea	|- ( ph -> Q e. A )
9	1	dalemrea	|- ( ph -> R e. A )
10	3 2 4	3dim3	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> E. c e. A -. c .<_ ( ( P .\/ Q ) .\/ R ) )
11	6 7 8 9 10	syl13anc	|- ( ph -> E. c e. A -. c .<_ ( ( P .\/ Q ) .\/ R ) )
12	5	breq2i	|- ( c .<_ Y <-> c .<_ ( ( P .\/ Q ) .\/ R ) )
13	12	notbii	|- ( -. c .<_ Y <-> -. c .<_ ( ( P .\/ Q ) .\/ R ) )
14	13	rexbii	|- ( E. c e. A -. c .<_ Y <-> E. c e. A -. c .<_ ( ( P .\/ Q ) .\/ R ) )
15	11 14	sylibr	|- ( ph -> E. c e. A -. c .<_ Y )